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Real flag manifolds and a construction of spaces over a polyhedron: Mathematical Investigations arising… Svensson, Anders Gunnar Stefan 1996

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REAL FLAG MANIFOLDS AND A CONSTRUCTION OF SPACES OVER A POLYHEDRON Mathematical Investigations arising from the Jahn-Teller Effect ANDERS GUNNAR STEFAN SVENSSON B.Sc. (Mathematics), The University of British Columbia, 1987 M.Sc. (Mathematics), The University of Toronto, 1989 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF by DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics THE UNIVERSITY OF BRITISH COLUMBIA July 1996 © Anders G. S. Svensson, 1996 in presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of MATTUghA A C T U S The University of British Columbia Vancouver, Canada Date /V-U^ \°(\l0 DE-6 (2/88) Abstract We examine a construction of topological spaces over an arbitrary polyhedron and show that it subsumes the lattice construction of R. R. Douglas and A. R. Rutherford. A Simplicial Approximation Theorem is proven for the general construction, for maps from a polyhedron to one of our spaces lying over another polyhedron. A special case of our construction (a slight generalization of the lattice construction) is examined and a class of locally trivial bundles is constructed. These are used to examine neighbourhood structure in the special case. We also enumerate exactly which spheres can be constructed by a lattice construction on a product of real orthogonal, complex unitary or quaternionic symplectic groups. The fundamental group of the real complete flag manifolds is determined following a detailed exposition of Clifford algebras. Appendices are provided on the diagonalization of quater-nionic Hermitean matrices and on a generalized mapping cylinder that can be regarded as an endofunctor on the category of locally trivial bundles over a fixed locally compact base. ii Table of Contents Abstract ii Table of Contents iii Acknowledgements iv Introduction 1 Chapter 0. Preliminaries 3 Chapter 1. Two Computations 10 A. Fundamental Group of the Complete Flags 10 1. Proper Group Actions and the Fundamental Group 11 2. Clifford Algebras 12 3. Clifford Groups 25 4. Addendum: Counting by Twos 26 B. Triviality of Certain Quotients of 0(4) over RP3 27 Chapter 2. Constructing Spaces over a Polyhedron 31 A. The Lattice Construction 31 B. Constructing Spheres 32 C. A More General Construction 35 Chapter 3. Lifting Simplicial Approximation 45 A. Lifting a Homotopy 45 B. Approximation by Simplicial Maps 49 C. An Example 51 Additional Structure 53 Chapter 4. Bundle Structure with a Topological Group 55 A. A Class of Bundles 55 B. Local Structure with a Compact Group 59 1. Strong Deformation Retractions above the Polyhedron 59 2. Fibre Structure of the Retracts 60 3. Locally Euclidean Structure with a Compact Lie Group 65 C. More Spheres 67 Counting Dimensions 71 Bibliography 78 Appendix A. Diagonalization in Herm(n, H) 81 Appendix B. Topologizing with Subsets 86 Appendix C. A Generalized Mapping Cylinder 89 Appendix D. Homotopy Groups of Flag Manifolds 93 Index 96 iii Acknowledgements It may well be that no man is an island but my land mass is particularly well populated. My parents, Karl and Gunilla Svensson, have given me boundless love and support over the years. I thank them for this and the wonderful life they've provided me with. This work could not have been completed without the constant support and encouragement of my research supervisor, Roy Douglas. Roy was also my friend and his untimely death just over two weeks ago is a personal as well as mathematical loss. I thank Sandy Rutherford for his help with many things. I'm also grateful to my supervisory committee for their advice and encouragement over the past few months: Professor K. Y. Lam, Professor E. Luft and Professor D. Rolfsen. My friends have always provided me with first-class distraction and I thank them all for this. For particularly effective distraction I thank Miguel Lopez and Laura Rivera, Peter and Claire Forster, Tony Varga and Susana Rendon, David and Teresa Ross, and Rob and Sandra North-cote. I gratefully acknowledge the financial support of the Natural Sciences and Engineering Re-search Council of Canada, The Department of Mathematics at UBC, Professor R. Douglas and Professor E. Luft. Anders Svensson October 8, 1996 Vancouver, Canada iv Introduction This work expands on some topological aspects of recent work by R. R. Douglas and A. R. Rutherford on the Jahn-Teller effect of molecular physics. It is not, however, a work on physics and can be read independently of any understanding of this subject. The Jahn-Teller effect is the name given to the physical phenomenon in which a polyatomic molecule distorts from a symmetric configuration so as to remove degeneracy in its electronic spectrum. Its study was initiated by the 1937 paper [JT] of H. A. Jahn and E. Teller, in which the authors studied the effects of symmetry on the electronic degeneracy of polyatomic molecules. They showed that, except for linear molecules, electronic degeneracy is unstable and is removed by distortion of the atomic nuclei within linear subspaces classified by the irreducible representations of the symmetry group of the molecular Hamiltonian; representations which also classify the electronic degeneracy. Since 1937, the physics community has produced an enormous volume of literature on the Jahn-Teller effect and its consequences, as evidenced by the equally enormous bibliographic review [B]. Few people have succeeded in predicting spectra for molecules exhibiting a Jahn-Teller effect and, until recently, the only such predictions have been for electronic doublet states in triangular molecules. In their papers [DR3] and [DR4], Douglas and Rutherford announce their prediction of spectra for triplet states in molecules exhibiting octahedral symmetry and quadruplet states in molecules exhibiting icosahedral symmetry. Their methods are highly geometric and involve a detailed dissection of Euclidean spaces of real symmetric and complex Hermitean matrices into regions of similar degeneracy, on which the eigenspace associated with a selected eigenvalue has constant dimension. This dissection quite naturally produces a decomposition of certain spheres above standard simplices, with the subset lying above an open simplex decomposing as a product of the open simplex with a flag manifold. The decomposition of spheres in turn generalizes readily to a construction which can be applied to any topological group together with a list of subgroups. Douglas coined the term lattice construction and studied some examples in [D]. The lattice construction can itself be viewed as a special case of a construction in which one builds a space over a polyhedron by gluing together spaces lying above the faces of the polyhedron. Chapter 2, 3 and 4 are devoted to different aspects of this construction. In chapter 2 we introduce the lattice construction and construct the real and complex spheres considered by Douglas and Rutherford, as well as quaternionic spheres. We then present our general construction as a functor on an appropriate category and show that the lattice construction is a special case. In chapter 3 we show that the classical Simplicial Approximation Theorem has a natural generalization to a statement about maps from a polyhedron to a class of our spaces lying 1 above another polyhedron. We apply this theorem to a construction of a space from a map between two arbitrary compact Hausdorff spaces. In chapter 4 we construct a class of bundles, using a locally compact Hausdorff group to shape a special case of our construction. We then use these bundles to examine the local structure in the case that the group is compact and the underlying simplicial complex is locally finite, giving a necessary condition for the construction of a topological manifold. Re-examining the spheres constructed in chapter 2, we show that these are not the only spheres constructed by the lattice construction. Indeed, in each of the real, complex and quaternionic cases, all but a finite number of spheres can be constructed from products of orthogonal, unitary or symplectic groups respectively. The real and complex flag manifolds are ubiquitous in the analyses of Douglas and Rutherford and some algebraic topology of these spaces can be found in [R]. In chapter 1 we determine the fundamental group of the real complete flags by examining their universal coverings by the spin groups, having been unable to find a reference in the literature. We also use covering space methods to identify a class of trivial bundles over the real projective space RP3. This class contains an JRF2-bundle whose triviality is required in [DR4]. 2 Chapter 0 Preliminaries We begin by summarizing some basic notational conventions that will be used throughout. • The ring of integers is denoted by The set of positive (strictly greater than zero) integers, or natural numbers, is denoted by N. The set of non-negative integers is denoted by No. If n £ N, the additive group of integers modulo n is denoted by En. • The real and complex fields are denoted by R and <C respectively. The division algebra of quaternions is denoted by M. The conjugate of an element x lying in either <C or M is denoted by x. • The standard basis vectors in Rn are denoted by e\ though en: e, is the tuple whose only non-zero component is a 1 in the i t h position. In chapter 1 we use ei,...,e n when regarding these as elements of a Clifford algebra on Rn. • Let n € N. If v = (yi,..., vn) and w = (wi,..., wn) are two elements of either <C" or iff" then their inner product is (v, iv) = 53"=i vlwi- We use the dot product notation v • w to denote the inner product in Rn. We denote the inner product preserving groups of n x n real orthogonal, complex unitary and quaternionic symplectic matrices by 0(n), U(n) and Sp(n) respectively. • The norm of an element x in JR, (C or H is denoted by \x\. If n > 1, the norm of an element v in Rn, <Dn or iff" is denoted by is denoted by ||u||. • If R is a ring and n is a non-negative integer then R(n) is the i?-algebra of n x n matrices having entries in R. If A € R(n) then AT denotes the transpose matrix. A is symmetric if AT = A. A is antisymmetric if AT = —A. • If A £ (C(n) or Jff(n) then A* denotes the conjugate transpose of A. If A G R(n) we let A* = AT. (Equivalently, we can endow R with the identity conjugation.) • If K = R, € or M and A G K(n) then A is Hermitean if A T = A. We denote the JK"-algebra of Hermitean matrices by Herm(n,K). 3 • The relation w denotes homeomorphism of topological spaces. The relation ~ denotes homotopy. • The relation = is used to denote equivalence in several algebraic categories: Isomorphism of groups, vector spaces, algebras, etc. • The term neighbourhood is used synonymously with open set in a topological space. That is, the elements of a topology are neighbourhoods.1 A function between two topological spaces is called a map if it is continuous. We will use the word function in this context only if continuity is either not assumed or not evident. • The relation = is occasionally used to introduce notation. For example, if 61 and 62 & r e two previously defined quantities and we write "... 61 = 62 = b ..." then the symbol b is being introduced as a synonym for the common value of 61 and 62- In chapter 1, = is also used to denote congruence modulo an integer, as in a = b mod n. • The relation X is used to denote the product in various categories. The relation © is used to denote the coproduct. (In particular, the disjoint union of topological spaces and the direct sum of real algebras.) • In addition to denoting the I T H homotopy group functor, the symbol TT{ is often used to denote a generic projection map from a cartesian product of spaces onto the i tb factor. • The symbol • is used to denote the end of a proof. It is also used to mark the end of a statement whose proof is immediate from preceding discussion. The symbol • is occasionally used within a proof to terminate the verification of some claim made during the course of the proof. We will construct many maps using the universal property of the identification. Recall that if / : X —> Y is a surjective map then Y has the identification topology by / if its topology consists of exactly those subsets of Y whose pre-image by / is an open subset of X, and in this case / is called an identification or identification map. An identification map / : X —• Y satisfies a universal property: If g : X —• Z is a map for which g(x) = g(x') whenever f(x) = f(x') then there is a unique map h : Y —• Z such that h ° / = g. That is, there is a unique map h which makes the following diagram commute. g X——^Z Given an equivalence relation R on a space X, the projection p : X —> X/R is an identification map when X/R is given the quotient topology. Thus, if / : X —• Y is an identification and R This differs from the use in [Br] for example. 4 is the equivalence relation on X defined by (x, x') £ R if and only if /(x) = /(x') then the unique map Y —• X/R which makes the diagram X Y X/R commute is a homeomorphism. If the equivalence relation R is the smallest equivalence relation containing some relation A C X x X 2 then there is the an alternate version of the universal property of the identification p : X —• X/R. For this, if we write R = (A) for this smallest equivalence containing A then (A) has the following characterization. 0.1 P ropos i t i on . Let x,x' 6 X. Then (x,x') € (A) if and only if either x' = x or there are n £ N, (xi,x[),..(xn, x'n) G A such that the following hold. 1. x £ {xi,x[}. 2. I f n > l then {xj,xj} D {xi+i,x'i+1} ^ 0 for each i € { 1,..., n — 1}. 3. x' € Proof . Let R C X X X be the set of pairs (x, x') such that either x' = x or there are n 6 N, (x\,x[),— ,(xn,x'n) 6 A such that 1, 2 and 3 hold. Then R is an equivalence relation and A C R. Thus, (A) C R. However, any equivalence relation containing A must also contain R so that R C (A) as well. • An easy consequence is then the following universal property for p : X —• X/(A). 0.2 P ropos i t i on . Let X be a topological space and let A C X x X be a relation on X. If a m a p g : X —• Z has g(x) = g(x') whenever (x,x') 6 A then there is a unique m a p h : X/(A) Z such that the following diagram commutes. 9 X/(A) That is, R is the intersection of all equivalence relations containing A. 5 Proof. Continuity is clear, so we need only demonstrate the existence of h. However, by Proposition 0.1, it is clear that the hypothesis implies that g(x) = g(x') whenever (a;, x') G (A). Thus, the existence of h follows from the standard universal property for p. u We will often employ the following result without extensive commentary. 0.3 Proposition. Let f : X —• Y be an identification and let Y0 C Y. If either (1) Y0 C Y is an open or closed subset or (2) f is an open or closed m a p then the subspace topology on Yo equals the identification topology by the m a p x i—• f(x) : / _ 1 ( V o ) —*• Yo. In other words, the map x •->• f(x) : / _ 1 (Y0) —• Yo is an identification. Proof. [Du], chapter VI, section 2.1. • We will say that a space X is locally compact if, given a point x G X and a neighbourhood U of x, there is a neighbourhood V of x whose closure is a compact subset of U. The following very useful result can be found in [Br] as proposition 4.3.2. Since the author there states the definition of local compactness slightly differently than do we (although our definition is equivalent to theirs), we include the proof. 0.4 Proposition. Let f : X —• Y be an identification and let B be locally compact. Then / x 1 : X x B - ^ Y x B i s a n identification. Proof. Let us write h = / X 1 and let U C Y X B be a subset for which h'1 (U) C X X B is open. We must show that U is open in Y x B. To see this, let (yo,b0) be any point in U, and let x0 G X such that /(xn) = yo- If Bo C B is defined by B0 = {beB\{x0,b)eh-1{U)} then BQ is open in B. For if b' G B then since h ~ l ( U ) C X X B is an open subset, there are open neighbourhoods V of x0 and W of V such that V x W C h~l(U), giving W C B0. Now, since B is locally compact, there is a neighbourhood V of bo whose closure V is a compact subset of BQ. Let W = { x G X | {x} x V C h~l (U) } and note that W is the largest subset of X having W X V C h~l(U). Since /^(/(W)) x v = fc-^Mw-* x v)) c /i"1^^-1^))) = / r 1 ^ ) , we have f~l{f{W)) C W so that / -1(/(W0) = W, W C /-1(/("^)) being the case for any W. Now, since (2/o, bo) G /(FT) x V = h ( W x V) C h ( W x V) C , it suffices to show that f ( W ) is an open subset of Y (since then f ( W ) x V is a neighbourhood of (j/o, M contained in U), and for this we must show that / _ 1 (f(W)) = W is an open subset of X. 6 However, if x 6 W then {x} xVch 1(?7) and, since V is compact, there is a neighbourhood W of x such that W' x V C /& _ 1(t/). 3 This neighbourhood is necessarily contained in W. m Many sources define locally compact to mean that each point has a neighbourhood whose closure is compact. If X is Hausdorff then the two definitions coincide and, except for in appendix C (the statement of Proposition C.4 in particular), all the locally compact spaces we consider will also be Hausdorff. As a general reference for topological facts we refer the reader to [Du]. Note that a locally compact space is assumed to be Hausdorff in this reference. We use the language of Category Theory throughout, although no specialized knowledge is required. Let us provide a brief introduction to the basic terminology. A category C consists of a class of objects and, for any two objects X and Y, a class of morphisms, or arrows, having source X and target Y. We write / : X —• Y if / is a morphism with source X and target Y. It is also required that there be a distinguished identity morphism lx '• X —+ X for each object X, as well as a rule of composition of morphisms which assigns to morphisms / : X —• Y and g : Y —» Z a morphism g » f : X —• Z satisfying the following two properties. • If / : X —> Y then / coincides with both l y ° / and f ° lx-• If / :W —• X, g : X —• Y and h : Y —> X then the compositions f ° (g ° h) and (/ ° g) ° h coincide. The first rule of Category Theory is that objects are unnecessary: A category is completely determined by its morphisms.4 Any set A with preordering -< (a reflexive, transitive relation) is the set of objects of a category in which there is a unique arrow with source a and target b if and only if a -< b. We will use this in chapter 2 to regard the face containment preordering as defining a category structure on a simplicial complex. A functor F from a category C to a category D assigns to each object X of C an object F ( X ) of D, and to each morphism / : X —• Y a morphism F(f) : F ( X ) —• F(Y), in such a way that the following hold. • For each object X of C, F(lx) coincides with IF(X)-For each point 6 € V, choose neighbourhoods W6" of x and B'b of b such that W^' x B'b C h 1(U). Let h,..., bn € B be such that U"=1 B'b. D V and set W = H"=1 Wb".. Still, it is common that one refer to well known categories by naming their objects rather than their morphisms. Thus, one speaks of the category Set of sets. A more precise way of referring to Set would be as the category of functions. 7 • If / : X -> Y and g : Y -* Z in C then F(f ° g) coincides with F(f) ° F(g). We write F : C —• D if F is a functor from C to D. 5 Functors from C to D are objects of the functor category D°, the morphisms in which are natural transformations. A natural transformation $ from F : C —* D to G : C D assigns to each object X of C an arrow $(X) : —• of D such that if / : X —• Y is an arrow of C then the following diagram commutes. F(X) — y - ± U F(Y) <?(/) $(Y) G(Y) The universal property of the identification is but one instance of a general phenomenon. If C is a category and F : C —* Set is a functor to the category of sets and functions then a universal element of F consists of an object c of C and an element x 6 F(c) such that if c' is some other object of C and x' € F(c') then there is a unique arrow / : c —• c' such that F(f)(x) = x'. For example, if / : X —• Y is an identification map then there is a functor F : Top —• Set from the category of topological spaces and maps which assigns to each space Z the set of maps g : X —* Z such that g(x) = g(x') whenever f(x) = f(x'). The space Y and map / then define a universal element of F. As another example, if V is a real vector spaces then the tensor product V ® V and the bilinear map (v, v') i-*- v <g» v' : V X V —• V ® V define a universal element of the functor which assigns to each real vector space W the set of all bilinear maps from V X V to W. More generally still, given a functor F :C —• D and an object d of Z), a universal arrow from d to F consists of an object c of C and an arrow / : d —> F(c) such that if c' and f : dF(c') is any other such pair then there is a unique arrow h ; c —• c' such that the following diagram commutes. <f F(c) (0.1) F(c') If we identify an element of a set X with a function from the one-point set into X then a universal element of a functor F : C —• Set is exactly a universal arrow from the one-point 5 Some authors refer to such an F as a covariant functor from C to D, and also define contravariant functors from C to D. These are just functors from C to Dop, where D o p is the opposite category of D, defined by letting the morphisms with source X and target V be the morphisms of D with source y and target X. 8 set to F. Colimits are another example of universal arrows from an object to a functor. Let J and C be categories and let A : C —• CJ be the diagonal functor that sends an object of C to the constant functor to that object,6 and an arrow / : c —• d to the natural transformation from A(c) to A(cQ whose component at every object of J is / . If F : J —• C is an object of CJ then a universal arrow from F to A is a colimit of F. By reversing arrows in (0.1), one can also define a universal arrow from F to d. By considering the diagonal functor A : C —+ CJ one is then lead to the notion of limit. We refer the reader to [M] for an in-depth exposition of Category Theory. That is, A(c) : J —* C is the functor which sends every morphism j —• j' in J to the identity morphism l c : c —• c in C. 9 Chapter 1 Two Computations Let K = R, <C or M, and let n,£>2 and n\,..., ne > 1 be integers with nx -\ \-ne = n. Let • F K ( » I > • • • , ne) denote the collection of tuples ( V * i , . . . , V~e) of mutually orthogonal subspaces of Kn having dim^ V{ = "i- 1 We can identify these with the homogeneous spaces 0(n)/(0(nx) x • • • x 0(ne)), U(n)/(U(ni) x • • • x U(ne)) and Sp(n)/(Sp(ni) x • • • x Sp(ne)) respectively, so the F]x(ni,.: .,ne) can be given the structure of real manifolds. These are the real, complex and quaternionic flag manifolds. If ni = • • • = ne = 1 we refer to . . . , 1) as a complete flag manifold. Let us write F(«i , . . . ,«*) = Fffi(n1;.. .,ne) for the real flag manifolds and F(n) = F(l,..., 1) for the real complete flag manifold of 1-dimensional subspaces of Rn. Since F(2) « RPl w 5 1, we have 7TiF(2) = ^. We compute 7T!F(n) for n > 3 in section A by examining the universal covering of F(n) by Spin(ra) after giving a detailed exposition of the definition of the spin groups via Clifford algebras. In section B we construct the universal covering of F(4) using the universal covering of 50(3) by SU(2) and identify a class of trivial bundles over RP3. A. Fundamental Group of the Complete Flags For n > 3, consider the universal covering of F(n) by Spin(n). Spin(ra) 2 : 1 SO(n) 2"-1 : 1 F{n) « 0(n) /0 ( l ) n = SO(n) / (0(l)n f)SO(n)) Here dim^ V denotes the dimension of the subspace V over the division ring K. 10 If Spin'(n) is the finite subgroup of Spin(n) which maps onto the subgroup 0(l)n n SO(n) C SO(n), then we have a continuous bijection from the orbit space Spin(ra)/Spin'(n) to F(n) defined by the universal property of the quotient. Spin(n) -+ SO{n) Spin(n) /Spin'(n) SO(n) / (0(l)n n SO(n)) ta F(n) This map is a homeomorphism, being a continuous bijection from a compact space to a Haus-dorff space. By standard results on proper group actions, which we review in the following subsection, it then follows that TTiF(n) ^ Spin'(ra).2 We need thus only examine the definition of the universal covering of SO(n) by Spin(n) to determine the subgroup Spin'(ra). Let us note for completeness that the situation for the complex and quaternionic complete flags is simpler than the real case. Indeed, we can compute the fundamental group of all the remaining flag manifolds by diagram chasing using the long exact homotopy sequences of various fibrations. These results, among others, are collected in appendix D. 1. Proper Group Actions and the Fundamental Group Suppose a topological group H acts continuously on a space X. A neighbourhood U C X is called proper if h U n U = 0 for all h € H — {1}. The group action is called proper if each point in X has a proper neighbourhood. Note, in particular, that a proper action must be free: h • x = x if and only if h = 1. 1.1 Proposition. Suppose a topological group H acts continuously on a space X. If H acts properly on X then the projection p : X —• X / H is a covering map. Proof. Since the map x H-• hx : X —• X is a homeomorphism for each h € H, if U C X is a proper neighbourhood then P~1(P(U))= |J hU is a disjoint union of open sets. For each h € H, the homeomorphism x >-> h~lx : X —> X is a covering transformation. Thus, to show that p\hu '• hU w p(U), it suffices to show this for h = 1. However, this is clear since p\u is a continuous, open bijection. • A finite group G acting freely on a Hausdorff space X acts properly. For if x 6 X then for each h € H — {1} we can choose neighbourhoods Uh of x and Vh of hx such that UhC\Vh = 0, and f| (uhnh-lvh) heH-{i} 2 Of course, n-.F(n) 9* 7TiSpin(n) 5=! 7r,50(n) for all i > 2. 11 is then a proper neighbourhood of x. Given a topological group G, a discrete subgroup H C G acts properly on G by multiplication. To see this, choose an open neighbourhood W of the identity element such that WnH = { 1 } . If x € G then, since the map (</', g) g'g'1 : G X G —• G is continuous, there is a neighbourhood V of x such that VV-1 = { f i f ' f i f - 1 | 5 , 5 ' € V} C IF. V is necessarily a proper neighbourhood of x since hV n V ^ 0 if and only if ^ G F F - 1 . Assume now that H acts properly on X. Given a covering transformation / € C o v ( X , X / H ) then, since the action of i? is free, there is a function <f>f : X —• H denned by the condition f(x) = 4>/(x) • x. In the special case of X = G, a group, and H a subgroup acting by multiplication then <f>j : G —• H is defined by 4>/(x) = f(x) • x~l. If, further, H is discrete then <j>j is the constant map <j>/(x) = <p/(l) = /(l), and each covering transformation is multiplication by an element of H. This function is necessarily the inverse of the obvious homomorphism H —> Cov(G,G/H). 1.2 Proposition. Let G be a group and let H C G be a discrete subgroup. Then the m a p is an isomorphism, u 1.3 Corollary. Let G be a simply connected topological group and let H C G be a discrete subgroup. Then iti{G/H) H. Proof. p:G^ G/H is a universal covering so TTI (G/H) ^ Cov(G, G/H). u 2. Clifford Algebras Let us now examine the definition of Spin(ra) via Clifford Algebras so as to identify Spin'(ra). The Clifford algebra of a pair (V, q) consisting of a real vector space V and a symmetric bilinear map q : V X V —> M. is the universal element of the functor from the category of real, associative, unitary algebras to the category of sets which assigns to each algebra A the set of those linear transformations <j> € hom(V, A ) satisfying <f>(v)2 = q(v,v) • 1. In other words, the Clifford algebra of V is a real algebra C(V, q) together with a linear map 9 : V —• C(V,q) satisfying 0(v)2 = q(v, v) • 1 for each v £ V, such that if <J>: V —• A is another such map then there is a unique algebra homomorphism / : C(V, q) —• A such that the diagram / •"•/(!) : C o v ( G , G / H ) ^ H V C(V,q) f + A commutes. 12 Other examples of universal algebras: • The tensor algebra of a real vector space, oo i=0 where <g>°V = R and = V, is universal for all linear maps / : V —• A. • The exterior algebra of a real vector space, A(V) = T(V)/I where I is the ideal generated by the elements v ® v with v € V, is universal for all linear maps / : V —• A satisfying f(v)2 = 0. In fact, the exterior algebra is a Clifford algebra: Take q identically zero. Let us write a • 1 = a for a multiple of the unit element of an algebra. We can construct the Clifford algebra as we do the exterior algebra: Let I be the ideal of T(V) generated by elements of the form v <g> v — q(v, v). Set C(V) = T(V)/I and define 6 : V —*• C(V, q) by composing the inclusion V T(V) with the projection T(V) -+ C(V, q). C(V,q) It is an easy exercise to show that 0 : V —* C(V, q) satisfies the universal property. We will write vt • • for the image in C(V, q) of an element v\ <g> • • • <g> Vk G T(V). In the finite-dimensional case, a basis for the Clifford algebra is given by the following result, which we state without proof.3 1.4 Theorem. Suppose that dim V = n and that e\,..., en is a basis for V having q(ei, ej) = 0 whenever i ^ j. Then the unit element 1 together with the elements ej, ---e^ € C(V,q), with t'i < • • • < it and 1 < k < n, form a basis for C(V, q). In particular, dim C(V, q) = 2". • Of course, the basis e\,...,en always exists since q is symmetric: Its matrix representation with respect to any basis can be diagonalized. Let us refer to such a basis as being orthogonal with respect to q or simply orthogonal when q is understood. We will identify R with the subalgebra R • 1 C C(V, q) and V with its isomorphic image 8(V)CC(V,q). 3 See, for example, [L]. (Chapter XIV, section 8.) 13 1.5 Proposition. Ifv,w£VC C(V,q) then vw + wv = 2q(v,w). In particular, vw + wv = 0 whenever q(v, w) — 0. Proof, (u + w)2 = v2 + vw + wv + w2 = q(v, v) + vw + wv + q(w, w) = q(v + w, v + w) = q(v, v) + 2q(v, w) + q(w, w) u 1.6 Corollary. If v, w G V C C(V, g) tiien vwv = 2?(u,«;)t;— In particular, vwv G V. Proof, vwv = (2q(v, w) — wv)v — 2q(v, w)v — wvv = 2q(v, w)v — q(v, v)w m If v G V, let i?9 i„ : V —> V be the linear transformation defined by Rq%vw = 2q(v, w)v — q(v, v)w. Thus, vwv = Rq<vw in C(V,q). We will write Rq>v = Rv when q is understood. The following properties are immediate. • Rvv = q(v, v)v • = — q(v, v)w whenever q{v, w) = 0 . R2v = q(v,v)2I • Rav = a2i2„ for all a G 1R The intuition for Rv comes from the following. 1.7 Proposition. Suppose V is a real inner product space with inner product g. If a G JR.— {0} and u,«; G V with (^u, u) = 1 tiien RagtVw = —a[w — 2g(v, w)v]. That is, Rag,v is the multiple —a of reflection across the plane perpendicular to v. m 1.8 Proposition. If v,w,w' G V then q(Rvw,Rvw') = q(v, v)2q(w, w'). Proof. q(Rvw,Rvw') = q(2q(v, w)v — q(v, v)w, 2q(v, w')v — q(v, v)w') = 2q(v, w) • 2q(v, w') • q(v, v) — 2q(v, w) • q(v, v) • q(v, w') — q(v, v) • 2q(v, w') • q(w, v) + q(v, v)2q(w, w') = q(v, v)2q(w, w') u 1.9 Corollary. If v, w,w' £V and q(v, v) G {1, -1} then q(Rvw, Rvw') = q(w, w'). u The corollary to the following identity will become useful. 1.10 Proposition. If v, w G V and a G R then Rv+aww = 2q(v + aw,w)v + [a2q(w,w) — q(v, v)]w. 14 Proof. Rv+aww = 2q(v + aw,w)(v + aw) — q(v + aw,v + aw)w = 2q(v + aw, w)v + [2aq(v + aw, w) — q(v + oil), v + aw)]w = 2q(v + aw, w)v + [q(v + aw, aw) + q(v + ait;, aw) — q(v + aw, u + aw)]w = 2q(y -f aw, w)v + [g(u + aw, aw) + a(u + aw, aw — (v + aw))]w — 2q(v + aw, w)v + [q(v + aw, aw) — q(y + aw, u)]w = 2q(v + aw, w)v + q(v + aw, aw — v)w = 2q(v + aw, w)v + [a2g(w, w) — q(y, v)]w m 1.11 Corollary. If a G {1, —1} and v, w 6 V" witA u) = g(w, w) then Rv+aww = aq(v + aw,v + aw)v. Proof. Rv+aww = 2q(v + aw,w)v = 2aq(v + aw,aw)v = aq(v + aw,v + aw)v m Let us henceforth assume that the vector space V is finite dimensional. We will find several occasions to use the fact that an epimorphism from one finite dimensional vector space to another of the same dimension is an isomorphism. There is an anti-automorphism of the tensor algebra T(V) which maps a product vi <g> • • • <g> Vk to the reversed product Vk <S> • • • <8> v\. Since an element of the form v <g> v — q(v, v) is mapped to itself by this morphism, we get an anti-endomorphism of the Clifford algebra which makes the following diagram commute. vi ® • • • <g> vk i-T(V) 0 C(V,q) — V l . . . V k , —>nv) 6 >Vk---V! = (Vi •••Vk)* Since this anti-endomorphism is onto, it is in fact an anti-automorphism. Next, let us define an endomorphism /? : C(V, q) —>• C(V, q) by applying the universal property of the Clifford algebra to the linear map — 6 : V —* C(V, q). C(V,q) C(V,q) Again, /3 is an isomorphism as it is clearly an epimorphism. In fact, since (52 ° 0 = 0 we have (32 = 1 by uniqueness of the algebra homomorphism in the universal property. 15 The automorphism 3 allows us to define a decomposition of C (V, q) as the sum of two vector subspaces: If Ce[V, q) = ker (/?-1) and C0(V, q) = ker (/3+1) then C{V, q) = Ce(V, q)+C0{V, q) and Ce(V, q) D C0(V, q) = {0}. Also, the correct relationships between products hold to make the decomposition a Z2 grading. Ce(V, q) • Ce(V, q) C Ce(V, q) C0(V, q) • C0(V, q) C Ce(V, q) Ce(V,q)-C0(V,q)cC0(V,q) C0(V, q) •Ce(V, q) C CQ(V, q) In particular, Ce(V, q) C C(V,q) is a subalgebra, the even Clifford algebra for the pair (V, q). The following follows easily from the corresponding result for the full Clifford algebra.4 1.12 Proposition. Suppose that dimV = n and that ei,...,e n is a basis for V having q(e{,ej) = 0 whenever i ^  j. Then the unit element 1 together with the elements • • •e^  G C(V, q), with ii < • • • < ik, 1 < k < n and k even, form a basis forCe(V, q), and dim Ce(V, q) = 2 n _ 1 . Also, the elements e,-, • • -e^  G C(V, q), with i\ < • • • < ik, 1 < k < n and k odd, form a basis for C0(V, q), and dim CQ(V, q) = 2n~l. m Note that the products e,ej, i < j, generate the even Clifford algebra. Let us now identify those elements of the Clifford algebra which either commute with each element of V C C(V,q) or which anticommute with each of V. Since C(V, q) is generated as an algebra by the unit element 1 and the elements of V, the former is the center subalgebra, Z(C(V, q)), while the latter is a vector subspace of C(V, q) which we will denote by Z(C(V, q)). If ei, , en is an orthogonal basis for V and if i\,...,ik G {l,...,ra} are distinct then, since the e{ anticommute in C(V, q), we have the following. ' (-l^eee^ •••eik if 1$ {ii,...,ik} < (-l)k~1 en e,-, •••eik if I G {h,...,ik} This identity has several immediate consequences. • If k < n then e<t •••e,-fc G Z(C(V,q)) if and only if k is even and g(e,j,etl) = q(.eik,eik) = 0 • If k < n then •••eik G Z(C(V,q)) if and only if k is odd and g(etl,etl) = • • • • en G Z(C(V, q)) if and only if n is odd or g = 0 • ei • • -en G Z(C(V, q)) if and only if ra is even or q — 0 These properties lead immediately to the equalities in the following result. 4 See the addendum to the current section (page 26) for the counting argument. 16 1.13 Proposition. Let e\,... ,e„ be an orthogonal basis for V. l.Ifq is nondegenerate then the following hold. R • 1 if n is even R • 1 + R • e\ • • • en ifnis odd R • e\ • • • en if n is even Z(C(V,q)) = Z(C(V,q))={ {{0} if n is odd 2. If q is degenerate then the following hold, where VQ C V is the subspace spanned by those ei having g(ej,ej) = 0. (Ce(V0,q) Z(C(V,q))={ if n is even CE(VQ, q) + R • e\ • • • en if n is odd ' C0(Vo,q) + R • e\ • • • en if n is even Z(C(V,q))={ [Co(V0,q) if n is odd Consequently, the subspace of C(V,q) spanned by e\ •••e„ is independent of the particular choice of the orthogonal basis. Proof. We need only demonstrate the uniqueness of the subspace spanned by t\ • • -en. Sup-pose u\,...,un is another orthogonal basis for V. Let us consider the case when q is nonde-generate. If n is even and then Z(C(V,q)) = ffi-ei •••c n = •••«„ and we are done. If n is odd then Z(C(V, q)) = R • 1 + R • ex • • • en = R • 1 + R • ui • • • un so that = a • 1 + b • e\ • • • en for some a,b £ R. Thus, a • 1 = u\ • • • un — b • e\ • • • en € C0{V, q), implying that a = 0 since C0(V, q) D R - 1 = {0}. Consequently, Ui • • -un — b-ei • • -e„, and 6 ^ 0 is necessary since u\ • • • un is a basis element. Thus, R • e\ • • - en = R • « x • • • un as claimed. Similarly, the case when q is degenerate is a consequence of the facts Ce(Vo,q) D Co(V, q) = {0} and Co(V0, q) fl CE(V, q) = {0}. • The Nondegenerate Case Suppose now that the symmetric, bilinear map q : V x V —• JR is nondegenerate. Then q is either positive definite or negative definite and we can choose e £ {1, —1} so that eq is an inner 17 product on V. We will use the terms orthogonal and orthonormal in relation to this inner product. Let Sq = { v € V \ q(v, v) = e} be the unit sphere in V. Since every element of V is a multiple of an element of Sq, this set generates C(V,q). Let C(V,q)* be the group of multiplicative units in C(V, q). We have V — {0} C C(V,q)* since v2 = q(v,v) ^ 0 for each v G V. Let U(V, q) be the subgroup of C(V, q)* generated by Sq. Note that -1 € U(V, q) since for any i 6 5, we have x • x = e and x • — x = — e both in U(V, q). U(V, q) necessarily contains all products x\-"Xk with x\,...,Xk £ Sq and k > 1. However, since xi • • -Xk{xi • • = £k € {1, —1}, all elements of U(V, q) are of this form. Thus, U(V,q) = {x1---xk\x1,...,xkeSq,k>l}. Now set Ue(V, q) = U(V,q) f)Ce(V, q), the subgroup of U(V, q) consisting of products of even numbers of elements from Sq. Clearly U(V, q) — Ue(V, q) — U(V, q) D C0(V, q) so that Ce(V, q) and C0(V, q) partition U(V, q). We have already seen that if ei,...,e n and u\,...,un are two orthogonal bases for V then e\ • • • en and u\- --un span the same subspace of C(V,q) so that ei • • - en — au\ • • • un for some non-zero a € JR. If these bases are both orthonormal then the equalities (ei•••en)(ei•••en)* = en = (aui • • •un)(oui • • - Un)* = a2en give a2 = 1. 1.14 Proposition. Let ei, . . . , en be an orthonormal basis for V. Then {1, —1} if n is even Z(C(V,q))nU(V,q) Z(C(V,q))nU(V,q)= { {1,—l,e, —e} if n is odd {e, —e) if n is even if n is odd wiiere e = e\ • • • er Proof. We have already determined, in Proposition 1.13, that Z(C(V,q)) is spanned by 1 if n is even and 1 and e if n is odd. Thus, the first equality is clear for even n. For odd n, no element a • 1 + b • e with a and 6 both non-zero can lie in U(V, q) since 1 € Ce(V,q) and e 6 C0(V,q): Each element of U(V, q) lies in either Ce(V, q) or C0(V, q) and such a sum lies in neither of these. The second equality follows immediately from our determination that Z(C(V, q)) is spanned by e if n is even and is the trivial subspace if n is odd. • 18 Now let Autq(V) consist of all those endomorphisms of V which preserve the norm (and, hence, also the inner product). Aut,(V) = { T e hom(V, V) \ q(Tv, Tv) = q(v, v) for all v G V } Since every such endomorphism is necessarily an isomorphism, Autg(V) is a subgroup of the group Aut(V) of automorphisms of V. Recall that if v G V then the endomorphism Rv : w *-* 2q(v, w)v — q(v, v)w : V —• V was defined so that vwv = Rv(w) in C(V, q). By Corollary 1.9, Rv G Autg(V) whenever v G Sq. Thus, there is a representation 7 : U(V, q) —• Autq(V) defined by y(u)v = uvu*. If u = xi • • -Xk then f(u) = RXl 0 . . . 0 RXk. For v € V — {0}, let us write Fv = Rv/r(v) where r : V — {0} —*• Sq is defined by r(v) = v/y/eq(v, v). Thus, Fv = Rv whenever v € Sq. The following properties of the function v I—• Fv are easy consequences of the four properties of v 1—• Rv stated on page 14. • Fvv = ev • Fvw = —ew whenever w is orthogonal to v •Fl=I • F a v = Fv for any a G R - {0} In addition, there is the following as a consequence of Corollary 1.11. • Fv+aww = aev whenever v, w € Sq, a € {1, —1} and t; + aw ^ 0 The following proposition will aid us in identifying the image of 7. 1.15 Proposition. Suppose that q is negative definite, or that q is positive definite and V is of even dimension. If v,w G Sq then there exists an automorphism T G im 7 such that Tv = w and Tz = z whenever z is orthogonal to both v and w. Proof. If v = w we can take T = I so let us assume v / w. Of course, Fv = j(v/r(v)) G im 7 for any v G V - {0}. 1. q negative definite. In this case e = — 1 and Fv-W has desired properties. 2. q is positive definite and V is of even dimension. In this case e = 1 and Fv-W has the following properties. • Fy-WW = -V • Fv-Wz = —z whenever z is orthogonal to v and w Thus, we can take T = — Fv-W once we show —I G im7. To show this, let e\,...,en be an orthonormal basis for V. Then, since n is even, y(ei • • -en) = F e „ o • • • 0 F e i = —/ and we are done. • 19 1.16 Proposition. Suppose that q is negative definite, or that q is positive definite and V is of even dimension. Then the representation 7 : U(V, q) —• Autq(V) is onto. Note that 7 cannot be onto if q is positive definite and V is of odd dimension. In this case it is easy to see that ~y(x) — Fx has determinant 1 for all x (E Sq.5 In particular, —I £ im 7. Proof. Suppose A € Aut,(V) -{/}. We will find T i , . . . , Te € im 7 such that A = Tt ° • • • ° Tx. Let ii to be the least integer such that Ae^ ^ e,-,. There is an automorphism Ti € im7 such that Tiejj = Ae{1 and Tiz — z whenever z is orthogonal to both and Ae,-,. Since, for i < i\, ej = Aei is orthogonal to both e{1 and Aei1, T\e{ = Ae{ for all i < tj. If ix = n we are done. Otherwise let us suppose we have chosen integers 1 < i\ < • • • < ik < n and automorphisms T\,...,Tk G im7 such that, (Tfc o . . . o T\)ei = Aei for all i < ik- Let ik+i be the least integer such that (Tk o . . . o Ti)ei t + 1 7^  Aej t + 1 . Necessarily z'fc+i > ik-There is an automorphism T +^i € im7 such that Tk+i((Tk o . . . o Ti)ej t + 1) = Aeik+1 and Tk+iz = z whenever 2 is orthogonal to both (T;t o . . . o Ti)e l j t + 1 and Aej f c + 1. Since, for * < (Tk o ... o Ti)ei — Aet is orthogonal to both (Tk o . . . o Ti)ej t + 1 and Ae,-fc+1, Tk+i({Tk o • • • o r ^ C i ) = (T f c + 1 o . . . o Tjei = Ae{ for all t < ik+1. Continuing in this way, we eventually get T i , . . . , Te as desired. • 1.17 Proposition. Let e\,.. .,en be an orthonormal basis for V and let e = e\ • • -en. Then the kernel of the representation 7 : U(V, q) —> Autq(V) is given by the following. ker 7 = < {1, —1, e, — e) if q is positive definite and V has odd dimension {1,-1} otherwise Proof. If u G U(V,q) then •y(u) = I if and only if ux(u*u) = xu for all x € Sq. Let us now consider the negative and positive definite cases separately. The results in both cases follow easily from our determination of Z(C(V, q)) f] U(V, q) and Z(C(V, q)) D U(V, q) in Proposition 1.14. 1. q is negative definite. In this case u*u = 1 for all u £ Ue(V,q) and u*u = —1 for all u e U(V,q)-Ue(V,q). Thus, kerT n Ue(V, q) = Z(C(V, q)) n Ue(V, q) = {L-1} and 5 Choose an orthonormal basis xi,...,x„ of V with £1 = x. Then Fxx\ = x\ and Fxxi = —Xi for all 1 < i < n so that det Fx - (-1)""1 = 1. 20 ker 7 n {U(V,q) - Ue(V, q)) = Z(C(V, q)) D {U(V, q) - Ue(V, q)) = 0. 2. q is positive definite. In this case u*u = 1 for all u € U(V, q) so that ker 7 = Z(C(V, q)) n U(V,q). m Clifford Algebras on Euclidean Space Let us now specialize to the cases (V, q) = (Rn,g) and (V, q) = (Rn, — g), where g(v, w) = v • w is the usual inner product on Rn. We will write r(Rn,+) = C(Rn,g) and T(Rn,-) = C(Rn,-g) as well as Te{Rn,+) = Ce(Rn,g) and Te(Rn, -) = C e(lR n , -g). Let e\,..., e„ now denote the standard basis for Rn. The following result shows that the algebras T(Rn, +) and T(Rn, —) are completely determined by TiR^-r), r(R\ -), T(R2, +) and T(R2,-). 1.18 Proposition. For every n > 1, there are the following algebra isomorphisms. r(Rn+2,+)^r(Rn,-)®r(R2,+) T{Rn+2,-)^T(Rn,+)®r(R2,-) Proof. We construct both isomorphisms using the universal property of Clifford algebras. The first isomorphism will be defined by the following diagram. e , - 1 - » ej <g> ei e2, 1 < i < n en+i 1 ® ei e n + 2 1 ® e2 r(iR n + 2 ,+) - r(jR n,-)®r(]R 2,+) To show that the above diagram does indeed define an algebra homomorphism / , which we will soon see to be an isomorphism, we need only show that the linear map <f> satisfies (j>{v)2 = v • v for each v € JR n + 2 . It is easy to check that each <£(e,)2 = 1 for each i and that (f>(ei)<j)(ej) — —4>{ej)(j){ei) whenever i ^  j. It follows that (j>(v)2 = v • v for all v 6 Rn+2 since cross terms in the square of / applied to a linear combination of the ej will cancel. Thus, / is uniquely defined by the universal property. Moreover, / is onto since /(en+ie,en+2) = ej <g> 1 for 1 < i < n. Thus, since the domain and range of / have the same dimension, / is an isomorphism. The second isomorphism is defined similarly. ej 1 — • ej ® eie2, 1 < i < n e n + i »-+ 1 <g> ei e n + 2 i-> 1 <S> e2 r {Rn+2,-)----- r (Rn, +) ® r(ia 2, -) 21 Verification that this diagram defines a homomorphism of algebras is, in this case, an easy exercise in checking that <j>(v)2 = — v • v for each v € R n + 2 using the same reasoning as above. Again, we get an isomorphism by reason of surjectivity and dimension. • Identical reasoning to that of the above can be used to construct several additional algebra isomorphisms. In each case we will require knowledge of the square of the element e\ • • • en in either F ( lR n ,+) or T(Rn,—). Indeed, if ii,...,ie G {l,...,n} are distinct integers then (e^  • • - 6 j J 2 is easily computed using anticommutation of the e,- in both Clifford algebras plus the fact that e? = 1 in r ( JR n ,+) and e? = -1 in r ( JR n , - ) . T(R",+): (eil---eie)2 = (-l)i«<-»> = r(JR",-): (eh...eit)2 = ( - l ) ^ ) = 1 if £ = 0 mod 4 or I = 1 mod 4 -1 if 1 = 2 mod 4 or £ = 3 mod 4 1 if e = 0 mod 4 or £ = 3 mod 4 -1 if £ = 1 mod 4 or £ = 2 mod 4 (1.1) (1.2) Thus, the following diagrams are seen to define algebra isomorphisms providing the stated condition is satisfied. = 0 mod 4 R n + 1 ej H-+ ej <g> 1, 1 < i < n en+x H->- t\ • • • en ® ei r( lR , l + 1 , +)) * r(lR", +) ® T(]R}, +) (1.3) n = 2 mod 4 n+l ej t-> ej ® 1, 1 < i < n en+i i-> ei • • -en ® ei r( ia n + 1 , +) ^ — ^ r(n n , +) ® r(iR1, -) (1.4) n = 2 mod 4 ej i — • ej ® 1, 1 < i < n en+i i-* ei • • -en ® ei r ( i a n + 1 , -)» w T(Rn, - ) ® T ^ 1 , +) (1.5) n = 0 mod 4 r ( i R n + 1 , - ) ej i — • ej <S> 1, 1 < i < n e n + i ei • • - en ® ex T ( iR ' l , - )®r ( lR 1 , - ) (1.6) 22 n = 2 mod 4 JR n + 2 r(iRn+2,+) e, i—• ig> 1, 1 < i < n en+i ei • • • e„ <g> ei e n+2 »-»• ei • • • en <g> e2 r(JR",+)®r(JR2,-) (1.7) n = 2 mod 4 Rn+2 e,- i-+ ej <8> 1, 1 < i < n T{Rn+2,-) en+i i-v ei • • - c„ ® ei e „+2 •-»• ei • • • e„ <g> e2 -*r(2Rn,-)<g>r(2R2,+) (1.8) The Clifford algebras r(JR1,+), rfJR1,-), r(JR2,+) and T(JR2,-) are identified by isomor-phisms constructed by the following diagrams. Here 2R(2) is the algebra of real 2x2 matri-ces and H is the quaternionic algebra spanned by 1, i, j and k with i2 — j2 = k2 = —1, ij = —ji — k, jk = —kj = i and ki = —ik = j. R1 ei >-> (1,-1) T(iR1,+) JR2 61 ~(o -i)'e2~(i J) T(JR2,+) -*JR(2) With these identifications, let us restate the isomorphisms of Proposition 1.18 and (1.3) through (1.8). r (2R n + 2 , +) S T(2Rn, -) <g> R(2) r(jRn+2,-)^r(jRn,+)®iff 23 r( jR n + 1 +) ^ r(jan,+)^  ${R®R) ^T(Rn,+)®T{Rn,+) r ( j R n + 1 -) ^r(Rn,-)$ ?> <c r ( j R n + 1 -)^r(Rn,-)$ ?> (R®R) r ( j R n , - ) © r ( j R n , - ) r{Rn+1 +) *r(Rn,+)s 5 € r{Rn+2 +) *r(Rn,+)$ $E T(Rn+2 -)<=r{Rn,-)$ $R(2) n = 0 mod 4 • n = 2 mod 4 The center subalgebras Z( r ( JR n ,+ ) ) and Z(V(Rn, -)) can be identified using the order of the element e\ • • -en since we already know by Proposition 1.13 that each algebra is spanned by 1 if n is even and 1 and e\ • • • en if n is odd. For odd n, we have an isomorphism from either center to R © R or C, depending on whether ex • • -en has order 2 or 4. The isomorphism sends e\ • • -en to either (1, —1) G R(B R or i G <C, as appropriate. Z(T(Rn,+)) = < Z(r(JRn,-)) £* < ' R if n is even R(& R if n = 1 mod 4 > <C if n = 3 mod 4 ' ft if ri is even < <C if n — 1 mod 4 k R&R if n 3 mod 4 For the even Clifford algebras, the elements e;en+i, with t < n, generate r e(JR n + 1,+) and Te(Rn+1,-) since e;en+1 • ejen+i = e^j • -e2n+1. The following diagrams define algebra isomorphisms. Rn T(Rn, —) ei *-* eien+i Te(Rn+1,+) (1.9) Rn r(Rn,-) ei e,en +i •»re(JtB+1,-) (1.10) 24 Spin Groups Most of the work has now been done in the more general setting. Let pin(n) = U(Rn, —g) and Spin(n) = Ue(Rn, -g)- We have Aut_5(JR") = 0(n) and the restriction of the representation 7 : pin(n) —> 0(n) to Spin(n) defines a representation ye : Spin(n) —> SO(n) with kernel {1,-1}. This representation double covers SO(n) and is a universal covering when n > 3. 3. Clifford Groups We are now in a position to identify the subgroup Spin'(ra).= j~1(SO(n) DO(l)n) = Spin(n) n 7 _ 1 (C(l) n ) C Spin(n). Set pin'(n) = 7 - 1 (0(l) n ). Since 7 maps a standard basis vector e,- to Fei, which is reflection across the plane perpendicular to e^ , pin'(ra) is exactly the subgroup of pin(n) generated by the standard basis vectors. Thus, pin'(n) is generated by e\,..., en with relations e\ — • • • = e2n = —1, and e,ej = — e^ e,- whenever i ^  j. This group is well-known: It is the Clifford group on n generators, Cn-6 Note that Cn has order 2 n + 1 . Indeed, each element of Cn has a unique representation as a product ee,-, • • -e^  with e £ {1, —1} and 1 < ii < • • • < it < n.7 Notice that Proposition 1.14 tells us that the center of Cn is {1, — 1, e, —e} if n = 2m +1 is odd, where e — e\ • • • e„. In this case (1.2) tells us that e has order 2 if n = 3 mod 4 and order 4 if n = 1 mod 4, giving Z ( C n ) — %2 x ^ 2 and Z ( C n ) — %>\ respectively. Since Proposition 1.14 also tells us that the center of Cn is {1,-1} if n is even, the claim on page 156 of [G], that there are isomorphisms Cn — Hn/2 if n is even and Cn — #(n-i)/2 x ^2 if n is odd, where Hn are finite Heisenberg groups over Z2, is false. Indeed, the author there first states that Z ( C n ) — Z2 x Z2 for all odd n and this, as we have just seen, is also false. We have not verified which of the claimed isomorphisms do exist, but certainly we cannot have both C\n — G and Cnn+i — G x Z2 regardless of the identity of the group G since Z ( C 4 „ + i ) ^ Z 4 while Z ( C 4 n x Z 2 ) Z 2 x Z 2 . Instead, regarding Cn as the subgroup of C n + i generated by a,... ,en £ Cn+i, there is a homomor-phism (a, (-1)*) *-* aek : Cim x Z 2 —• C 2 m + i if m is odd and a homomorphism (a, ih) !-• ae* : C 2 m x Z 4 —» C2m+i if m is even, regarding Z2 and Z 4 as the multiplicative subgroups {1,-1} and {1,-1, i, —i} of S 1 C <C. Both of these homomorphisms are onto since (en • • • ei)e = (—l)"en+i, and their kernels are {(1,1)} and K = {(1,1), (—1, — 1)} respectively. Thus, C2m+i — C2m x Z2 if m is odd and C2m+i — (^m x Z4)/AT if m is even. Note that C\ and C 2 are thus easily recognizable as Z 4 and Qs = { ± 1 , ±ij ± i j C M respectively. 25 We can now identify Spin'(n) by examining the Clifford algebra isomorphism (1.10). R n T ( R n , - ) T e ( R n + \ - ) Since this isomorphism maps pin'(n) C r(JRn, —) onto Spin'(ra + 1) we have Spin'(ra + 1) = pin'(n) = Cn for all n > 1. Thus, Spin'(n) = C n _ i for all n > 2 and we have found the fundamental group of all the real complete flags. 1.19 Proposition. 7 ^ ( 2 ) = Z and TTiF(n) ^ C„_i for all n > 3. • 4. Addendum: Counting by Twos Given an integer n > 1, define integers de(n) and d0(n) as follows. de(n) = < if n is even if n is odd dQ(n) = < .GK)+-G) if n is even if n is odd If V is a vector space of dimension n then any Clifford algebra C(V, q) has dim Ce(V, q) = de(n) and d\mC0(V,q) = d0(n). If ra is odd then, since (£) = (n2fc)> w e n a v e ^ » ) = ( ; ) + ( : ) + - + ( : ) = 2 « so that de(n) = 2 n _ 1 . If n > 4 is even, apply the identity (™) + (^) = (7+1) with m + l = n and £+1 = 2,...,« — 2 to get ^ = ( : ) < > ( T ) + - - < M : > C : ) so that de(n) = 2n~1 in this case as well. Clearly this also holds when n = 2. Since de(n) + d0(n) =2n, we also have d0(n) = 2 n _ 1 for all n. 26 B. Triviality of Certain Quotients of 0(4) over JRP 3 In addition to their coverings by Spin (3) and Spin (4) respectively, universal coverings of F(3) and F(4) can be constructed using the universal covering of 50(3) by SU(2). SU{2) « 5 3 2 : 1 50(3) 4 : 1 F(3) « 0(3) /0(1) 3 = 50(3) / (0(1)3 n 50(3)) 5 3 x SU{2) » 5 3 x 5 3 2 : 1 50(4) « 5 3 x 50(3) 8 : 1 F(4) « 0(4) /O(l ) 4 = 50(4) / (O(l)4 D 50(4)) Let us examine these universal coverings to determine the action of the group of covering transformations on 5 3 and 5 3 x 5 3 respectively. First, recall the construction of the covering of 50(3) by SU(2). We have of topological groups, where we regard 5 3 C iff as the unit quaternions and <C as a subalgebra of H in the usual way. Embedding JR3 in If as the imaginary quaternions, conjugation by an element of 5 3 defines an element of 50(3) and the homomorphism 7 : 5 3 —• 50(3) which maps u € 5 3 to conjugation by u is a 2-fold universal covering. It is clear that 7"1(0(1)3 n 50(3)) = Qs C 5 3 where Q 8 = {±1, ±i, ±j, ±k} C M is the order 8 quaternionic group. Since the covering 7 is a homomorphism of groups the following diagram /) 6 C , H 2 + l/5|2 = 1 as well as an isomorphism 27 is easily seen to define a map S3/Qs —• F(3) by the universal property of quotients. 5 3 • 50 (3) 5 3 j Qs • 50(3) I (O(l)3 n 50(3)) •« F(3) Further, this map is a homeomorphism being a continuous bijection from a compact space to a Hausdorff space. Thus, F(3) « S3/Q8, 7rxF(3) = Q8 and each covering transformation of 5 over F(3) is multiplication by an element of Q%. To examine the covering of F(4) let us first construct a homeomorphism 50(4) —• 5 3 x 50(3). Identifying the underlying vector space of the quaternionic algebra H with R4 in the usual way (1 *-+ i\, i *-+ e~2, j £3, k £4) and 50(3) with the subgroup of 50(4) consisting of those transformations which leave t\ fixed, the principal bundle 50(3) < .50(4) A S3 Aex has s : x 1 — • (x \ xi \ xj \ xk) : S3 —* 50(4) as a section. It is easily checked that (x \ xi | xj \ xk) is indeed an element of 50(4): The columns of this matrix are orthonormal and it has deter-minant equal to \x\2 = 1. Thus, the map / : A H (Aii,s(Aii)~1 A) : 50(4) -» 5 3 x 50(3) is a homeomorphism with inverse (x,R) >-»• s(x)R. Notice that s(-x) = —s(x) for all x G 5 3. 1.20 Proposition. Let H be any subgroup of 0(3) which contains —I. Then there is a homeomorphism f : 50(4) / (O(l) x H) n 50(4) 5 3 x 50(3) / {1,-l}x(ffn 50(3)) whicii makes the diagram 50 (4) • 5 3 x 50(3) 50(4) / (0(1) x H) D 50(4) ^ S3 x 50(3) / {1, -1} x (H n 50(3)) II • / II 0(4) / (0(1) x H) RP3 x (0(3) /H) RP3 28 commute, where n : 0(4) /0(1) x H —• RP3 : [A] [Ae{[ and nx is projection onto the first factor. Proof. To demonstrate the existence of the homeomorphism / , we let our subgroups of 50(4) and 5 3 x 50(3) act by multiplication on the right. 1. / is well-defined. If A, Be 50(4) lie in the same (0(1) xH)C) 50(4) orbit then A = BD for some D € (0(1) X H) (~l 50(4). Now, either Dei = ex or Dit = -ii. In the first case we have D € H n 50(3) and f(A) = f(BD) = (BDe1,s(BDel)-1BD) = (Bel,s(Be1)-1BD) = f(B) • (1,D) so that f(A) and f(B) lie in the same {1, -1} x (H n 50(3)) orbit of 5 3 x 50(3). In the second case we have — D 6 H D 50(3) and f(A) = f(BD) = (BDe1,s(BDe1)-1BD) = (-Bei, si-Be!)-1 BD) = (-B(et), -s(Bh)-lBD) = f(B) • (-1, -D) so that again f(A) and f(B) lie in the same orbit. • 2. / is injective. Suppose f(A) = f(B) • (s, D) for some (e, D) £ {1, -1} x (H D 50(3)). Applying f~l we have A = r1 (f(B) • (e, D)) = f~1 (sBex, s(Bel)~1BD) = s(eBii)s(Bei)~1BD = £s(Be1)s(Bii)-1BD = sBD — B - eD so that A and B lie in the same (0(1) x H) D 50(4) orbit of 50(4). • Clearly / is surjective. Thus, / is homeomorphism being a continuous bijection from a compact space to a Hausdorff space. Commutativity of the triangle is clear since / maps A G 50(4) onto a pair whose first compo-nent is Aei. • 1.21 Corollary. Tiie following bundles are trivial. F(l,l,l)< ^(1,1,1,1) [A] RP2< ^(1,1,2) [A] RP3 [Ae^ RP3 [Aei] • 29 1.22 Corollary. F(4) « RP3 x F(3) « 5 3 x 5 3 /{ l , - l} x Q 8 . TAus, T T I F ( 4 ) ^ Z 2 X Q 8 and eacA covering transformation of S3 X 5 3 over F(4) is multiplication by an element of {1,-1} x Q 8 . Proof. We have just demonstrated F(4) « JRP3 x F(3). Since the universal covering 1 x 7 : S 3 x S3 —»• S 3 X 50(3) is a group homomorphism the following diagram is easily seen to define a mapping between the coset spaces in its bottom row by the universal property of quotients. S 3 x S3 1^-1 • S3 x 50(3) 5 3 x 5 3 / {1, -1} x Qs • 5 3 x 50(3) / {1, -1} x (0(1)3 n 50(3)) » RP3 x F(3) Being a continuous bijection from a compact space to a Hausdorff space, this map is a homeo-morphism. • 30 Chapter 2 Constructing Spaces over a Polyhedron In this chapter we construct topological spaces lying over an arbitrary polyhedron. The moti-vation for the spaces we construct here is the lattice construction of R. R. Douglas and A. R. Rutherford, whose interest stems primarily from the novel decompositions of certain spheres of real and complex matrices obtained by Douglas and Rutherford in their work on the Jahn-Teller effect.1 The lattice construction is presented in section A and spheres are constructed in this manner in section B. Intuitively, the construction presented in section C captures the idea of replacing the faces of a polyhedron by spaces lying above these faces. We define a category on which our construction is realized as a functor to the topological category and show that this construction subsumes the lattice construction. The construction is presented in a slightly more general form than we actually require in the following chapters since doing so simplifies much of the required notation. A. The Lattice Construction Let A n denote the standard n-simplex in JK n + 1 . A„ = { (60, • • •, 6B) G Rn+1 \bo,...,bn>0,bo + --- + bn = l} As defined in [D], the lattice construction defines a topological space X and a map 3 : X —• A„ from a list Go,.. . , Gn of closed subgroups of a compact Lie group G. The space X is defined as the quotient of G X A„ by the equivalence relation that identifies (g,b) with (g',b') if and We recently learned of the 1974 paper [Ma], in which the author constructs imbeddings of real, complex, quaternionic and Cayley projective planes in spheres of dimension 4, 7, 13 and 25 (respectively) using methods similar to those of Douglas and Rutherford. The decomposition there corresponds to the lattice construction over a 1-simplex. 31 only if b' = 6 = (60,..., bn) and g 1g' £ f\ >o a n < ^ ^  l s * n e m a P induced by the projection G x A n —• A n onto the second factor. Certain spheres can be constructed in this manner by examining the conjugation action of a real orthogonal, complex unitary or quaternionic symplectic group on, respectively, a Euclidean space of real symmetric, complex Hermitean or quaternionic Hermitean matrices. We will see in chapter 4 that many more spheres can be constructed as a lattice construction on products of orthogonal, unitary or symplectic matrices. B. Constructing Spheres By the well known 1877 theorem of Frobenius, there are only three finite dimensional, asso-ciative division algebras over R: R itself, the complex field <C and the quaternion algebra iff.2 Let K denote any of these three and let 0(n) C K(n) denote the corresponding group of inner product preserving matrices. That is, 0(n) is either the orthogonal group 0(n), the unitary group U(n) or the symplectic group Sp(n). Of course, 0(n) acts on Herm(n, JK") by conjugation: P • A = PAP*. Matrices in Herm(n,iK") can be diagonalized over R: If A £ Herm(n, IK") then there is a real diagonal matrix D and a P £ 0(n) such that A = PDP*.3 Moreover, the diagonal entries of D are unique up to a permutation. Let us endow Herm(n, K) with the Hilbert-Schmidt inner product, (A, B) = trAB. Let Hermo(n, K) C Herm(n,JK') denote the real (codimension 1) subspace of trace zero matrices. 2.1 Proposition. The fixed point set of the conjugation action of 0(n) on Herm(n, K) is RI, the set of real multiples of the identity matrix. Proof. Let A £ Herm(n, K) be a fixed point of the action. If A is singular then there is a non-zero unit vector x £ ker A. Since APx = PAx = 0 for all P £ 0(n) and since Px ranges over all elements of the unit sphere in Kn as P £ 0(n), we have ker A = Kn. If A is invertible, choose an eigenvector A £ R and consider A — XI. m For n > 2, let us define a function S : Herm(n, K)-RI —» A„_ 2 as follows. If A € Herm(n, K)-RI has eigenvalues Ai < • • • < A„ then HA) = —1— (A2 - A l t . . . , A„ - An_!). (2.1) A n — A i See, for example, [E] for a modern exposition. (Chapter 8, section 2.) The quaternionic case is treated in appendix A. 32 Notice that 8 has the following properties. • 6(PAP*) = 6(A) for each P G C(n). • 8(aoA -f a\I) = 8(A) whenever ao,ax G R with ao ^  0. Now, each element of Herm(n, K) — RI lies in the orbit of a real diagonal matrix. Let D = (^ 1^ 1 | • • • | A ne n ) be the unique such matrix such that Ai < • • • < A n, having Ai = • • • = Ajj < • • • < Ajj^ —i-tjt+i = • • • = A n and A n - Ai > 0. If P G 0(n) and 8(D) = (b0,6n_2) then PDP^ = D if and only if P G 0(h) x • • • x 0(ik) xO(n-ii ik) = 0(£) x 0(n - f) = p| 0(e + 1) x 0(n - £ - 1) (2.2) be>0 so that the stabilizer of each element of Herm(ra, K) — RI is conjugate to one of these subgroups. There is a homeomorphism A*-+ (A - (tr A)I/n,tt A) : Herm(n, K) - • Herm0(n, K) x ]R with inverse (A, r) i—• A + rZ/n under which the fixed point set of the 0(n) action is mapped to the line {0} X R. There is also a homeomorphism A ^ (A/\\A\\, \\A\\) : Eeim0(n,K) - {0} -> S(n,K) x (0,oo) where S(n, K) C Hermo(n, K) is the unit sphere with respect to the Hilbert-Schmidt norm. Since (counting components of a real basis for K along diagonals of matrices in Herm(n, JK")) dim Herm(n, R) - n + (n - 1) H h 1 = n(n + l)/2 dim Herm(n, €) = n + 2((n - 1) H + l) = n + n(n - 1) = n2 and dim Herm(n, H) = n + 4((n - 1) + • • • + l) = n + 2n(n - 1) = n(2n - 1) as real vector spaces, and since Hermo(ra,K) C Herm(ra,K) has codimension 1, the spheres S(n,R), 5(n, €) and S(n,H) have dimensions n(n + l)/2 - 2, ra2 - 2 and n(2n - 1) - 2 respectively.4 4 To elucidate only the real case, the isometry n n — 1 A = (atj) i-f ^aiie< + 51 z3a,'je'"-('-1)'/2+J—' : H e r m(">K) K"( n + 1 )/ 2 • = 1 i=l i>«' maps 5(n,5l) onto an ellipsoid in the n(n + l)/2 — 1 dimensional image of Hermo(n, E). 33 Let n : Herm(ra, JFQ — RI —+ S(n, K) denote the retraction defined by the first factor of the composition homeomorphism Herm(n, K) — RI —y S(n, K) X (0, oo) X R. _ A-(tvA)I/n " ( A ) - | | A - ( t r , 4 ) / / » | | ( 2 > 3 ) Notice that /J has the following properties. • n(PAP^) = P/i(A)F t for each P £ 0(n). • n(a<}A + ail) — n(A) whenever ao, ai € R with ao 7^  0. . 8 ( A ) = S ( J M ( A ) ) . • n(A) = A if A £ S(n,K). 2.2 Proposition. Let A,B £ Herm(n,JK') — RI. Then p(A) and p(B) lie in the same 0(n) orbit ofS(n,K) if and only if 5(A) = 8(B). Before proving this statement, define a map c : A n _ 2 —• Herm(n, K) — RI as follows. <(&o, • • •,bn.2) = (Oh I b0i2 I (60 + h)e3 I • • • I (60 + • • • + 6«-2)c„ = len) (2.4) If A e Herm(n, K) — RI has eigenvalues Ai < • • • < A n then (A„ - \i)s(8(A)) + XiI=( Xiei I ••• I A ne n ) so that this matrix lies in the same 0(n) orbit as A. Thus, p(A) and n((\n — \i)s(8(A))+\iI) = n(s(8(A))) lie in the same 0(n) orbit of S(n, K). Note that 8 o c = 1. Proof of 2.2. If n(A) = Pp(B)P^ for some P £ 0(n) then 8(A) = S(ji(A)) = 8(Pfi(B)P^) = 6(n(B)) = 8(B). If 8(A) = 8(B) then n(s(8(A))) = n(<;(8(B))), which lies in the same 0(n) orbit of S(n,K) as both n(A) and n(B). u 2.3 Corollary. Let A,B£ S(n, K). Then A and B lie in the same 0(n) orbit if and only if 8(A) = 8(B). u We can now exhibit 5(n, K) as a lattice construction. Consider G = 0(n) together with the n — 1 closed subgroups Go = 0(1) x 0(n - 1),..., Gk-i = 0(k) x 0(n -* ) , . . . , G n _ 2 = 0(n - 1) x 0(1), where 0(k) x 0(n — k) is identified with the subgroup of 0(n) consisting of those matrices whose only non-zero entries lie in the upper left k x k and lower right (n — k) x (n — k) blocks. 34 Let the space X and the map 3 : X —• A N _ 2 be the result of the lattice construction on 0(n) and the aforementioned n — 1 subgroups. There is the following commutative triangle. (P, b) i • /j(P*(6)pt) = P/i(c(6))PT 0(n) x A n _ 2 ^5(n,JK) A n _ 2 By definition, two pairs (P,b),(Q,b) £ 0(n) x A N _ 2 represent the same point in X if and only if Qtp e f \ > o G » - ' a n d t h i s h o l d s 5 f a n d o n l y i f Q^PvteityP^Q = /*(*(*)) by (2.2) since £(/i(c(6))) = <5(c(i)) = b. Thus, the map (P, 6) i—• /z(Pc(6)pt) induces an injective map £ : X —• 5(n, IK). Moreover, £ is surjective since if D = (Aie"i| ••• | A ne n) is a diagonal element of S(n, JK") then D = (A„ - Xi)<;(S(D)) + Ax/ = fi(D) = r[<(8{D))) = *[M(0)]. Since X is compact and S(n, K) is Hausdorff, the continuous bijection £ is a homeomorphism. As an interesting consequence, notice that the function S : Herm(n, JK") — RI —• A N _ 2 is continuous because of the following commutative diagram. Herm(n,JK") - RI—^5(n,JK") C. A More General Construction Let K be a simplicial complex: A set of finite non-empty subsets, called simplices, of some set Vert(K) of vertices, having the property that (1) every subset consisting of a single vertex is a simplex and (2) any non-empty subset of a simplex is itself a simplex. We will often identify a vertex with the one-element simplex containing it. Let \K\ denote the set of all functions b : Vert(X) —• [0,1] such that 6 - 1 (0,1] € K and St,evert(/c) Hu) = 1- F° r e a c n s £ the closed simplex |s| = {6G|/C||6- 1(0,l]Cs} has the metric topology of a standard Euclidean simplex. The realization of K is the set \K\ endowed with the largest topology for which the inclusions |s| <—>• \K\ are continuous. We refer the reader to chapter 3 of [S] for introductory material on simplicial complexes. 35 If u C Vert(K) is any non-empty subset, let \u\ = {b £ \K\ \ b 1(0,1] C u}. The subset u generates a subcomplex of K, u = {s £ K \ s C u}. Note that |u| = {b £ \K\ | 6_1(0,1] £ *} = U.6*W. We want to replace the faces of \K\ by spaces lying over them. In order to show that our defini-tion constructs a space homeomorphic to the lattice construction in the appropriate situation, it will be convenient to define our spaces in terms of an order-preserving function a from a par-tially ordered set A to the set of non-empty subsets of Vert(/C) partially ordered by containment (rather than inclusion). Given such a function, we let \a\ = |<r(a)| and a — {s£K\sC <j(a) }. Note that \a\ = {b £ \K\ | 6_1(0,1] £ a} = \Ja€i \s\. We also require that a satisfy the following property. If s £ K then the set { a £ A \ s £ a} C A is nonempty and contains a lower bound. We will denote this (necessarily unique) lower bound by T(s). Further, we will write a —• a' when a precedes a' in A, viewing the partial ordering as a category whenever convenient. Now, suppose we a given the following data. 1. A collection X = { Xa | a £ A } of spaces. 2. A collection fi — {/3a : Xa —• |a| | a £ A} of surjective maps such that, for each a £ A, the topology on Xa is that generated by the collection {/S l^s] | s £ a} of closed subsets.5 3. A collection / = { fa->a' '• P a 1 \a'\ ~^ Xa' \ a, a' £ A; a ^ a' } of maps such that fa-+a is the identity map on Xa and the diagrams Note that 2 is automatic when a is a locally finite complex. In particular, 2 holds when a is finite, and this is the case whenever a(a) £ K. Note also that the assignment of the space Xa to the element a £ A and of the map /a->a' to the arrow a —> a' defines a functor from A to the category Par of spaces and partial maps. Indeed, the triple (X,[3,f) is an object in a category which we denote by Glue^,6 and the assignment of the aforementioned functor to That is, U C Xa is open if and only if U fl/?^1 \s\ is open in (3a1 \s\ for each s £ a. Appendix B collects the results we require on this topology. This might more properly be denoted Glue<7, but we prefer to emphasize the role of the partially ordered set A since we are always dealing with a single cr. 36 this pair defines a functor from Glue^ into the functor category Par"4. For the morphisms from (X,3,f) to (Y, 7,5) in Glue A, we take collections <j> = {<f>a : Xa —• Ya \ a € A} such that the diagrams ft" VI 7a~Vl fa—>a Xai 4>a> commute whenever a £ A and a —• a' respectively. In the case that A C K, partially ordered by face containment, and a(a) = a, the category Glue A contains an embedded copy of the functor category Top"4. We can define a functor from Top A to Glue,4 by sending F e TopA to the triple (X,3,f), where Xs = F{s) X |s|, (3s(x,b) = b and (x, b) = (F(s —+ t)(x), b), and sending a natural transformation $ : F —• G to the collection { $ s X 1 : F(s) x |s| —* G(s) x |s| | s G A }. This may not work for an arbitrary A because of the restriction of property 2 in the definition of Glue^. Given an object X = (X,3,f) in Glue^, define QAX to be the quotient of the coproduct QaeA-Xa by the smallest equivalence relation that identifies x G 3~l\a'\ with fa^>a'{x) when-ever a —• a'.7 The maps 3a : X a —• |a| and the universal property of the coproduct provides an obvious map © a e ^ ^ a the following diagram. \K\ which we can use to construct a map 3x '• QAX —• \K\ by QAX The assignment X is the object map of a functor QA '• Glue 4 —• Top. For if (j>: X —• Y = (Y,y,g) is a morphism in Glue,4 then the map QA4> '• QAX —> QA W is uniquely defined by the following diagram. © 0 4>a s r . a£A QAX a£A QA4> QAY 7 Thus, QAX is the colimit space in Par 7 1 of the aforementioned functor determined by X and / . 37 Since f3y ° GA<J> = Px, the assignment X *-* Px is the object map of a functor from Glue,4 to the category Top \-\K\ of spaces over \K\. If b € \K\, let us denote the carrier of b by c(6) or just cb: cb = {v£ Vevt(K) I b{v) > 0} is the unique smallest simplex of K having 6 € |c6|. Note that if a € A then \a\ = {b G \K\ \ cb £ a}. Thus, if b 6 \a\ then a —• T(cb). The open simplex determined by s G K is the subset (s) = { ft £ |.Kj I cb = s} C |Kj. It is an open subset of \s\ but not necessarily of \K\. 2.4 Proposition. Let X = (X,f3,f) be an object in Glue^. Two points x € Xa and x' € Xai represent the same point in GAX if and only if f3a(x) = f3a'{x') = b and fA-*T(cb)(x) = fa'->T(cb)(x')-Proof. If / a_ >T(cb) ( x ) = /a'->T(cb) i x ' ) then x and x' represent the same point since these represent the same point as /0_,T(C6) i x ) and / 0 '-»T(cfc) (x') respectively. Conversely, suppose x and x' represent the same point in GA%- Applying Px to this common point immediately gives f3a(x) = f3a>(x'). Since they represent the same point in GAX as x and x' respectively, / A - » T ( c6 ) (x) and /0'-+T(C6) (x>) represent this same point. If these points are distinct then, by Proposition 0.1, we necessarily have sequences ax —* a[,..., an —• a'n and xi e Pa* lail) • • •, xi £ P a l lanl s u c n that the following hold. (1) /a-T(cb)(z) € {xi . /ox-alC^l)}-(2) {xt, fa.^a..(xi)} D {xi+1, fa.+1^a'.+i(xi+1)} for each i€ { l , . . . , n - 1}. (3) /a'-T(cb) (*') G i > n , / a „ - + a ' „ ( Z n ) }• Note that aj —• T(c6) for each i since 6 = [3ai(xi) € |a |^. Let us show that the assumption that /o->T(c6) (x) and / a ' - > T ( c 6 ' ) are distinct leads to a contradiction. First, we prove by induction that / a j - » T ( c 6 ) (xi) — /a-fT(cb) (x) f° r all i € { 1,..., n }. For i = 1, necessarily a[ = T(cb) and fai-+a\(xi) = fA-*T(cb){x) by (1), for if xx - fa^r{cb)(x) then ai = T(cb) so that ai = T(c6) as well. Similarly a'n = T(cb) and fAN-,T(cb)(xn) = fA'-*T(cb){x') by (3). Now, suppose fAI-*T(cb)ixi) = fa-*T(cb)(x) f° r some i > 1. By (2) there are four possibilities: a{ = ai+i and x,- = xi+1, ai = a'i+1 and x{ = / 0 , - + 1 - » o ; . + 1 a\ = o-i+i and /^-^.(x;) = x i + 1 , or a- = aj + 1 and / a j - 0 j ( * i ) = /a i + 1 -+a ; + 1 (z.+i)- In each case, / a , + j - T ( c 6 ) (zt+i) = /a->T(cb)(z) follows easily using the composition property of the maps in the collection '/. Thus, we have, in particular, / a n - > T(cb) (xn) = fA->T(cb) (x)- Since, as already noted, a'n = T(cb) and / a „ - T ( c 6 ) i x n ) = fa'-*T(cb) (x>), we therefore have / a - T ( c b ) ( x ) = /a'-T(cb) (x>), contradicting the assumption that these pairs be distinct. • 38 Because of the lower bound condition we have imposed on A, there is a functor TA '• Glue,4 —• Gluejc which is defined on objects by TA{X,0, f) = (X,j3,f), where Xs, 0S : Xs —• |s| and fa->t : Xs —y Xt are defined as follows. PS(V) = PT(.){V) fs^t(y) = /T(,)-»T(«) ( y ) Now, if (X,0,f) = X then a simple diagram chase reveals that the following diagram defines the component at X of a natural transformation GK ° TA QA-X T(s) GK(?AX) a€A PA -^GAX We claim that i is a homeomorphism. It is surjective since a point x € /3~11sj represents the same point in GAX as fa-tT{a)(x) G XA, and an application of Proposition 2.4 shows that it is injective. We need only show that i is a closed map. Since the topology on Xa is generated by the collection {ft"1 ]^ | s € d} of closed subsets, this is a consequence of the following. 2.5 Claim. If s € a and U C GK{FAX) then p^{i{U))n p?\s\ = / " ^ ( p - 1 ^ ) n XA). Proof. Suppose x € P ^ 1 n ^~x|s|. Then PA(X) = *(p(j/)) ^or s o m e V € p~l(U). Since x represents the same point in GA% as / a _,x (s) (*) £ * ( p ( / o - T ( . ) ( * ) ) ) = P A (/a-T( * ) ( * ) ) = P A ( Z ) = * ( p ( y ) ) which implies that p ( / a - » T ( s ) (* ) ) = p(y) G ^  since i is injective. Thus, / a - * - ^ ) ^ ) € D X 5 so that x <E / ^ T ( , ) ( P _ 1 ( y ) n ^ ) ' Conversely, suppose x G / ^ j ( 3 ) ( p - 1 (U) f)XS). Since x represents the same point in GAX as / o - T ( s ) ( « ) . P A ( * ) = P i 4 ( / a - » T ( * ) ( « ) ) = <(p(/.-T(.)(*)))€i(lO 39 so that x G P^iHU)). x G ft1^! is clear since 0a(x) = 0T{s){fa-*T(s)(x)) and /,^T(,)(i) G •X^ - • Thus, we have proven the following. 2.6 Proposition. The functors GA and GK 0 are equivalent, m As alluded to earlier, the point of introducing A is to show that under the appropriate circum-stances the space GKX is a lattice construction. For this we will apply the previous proposition with A = Ko, where KQ is the partially ordered set constructed from K by adjoining a single element, call it 0, as an initial object. That is we set Ko = K U {0} and require that 0 —• s for each s G K. We define the function a from Ko to the collection of non-empty subsets of Vert(iY) by a(s) = s for all s G K and CT(0) = Vert(K) (so that |0| = \K\). Given an arbitrary functor F : Ko —>• Top, the definition of an object X = (X, 0, f) of Glue/<-0 by Xs = F(s) x |s|, 0s(x, b) = b and / s_> t(£, b) = (F(s —• t)(a;), 6) is flawed unless the topology of space F(0) X \K\ is generated by the collection { F(0) x |s| | s G K} of closed subsets. This is certainly the case when K is locally finite. By Proposition B.5, F(0) locally compact Hausdorff is also sufficient. Now, if G is a topological group and /G/K = { Gs C G \ s G K } is a lattice of subgroups, indexed by K and having Gs C Gt whenever s —there is an obvious functor F defined by letting its application to a non-identity arrow s —• t in Ko be one of the left coset projections G —• G/Gt or G/Gs —* G/Gt, respective of whether or not s is the initial object in Ko- Thus, the pair (G,/G/K) defines an object X in Glue/<0 as above when either K is locally finite or G is locally compact Hausdorff. For the lattice construction, the lattice /G/K is constructed from a collection {Hv \ v G Vert(K) } of subgroups of G, indexed by the vertices of K, by setting Gs = f)vesHv. The space defined by the lattice construction is then the application to the object X of the functor CK ' Glue^0 —> Top defined as follows. If X = (X,0,f) is any object in Glue^o then the space CKX is defined as the quotient of Xo by the equivalence relation which identifies x and x' if and only if 0o{%) = A)(x') — & a n d fo-+cb{x) = fo->cb(x')-2.7 Proposition. The functors CK and Gi<0 are equivalent on the full subcategory of Glue/<0 consisting of those triples (X, 0, /) satisfying the following two properties. 1. fo->s : /3o~11s| —* Xs is either a closed map for each s G K or an open map for each s<=K. 2. The map x H-> f0^s(x) : /30~1(s) —• 0'1 (s) is surjective for each s G K. Consequently, CK is equivalent to GK ° F K 0 on the same subcategory. Note that both properties 1 and 2 are satisfied for the objects of Glue/<0 defined, as above, by a group and lattice of subgroups indexed by K. In this case, the map /o-^ : G x \s\ —• G/Gs x |s| is the product of two open maps: The coset projection G —• G/Gs and the identity map on the closed simplex \s\. 40 Proof. The component of the desired equivalence at X = (X, /?, /) is defined by the following diagram. X0< • © X , s e K 0 cKx — r - > gKox 3 That j is indeed the component of a natural transformation is a simple diagram chase. An application of Proposition 2.4 shows that j is injective. The hypothesis that x H-> /0_ts(x) : Po1 (s) ~¥ 071 (s)1S surjective for each s £ K implies that j is surjective since a point x £ /SQ - 1 (s) represents the same point in GK0X as /o-» s(x) £ /^(s). Since fo-,s : —* Xs is either a closed map for each s £ K or an open map for each s £ K, the following identity shows that j is closed or open (respectively). P o 1 (j(u)) n [3-1 \s\ = IJ f~it (fo^t (q-Hu) n flr1 \t\)) (a € K0,s £ K,a—> s) To see this, suppose x £ Po^ OX^O) D ]s]- Let 6 = (3a(x) and consider fa-tcb(x) £ (3~^(cb). By hypothesis, fa->cb(x) = /o-»cb(#o) for some io € 1 j«j. We claim that xo £ g_1(t/). Since Po(x) = po(xo) £ j{U), there is a point z £ g _ 1 (£/) such that po(xo) = j(q(z)) = Po(z). Thus, /o->c&(zo) = fo-+cb(z) by Proposition 2.4 so that q(x0) = q{z) £ U, giving x0 £ g -1(f/). Thus, x£fa-*cb(f0^b(q-1(U)n/30-1\cb\)). Conversely, if x £ f~^t (/o-^g - 1 (U) D (5QX\t |)) for some £ € s then / a- t(x) = /o-t(x0) for some x0 € g_1(£0 n /JQ - 1 |*|. Thus, /?a(x) = /30(x0) so that x £ /5 ~x I * I C Also, j(q(x0)) =po(x0) =p0(x) so that x £ p^iJiU)) (~l / J " 1 ^ . • In the remaining two chapters we will examine two aspects of the spaces GK%. In chapter 3 we work directly with the construction of Gi<X. In chapter 4 we work instead with the construction of £j<X since Proposition 2.7 guarantees that £ ^ 1 « Gi<X in the situation we place ourselves in. Thus, the next two results collect a few properties of the identification maps P '• ©»eic x« —*• Gi<X and q : X0 -> £j<X. 2.8 Proposition. Let X = (X,3,f) be an object in Glue^ and let p : ®seKXs —> GK% be the identification map. 1. IfU C 0 s e / < ^ and s £ K then there is the following equality. p-'(#))nx s= U (J fZAft-ftfrWnU)) (2.5) 2. If each member of f is an open map then p is a open map. 41 3. If K is locally finite and each member of f is a closed map then p is a closed map. 4. If p is either an open map or a closed map then the map x t-* p(x) : ft"1^) —• ft^ys) is a homeomorphism for each s G K. 5. IfUc GKX andtes then p~l (U) D 8jl\t\ = f£t (p~l (U) n Xt). 6. If s G K then the map x i—• p(x) : 0 < e 5 Xt —• /S^ t1 |s| is an identification. Proof. 1. This is an almost immediate consequence of Proposition 2.4. If x lies in the left hand side of (2.5) then p(x) = p(x0) for some x0 G U. Suppose x G U fl Xt. Then we necessarily have 3a(x) = /?<(x0) = b G \sf)t\ and f3->cb(x) = ft->cb(xo)- In particular, x G /^ c j ) (/t-+c6(/9r 1 | c ^l Fl U)), which is a subset of the right hand side of (2.5) since cb is a face of both s and t. Conversely, if x is an element of the right hand side of (2.5) then f3->t'(x) = /t->t'(zo) for some xo G ft-1|£'| l~l U, where t G K and t' G sni. Thus, p(x) = p(x0) so that x G P~l{p{U)) D Xs. 2. If U C 0 s e x Xs is an open set then p~1(p(U)) D Xs is a union of open sets by (2.5). 3. If U C 0 s e x ^« i s a c l ° s e ( I set and /<" is locally finite then the union in (2.5) is finite, so p~1(p(U)) D Xa is a finite union of closed sets. 4. If p is an open or closed map then x H-• p(x) : p~1(A) —* A is an identification for any A C GKX. Taking A = /^(s), we have p~l(A) = /?71(s). Since p is injective on this subset by Proposition 2.4, the map x ^ p(x) : ft"1(s) —• ft^5) is a homeomorphism. 5. If x G p~l{U) fl ji| then x represents the same point in U as (x) G Xt- Thus, p" 1 ([On/?,"1 |*l C / ^ ( p - ' t ^ n l t ) . Conversely, if x G f£t {p~l{U) n Xt) then x G ft-1!*! and x = /s-+t(xo) for some Xo € P-1(C/) fl Xt. Since x and xo represent the same point in GKX, necessarily x G p" 1^) n ft"1!^. Thus, f£t (p"1 (U) n X t) C p" 1 (tf) n ft"1 6. Since ft^N is a closed subset of GKX, the maps x i-* p(x) : p — 1 —• Z? 1^ js| is an identification. We have p " W W ) = ©ft"11^ 1 = ©ft_1Nnt| tei< <£/<• and, if U C ft^1 |s| and t £ K with s n t ^ 0 and t £ s, p-1 (U) n ft"11« n t| = (p"1 (U) n x , m ) by 5. Thus, p~l{U) is open in p~l(3xl\s\) if and only if p~l(U) n 0 ( 6 5 X t is open in 0 t e 5 ^ < . It follows that the map x \-+ p(x) : 0 t e s X < —• ft^M is an identification. • 42 2.9 Proposition. Let X = (X,fi, f) be an object in Glue^o and let q : X0 —• £ K % be the identification map. 1. IfU C Xo and s £ K then there is the following equality. q-'iqiU)) n/3^\s\ = (J f0~*t ( / o ^ ^ 1 \t\ n U)) (2.6) tea 2. If the map /o-»s : PQ1 \S\ Xs is an open map for each s £ K then q is a open map. 3. If the map fo-*s '• Po*\s\ ~* Xa is a closed map for each s £ K then q is a closed map. 4. If q is either an open map or a closed map, s £ K and the map x i—• fo-+s(x) : /30~1(s) —• /?7X(S) is an identification then the map x i—> q(x) : PQ1(S) —• (5XX (s) induces a homeo-morphism Pxl(s) -+ f3s~1{s). 5. Ifs£K then the map x *—> g(x) : f30 l\s\ —• / J ^ 1 ^ is an identification. Proof. 1. If x lies in the left hand side of (2.6) then q(x) = q(x0) for some xo £ U. We then have Po(x) = Po(xo) = b £ \s\ and fo->cb(x) = /o->cb (#<))• In particular, x £ fo-tcbifo^cbiPo1^] n [/)), which is a subset of the right hand side of (2.6) since cb £ cs. Conversely, if x is an element of the right hand side of (2.6) then /o_»t(x) = /o-»t(^o) for some xo £ AT1!*! U, where t £ s. Thus, PQ(X) £ P$l\s\ and q(x) = q(xo) so that x£q-1(q(U))n/3o1\s\. 2 and 3 follow immediately from (2.6). 4. Since q is either an open or closed map, the map x i—• g(x) : q~1((3x1(s)) —* Px* (s) is an identification. We have q~1(Px1(s)) ~ Po* (s) a n <I * w o point x,x' £ PQ1(S) represent the same point in P^is) if and only of /0_>s(x) = fo-,s(x'). Thus, the map x i-> /0_+s(x) : /?0_1(s) ~* dg1^) induces a bijection P^is) —* PT1^)-Po1^ q Since x i—• /o_>s(x) : PQ1(S) —• /?71(s) 1 S a n identification, this bijection is a homeomorphism. 5. l s a closed subset of £ ^ X . • Finally, let us introduce some new notation which we will use in the next two chapters to discuss the spheres constructed earlier. Let Sn be the simplicial complex consisting of all non-empty subsets of the vertex set Vert(£n) = {0,1,..., n}. That is, Sn is the simplicial complex underlying the standard simplex A n . Let 0(n) be either the dimension n(n — l)/2 orthogonal /o-fs 43 group 0(n), the dimension n2 unitary group U(n) or the dimension ra(2ra + l) symplectic group Sp(n). For each integer i with 0 < i < n — 2, let 0(n, i) = 0{i + 1) x 0{n - i - 1) C 0(n) be the subgroup of matrices whose only non-zero entries lie in the upper (i + 1) x (i + 1) and lower (n - i - 1) x (n - i - 1) blocks. For each s € £ n _ 2 let Os(n) = P| { C?(n, i) C O(n) | i € Vert((Sn_2) - s } , where an empty intersection is taken to be 0(n). Define a lattice /03(n)/s = {0s,t(n)\tes} of subgroups of Os(n) by 0Sit{n) = Os(n) fl ("],<=< 0(n, i). Notice that C?5)tlUt2(n) = Os-tut2(n) whenever t\ and £2 are disjoint faces of s. If s is the (n — 2)-simplex of <5„_2 then s = (5n_2, = 0(n) and we will write /Os(n)/-a = /0(n)/sn_2. The space £<5„_ 2 (C?(n), /0(n)/sn_2) is exactly the lattice construction of spheres presented earlier. 44 Chapter 3 Lifting Simplicial Approximation Let C H be the category of compact Hausdorff spaces. If we restrict our attention to the em-bedded subcategory C H X of G l u e ^ , the spaces Gi<X retain enough of the structure of the underlying polyhedron that we can prove a generalization of the classical Simplicial Approxi-mation Theorem, for maps from a polyhedron. After presenting this result, we apply it to a space constructed over an rc-simplex from an arbitrary map between compact Hausdorff spaces, resulting in a statement on the connectivity of the construction. We then show that this result can be improved if the map in question is locally trivial. Throughout this chapter we use the embedding of TopK —• Glue/< defined in the previous chapter (page 37) to identify TopK with its embedded image in G l u e / < , writing GKX whenever X is a functor in TopK. A. Lifting a Homotopy Let us begin by making the following observation. 3.1 Proposition. Let p : X —> Y and f : Z —> Y be any maps with p surjective. Consider the following pullback square. P = {(z,x)£ ZxX\ f{z)=p(x)} Let 7T : P —> Y be the diagonal composition in this diagram. If X and Z are compact and Y is Hausdorff then the map (z,x) H-> f(z) = p{x) : n~1(f(Z)) —> f(Z) is an identification. Proof. The map in question is a continuous surjection from a compact space to a Hausdorff space. • 45 We want to apply this result when p the identification ® s 6 x X{S) x l sl ~~f GKX = GK(TKX), so let us first show that GKX is Hausdorff whenever X is an object in CH.K. 3.2 Proposition. Let X be an object in Top^. TAen Gi<X is a Hausdorff space if and only if X(s) is Hausdorff for each s G K. To prove this, we employ the following. 3.3 Proposition. Let X be any space and suppose there is a map (3 : X —> Y with Y Hausdorff. The following are equivalent. 1. X is Hausdorff. 2. For any two distinct points x, x' G X with fi(x) = /3(x') = y, there is an open neighbour-hood Uofy,a Hausdorff space Z and a map h : (U) —• Z such that h(x) ^ h(x'). Proof. If X is Hausdorff then for any x,x' € X we can choose U = Y and h = 1 : X —• X, so only the opposite direction requires proof. Thus, assume 2 holds and let x, x' G X be distinct points. If fi(x) ^ P(x') then, since Y is Hausdorff, there are disjoint neighbourhoods V, V C Y of (5(x) and 0(x') respectively. Thus, /3 -1(V) and /3 _ 1(V) are disjoint neighbourhoods of x and x' respectively. If /?(#) = P(x') then, by the hypothesis, there is an open neighbourhood U of this point, a Hausdorff space Z and a map h : /3-1(C7) —> Z such that h(x) ^ h(x'). Since Z is Hausdorff, there are disjoint neighbourhoods V, V C Z of h(x) and h(x') respectively. Since f3~l{U) C X is open, h~l(V) and h~l{V') are the required disjoint neighbourhoods of x and x'. • We require more notation before proving Proposition 3.2. The open star of a vertex v of K is the open subset St(u) = { & € | # | | 6 ( « ) > 0 } of \K\. If s is a simplex of K, let Wa = (\es^(u)' Proof of 3.2. If GKX is Hausdorff and s € K, choose a point 6 £ (s). Then we have a map h : X(s) —• GKX defined by the following diagram. a; i > (a;, 6) X W « • © * ( * ) x | « | GKX Since 6 G (s), h is an injection by Proposition 2.4. Thus, Hausdorff implies the same for X(s). 46 Conversely, suppose each X(s) is Hausdorff. Denote the identification ® A E K X(s)x \s\ -*• GKX by p. Since \K\ is a Hausdorff space,1, we can apply Proposition 3.3 with 0 : GKX —> \K\ the canonical map onto the underlying polyhedron. Thus, let (x,b) G X(s) x |s| and (x',b) G X(s') x represent distinct points in GKX which are mapped to the same point by 0. We need to exhibit a neighbourhood U of b, a Hausdorff space Z and a map h : 0~x (U) —• Z such that h(p(x, b)) ^  h(p(x', b)). If u is any simplex of K then, since 0~1(WU) is an open subset of GKX, the map (y, a) >-* p(y, a) : p~l {0~l (Wu)) —• (VFu) is an identification. Also, \t\ (1 Wu # 0 if and only if t —• u. Thus, p- 1 (/?- 1 (W u ) )=0XWx(|f|nw u ) . We can now define a map hu : /3 - 1(Wu) —• X s by the following diagram. X{t) x (\t\nwu) @x(t)x(\t\nwu) t—tu To complete the proof, take U = Wcb, Z = Xcb and h = hcb : 0~l(Wcb) —+ X(cb). Since p(x,b) and p(x',b) are distinct points in —»• cb)(x) and X(s' —> c6)(x') are distinct by Proposition 2.4. Thus, hcb(p(x,b)) ^  hcb(p(x',b)) as required. • Now, suppose we are given a map / : Z —• fel, where Z is compact, X is an object in CHK and K is finite. The pullback square 7T then satisfies the hypotheses of Proposition 3.1. Let h : Z X I —*• \K\ be a homotopy satisfying h(z,0) = 0(f{z)) and h(z,t) G \c(0(f(z))) | for all z G Z and i 6 7. Since 7 is locally compact, the map (z,x,b,t) i—• ( / ( ,?) ,£) == : 7r x J —• f{Z) X I is an identification. 1 [S] page 111. 47 Thus, the following diagram defines a deformation D : f(Z) x I —* GKX of f(Z) in GKX and, then, a homotopy H : Z x I —• GKX. (z, x, b, t) i (x,h(z,t)) 0X( S)x|s| s£K Zxl Note that 0 ° H = h and that if (z, 0) = f(z) for all z £ Z. We have thus proven the following. 3.4 Proposition. Let j': Z —• £.K^, where Z is compact, X is an object in CHK and K is finite. Let h : Z X I ->• \K\ be a homotopy with h(z,0) = f3(f(z)) and h(z,t) £ |c(/3(/(z)))| for all z £ Z and t £ I. Then there is homotopy H : Z x I —• GKX such that j3 ° H = h and H(z, 0) = f(z) for all z £ Z, and having H(z, t) = p(x, h(z, t)) whenever f(z) = p(x, b). u In other words, we have verified the existence of a homotopy H which makes the following diagram commute. z Z — > Q K X M ) Zxl Of course, the map /3 is almost certainly not a fibration.2 The existence of the lifting H depends on the hypothesis that the path t i-+ h(z, t) : I —• \K\ remains inside the closed simplex |c(/?(/(2)))|. We can easily remove the restriction that K be finite in Proposition 3.4. 3.5 Proposition. Let f : Z —> GKX, where Z is compact and X is an object in CH/C. Let h:Z*I-*\K\ be a homotopy with h(z,0) = (3(f(z)) and h(z,t) £ \c(3(f(z)))\ for all z £ Z and tel. Then there is homotopy H : Z x I —• GKX such that 3 ° H = h and H(z, 0) = f(z) for all z e Z, and having H(z, t) = p(x, h(z, t)) whenever f(z) — p(x, b). Proof. Since P{f(Z)) C ]K\ is compact, P{f(Z)) C \K'\ for some finite subcomplex K' C K.3 If X' is the restriction of X to K' then consider the map i : GK'X' —> GKX defined by the The fibres over different points in \K\ may have different homotopy types: See Proposition 2.8. [S], page 113. 48 following diagram. 0 sen1 P GK'X' GKX I An application of Proposition 2.4 shows that i is injective. Thus, since GK'X' is compact and GKX is Hausdorff, i is an embedding onto its image (which is 3~l\K'\). Apply Proposition 3.4 to the map z \—• i'_1(/(z)) : Z —• GK'X' and homotopy (z,t) t - > h(z, t) : Z X I —• |K'| to get a lifting #' :Z x 7 -+ JC, and set if = i ° H. u Let / : |L| —• where X and if are simplicial complexes. A simplicial map <j> : L —• Jv" is called a simplicial approximation to / if the induced map |<£| : \L\ —*• |7C| satisfies the condition that |<^ |(a) € |s| whenever /(a) G \s\. Because of this condition, if <f> exists then the map is a well-defined homotopy h : \L\ x I —• \K\.4 Moreover, this homotopy has h(a,t) G |s| whenever /(a) G or, equivalently, h(a, t) G |c(/(a))|. Note also that h is constant on the set of points where / and |<^>| agree. Recall that a subdivision of L is a simplicial complex V such that (1) the vertices of V are points in (2) if s' is a simplex of V then there is a simplex s of L such that s' C |s| and (3) the linear map sending a G \L'\ to the point YlveL1 a ( v ) v € \L\ is a homeomorphism. We will identify \L'\ with \L\ by the homeomorphism of (3), implicitly composing with either this map or its inverse whenever context demands. The classical Simplicial Approximation Theorem5 states that there exists a subdivision V of L and a simplicial approximation <f> : V —> K to / : \L'\ —• \K\. As noted above, this simplicial approximation is necessarily homotopic to / by a homotopy h satisfying h(a, t) G |c(/(a))|. Let us call a map ip : \L\ —• GKX simplicial if there is a simplicial map (j> : L —> K such that Q o ip = 1^ 1, where 3 : GKX —• \K\ is the map onto the underlying polyhedron. Call it a simplicial approximation to a map / : \L\ —»• GKX if ^ is a simplicial approximation to 3 » f. 3.6 P ropos i t i on . Let K and L be simplicial complexes and suppose f : \L\ —* GKX, where X is an object in C H ^ . Then there is a subdivision V of L and a simplicial approximation Continuity follows from the fact that / i | | s | X / is continuous for each s E L since, by Proposition B.5, the topology of \L\ x I is generated by the collection {\s\ x I \ s £ L}. See [S], page 128, or [Mu], page 95 (Theorem 16.5). B. Approximation by Simplicial Maps (a,t)~(l-t)f(a)+t\<f>\{a) (3.1) 49 ip : \L'\ —> QKX to f. Moreover, if A is the set of points in \L'\ where f3 ° f and 0 ° tb agree then f ~ tb rel 0~l(A). Proof. By the Simplicial Approximation Theorem, there is a subdivision V of L and a sim-plicial approximation <j> : V —• K to 0 ° / . Let h : \L'\ x I —• \K\ be the homotopy defined, as in (3.1), by h(a,t) = (1 - t)0(f(a)) + t\<j>\(a). If s is a simplex of V, apply Proposition 3.5 to the map / | | s | and the homotopy / i | | s | X / to get a homotopy Hs : |s| x I —• GKX such that 0(Hs(a,t)) = h(a, t) and Hs(a, 0) = /(a?), and having Hs(a,t) = p(x,/i(a, £)) whenever /(a) = p(x,b). Because of this last property, the function H : \L'\ X I —> GKX defined by H\\s\xl = Hs is well defined. It is continuous since each H\\s\xi is and the topology on \L'\ X I is generated by the collection { |s| x I | s € L'} (by Proposition B.5). Since 0(xj)(a)) = h(a, 1) = |<^ |(o;), H is a homotopy from / to a simplicial map. The homotopy is relative 0_1 (A) since H(a, t) = p(x, h(a, t)) whenever f(a) = p(x, b) and since the homotopy h is relative the set A. u As with ordinary simplicial approximation, Proposition 3.6 can be used to infer results on the vanishing of the homotopy groups of GKX. A map / : SN —• GKX is homotopic to a map whose image is contained in 0~l\K^ |, where is the n-skeleton of K. If we can determine that 7 r n A = 0 for some A C GKX containing 0~L\K^\ then necessarily TTUGKX = 0. We examine an example where we can do this in the next section. Let us try to replace the polyhedron \L\ in Proposition 3.6 by a space GiX lying above it, where Y is an object in C H ^ . Call a map ip : QiX —* GKX simplicial if there is a simplicial map <j> : L —• K such that the following diagram commutes, the vertical maps being the canonical maps onto the underlying polyhedron. GLY GKX 0y 0x Call ib a simplicial approximation to a map / : GLY —> GKX if 0x{ib{oi)) = \<f>\(0Y(<*)) G |s| whenever 0x{f{&)) S \s\. As above, a simplicial approximation is homotopic to the map it is approximating. The notion of subdivision can also be extended to GLY. Suppose V is a subdivision of L. If s' is a simplex of V then, by the definition of subdivision, there is a simplex s of L such that s' C |s|. Since any non-empty intersection of simplices is itself a simplex, there is a smallest such simplex of L. Denote this simplex by [s']. We can then define a functor Y' : V —> C H by Y'(s —> t) = Y([s] —» [t]). The identity maps Y'(s) —+ Y([s]) then induce a homeomorphism from GL'Y' to GLY, and we refer to GL<Y' as the subdivision of GLY over V. Now, one can try to prove the statement that given a map / : GLY —* GKX, where Y is an object of C H L and X is an object in C H K , then there is a subdivision V of L and a simplicial 50 approximation tb : GL'Y' —+ GKX to / . However, this is false in general as the following example shows. As we saw in the previous chapter, S 4 can be constructed as a lattice construction over the standard 1-simplex. Consider the maps from S 4 over a standard 1-simplex to S2, triangulated as the boundary of a standard 3-simplex.6 A simplicial (in our newly defined sense) map S 4 —• S2 necessarily maps into the 1-skeleton of S2. In particular, such a map is not onto and is therefore null-homotopic. If every map from S4 to S2 had a simplicial approximation (to which it would be homotopic) then every map from S4 to S2 would be null-homotopic, and this is certainly not true as TT^S2 = Z 2 . C. An Example Let Dn denote the closed n-ball with interior Bn and bounding sphere SN~ I. Let / : X —> Y be any map with X and Y compact Hausdorff. Define the space Dnf to be the quotient of the disjoint union (X x Dn) © (Y x 5 n _ 1 ) by the smallest equivalence relation that identifies (x, 6) and (/(a;),b) whenever b G 5 n _ 1 . Let pj denote the quotient map. There is a map qf : Dnf —• Y induced by the maps (x, b) H-> /(X) : X X Dn —• Y and (y, b) !->• y : Y x 5 n _ 1 —• Y. Similarly, there is a map 0j : Dnf —• Dn induced by the maps (x, 6) i-> 6 : X x Dn —> Dn and (y,b) !->• b : Y x 5 n _ 1 —• Dn. Moreover, an application of Proposition 3.3 to this map shows that Dnf is a Hausdorff space. To see this, suppose z and z' are two distinct points in Dnf with 0/(z) = 0f(z') = b. If b e 5 n _ 1 C Dn then the map q : D n / —• Y has g/(z) 7^  <lj{z') since 2 = pf(qf(z),b) and 2' = pf(qf(z'),b). If b £ Bn then the map a H-* p(a) : p~1(0J1(Bn)) —> 0J1(Bn) is an identification since 0J1(Bn) is an open subset of Dnf. Thus, since p-1(/371(JBn)) = X x Bn, the map on (x, 6) x : X x Bn -> X induces a map q'j : 0~}l{Bn) X x Bn having ^ (p;(x,6)) = x. Necessarily g}(z) ^ (^2') since 2 = pf(q'j(z), b) and z' = pf(q'j(z'), b). Note that if / is surjective then Dnf is homeomorphic to the quotient of X x Dn by the equivalence relation that identifies (x,6) with (x',6') if and only if b' = b € 5" - 1 and /(x') = Note that GKX nt \K\ whenever K is locally finite and X is the constant functor to a 1-point space. F(l,2) F( l , l , l ) 51 f(x): The composition X xDn [X x Dn)@ (Y x 5 n _ 1 ) factors through a bijection from this compact quotient onto the Hausdorff space Dnf. If * is the constant map onto a one point space, Dn* is homeomorphic to the join 5 n _ 1 • X. To see this, first recall that the join Y • Z of two spaces Y and Z is the quotient of Y x I x Z by the smallest equivalence relation that identifies each of the subspaces Y x {0} x {2} (z € Z) and {y} x {1} x Z (y € Y) to distinct points. The map (b,t,x) ^ : S n _ 1 xIxX^ Dn xX then induces a homeomorphism from 5 n _ 1 • X to the quotient of Dn x X by the smallest equivalence relation that identifies {b} x X to a point for each b £ 5" _ 1, and this latter quotient is homeomorphic to Dn* as previously indicated. Since *X is homeomorphic to the ra-fold suspension of X, Dn* is (n — l)-connected. It is not hard to see that Dnf is homeomorphic to a space in the image of Q$n. Indeed, if d8n denotes the subcomplex of Sn consisting only of the proper faces of the n-simplex in 6n, choose a triangulation £ : (\8n\, \dSn\) —• (Bn,Sn~1). Define a functor Z : Sn —• C H by the following. ' 1 : X X ifte8n-d5n Z(s t) = I 1 : Y Y \fs£d6n , / : I - > y otherwise The maps {x,b) ^ (x,£{b)) : X x \5n\ - f X x Bn and [y,b) i - » (y,f(6)) : y x |s| - f y x induce a map 0 s £ i n (^s) X |s| —>• (X x Bn) © (Y x 5 n _ 1 ) which, in turn, induces a bijection i:GSnZ->Dnf. © Z(s) x s 4 ( i x B " ) ® ( y x 5 n _ 1 ) GsZ P Dnf Since ^j„Z is compact and Dnf is Hausdorff, i is a homeomorphism. Now, consider maps from a fc-sphere into Dnf. Each map from Sk into Dnf is homotopic to a map into /Jj^l^n^l- Since the homeomorphism i : G&nZ —v maps ^ 1 | ^ n _ 1 ) | onto 0'j1(Sn~1) RiYx 5" _ 1, and since Sn~l is (n-2)-connected, the following result is immediate. 3.7 Proposition. For any k < n - 2, if nkY = 0 then nkDnf = 0. 52 Additional Structure Proposition 3.7 imposes no constraints on the map / other than that it be a map between compact Hausdorff spaces. Let us now assume that / is locally trivial with fibre F. 3.8 Proposition. Let * denote the constant map from F to a 1-point space. Then the map qj : Dnf —*• Y is locally trivial with fibre Dn*. Proof. Given a trivialization <f> = (/, V>) : / - 1 {U) —• U X F, let us construct a trivialization —> U X Dn*. First, choose any point a G F and define a map pJ1 (qj1 (£/)) —• U X Dn* by the following diagram ( a ; , 6). »(/(*), ^(x), b) /_1(C7) x Dn • U x F x Dn P~s\<lj-\U)) = (f-^U) x Dn) ®(Ux S""1) • U x Dn* U x 5 n _ 1 -+U x F x Dn • (y,a,b) Since qj (U) is an open subset of Dnf, the map a t-+ pj(a) : pJ (qj (U)) —• qj (U) is an identification. Thus, since any two pairs in F x 5 n _ 1 represent the same point in Dn*, the above map factors through this identification to define a map £ : q~jl(U) —• U x D n *. £/ x Dn* As noted previously, since * is a surjective map, Dn* is homeomorphic to a quotient of F x Dn. Identifying Dn* with this quotient, the inverse of £ is defined by the following diagram. (if, a , 6) •— U x F x Dn l x p , U x Dn* r 1 53 ?7 (u) Here the map 1 x p„ is an identification since U is locally compact. Thus, the existence of C, follows by the universal property of the identification. • 3.9 Corollary. Tfie map q : Dnf —• Y induces isomorphisms KkDnf = 7TfcY for all k < n — 1 and an epimorphism TrnDnf -» nnY. In particular, if TtkY = 0 for some k < n — 1 then itkDnf = 0, improving the result of Proposition 3.7.7 Proof. The n-fold suspension of F is (ra — l)-connected. Apply the long exact homotopy sequence of the fibration. • Is there an example (for any n) of a map / such that 7r„_iY = 0 but nn-iDnf ^ 0? 54 Chapter 4 Bundle Structure with a Topological Group As discussed in chapter 2, a group G together with a lattice /G/K = { G„ | s € K } of subgroups determines an object in Glue#0 whenever either G is locally compact Hausdorff or K is locally finite. If this object is the triple G = (X,(3,f) then, setting Go = {1}, the spaces Xs and the maps 0„ : X„ —• \K\ and fa_>t : ft"11*1 — y xt are defined by Xs = G / G s X \s\, 0a(gGa, b) = 6 and fa-tt(gGa,b) = (gGt,b). Pairs ( G , / G/K) are the objects of a category in which a morphism from (G,/G/K) to (H,/H/K) is a homomorphism f : G —+ H such that f(Gs) C # s for each s € X, and the construction of <G from the pair (G,/G/K) defines an embedding of this category into the category Glue#0. Let us identify the embedded image with the category itself and write Q K O G = GKQ ( G , /G/K) and C K G = C K (G, /G/K). We see in the proof of Proposition 2.7 that the inclusion G X \K\ <^-> ©se/<0 G/Ga x \s\ induces a homeomorphism JC/<(G, /G/K) —*• GKQ{G, /G/K). Now, if H is another subgroup of G, let us write /G/K (~l H = {GaC\H \ s € K }. In this chapter we examine the space C K ( G , / G / K n i?) in the case that (1) G is locally compact Hausdorff, (2) each of the members of /G/K is a closed subgroup and (3) H C G is a closed subgroup for which the coset projection G —> G/H is locally trivial. There is then a locally trivial map from CK{G,/G/KH H) to G/H. This will provide us with examples of lattice construction in which the space constructed is not simply connected. It will also give information on the structure of neighbourhoods in £ K { G , /G/K) in the case that G is compact and K is locally finite. In section C, the spheres originally constructed by R. R. Douglas and A. R. Rutherford are re-examined, and many more spheres are constructed. In each of the real, complex and quater-nionic cases we construct a finite number of spheres over a simplex of a given dimension, but all but a finite number of spheres in totality. Exactly which spheres are constructed is enumerated. A. A Class of Bundles We begin with the following fact. 55 4.1 Proposition. Let G be a locally compact Hausdorff group and let H',H C G be sub-groups with H' C H and H closed. Let p : G/H' —• G/H be the coset projection. If the coset projection G —> G/H has a local section <; : U —> G on the neighbourhood U C G/H then the map 9H' ~ {gH^{gH)-lgH') : p~l{U) - U x H/H' (4.1) is a homeomorphism. In particular, taking H' = {1}, the map 9^{gH,<:{gH)-lg):p-l{U) ^UxH is a homeomorphism. Proof. Let us denote the coset projection G —• G/H by pjj. Since U is an open subset of G/H, the identification G —• G/H' restricts to an identification P~H(U) —• p-1([7). Thus, the following diagram defines the map (4.1). 9' •+.(gHM9H)-1g) P H ( U ) >UxH P~l{U) *U x H/H' To construct the inverse map, since G is locally compact Hausdorff and H C G is a closed subgroup, the coset space G/H is locally compact Hausdorff.1 Since an open subset of a locally compact space is itself locally compact, the product of the identity map on U with the identification H —* H/H' is an identification. Thus, the following diagram defines a map U xH/H' ^p~l{U). (gH,h), >s(gH)h U xH >PH(U) U xH/H' tp'^U) It is easy to check that this map is the inverse of (4.1). • We can now apply the above result to spaces of the form C R { G , /G/K n H). In this situation, the map (g,b) ^ gH : G x\K\-> G/H then induces a map n : C K { G , / G / K (~\ H) —* G/H. See [MZ], page 52. Note that the authors here assume that a topological space satisfies the To separation axiom by definition. As discussed on pages 20 and 25 of [MZ], this implies the Hausdorff property in a topological group. 56 4.2 Proposition. Let G be a locally compact Hausdorff group, H C G a closed subgroup such that the coset projection pu : G —• G/H is locally trivial, and /G/K = { G S \ s G K } a lattice of closed subgroups of G . Then the map n : C K ( G , /G/K H H) —* G / i f is locally trivial with fibre CK(H, /G/K n if). Proof. Let qG : G x \K\ •-• £ K ( G , / G / K D ii") and qH - H x \K\ -+ £*(ff, /G/K n if) be the identification maps, and let £/ C G / i f be any neighbourhood on which pn '• G —• G/H has a section c : t/ — + G . Let us also write Hs = G S n if and /if//c = / G / / < : Pi if. Since 7r _ 1(C/) is an open subset of £/<•((?,/H/K), the map (g,b) —• qo(g,b) : o^ j1 (^ 0) "~* w~l(U) is an identification. Now, if s G ff then two pairs (g, b) and (g1, b') in p^1 (U) X (s) C p^1 (ff) X |if| = OQ 1(7r - 1 (f7)) represent the same point in n~x(U) if and only if b' = 6 and <7ifs = g'Hs. Thus, the pairs (<;(gH)g,b) and (s(g'H)g',b') in ff x |f<"| represent the same point in £j<(H,/H/K), and the following diagram defines a map n~x{U) —> U x £ # ( i f , / H / K ) -(g,b)>— p-„l{U)x\K\ <1G ,-1 [gH,c;{gH)-xg,b) -^U xH x\K\ 1 X qH UxCK(H,/H/K) (4.2) The topmost map in this diagram is just the product of the identity map on U with the homeomorphism pJjX(U) —+ U X if provided by Proposition 4.1. In fact, it is clear that two pairs (g,b) and {g',b') represent the same point in 7r - 1(f/) if and only if (gH,q(gH)~lg,b) and (g'H,s(g'H)~1g',b') represent the same point in U X CK(H, /H/K)- Thus, since U is locally compact, the product of the identity map on U with the identification qn is again an identification, and we can reverse the topmost horizontal map in the previous diagram to define the inverse of our map n~x(U) —> U x C K { H , / H/K)-(gH,h,b)> (<;(gH)h,b) UxHx \K\ 1 x qH UxCK{H,/H/K) PH\U)X\K\ • TT~X (U) Of course, the previous result leads to a long exact homotopy sequence. Let us continue to write H 3 = G S C \ if and /H/K = /G/K fl if as in the just-completed proof of Proposition 4.2. • 7rKCK (ff, /H/K) • n K C K (G, / H / K ) nKG/H • 7r*-i C K (if, I E IK) If C K ( H , /H/K) is contractible then it follows that n : C K { G , /H/K) —* G / H is a weak homo-topy equivalence; that is, that it induces isomorphisms of homotopy groups. 57 Consider the case where K = 5n is the simplicial complex of all subsets of the vertex set {0,..., n}. If v is a vertex of 6n then there is a retraction r : \8n\ x I —• \Sn\ defined by sliding a point in \Sn\ along the line segment joining it to v. r(a,t) = (1 - t)a + tv • Here we are identifying the vertex v of the simplicial complex 6n with the obvious element of the polyhedron \Sn\. In general the map 1 x r : H x \5n\ —• H x \5n\ will not induce a map Cg„ (H, /H/$n) —• £&n (H, /H/$n) for the reason that if s G Sn is a simplex not containing v then two pairs (h,b) and (h',b) in H x (s) represent the same point in Csn(H, /H/$n) if and only if hHs = h'Hs while the pairs (h, r(b,1/2)) and (h1, r(b, 1/2)) represent the same point if and only if hHsU{vy — h'HsU{vy. Since S U { D } —• s implies that HaU{vy C Hs, a necessary and sufficient condition for the map 1 X r : H X \8n\ —• if x |£ n | to induce a map jCsn(H, /H/sn) —> Csn{H, /H/in) is the following: If s is a simplex of <Sn not containing v then ^«u{i} = ^s- 1° this case, the induced map is a retraction onto Q~l(v) K, H/H^vy. 4.3 Proposition. Let G be a locally compact Hausdorff group, /G/s„ = { Ga \ s G Sn } a lattice of closed subgroups of G, and H C G a closed subgroup such that the coset projection G —• G/H is locally trivial and H C G{vy for some vertex v of Sn and. The following are true. 1. IfCSn (H, /G/s„ n H) is contractible then the map n : £Sn {G, /G/&n n H) -+ G/H is a weak homotopy equivalence. 2. IfGsU{vy D H = Gs f l H for each simplex s not containing v then £$n (H, /G/sn f l H) is contractible. 3. IfGsC\H = C\wes G{w} H H for each s G 5n then GsU^vy D H = G s fl H for each simplex s not containing v. Note G3 = f \e s G{vy is always the case for the lattice construction so that the hypothesis of 3 and, thus, 2 and 1 are satisfied in this case. Proof. 1 follows from the long exact homotopy sequence of the bundle C$n (H, /G/$n fl H) <-^> C&n (G, /G/Sn n H) -»• G/H as noted above. For 2, the preceding discussion showed that there is a retraction of £$„ {H, /G/sn fl H) onto /3_1(u) « H/(G{vy (~1 H) = H/H, a one point space. 3 is clear since h C G^vy. m Let us state the conclusion of Proposition 4.3 in the case G = 0(n) and /G/s„_2 = /0(n)/sn_2. If H C 0(n) is a closed subgroup then with H C 0(n, i) = 0(i +1) x 0(n - i - 1) for some 0 < i < n — 2 then there is a weak homotopy equivalence £s„_ 2 (0(n), /0(n)/sn_2 CiH) —> 0(n)/H. Notice that if H = 0{ni) x • • • x 0(rik), with k > 2 and ni H h nk = n, then 0(n)/H is a real, complex or quaternionic flag manifold. 58 B. Local Structure with a Compact Group Let us now use Proposition 4.2 to decompose certain neighbourhoods in C K { G , /G/K) in the case that G is compact, K is locally finite, and each of the coset projections G —» G/Ga is locally trivial. This is the case when, for example, G is a Lie group. Let q : G x \ K \ CK{G,/G/K) be the identification map and let 0 : CK(G,/G/K) -*• \ K \ be the map induced by the projection G x \ K \ —> \ K \ . Note that q is an open map by statement 2 of Proposition 2.9. Thus, the map a t-+ q(a) : q-1(A) —• A is an identification for any subset •<4 C C K ( G , /G/K)- We will use this fact repeatedly (and sometimes without mention) to define maps on subsets of CK(G,/G/K)-1. Strong Deformation Retractions above the Polyhedron Let us start by defining two open subsets of \ K \ for each s £ K . U3 = (J St(u) W3 = f | St(«) Define a strong deformation retraction hs : U3 X I —• C/s as follows. Here we use the same letter to denote a vertex of K and the function in \ K \ which is 1 at this vertex and 0 at all others. Since ha((UsC\\t\) x I) C U3n\t\ for any t £ K,h3 induces a map Ha : ^_1(L7a) x I 0~1{U3). GxU3xI 1 X h' ) G xUs qxl r1(U3)xI——^l3-1(U3) tl3 Notice that 7fs maps 0~l(U3) X {1} and 0~X(W3) X {1} onto and /3_1(s) respectively since /is(t/3 x {1}) = \s\ and ^(W, x {1}) = (s). Let rs :US —+ \s\ and i?5 : (Us) —>• denote the retracts associated with the retractions h3 and 7fs respectively. r,(6) = ^(6,1) Ra(a) = H.(a,l) We will show that the map a t—>• i?s(a) : /3 -1(iy3) —• /3-1(s) is locally trivial. Notice that statement 4 of Proposition 2.9 gives a homeomorphism q(g,b) >—• (gGs,b) : 0~1(s) —• G/G3 x (s). 59 9 2. Fibre Structure of the Retracts The simplex s determines a collection of complementary simplices. sc = { t G K | s n t = 0, s U t G K } This collection is either empty or a finite (since K is locally finite) subcomplex of K. The former holds if and only if s is not a proper face of any simplex or, equivalently, Ws = (s). In this case we have /3-1(Ws) = /3_1(s) « (s) X G/Gs. Let us now assume that sc is not empty. Define a lattice of subgroups of G, indexed by sc, by /Ga/ac = {GaUt \ t G sc}. Note that Gsut C Gs since s\Jt —> s in K. Thus, by Proposition 4.2, the map (g, 6) i—> gGa : G x |sc| —• G/Ga induces a locally trivial map ivac : Cac(G, /Ga/3c) —* G/Gs, having fibre Cac (Ga, /Gs/ac). There is an embedding of C3c(G, /Gs/Sc) in £K(G, /G/K). TO construct this embedding, define two subsets of \K\. Fpa={be\K\\llv(.sb{v) = l/2}cUa F: = {beUa\ra(b) = b3} cw3 Here b3 = l/||s|| • $^„ e s « is the barycenter of the closed face |s| C \K\, where ||s|| is the number of vertices contained in s. Now, because the union of any member of sc and any face of s is a member of K, the function 6/2 + b3/2 : Vert(.R') —• [0,1] is a member of |/<"| for any b € |sc|. That is, the inverse image, by this function, of the half-open interval (0,1] is a simplex of K. Moreover, it is clear that this function lies in the subset D F° C \K\ and that the map b —• 6/2 + b3/2 :\sc\ —• Ff 0 Ff is a homeomorphism. Also, b e (t) if and only if 6/2 + 6s/2 € (slit). Thus, the map (ff,6) (g,6/2+ 6s/2) : G x |sc| - f G x (Ff nFf) induces a bijective map T : £y= (G, /G,/,.) - » ( F f n Ff). (g, b). • (ff, 6/2 + 6,/2) G x |sc| • G X (Ff fi Ff) 3-1/ (4.3) 9 A* (G, / G , / S c ) (Ff n Ff) Since the rightmost map in the above diagram is an identification as well, we can simply reverse the topmost arrow in this diagram to define the inverse map /3 - 1(Ff Cl Ff) —> Csc (G, /G3/3c). Now, since s is a face of the carrier of 6 for each 6 G VFS — c6 —* s in symbols — the map {9,b) >-f gG„ : G x iy s —• G / G s induces a map 7r s : /J""^^) —» G / G s , and this map clearly 60 makes the following diagram commute. 9a«=(5,6)l •q(g,b/2 + ba/2) £s c {G, /Ga/s c) p-HFfnFf) G/Ga Thus, na\f}-i(F?c\Ff) l s locally trivial with fibre Ca<=(Ga,/Ga/ac) since n3c is. Note that if 6 € Ff then £ v € j s b{v)v = rs(b)/2. Thus, the function b-ra(b)/2+ba/2 is a member of \K\ if and only of 6-1(0,1] U s G K, and in this case it is clearly a member of the subset Ff n Ff. If b G Ff D Wa then s C 6 _ 1 (0,1], and there is a map 7 : Ff n VTS (Ff n Ff) x (s) defined by 7 = (71, r s), where 71(6) = b-ra(b)/2 + ba/2. It is clear that 7 is a homeomorphism, the inverse map being (a, 6) i—>• 6 + a/2 — ba/2. We can now use 7 to define a homeomorphism on 3~l (FfC\Wa). If 6 G Ff fW s then 71(6) G (cb). Thus, the following diagram defines a bijective map T : (Ff n WV) ->• (Ff n Ff) X (s). M) ' (g,7l(b),ra(b)) G x (Ff n WV) 9 1 x 7 G x (Ff n F f ) x (*) g x 1 (4.4) / r ^ F f nFf ) x ( s ) Again, since (s) is locally compact, the rightmost map above is an identification so we can reverse the arrows in the above diagram to define an inverse map, showing that T is a homeo-morphism. This homeomorphism evidently makes the following diagram commute. l(9, b) 1 (?(0,7i (6) W&)) Ra B-^FfnF^xis) 7TS X 1 G/Gax{s) q(g,rs{b))< (9Gs,ra(b)) Thus, the map a t-+ R , ( a ) : /? _ 1(Ff nWs) -+ («) is locally trivial with fibre £sc (G,/Gs/Sc) since na\p-i(F?nF?) 1S- Denote this map by Ra. We now want to exhibit d~1(Wa) as the quotient of /3 _ 1(Ff n Ws) X [0,1) by the equivalence relation that identifies (a, t) and («',£') if and only if t' = t = 0 and Ra(o) = Ra(a'). We 61 denote this quotient space by QRa. Since Ra is locally trivial with fibre Cac (Ga, /Ga/ac), it follows that the map QRa —* 3~l(s) induced by the composition of Ra with the projection P~x(Ff D Wa) x [0,1) -> 3~x(Ff n W,) onto the first factor is locally trivial with fibre the open cone2 on £sc (Ga, /Gs/Sc). A rather more general version of this statement can be found in appendix C. To exhibit (Wa) as desired, let us first define two more subcomplexes of K determined by s. The simplicial neighbourhood of s is the subcomplex s = {t £ K | t U s' £ K for some s' £ s } of all simplices which are faces of a simplex intersecting s. Note that, because K is locally finite, s is a finite subcomplex of K. The link of s is the subcomplex s x = {tes\tns = $} of those simplices in s which do not intersect s. Note that |s| = Us U |s x| and Us D Is1! = 0. There is a surjective map <j> : Ff X I —• \s\ defined by sliding b £ Ff along the line segment joining rs(b) £ |s| and 2b— rs(b) £ {s^l-<j>(b,t) = (l-t)rs(b) + t(2b-rs(b)) = (1-t) -2^2b(v)v + t -2j2b(v)v v ^ 3 « € V e r t ( s x ) Notice that (f>~1 (Ws) = (Ff D Ws) X [0,1) and that r3(<f>(b, t)) = rs(b) for each t £ [0,1). Since <j>(b,t) £ |c6| for all b £ Ff and t £ 7, <f> induces a surjective map $ : /?_1(.F,f) x J —• 0 G x Ff x I -q X 1 fi-l{F>)xI-Gx\s\ Because >^ has the corresponding property, we have $ x(/3 = 3 1(Ff n ltf^ ) x [0,1). Now, the map (a, £) H+ £) : 3~l (Fff)Wa) x [0,1) -+ 3~l (Wa) factors through the projection p-x(Ff n Wa) x [0,1) -> Q £ s to define a map 6 : QRa -> 3-l(Wa). [a,t] (4.5) 2 The open cone of a space X is here the quotient of X x [0,1) which identifies the subspace X x {0} to a point. 62 We claim that 0 is a homeomorphism and show this in two steps. 1. 0 is a bijection. It is clearly a surjection since 4> is. We can see it is an injection by considering separately the disjoint images of 0~l{Ff n Wa) x {0} and 0~l{Ff n Wa) x (0,1) in QRS. On the first set, two pairs (a, 0) and (o/,0) represent the same point in QRa if and only if 0[a, 0] = Ra{a) = Ra(a') = 0[o/,O]. On the second set, we can construct a partial inverse to 0. For this, first define a partial inverse ib : Us - \s\ —• Ff x (0,1) to <j> : Ff x I —• \s\. If b G Us - \s\ then we can write 6 = (1 - e) • rs(b) + e • (b - (1 - e)r.a(b))/e, where e = 1 - £ „ G s b(v) G (0,1). Thus, if we define V>(6) = (r.(6)/2 + (6 - (1 - e)r.(4))/2e , 1 - e) \ ' "€* veVeit(s*-) ) then ib o <f> = 1 and (f> ° ib = \. Now, since ib(b) G |t| x (0,1) whenever b G we can define a partial inverse * : - |s|) -> /3_1(FSP) x (0,1) to $ : /^(Ff) x / - » 0~l\s\ by the following diagram. G x (Ua - \s\) - l ^ t G x Ff x (0,1) g x 1 / T 1 (C/s - — ^ — » ( F f ) x (0,1) Composing the map a i - f *(o;) : fi~l{Wa - (s)) -* /3_ 1(Ff D Wa) X (0,1) with the projection into QRa gives the desired partial inverse to 0. • 2. 0 is a closed map. Because s is finite, so that |s| is compact, the closed subset Ff C |s| is compact. Thus, 0-l(Ff) = q(G X Ff) is compact since G is. It follows that $ : 0~l{Ff) x I 0~l\s\ is a closed map since c C K ( G , /G/K) is Hausdorff.3 Given any closed map / : X —• Y and a subset B C Y , the map x H-> f(x) : f~l{B) -» B is also closed since / (An / _ 1 ( f i ) ) = / ( A ) n B . Thus, the map (a, t) ^ $(a, t) : (Ff n Ws) x [0,1) -» is a closed map. That 0 is a closed map now follows from (4.5): The image by 0 of a closed set is equal to the image by (a, t) i-> $(a, t) : / T 1 (Ff n Ws) x [0,1) - f (W,) of a closed set. • Finally, as already noted, the composition of Rs : (3~l{Wa) —*• 0~1{s) with the projection 0-x(Ff n Wa) X [0,1) 0~l{Ff n Wa) onto the first factor induces a map QRa - f /3_1(s) which is locally trivial with fibre the open cone on C a c ( G A , / G a / a c ) . This map makes the 3 That CK(G, /G/K) is a Hausdorff space follows from Proposition 3.2 since Gs C G is closed for each s G K: G/Gs and, thus, G/Ga x |s| is Hausdorff for each s G K. 63 following diagram commute. [q(g,b),t]> •q(g,<t>{b,t)) q(9,ra(b)) = q{g,ra{<f>(b,t))) Thus, the map a H - > Rs(O) : (5~L{WA) — • 0~1(s) is locally trivial with fibre the open cone on £ « c ( G s , / G s / S c ) . Let us explicitly write down a trivialization. If £ = (Rs,^2) '• Rj1W) —» V X £ s c ( G S , /Ga/Sc) is a trivialization of R A : /? - 1(Ff fl W S ) —• 0~1(s) then, as in Appendix C, the map [a,t] i—• (Ra(a), [^(a), t]) is a trivialization of the induced map Q R S —> V, where square brackets denote the appropriate equivalence classes. [6(a), t]« (fl.fa), [6(a),«]) If U C G / G S is a neighbourhood on which we have a section c : U —• G of the coset projection G —+ G / G s , the form of the map £2 can be extracted from the following diagram, which summarizes (4.2), (4.3) and (4.4). q(g,b) q(g,b) (q{g,b - r.(b)/2 + bs/2) , r.(bj) / T ^ F f n F°s) xfc> T" 1 x 1 (qsc (g, 2b - rs(b)) , rs(b)) Csc (G, /G./,e) X <*) G/G sx(s> (flfG.,6) U x (s) (M<7, &),&') U x Cso(Ga,/G3/ac) x (s) (sG s ,a s ^( 5 G , )-V &),&') 64 Thus, if p denotes the homeomorphism q(g,b) i — • (gGs,b) : 0 1(s) —* G / G A x (s), we define £ above the open subset V = p~l{U X (s)) C /3_1(s). t{q{9,b)) = (q(g,rs(b)), qS'{<;{gGs)-lg,2b- rs{b))) 3. Locally Euclidean Structure with a Compact Lie Group When G is a compact group, we have shown that the neighbourhood 0~L(WA) in £/<(G, /G/K) either decomposes as a product or is the total space of a bundle over 0~1(s). sc = 0 : / T 1 (W.) = / T 1 (s) « G/Gs x (s) sc^<b: CCsc(Gs,/Gs/sc)< >0-\Wa) Ra p-1(s)^G/Gsx (s) 4.4 Proposition. Let G be a compact Lie group and let /G/K = {Gs \ s G K } be a lattice of closed subgroups indexed by a locally finite simplicial complex K. Let s G K and /Ga/ac = {GsUt\tesc}. 1. If sc = 0 then each point in (s) has a Euclidean neighbourhood of dimension dim G — dim Gs + dim s. 2. If sc 7^  0 then a point in Aas a Euclidean neighbourhood only if £S<=(GS, /Gs/Sc) has the reduced homology of a sphere. Moreover, the dimension of such a neighbourhood must be dim G — dim Gs+dim s+d+1, where d is the dimension of the homology sphere. To prove this we need the following fact. 4.5 Proposition. Let X be any space and let XQ G X be a closed point. Then Hk(X xR,X xR - (x0,0)) ^ Hk-X{X,X- as0) • Conseguentfy, xRn,X xRn - (a>0>0)) = i f f c - n ( ^ , ^ - a:o)-Proof. The second statement follows inductively from the first. For the first statement, consider the long exact homology sequence of the triple (X X R, X X R — (XQ, 0), X X R — XQ X [0,oo)). • Hk (X x R - [x0,0), X x R - x0 x [0, oo)) • Hk(X xR,X XR-XQX [ 0 , O O ) ) • Hk(X xR, X xR-(xo,0)) H k-i (X xR - (x0,0), X x R - x0 x [0, oo)) • • • • (4.6) 65 Since X x (-00,0] - (x0,0) C X x JR - x0 x [0,00) is closed in X x iR - (x0,0) (a consequence of x being closed in X) and X x iR - x0 x [0,00) is open, we can excise X x (—00,0] - (x0, 0) from the pair (X x R - (x0,0), X x R - x0 x [0,00)) to get Hk(X xR - (x0,0), X x R- x0 x [0,oo)) = fffc(X x(0,oo) , lx (0,oo) - x0 x (0,oo)) ^ 7 f f c ( X , X - x 0 ) . Also, the projection X X R —* X onto the first factor is a homotopy equivalence (X X R, X X R—XQ X [0,00)) —• (X, X), the map x i-> (x, —1) being an inverse. Thus, (4.6) gives the desired isomorphism. • Proof of 4.4. 1. We have /9_1(s) = Q-l{Ws) « G/Gs x (s), which is locally Euclidean since (s) is homeomorphic to a Euclidean space and the homogeneous space G/Gs is locally Euclidean. 2. As previously, let p : /3_1(s) —»• G/Gs X (s) be the homeomorphism q(g,b) 1-+ (gGs,b). If x € /3_1(s), we've seen there is a neighbourhood U C G/Gs such that x 6 x (s)) and such that /?-1(C/ x (s)) is a trivialization neighbourhood for a 1—• Rs(a) : @~l(Wa) —• Q~l(s). e(g(5,6), i). • {gG„ r.(6), 6)), t]) RJ^p-^U x (s)))nr1(Ws)^-^Ux (s) x CCsc(Gs,/Gs/sc) Rs p~l{U x <s» > U x (s) Since x G 0~l{s), the third component of its image under the above trivialization is the vertex, *, of the open cone. Now, since G/Gs is locally Euclidean, we can choose U homeomorphic to a Euclidean space. Suppose U and (s) have dimensions n = dim G — dim Gs and m = dim s respectively so that U X (s) « JR n + m . Moreover, we can choose the homeomorphism so that it maps the appropriate components of the image of x (under the above trivialization) to the origin in i R n + m . RJ^p-^U x (s^nff-HWs) * Rn+m x C£sc(Gs,/Gs/sc) (4.7) x 1 • (0, *) We now compute the local homology of C K ( G , / G / K ) at x. Since £K(G, /G/K) is a Hausdorff space, Hk(£K{G, /G/K),CK(G, /G/K) - x) £ Hk{V,V-x) for any neighbourhood V of x. If we choose V = Rj1(p~1(U X (s))) n d~l{Ws) then by (4.7) and Proposition 4.5 we have Hk{CK(G,/G/K),CK(G,/G/K) -x)* Hk.n.m{CCsc(Gs,/Gs/sc),CCsc(Gs,/Gs/s.) - *) — Hk-n-rn-\£>sc {Gs, / G s / s c ) , 66 the second isomorphism following from the long exact homology sequence of the pair since the open cone is contractible and CCac(Ga, /Ga/ac) — * « £ S C ( G J , / G S / J « ) X (0,1). On the other hand, if we choose the neighbourhood V to be homeomorphic to an open ball in some Euclidean space then we have Hk(V, V - x) = Hk{BN, BN - 0) £ Hk_iSN~l, where N is the dimension of the Euclidean space. Thus, Hk-n-m-l£sc(Ga, /Ga/ac) = Hk-\SN~l so that Cac(Ga, /Ga/ac) is a reduced homology sphere of dimension d = N — n — m — 1 = N - dim G + dim Ga - dim s — 1, giving N = dim G — dim Ga + dim s + d + 1 as required. • It is unclear whether or not the homology spheres of Proposition 4.4 are necessarily homeo-morphic to spheres. In the case of the spheres Cs„_2 (0(n), /0(n)/sn_2) this is indeed the case as we will now proceed to show. C. More Spheres Let K and L be any two simplicial complexes, not necessarily locally finite. Without loss of generality, the vertex sets of these two are disjoint. The join of K and L is the simplicial complex K * L defined as follows. K*L = KULl){sl)t\se K, te L} (4.8) Thus, K-kL consists of all nonempty subsets u C Vert(K) UVert(L) such that uD Vert(K) and u fl Vert(iz) are either empty or simplices of K and L respectively. Notice that the union (4.8) is disjoint: A simplex of u € K it L is either a simplex of K, a simplex of L or can be written as s U t where s = u n Vert(ii') £ K and t = « f l Vert(L) G L. Suppose now that we have two topological groups G and H together with lattices / G / K = { G A C G | s G K} and /H/L = {Ht C H \ t G L} indexed by K and L respectively. Define the join of / G / K and /H/L to be the lattice / G / K * / H / L = { A U C G X H \ U £ K-kL}, where ' GuxH if u G K, A U = \ G X H u \iueL, (4.9) TGsxHt if s = u n Vert (If) ^ 0 and t = u n Vert(L) # 0. We can embed the realizations |/f | and \L\ as subsets of |/f*Z/| in the obvious way, as subspaces whose member functions are identically zero on the vertices of L and K respectively. With these embeddings in mind, we define a map 0 : \K\ xix \L\ —* \K*L\ by 0(b,t,c) = tb+(l — t)c. This map is indeed a well-defined member of \K*L\ since 0(b, t, c)-1(0,1] G K-kL for all t G / , the union of a simplex of K with a simplex of L being a simplex of K • L. Notice that 0 has the following properties. 0{\K\ x {0} x |L|) = \L\ (4.10) 0{\K\ x {1} x \L\) = \K\ (4.11) 0{\K\ x (0,1) x |L|) = \KkL\ - {\K\ U \L\) (4.12) 67 (4.10) and (4.11) are clear. For (4.12), if a G \K*L\ - (\K\U\L\) then we have a = tb + (l-t)c where t = -£veKa(v) G (0,1), b = l / i -Evg* G |ff | and c = 1 / (1 - t ) -E„ e z , G This shows that 0 is injective as well as surjective on \K\ X (0,1) X \L\. 4.6 Proposition. Let K and L be finite simplicial complexes. Let G and H be two compact groups with lattices /G/K = {GS C G \ s G K) and /H/L = { HT C H \ t G L) of closed subgroups. Then the map (g,b,t,h,c)*-+ (g,h,9(b,t,c)) : G x \K\ x I x H x \L\ -> G x H x (4.13) induces a homeomorphism £K(G, /G/K) *£L(H, /H/L) £K*L{G X H,/G/K * / H / L ) -Proof. Let qx, qL and q denote the identifications G X \K\ —* CK(G,/G/K), H X |Z| —> £ L ( H , /H/i) and G x H x \K * L\ ^ £K*L(G X if, /G/K * /H/L) respectively. Since CK(G,/G/K) * £>L(H, /H/L) is compact and £K*L(G x H, /G/K * /H/L) is Hausdorff, we need only demonstrate that (4.13) induces a continuous bijection. - (0,M(M,-e)) G x H x \K * L\ q £K*L(GXH,/G/K*/H/L) (g, b, t, h, c) i G x \K\ x I x H x \L\ QK x 1 x gL -CK(C, /G/K) xix £L(H, /H/L) £K(G,/G/K)*£L(H,/H/L) Since G X \K\ X I X H x |L| is compact and £K{G, /G/K)*£L{H, /H/L) is Hausdorff (being a join of Hausdorff spaces), the composition of the product map qu x 1 x qi with the quotient map into the join are closed maps. Thus, it is an identification and we need only show that two two tuples (g, 6, t, h, c) and (g1, b', t', h', c') in G x \K\ x I x H x \L\ represent the same point in £K(G,/G/k)*£L(H,/H/L) if and only if (g,h,0(b,t,c)) and [g',h',9(b',t',c')) represent the same point in £K*L(G x H, /G/K * /H/L)-Writing /G/K*/H/L = { Au | u G K * L } as in (4.9), (g,h,0(b,t,c)) and (g\h',0(b',t',c')) represent the same point in £K*L{G X H, /G/K * /H/L) if and only if the following holds. (1) 9(b, t, c) = 9(b', t', c') = a and [g'g-x,h'h~l) G ACQ. By (4.10), (4.11) and (4.12), if 0(b,t,c) = 0(b',t',c') then either t = t' = 0, t = t' = 1 or t, t' G (0,1). Thus, by the definition of 9 and since, as previously remarked, 9 is injective on \K\ x (0,1) x 9(b, t, c) = 9(b', t', c') if and only if one of the following hold. (2') t = t' = 0 and c = c'. 68 (3') t = t' = 1 and 6 = b'. (4') t = t' € (0,1), b = b' and c = c'. Therefore, by the definition of Aca and since c(6(b,t, c)) = c6Ucc whenever t € (0,1), (1) holds if and only if one of the following hold. (2) t = t' = 0, c = c' and h'h'1 6 77 c c. (3) t = t' = 1 and 6 = 6' and fir'sT1 £ Gc&. (4) t = t' £ (0,1), 6 = 6', c = c', € Gcb and h'h~l £ H C C . These are exactly the conditions under which (g,b,t,h,c) and (g',b',t',h',c') represent the same point in CK{G, /G/K) * C L { H , /H/L). m We now apply Proposition 4.6 to the spaces C-s(Os{n),/Os(n)/-s), s G &n-2-4.7 Proposition. Let n > 2 and let s € 5n-2- Then Cs(Oa(n),/Os(ri)/s) is homeomorphic to a sphere of dimension dim Os(n) — n • dim 0(1) + dim s. Proof. It will be convenient to extend our notation slightly. If k is an integer and lying between 0 and n — 2 inclusive and if a : {0, ...,&}—• {0,..., n — 2} is an injective function with which we embed Sk in Sn-2, let Oa(n) = C?im^ (n) and /Oa(n)/sk = { CV.tfn) | t £ 6k }, where C)^(n) = C? ?(n)nni e < 0(n, u(i)). We will show that C&K (Oa(n), /Oa(n)/sk) is homeomorphic to a sphere of dimension dim Os(n) — n • dim 0(1) + k for each n > 2 and 0 < < n — 2. We can easily construct a permutation r of {0,.. ,,k} for which the composition a <> r is order-preserving. This permutation is then a simplicial isomorphism r : <Sfc —• Sk. Since 0a.T(n) = C?^  (n) we then have a homeomorphism I X \r\ :Oa.T{n) x \&k\^Oain) x |r | , and this map induces a homeomorphism £$k (Oa.T(n),/O^.i-(n)/st) —> (^(n),/O a(n)/i k). We can therefore assume that a is order-preserving. Thus, 0(n, o~(0)),..., 0(n, is just a subsequence of 0(n, 0),..., 0(n, n — 2). The proof is now by induction on n. If n = 2 then jb = 0 and C5k{Oa{n)JOa{n)/sk) = 0(2)/(0(l) x 0(1)), which is either 0(2)/(0(1) x 0(1)) = JRP1 w 5 1, f/(2)/(C/(l) x U(l)) = CP1 w 5 2 or 5p(2)/(5p(l) x5p(l)) = JfP 1 « S4. In the first case,4 dim 0a(n)-ra-dim 0(1) + = dim 0(2) = 1. In the second case, dim Oa(n) - n • dim O(l) + k = dim 17(2) - 2 = 2. In the third case, dim Oa(n) — n • dim O(l) + k = dim 5p(2) — 6 = 4. Thus, the dimensions are as claimed. 4 Recall that dim O(n) = n(n — l)/2, dimC/(n) = n2 and dim Sp(n) = n(2n + 1). 69 Assume now that n > 2 and that Csk{Oa(n')JOCT(n')/sk) is homeomorphic to a sphere of dimension dim Os{n) — n • dim 0(1) + k for each n' < n, 0 < k < n' — 2 and injective a : {0,...,&}-+{0,...,n'-2}. If A; = n — 2 then 0^ (n) = 0(n), and £«„_2 (0(rc), /0(n)/sn_2) is one of the spheres constructed in chapter 2. Since dim0(n) — n • dim0(1) + k = n(n — l)/2 — n-O + n — 2 — n(n +l)/2 — 2, dim U(n) — n • dim U(l) + k = n2-n-l + n — 2 = n2 — 2 and dim 5p(n) — n • dim 5p(l) + k — n(2n + 1) — n-3 + ra — 2 = n(2n — 1) — 2, the formula for the dimension of the sphere is again correct. (See page 33.) If k < n — 2, write {0,..., n — 2} — im a = {ii,..., in_2_fc}, where »i < •• • < in_2-fc. Then Oa(n) = Oiix + 1) x 0(i2 - h) x • • • x 0(in-2-k - » „_2 - f c - i ) x 0(n - in-2-k - 1), (4.14) with the obvious interpretation if n — 2 — k = 1 or 2. We have three cases to consider. 1. ti = 0. We then have 0-„(n) = 0(1) x Oa.{n - 1), where a' : {0,..., k} - f {0,..., n - 3} is defined by c'(j') = — 1. Since each element of the lattice /Oa{n)/sk is of the form O(l) x H, the product of the projection O(l) X Oa< (n-1) —• Oa'(n-l) with the identity map on \8k\ induces a homeomorphism between C&k (Oa(n), /Oa(n)/sk) and £$k (Oai(n-l), /Oai(n-l)/sk). By the inductive hypothesis, the latter isasphere of dimension dim 0„i(n — l)-(n-1)-dim C(l)+k — dim Oa (TI) - TI • dim 0(1) + k. • 2. in-k-2 = n — 2. In this case Oa(n) = 0a.(n - 1) X 0(1), where a' : {0,..., k} -> {0,..., n - 3} is defined by a ' U ) = ati)- As in the previous case, we can use projection Oa(n) —> 0c t»(TI — 1) to define a homeomorphism between Cslc(0(T(n),/Oa(n)/sk) and Csk(Oa<(n- 1),/Oa<(n- l)/sk). This is a again sphere of the claimed dimension by the inductive hypothesis. • 3. ii > 0 and in-2-k < n - 2. Define two simplices of 8k by sx = cr_1{0,..., ix} and s2 = <T _ 1{Z'I, . . . , n — 2}. These are indeed nonempty sets since {0,n —2} C imc Moreover, since i\ ^ imcr, si and S2 are disjoint. Since their union is Vert(5fc) = {0,...,k}, it follows that sx • s2 = 8k- Note that sx is an (ix — l)-simplex and s2 is a (k — i'i)-simplex. Define ax : {0,..., ii - 1} —> {0,..., ii + 1} and a2 : {0,..., k - ii} —• {0,..., n - ix - 1} by <7i(j) = o-(j') and a2(j) = a(h + j) - u - 1 . Then Oa (n) = Oax (h + 1) x 0Va (n - h - 1) = 0(h + l)x O02(n- ii-1) and, if we let 0 C l t 9 ( i i + 1) = 0 f f l l 0 ( l i + 1) and 0n_ i l_1(O-2,0) = 0 a 2(n - h - 1), 0<r,t(") = 0,2,<nSl(*'i + 1) X 0C T 2,< n s 2(n — *i — 1) 70 for each t G 5k. Thus, /0a(n)/5k - /0<,2(ii + l)/*^^ * /Oa2(n - ii - l)/sk-il, and we get C&k {00{n),/00{n)/ih) = Cs^iO^ih + 1) x 0„ 2 (n - h - 1), /Offl(ii + I)/*,., */Oa2{n — i\ — l ) ^ ) « £ S l (0,, (ii + l),/Oai(t'i + l)/sil_1) • £ 5 2 (0 f f 2 (n - n - 1), /0 f f 3 (n - ii - 1)/Sk_h) by Proposition 4.6. By our inductive hypothesis, the two factors of the above join are spheres of dimension dim 0<Ti (i\ + 1) — (ii + 1) • dim 0(1) + dim si and dim 0 f f 2(n — ii — 1) — (n — ii - 1) - dim (9(1) + dims2 respectively. Thus, since 5m> • S7712 « S^i+^+i, dim(9(7(n) = dim Oai (ii + 1) + dim Oa2 (n — i\ — 1) and dim si + dim s2 = k — 1, their join is a sphere of dimension (dim Oax (ii + 1) - (ii + 1) • dim 0(1) + dim si) + (dim O0, (n - ii - 1) - (n - *i - 1) • dim 0(1) + dim s2) + 1 = dim Oa(n) - n • dim 0(1) + k . u Let us apply 4.7 to the homology spheres in the second statement of Proposition 4.4 in the case K = u for some u G £ n - 2 , G = Ou{n) and /G/u = /0 U («) /«• If s G ii is a proper face of u then Gs = Ou{n) n n,-e. 0(n, i) = 0u,,(n) and, letting w = u — s, /Gs/3c = {GsUt\t£sc} = {Ou,sut(n)\tesc} = {0w,t(n) | t G w} = /Ow(n)U so that Csc(GJGs/so) - £^(0(n),/O w(n)/^) and Cac(GsJGa/ac) = C^(Ow(n),/O^n)/^). Thus, in this case the homology spheres Cac (G s, /Gs/ac) are in fact homeomorphic to spheres. Counting Dimensions We can enumerate exactly which spheres are constructed in Proposition 4.7. Let Z>R, V<C and Pjjf be the sets of dimensions of the spheres constructed in the real, complex and quaternionic cases respectively. We then have Z>H = { dim Os(n) + dim s | s G £ „ _ 2 , n > 2 } , Vc = { dim Ua (n) — n + dim s | s G <5n-2, n > 2 } , and 71 Dm = { dim Sps(n) - 3n + dim s | s € <S„_2, n > 2 } since dim 0(1) = 0, dim U(l) = 1 and dim5p(l) = 3. Let us also consider those dimensions which are constructed over an JV-simplex for some N £ No. VR(N) = {dimOs(n) + N \ s e £ n_ 2, n > 2, dims = N} VC(N) = { dim Us(n) - n + TV | s e S„- 2 , « > 2, dim s = N } £>w (iV) = { dim Sps(n) -Sn + N\ se <5n_2, n > 2, dim s = N } Of course, VK = U ^ = 0 ^K(iV) for K = R, € and Iff. 4.8 P ropos i t i on . V]R(N), T>C{N) and VH(N) are finite sets for each N e No, and there are the following equalities. minX>]R(Ar) = 2JV + 1 maxVR{N) = {N + 2) (N + l)/2 + N minVc{N) = SN + 2 maxVc(N) = (N + 2) [N + 1) + N mmVH{N) = 5iV + 4 maxP^CiV) = 2(iV + 2)(iV + 1) + iV 4.9 P ropos i t i on . There are the the following equalities. 1. VR = N - {2}. 2. VC = N - {1,3,4,6,9,12}. 3. VH = N- {1,2,3,5,6,7,8,10,11,12,15,16,17,20,21,22,25,30,35}. To prove 4.8 and 4.9, note that for each integer 2 < k < n, there is the following one to one correspondence between the (k — 2)-simplices of <5n_2 and the collection of ^ -tuples of positive integers summing to n. ?!<•••< ik-i {h, ...,ik-i}i • (ii + l , i 2 - i i , . . .,ifc_i - ik-2,n - i*-i - 1) Sn-2 < • { (»i, • • -,Tik) e Nk I k > 2, ni -\ h nk - n} {ni - 1,..., ni H h nk-i - 1} < 1 (m,..., nk) If we let the empty set correspond to the "1-tuple" n we then have a one to one correspondence between £„_ 2 U {0} and the set of tuples as above, but with k > 1 rather than k > 2. Now, if s € $n-2 then the complement of s in the vertex set of <Sn_2 is either empty or a proper face in Sn-2, and this corresponds to some &-tuple (ni,..., nk) ^ (1,..., 1) = 1. Moreover, Oa{n) = 0(ni) x---xO(nk) 72 (with the obvious interpretation if k = 1) and dim s = n — k — 1 = EjLi nj — k — l . 5 Proof of 4.8. Since dim 0(1) = £(£ - l)/2, dim U(£) = £2 and dim Sp(£) = £(2£ + 1), and because of the aforementioned correspondence, we have the following equalities. VB(N) = {d\mOs(n) + N \ s e <S„-2, n > 2, dims = N} k 3=1 keN, (m, ...,nk)eNk - {!}, J2ni - k - 1 = N i=i VC(N) = {dimUa(n) - n + N \ s£ <S„_2, n > 2, dims = TV} = { 5>i " n i ) + N k £ N, (m,...,n f c) e W f c — f 1>, ^ r i j — k — 1 = N K 3=1 keN,(m,...,nk)eNk- {i}, J2n3 -  k -  1 =  N 3 = 1 VH(N) = { dim Sp3(n) - 3n + N | s e <5„-2, n > 2, dim s = N} = { E ( » i ( 2 » i + l ) -3n i ) + iv * G W, (ni,..., n*) € Nk - {1}, nJ ~ k " 1 = N K = {j2^j(nj-l) + N 3 = 1 3 = 1 keN, (ni,.. .,nk) e Nk - {1}, ^ rij - k - 1 = N 3 = 1 (4.15) (4.16) (4.17) Notice that any nj = 1 does not contribute to the sums in (4.15), (4.16) and (4.17). Since the constraint ]Cj=i rij—k—1 — N is still satisfied if we omit these (the value of k decreasing by 1 for each rij omitted), it suffices to consider sums with each rij € W2, where N2 = {n£N\n>2}. 3 = 1 K VR(N) = {J2nj(nj-l)/2 + N k keN, {nu...,nk)e N*,J2nj-k-l = N} (4-18) 3 = 1 5 If < is the complement of s in the vertex set of Sn-2 then dim< = k — 2. Since dims + dim* = dim<5„_2 — 1, dims = (n — 2) — 1 — (A; — 2) = n — k — 1. 73 K k keN, (n1,...,nk)eNf,'%2nj-k-l = N} (4.19) 3=1 i, .., nk) e N$, 3 = 1 k VH(N) = {j2^j(nj-l) + N 3 = 1 keN, (n1,...,nk)eN%,J2n3-k-1 = N } (4-20) 3 = 1 We need now only find the minimum and maximum integers occurring in (4.18), (4.19) and (4.20). Let us write K for any of R, <D or H and suppose k e N, (nx,..., nk) G N2 and E j L i nj ~ k — 1 = N so that (n\,.. .,nk) determines an element of T>K{N). If i G {l,...,k} and rij > 3, let a e N2 and 6 G IV be any two elements whose sum is rij. Then the (k + l)-tuple (mi,..., mk+i) = (ni,..., rij_i, a, b+ 1, n , + i , . . . , nk) (with the obvious interpretation if i = 1 or &) determines another element of VJH- Moreover, starting with the 1-tuple rii = N + 2, every (ni,..., nk) e N$ satisfying J2j=i nj - k — 1 — N can be obtained in this manner, and from such a tuple the (N + l)-tuple (2,..., 2) can be obtained. Thus, since k k+1 X^'K' - 1) - Ylmi(mi ~ 1) = ra«'(ni K a - f) + (b + J = 1 j = 1 =(a + 6)(a + 6 - l ) - ( a 2 - a + 62 + 6) = (a2 + 2ab + b2 - a - b) - a2 + a - b2 - b = 2ab - 2b = 2b(a - 1) > 0 for any a G N2 and b G N, the minimum values of the sums in (4.18), (4.19) and (4.20) are attained when k = N + 1 and -m = • • • = rajv+i = 2, and the maximum values are attained when k = 1 and = N + 2. Thus, (4.18), (4.19) and (4.20) give the following minimum and maximum values. minVR(N) = {N + 1) + N maxPR(iV) = (N + 2)(N + l)/2 + N = 2N+1 mmV€{N) = 2(N + l) + N maxPc(iV) = (N + 2)(N + 1) + N = SN + 2 min VH(N) = 4(N + l) + N m&xVH(N) = 2(N + 2)(N + 1) + iV = 5iV + 4 • 74 (4.21) Proof of 4.9. Since VK = U;v=o V K { N ) for K = R , € and iff, the following equalities are a direct consequence of (4.18), (4.19) and (4.20). fc fe j=i j=i fc = { ^ n i ( « i + l ) / 2 - A - l | K I V , (ni,...,nfc) € W2fc } fc fc Vc = { Y^nAni ~ - 1 fc € W, (m,...,nfc) € W2fc} ' " 3 = 1 keN, {nu...,nk) G w2fc} K fc I > H = { J^2n i(n i-l) + 5^n,--fc-l|fc€-W, ( « i , . . . , nfc) € W2fc J 3 = 1 3 = 1 fc = { ^ 2 n i ( 2 N 3 - 1 ) - K - 1 k <E N, (nu...,nk) e 3=1 { t n ) - k - l 3 = 1 (4.22) (4.23) 3 = 1 We now consider our three cases separately. l.VR = N- {2}. Taking k = 1 and ni = 2, we get 1 6 PJR. Taking = 2 and ni = n 2 = 2, we get 4 € £ > J R . Now, if a € VR write a = Sj=i n i ( n i ) / 2 - fc — 1, where k e N and (ni,..., nfc) € W2 . Setting rijfc+i = 2, we then have fc+i 3=1 J2 njinj + l)/2 - (fc + 1) - 1 = ^rij(nj + l)/2 + 3 - (fc + 1) - 1 fc = nj- (T»J- + 1)/2 - fc - 1 + 2 = a + 2 so that a + 2eVR. Thus, {1,3,5,...} U {4,6,8,...} = N - {2} C 2>K. To see that 2 g £>R, if ( m , . . . , nfc) € W2fc then iV = £ * = 1 rij{nj + l)/2 - fc - 1 > 2fc - 1. Thus, N = 2 only if fc = 1. In this case N — n\(n\ + l)/2 — 2, and this sum is equal to 1 if n\ = 2 and greater than or equal to 4 if ni > 3. • 2. Vc = N - {1,3,4,6,9,12}. Taking fc = 1 and n\ = 2, we get 2 6 Vc- Taking fc = 1 and rii = 3, we get 7 6 T>c- Taking fc = 2 and n\ = n 2 = 3, we get 15 € X><c. Now, if a e 2><o write a = 5Zj = 1 ri2 — fc — 1, where 75 k G N and {n\,..., nk) G . Setting nk+\ = 2, we then have 5>3_(*+i)-i = X>3+4-(*+i)-i j=i j=i fc = ^ n 2 - f c - l + 3 = a + 3 i=i so that a+3 G £><D- Thus, {2,5,8,.. .}U{7,10,13,.. .}U{15,18,21,...} = N-{1,3,4,6,9,12} C To see that equality holds, if ( « ! , . . . , nk) G N% then AT = £ * = 1 T i 2 ; - - / c - l = 4A;-A;-l>3A:-l. Thus, N < 12 only if A; < 4. Now, we can clearly assume ni < • • • < nk for every tuple (rii,..., nk) G iW* defining a sum in (4.22), and there are only a finite number of such tuples for which the corresponding element of T>c is at most 12. We enumerate these below. k (ni,..., nk) E j U n) ~ k ~ 1 1 2 2 3 7 2 (2,2) 5 (2,3) 10 3 (2,2,2) 8 4 (2,2,2,2) 11 3. VM = N - {1,2,3,5,6,7,8,10,11,12,15,16,17,20,21,22,25,30,35}. Taking k = 1 and ri\ = 2, 3 or 4, we get 4,13,26 G DJJ- Taking k — 2 and ni = n 2 = 3, we get 27 € X>H- Taking k = 2, rax = 3 and n 2 = 4, we get 40 € X>H- Now, if a € T>u write a = ^2j=i (2ra2 — 7 i j ) — k — 1, where k £ N and (ni,..., njt) £ W2fc- Setting rifc+i = 2, we then have fc+l fc nj(2nj - !) - (k + 1) - 1 = raJ'(2nJ - 1) + 6 - (* + 1) - 1 j=i j=i fc = ^ n,-(2nj - l ) - f c - l + 5 = a + 5 i=i so that a + 5 G X>H- Thus, {4,9,14,...} U {13,18,23,...} U {26,31,36,...} U {27,32,37,...} U {40,45,50,...} = N- {1,2,3,5,6,7,8,10,11,12,15,16,17,20,21,22,25,30,35} C Vm. To see that equality holds, if (m,...,nk) G N$ then A^  = £ * = 1 rij(2nj-l)-k-l >6k-k-l = 5k — 1. Thus, N <35 only if k < 7. As in the previous case, we can assume ni < • • • < nk for 76 every tuple (nj,..., n^ ) € i/V"^  defining a sum in (4.23), and there are only a finite number of such tuples for which the corresponding element of T>M is at most 35. k (ni,...,n f c) E * = i n j ( 2 » j - l ) - * - l 1 2 4 3 13 4 26 2 (2,2) 9 (2,3) 18 (2,4) 31 (3,3) 27 3 (2,2,2) 14 (2,2,3) 23 (2,3,3) 32 4 (2,2,2,2) 19 (2,2,2,3) 28 5 2^^  2^  2} 2) ^ ) 24 (2^  2,2j 2) 3) 33 6 (2,2,2,2,2,2) 29 7 (2 ^  2 ^  2^  2j 2 ^  2^  2^  34 77 Bibliography [A] M. F. Atiyah, R. Bott and A. Shapiro. Clifford Modules. Topology 3, 3-38 (1964). [B] I. B. Bersuker. The Jahn-Teller Effect: A Bibliographic Review. Plenum, 1984. [Br] R. Brown. Topology: A Geometric Account of General Topology, Homotopy Types and the Fundamental Groupoid. Ellis Horwood, 1988. [D] R. R. Douglas. Department of Mathematics, University of British Columbia. A canonical construction yielding a global view of twistor theory. Mathematical Physics Electronic Journal 2, No. 2 (1996). Access to the journal is through a variety of electronic means. • Electronic mail: mpej@math.utexas.edu • Anonymous ftp: ftp.ma.utexas.edu/pub/mpej • Gopher: gopher.ma.utexas.edu 70 • WWW: http://www.ma.utexas.edu/mpej/MPEJ.html http://mpej.unige.ch/mpej/MPEJ.html [DR1] R. R. Douglas and A. R. Rutherford. Berry Phase and Compact Transformation Groups. In Mathematical Quantum Field Theory and Related Topics, Proceedings of the Montreal Conference. J. S. Feldman and L. Rosen (eds). Canadian Mathematical Society, 1988. [DR2] R. R. Douglas and A. R. Rutherford. Department of Mathematics, University of British Columbia. The Geometry of Degeneracy. Manuscript in preparation, 1995. [DR3] R. R. Douglas and A. R. Rutherford. Department of Mathematics, University of British Columbia. Pseudorotations in molecules I: Electronic triplets. Preprint, 1995. [DR4] R. R. Douglas and A. R. Rutherford. Department of Mathematics, University of British Columbia. Pseudorotations in molecules II: Electronic quadruplets. Preprint, 1995. 78 J. Dugundji. Topology. Allyn and Bacon, 1966. H.-D. Ebbinghaus, H. Hermes, F. Hirzebruch, M. Koecher, K. Mainzer, J. Neukirch, A. Prestel and R. Remmert. Numbers. Springer-Verlag, 1990. David Gurarie. Symmetries and Laplacians: Introduction to Harmonic Analysis, Group Representations and Applications. North-Holland, 1992. T. W. Hungerford. Algebra. Springer-Verlag, 1974. N. Jacobson. Normal semi-linear transformations. American Journal of Mathematics 61, 45-58 (1939). H. A. Jahn and E. Teller. Stability of polyatomic molecules in degenerate electronic states I — Orbital degeneracy. Proceedings of the Royal Society (London) A161, 220-235 (1937). S. Lang. Algebra. Addison-Wesley, second edition 1984. S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag, 1977 W. S. Massey. Imbeddings of projective planes and related manifolds in spheres. Indiana University Mathematics Journal 23 9, 791-812 (1974). D. Montgomery and L. Zippin. Topological Transformation Groups. Interscience, 1955 J. R. Munkres. Elements of Algebraic Topology. Addison-Wesley, 1984 79 A. R. Rutherford. Holonomy in Quantum Physics. Ph.D. Thesis, University of British Columbia, 1989. E. H. Spanier. Algebraic Topology. Springer-Verlag, 1966. N. [E.] Steenrod. The Topology of Fibre Bundles. Princeton University Press, 1951. N. E. Steenrod. A Convenient Category of Topological Spaces. Michigan Mathematics Journal 14, 133-152 (1967). F. W. Warner. Foundations of Differentiable Manifolds and Lie Groups. Springer-Verlag, 1983. 80 Appendix A Diagonalization in Herm(n, M) The Principal Axis Theorem states that every symmetric matrix with entries lying in the real number field is orthogonally diagonalizable. That is, if A G Herm(n, R) then there is an orthogonal matrix P G 0(n) and a diagonal matrix D such that A = PDPT, where PT is the transpose of P. Of course, the diagonal elements of D are then just the (real) eigenvalues of A, repeated with the dimension of the corresponding eigenspace, and the columns of P are eigenvectors of A, forming a basis for Rn. The corresponding statement for complex Hermitean matrices is also well known: If A G Herm(n, <C) then there is a unitary matrix P G U(n) and a diagonal matrix D such that A = PDP^, where P* is the conjugate transpose of P. Less well known is the fact that one can also diagonalize Hermitean matrices having entries in the quaternion algebra iff: If A G Herm(ra, Jff) then there is a symplectic matrix P G Sp(n) and a real diagonal matrix D such that A = PDP^. This is a consequence of Theorem 13 in the 1939 paper [J], but the approach we take here differs from that of this reference. We will show that diagonalizability in Herm(n, Iff) is a consequence of diagonalizability in Herm(n, R). The method of proof can be adapted to show that diagonalizability in Herm(n, <C) is also a consequence of diagonalizability in Herm(n, R) but we will restrict our attention to Herm(n, Iff). Recall that Iff has a basis consisting a multiplicative unit 1 and the three imaginary quaternions i, j and k, with 1 generating (as a real subspace) the center subalgebra R C Iff, i2 = j2 = k2 = —1, ij = —ji = k, jk = —kj = i and ki = —ik = j. Also, the symplectic group Sp(n) consists of those matrices P G Iff (n) satisfying PP* = P*P = I, where is now the conjugate transpose in H(n). Equivalently, Sp(n) consists of those P satisfying (Pu, Pw) = (v, w) for all v, w G Iff", where the inner product (•, •) is defined by (0, w) = E?=i Viw\ when v = («!,..., vn) and w = (u>i,..., wn), Vi being the conjugate in M. Suppose now that A G Jff(n) and that A = PDP^ for some P G Sp(n) and real diagonal D G lff(n). If Vi G iff" is the Vth column of P and A,- G R is the entry of D lying in row and column i then Avi = PDP^Vi = PDii = P • A.ej = iJjA,. That is, the column Vi of 81 P is an eigenvector of A corresponding to the right eigenvalue A;.1 Thus, the problem of diagonalization in H(n) is one of finding solutions to the equation Av = CA. As in the complex case, it is easy to see that {Av, w) = (v, A^w), for any A G H(n) and v, w G Mn. Moreover, (va, w) = a • (v, w) and (v, wa) = (v, w) • a for any a G M. Thus, there is the following. A . l Proposition. The right eigenvalues of a quaternionic Hermitean matrix are real. Of course, any real right eigenvalue is also a left eigenvalue and vice versa. Proof. If A € Herm(n, H) and Av = vX then so that (A — A) • (v, v) = 0, (v, v) = | |u|| 2 being a real number (which commutes with A). Thus, if v 7^  0 then A = A. • The situation is different for left eigenvalues. For example, the matrix so that —k is a left eigenvalue. A.2 Proposition. Let A G Herm(n, H) and P G Sp(n). The following are equivalent. 1. The columns of P are eigenvectors of A corresponding to real eigenvalues. 2. P^AP is a diagonal matrix. 3. P^AP is a real diagonal matrix. Proof. 2 and 3 are equivalent since P^AP is Hermitean. That 3 follows from 1 is clear. If 3 holds, let D = P^AP. Then AP = PD, and this says exactly that the columns of P are eigenvectors of A with the eigenvalues being the diagonal entries of D. u We find the real eigenvalues of an A G Herm(n, H) by constructing a real symmetric matrix in Herm(4n, R) which has the same real eigenvalues as A. Let us write e\ = 1, Ei = i, £3 = j Since the multiplication in M is not commutative, we must distinguish between right eigenvalues satisfying Av = vX and left eigenvalues satisfying Av = Xv. (Av, v) = (vX, v) = X • (v, v) = (v, Av) = (v, vX) = (v, v) • X has 82 and £4 = k. Then the elements £^em, where £ £ {1,2,3,4} and m £ {l,...,n}, form a basis for Hn as a real vector space, and we define a real vector space isomorphism $ : iff" —• R4n by $(£eim) = e^m_i)+g. Use this isomorphism to define another real vector space isomorphism * : Jff(n) ->• R(4n) by #(A) • v = $(A • $ _ 1 («;)) . It is clear that an element v £ Rin is an eigenvector of \P(A) corresponding to an eigenvalue A € if and only if £ iff" is an eigenvector of A corresponding to the same real eigenvalue. Thus, the existence of real eigenvalues of quaternionic Hermitian matrices is a consequence of their existence for real symmetric matrices and the following. A.3 Proposition. Let A £ H(n). Then #(A) £ Herm(4n, iR) if and only if A £ Herm(n, If). Proof. Let A[k, k'] £ iff be the entry of A lying in the row k and column k'. If i and j are two integers lying between 1 and 4n inclusive, then the entry of \&(A) lying in row i and column j is the dot product e,- • \T/(A)ej. If we write i = 4(m — 1) + £ and j = 4(m' — 1) + £' respectively, where £,£' £ {1,2,3,4} and m,m' £ {1,.. . ,n}, and if A[fc,A:'] = Ylr=i o-[k,k',r]£r with a[k, k', r] £ R, we then have the following equalities. 9{A)ij = *(A)e 4 ( m ,_ 1 ) + f = ¥ (A) •*(£/. em.) = $(A • et>em>) = $ ( £ f c A[k, m'}et,ek) = ^2^(A[k,m']ee<ek) k = y~]$(a[k,m',r]£r£eik) k,r = ^2a[k,m',r]$(£r£i,ek) (A.l) k,r Now, since i = 4(m - 1) + £ and &(£sek) = e4(k_i)+s, ii • &(£r£{>ik) ^ 0 if and only if k = m and r £ {1,2,3,4} is the unique integer such that £r£v £ {£e, —£e}- Indeed, if we write £r£t> — fise, where p £ {—1,1}, then i{ • Aij = a[m,m',r] • p. Interchanging the roles of m and m' and of £ and £', we also have ij • Aii = a[m',m,r'] • / / , where r' G {1,2,3,4} and n' G {—1,1} are the unique integers such that £r>£t = p'sp- Of course, since we already have £r££i = fi£f, necessarily £r£g = H£2£p so that r' = r and fi' = fi or — fi respective of whether £2r = £i or -£ i ; that is, of whether r = 1 or r £ {2,3,4}. Thus, we have ii • Aij = fi • a[m, m', r] and { fi • a[m', m, 1] if r = 1, -fi-a[m',m,r] ii r £ {2,3,4}. It is clear then that \P(A) G Herm(4n, JR) if and only if A £ Herm(n, iff). • 83 Although we won't use this fact, it is not difficult to verify that (A.l) is equivalent to the statement that the matrix \P(A) is obtained from A G Jff(rc) by replacing each entry ae\ +6^ 2 + c£3 + dei = a + bi + cj + dk G Jff by the antisymmetric matrix fa -b —c -d\ b a -d c c d a -b \d —c b a) G JR(4). The notion of dimension is well defined for i?-modules in the case that R is a division ring, the dimension of a submodule being the number of elements in a basis.2 Since iff is a division ring and since iff" has the structure of a right Jff-module we therefore can speak of the dimension of a submodule W C iffn and we write dimjj W for this integer. For a right eigenvalue of an matrix AG Jff (n) the eigenspace E\ = { v G iff" | Av = vX } is a submodule of iff" and we can therefore speak of dimjfj E\.3 Of course, dimjfj- iff" = n. A . 4 Proposition. Let A G Herm(n, iff). Then there is a symplectic matrix P G Sp(n) and a real diagonal matrix D such that A = PDP^. Moreover, D is unique up to a permutation of its diagonal entries and every real eigenvalue A of A occurs as one of these entries with multiplicity dimjff Proof. If P G Sp(n) exists as claimed then its columns form a basis for Jff" as a right Jff-module. For these columns are mutually orthogonal and therefore linearly independent, and there are n = dimu Jff" of them. Denote these columns by vx,.. ,,vn and let A j , . . . , A n G JK be the respective eigenvalues of A to which they correspond, these being the diagonal entries of D. If A is any real eigenvalue of A and v G Jff" is a corresponding eigenvector then we have v = victi + h vnan for some a\,..., an G Jff, not all zero. Since Av = vX = (viai H h vnan)X = viXxai -) + vnXnan we have vi(X — Xi)ax + \-vn(X — A n )a n = 0. Thus, A G {Ai,..., An} since at least one of a i , . . . , an is non-zero. Moreover, dimu Ex = dimn ker (A — XI) = dimn ker (D — XI) and this number is exactly the number of times that A occurs on the diagonal of D. It remains only to demonstrate the existence of P. Let A i , . . . , A* G R be the distinct eigenvalues of *(A) G Herm(4ra, Jft) and let Ex,..., Et C JK4n be the corresponding eigenspaces, which are mutually orthogonal and whose sum is all of JR4". 2 See, for example, [H]. (Chapter IV, section 2.) 3 Note that the eigenspace of a left eigenvalue need not be a submodule (since multiplication in Jff is not commutative) although it is a subspace of H" when this is regarded as a real vector space. 84 As we noted previously, an element v G R4n is an eigenvector of Si! (A) corresponding to an eigenvalue A G R if and only if G Hn is an eigenvector of A corresponding to the same real eigenvalue. Thus, E[ = (E\),..., E\ = 4>-1 (Eg) are eigenspaces of A. Moreover, as in the real and complex cases, these eigenspaces are mutually orthogonal since if v, v' G Iff" are eigenvectors corresponding to distinct real eigenvalues A, A' G R respectively then (Av,v') = X- (v,v'} = (v, Av') = A' • (v, v') so that necessarily (v, v') = 0. Choose ui G iff" to be any eigenvector of A corresponding to one of Ai,... ,A^. Suppose now that there are mutually orthogonal eigenvectors v\,...,vm G Iff", with m < n and each eigenvector corresponding to one of the eigenvalues A i , . . . , A^ . Let W C Rin be the subspace spanned by the elements $(vre) G 1ft4", where r G {l,....,m} and e G {l,i,j,k}. W is a proper subspace since m < n. Since Ex + \- E( = R4n we can choose a non-zero element w G Ei U • • • U Et which does not lie in W. If v = <fr-1 (w) then v is an eigenvector of A and we can define um+i G Mn by subtracting off the components along each of V\,.. .,vm. vm+1 = v- ui(ui,i;)/||ui||2 vm(vm,v)/\\vm\\2 Note that vm+i 7^  0 since otherwise <&(u) G W. Note also that vm+i is an eigenvector of A since (vr,v) = 0 unless vr corresponds to the same real eigenvalue of A as does v. Thus, we have mutually orthogonal eigenvectors vi,.. .,vm+i G Iff" of A, each eigenvector corresponding to one of the eigenvalues Ai,...,A^. Starting with vx we can therefore construct mutually orthogonal eigenvectors vi,..., vn G Iff". If we normalize each of these eigenvectors by dividing through by their norms and if we let P be a matrix having the normalized eigenvectors as columns then P G Sp(n) and D = P^AP is real diagonal. • 85 Appendix B Topologizing with Subsets In this appendix we collect some results on the construction of a topology on a set from a collection of topologized subsets. These have the misfortune of having often been saddled with the rather nondescriptive and overused adjective "weak". Our goal is Proposition B.5, which we employ in chapters 2 and 3. Given a topology T on a set X and a subset Y C X, let Y fl T denote the subspace topology induced on V: Y f)T — {Y C\ U \ U £ T}. Let T° denote the collection of closed sets: TC = {UC = X -U\U GT>. Now, let X be any set and suppose s/ is a collection of topologized subsets. If A € s/, we will write 2TA for the topology on A. The collection & = { U C X | A n U £ &A for all A £ si } is a topology on X, the largest topology for which the inclusion A «-+ X is a continuous function for each A £ si'. We will refer to & as the topology generated by s/. We clearly have A fl & C &A for any A £ si but the converse need not hold. For example, A D B £ 2TA m a y fail for some B £ s/ distinct from A.1 B . l Lemma. The following are equivalent. 1. For all A,B £ s/, (Ar\B)nSA = (AnB)n^B and ADB £ 3TB. 2. For all A £ s/, A £ ST and A n & = STA-Proof. (1 => 2) Let A £ sf. A £ ST is clear from the definition of 9 since B n A £ 3TB for all B £ si'. As already noted, we always have A n & C &A- Suppose U £ SA- Then for any B £ s/ we have B D U = (Af) B) DU £ (An B) D SA = (An B) C) &B- Since AnB £ &B, necessarily BnU £ STB. Thus, U £ 2F as required. 1 Take X = R and ^  = {A, B}, where A = (—1,1) and B = [0,1) are given Euclidean topologies. 86 (2=> 1) Let A,B £ si. Since i n ^ = ^ and B 6 ^ , necessarily An B £ ^4. Also, A f 1 ^ = ^ i gives ( A n B ) f l ^ i = ( A n B ) n 5\ Similarly, (73 n A) n 3TB = (B n A) n ^ \ Thus, (A D B) D 5^ = (A n JE?) ("I &B as required. • B.2 Lemma. The following are equivalent. 1. For all A,B € £/, {An B) n = (An B) n and An B € 2T£. 2. For all A £ si, A £ STC and A n & c = Proof. Work with closed sets rather than open sets, replacing each occurence of a topology in the proof of Lemma B.l by its complement. • Suppose now that the set X is a topological space with topology 3?x and that, for each A £ si, the topology 3?A is the subspace topology: 2?A = A n STX. As before, si generates a topology ST = { U C X I A n C/ e 2fA for all A £ ^ } on X. We clearly have ^ C &, but the converse need not hold. For example, take X = { 1,2,3 }, 3TX = { 0, {1}, {2}, {1,2},X} and ^  = {A}, where A = {1, 2}. Since A has the discrete topology, so does X. Recall that a collection of subsets of a space is called locally finite if each point in the space has a neighbourhood that intersects at most finitely many members of the collection. Recall also that the union of a locally finite collection of closed sets is closed. B.3 Lemma. The following statements hold. 1. If si generates 2?x then either \J si = X or any subset of X — \J si is open in X. 2. If each A £ si is open in X and either \Jsi = X or any subset of X — \Jsi is open in X then si generates SFX-3. If si is locally finite, each A £ si is closed in X and \Js/ = X then si generates &x-Moreover, ifY is any space then the collection { y x A | A G ^ } generates the product topology onYxX. Proof. 1. If (Jsi 7^  X then any subset which does not intersect [jsi lies in ST. 2. Let U £ ST and write JJ = (U n (J si) U (U - \J si). By hypothesis, the second term in this union is open in X. As for the first, for any A £ si, Anil £ &A since U £ &, and A £ 3TX by hypothesis. Thus, A D U £ &x for all A £ si so that [JAet/(U n A) = U n\J si is also open in X. 3. Let K £ STC. Since K £ 3TC\ we have AnK £ ST* for each A £ si. Since each A £ si is closed in X, we therefore have AnK closed in X for each A £ si. Thus, since the collection { A D K | A e ^ } i s locally finite, K = \JAee,(A f~l K) is closed in X. The second statement follows similarly since {Y"xA |Ae .e^}is also locally finite. • 87 Let us now suppose that our space X is Hausdorff. If Jff is the collection of all compact subsets of X, define yet another topology on X. 3TCG = { U C X | Uc D K e &K for all K eJXT} We clearly have Six C ^CG- X is said to be compactly generated if JXf generates Six • Note that if si C then ^CG C ^  so that Six C ^CG C &. In particular, if ^ generates then X is compactly generated. We are now in a position to prove our desired result. B.4 Proposition. Suppose the topology of a Hausdorff space X is generated by a collection si of compact subsets. Suppose further that any compact subset of X is contained in a finite union of members of si. IfY is locally compact Hausdorff then the product topology on X xY is generated by{AxY\A£s>/}. Proof. As noted above, X is compactly generated since si generates S?x • Thus, since Y is locally compact Hausdorff, Theorem 4.3 in [St2] tells us that X x Y is compactly generated. Let & be the topology on X X Y generated by sixY = {AxY\A£ si}. As always, &XxY C Si. To demonstrate the reverse inclusion, let C € Sfc. We will show that C € &xxY by showing that C fl K is closed in K for each compact K C X x Y. Given such a K, consider its image under the projection X xY —y X onto the first factor. Since this image is compact, our hypotheses imply that it is contained in a finite union of members of &/. Thus, K is contained in a finite union of members of s/ x V, say K C Ui=i A{ X Y. Since C € necessarily Cfl (A X Y) is closed in A X Y for each A € s/. Since A X Y is itself closed in X x Y, C D (A x Y) is closed in X x Y for each A € si. Thus, U,*U C n (A< x V) is closed in X X Y so that \Jk=1 C (~l (Ai X Y) n K = C n K is closed in K. • B.5 Proposition. Let K be a simplicial complex. If either K is locally finite or Z is locally compact Hausdorff then the product topology on Z X \K\ is generated by { Z X \s\ \ s € K }. Proof. If K is locally finite then { |s| | s G K } is locally finite since the open star of any vertex intersects only finitely many closed simplices in a nonempty set, and the result follows from Lemma B.3. If Z is locally compact Hausdorff, apply Proposition B-4. By definition, the topology of \K\ is generated by { |s| | s £ K}. It is Hausdorff by Theorem 17 on page 111 of [S]. Every compact subset of |.Kj is contained in a finite union of closed simplices by Corollary 19 on page 113 of the same reference. • 88 Appendix C A Generalized Mapping Cylinder In this appendix we define a generalization of the ordinary mapping cylinder and show that this generalization is an endofunctor on the category of-locally trivial bundles over a fixed locally compact space. Given a collection SC of topological spaces, let us denote their disjoint union by (&3C'. If SC = {Xi,..., Xn} is a finite collection, we will write @SC = X\@ • • - ® X n . If X\ = • • • = Xn = X, we may write X\ © • • • © Xn = © n X . Of course, the disjoint union is the coproduct in the topological category and there is the usual universal property. If in denotes the discrete space {1,..., n) then it is easy to see that X x si satisfies the universal property of ©"X so that ®nX « X x 21. Fix a tuple (K; J i , . . . , Jn) of spaces with J\,..., Jn C K. For any map / : X —• Y, define the mapping cylinder of / on (K; J\,..., Jn) to be the colimit space of the following diagram. We will denote this colimit by M(K; J\,..., Jn)f and write M(K; J\,..., Jn) f = Mf when the tuple (K; J i , . . . . , Jn) is understood. Explicitly, let us take mapping cylinder to be the quotient of (X x K) © (Y x in) by the smallest equivalence relation that identifies points (x, s) 6 X x K and (/(x), i) € Y x m whenever s 6 Ji- Note that the ordinary mapping cylinder is the mapping cylinder on (I; {0}) Let us also define the stunted mapping cylinder of / on (K; J i , . . . , Jn) to be the identification space Q(K; J i , . . . , Jn)f = Qf of X x K by the smallest equivalence relation which identifies pairs (x,s) and (x',s') whenever /(x) = f(x') and s, s' € J,- for some i. Thus, by Proposition Y x in XxK n 89 0.2, a map g : XxK —> Z has a unique factorization through Qf if and only if g(x, s) = g(x', s') for all x, x' G X and s, s' € Ji, i = 1,..., ra. By the universal property of the identification, the map (x,s) •—• [x, s] : X X 7f —• Mf factors through the identification X x K —> Q / to give a map i : [x,s] t - > [x, s] : Q / —> M / . This is clearly an injection. C . l Proposition. If f : X —* Y is an identification then i: Qf —* M / is a homeomorphism. Proof. For each i = 1,..., ra, choose an element a, S Ji and define a map hi : Y —* Qf by the universal property of the identification / and the following diagram. Define h :Y x m —> Qf by h(y, i) = hi(y). Then, since h(f(x),i) = [x, ai\ G Qf for each z, the universal property of the colimit gives a map k : Mf -+ Qf which makes the following diagram commute. X xJi Since fc([x,s]) = [x,s] whenever (x,s) £ X x K, k is the inverse of i. u C.2 Corollary. If p : E —> B is a bundle map then Qp w Mp. • Next, define the generalized cone of a space X on (K; J i , . . . , J n) to be C(K; J i , . . . , J n )X = CX = Q*, where * is a map to a one point space. That is, CX is the quotient of X x K by the smallest equivalence relation that identifies points (x, s) and (x', s') whenever s, s' G Ji for some i. Note that the ordinary cone is the generalized cone on (7; {0}) and that the suspension is the generalized cone on (7; {0}, {1}). 90 If ib = (p, <h) : E —• B X F is a bundle trivialization then, by the universal property of the identification, we have a maps p : Qp —• B and rj> : Qp —• B x CF which make the following diagram commute. E x K ^ X 1 • BxFxK (b, f, s) Qp = >BxCF (6, [/,«]) C.3 Proposition. Suppose p : E —> B is the bundle map of a trivial bundle with fibre F. If B is locally compact then p : Qp —> B is the bundle map of a trivial bundle with fibre CF. Proof. Since B is locally compact, the map (b, / , s) \-> (6, [/, s]) : B X F x K —• B X CF in the previous diagram is an identification. Thus, since ib : E —v 5 X F is a homeomorphism and since two points in £ X if project to the same point in Qp if and only if their images (by ^xl) in B X F X K project to the same point in B X CF, we can construct a map B X CF —> Qp from ib-1 X 1 using the universal property of the identification. E x K BxFxK Qp< B x CF The constructed map is clearly the inverse of tb. m For non-trivial bundles, we use the above result on a trivialization neighbourhood. C .4 Proposition. Let p : E —> B be locally trivial with fibre F. If B is locally compact then p : Qp —> B is locally trivial with fibre CF. Before proving this, let us first recall a topological result. C.5 Proposition. Let R be an equivalence relation on a space X and letir:X—y X/R be the identification. If B C X/R is either open or closed then the map [x] n(x) : w~1(B)/Ro —> B is a homeomorphism, where Ro is the equivalence relation on n~1(B) induced by R. Proof. [Du], chapter VI, section 2. • Proof of Proposition C .4. Let U C B be a trivialization neighbourhood and let ip : p~l(U) —* U x F be a local trivialization. If pu • x H-> p{x) : p_1(C/) —• U then, since an open subset of a locally compact space is locally compact, Proposition C.3 gives a trivialization 91 $ : Qpu -+UxCF over U. ~ Qpu • UxCF U To complete the proof, note that Proposition C.5 provides a homeomorphism Qpu —* p~l (U) over U since p_1(C/) C Qp is open, 7r~1(p_1(C/)) = p _ 1 (£ / ) x Jf where 7r : E x K —* Qp is the identification, and Qpu is exactly p_1(C/) X K modulo the equivalence relation induced by that on E x K. • 92 Appendix D Homotopy Groups of Flag Manifolds The fundamental group of the real complete flag manifolds was computed in chapter 1, where we showed that niF(n) was a Clifford group on n — 1 generators whenever n > 3. Diagram chasing with the long exact homotopy sequence of various fibrations can be used to compute the fundamental group of the remaining real flag manifolds as well as the complex and quaternionic flag manifolds. Many higher homotopy groups can also be computed in the same manner, and we collect these results here. As in the real case, we write F<c(n) = F<c(l,...,l) and F]fi(n) = FH(1, . . . , 1) for the complex and quaternionic complete flag manifolds. As previously, we will use the notation 0(n) to denote 0(n), U(n) and Sp(n) in the real, complex and quaternionic cases respectively. For any given n we can include 0(n) in 0(n + 1) as the subgroup of matrices which leave the standard basis vector en+i fixed. We can thus regard these groups as forming an infinite chain 0(1) C 0(2) C • • • of inclusions. We will also use the notation V{n,k) = 0(n + k)/0(k) for the Stiefel Manifolds Vn(n, k) = 0(n + k)/0(k), Vc(n, k) = U(n + k)/U(n) and Vu(n, k) = Sp(n + k)/Sp(k), as well as T(ni,..., ne) for the respective flag manifolds. Stability of 7TfcO(n) Each of the principal bundles 0(n)< >0{n + l) U{n)< >U(n + l) Sp{n)^ >Sp(n+l) leads to a stability theorem of homotopy groups when we apply the long exact homotopy sequence of the fibration. Indeed, these three sequences have the following portions respectively. nk+1Sn >7rkO{n) •TTfcOfn+l) •TTfcS" n k + 1 S 2 n + 1 • nkU(n) > irkU{n + 1) • 7 r f c 5 2 n + 1 7r f c + 15 4"+ 3 • nkSp(n) • nkSp(n + 1) • 7 r f c 5 4 n + 3 93 The connectivity of the spheres then leads immediately to the following three statements. D . l Proposition. The following are true. 1. The inclusion 0(n) <-»• 0 ( n + 1) induces an epimorphism nkO(n) -» KkO(n + 1) if k = n — 1 and an isomorphism iTkO(n) = TTkO(n + 1) if k < n — 1. 2. The inclusion U(n) <->• C/(ra+l) induces an epimorphism iXkU(n) -» n k U ( n + l ) ifk = 2n and an isomorphism 7Tfc£/(n) = TTkU(n + 1) if < 2n. 3. The inclusion Sp(n) «-»• 5p(n + 1) induces an epimorphism TTkSp(n) -» iTkSp(n + 1) if A; = An -f- 2 and an isomorphism nkSp(n) = nkSp(n + 1) whenever A; < 4ra + 2. • Thus, for any given A;, the groups iTkO(n), irkU(n) and TTkSp(n) become stable for sufficiently large n. Connectedness of the Stiefel Manifolds Stability of the orthogonal, unitary and symplectic groups leads to statements about the con-nectivity of the Stiefel Manifolds V(n, k). Indeed, the principal bundle 0(k) < • 0 ( n + k) V{n,k) gives rise to the following portion of the long exact homotopy sequence of the fibration. 0{n + k) — —• nt+i V(ra, k) • neO(k) • ixeO(n + k) The leftmost homomorphism of this sequence being an epimorphism and the rightmost being a monomorphism is then a necessary and sufficient condition for 7 r ^ i V(n, k) = 0. By Proposition D.l, this is the case when ^+l<fc — l in the real case, £ + 1 < 2k in the complex case and £ + 1 < 4k + 2 in the quaternionic case. Thus, we have the following results. D.2 Proposition. The following are true. 1. VR.(TI, k) is (k — l)-connected. 2. V(c(n, k) is (2k)-connected. 3. V$i(n, k) is (4k + 2)-connected, m 94 Homotopy Groups of Flags Connectedness of the Stiefel Manifolds now leads to the homotopy groups of many flag mani-folds by consideration of the long exact homotopy sequence of the following principal bundle. 0(n{) x • • • x 0(ne) < > V(ni + • • • + ne, k) (D.l) J:(nu...,nl,k) The following result is an immediate consequence of Proposition D.2. D.3 Proposition. FR(n\,..., ne, k), Fc(n\,.. .,ne,k) and FH(»I , .. .,ne,k) are all path con-nected and the following are true. 1. Ifl<i<k-1 then 7 r , F ]R ( n i , . ..,ne,k) = 7T;_iO(ni) x • • • x 7rj_xO(n f). 2. Ifl<i<2k then 7 r , F c ( n i , . . .,ne,k) = 7T;_i[/(ni) x • • • x 7ri_1C/(n^). 3. Ifl<i<4k + 2 then KiFjH(ni, ...,ni,k) = 7r,_i5p(ni) x • • • x ^_i5p(n^). • Note that we do indeed have a group isomorphism when i = 1: Since the bundle (D.l) is principal, the function • • •, ne, k) —• TToO(nv) x • • • x 7roO(ra^ ) in the long exact homotopy sequence is a group homomorphism. (See [Stl], page 93.) Of course, the position of the integer k in Proposition D.3 is immaterial and we can make corresponding statements about isomorphisms between TTiT(n\,..., ne) and a product of all but one of 7r;_i0(ni),..., Ki_iO(ne) when £>2. Since we are guaranteed that at least one of n i , . . . , ne be greater than or equal to 2 if nx -\ 1- ne > I, there is the following consequence for the real flags. D.4 Corollary. If nx + • • • + nt > I then ^F^ni,..., ne) = ©£_1^2- • Since U(l) « S1 and Sp(l) « S3, there are also the following two consequences of Proposition D.3 for the complex and quaternionic complete flags. D.5 Corollary. Fc{n) is 1-connected and 7 r 2 F c ( n ) = © n - 1 0 for all n>2.u D.6 Corollary. FH(W) is 3-connected and 7 r 4 F H ( n ) = (Bn~lZ for all n>2.m 95 Index Notation O(n) 3 U(n) 3 Sp(n) 3 (A) 5 ^(ni . . . . .n / ) 10 F(n) 10 C(V,q) 12 Rq,v 14 Ce(V,q) 16 C0(V,q) 16 Z(C(V,q)) 16 Z(C(V,q)) 16 Sq 18 c(v,gy i s U(V,q) 18 Ue(V,q) 18 Aut,(V) 19 Fv 19 T(Rn,+) 21 T{Rn,-) 21 T e { R n , + ) 21 r e ( K " , - ) 21 pin(n) 25 Spin(n) 25 Cn 25 Qs • 27 A „ 31 S(n,K) 33 Vert(A') 35 \s\ 35 \K\ 35 |«| 36 u 36 T(s) . . .36 Glue^ 36 QA 37 cb 38 (s) 38 7A • 39 Ko 40 /G/K 40 C K 40 6n 43 0(n) 43 0(n,i) 44 Os{n) 44 0.,t(n) 44 St(v) 46 Dnf 51 Y*Z 52 CK 55 U. 59 Wa 59 sc 60 Ff 60 Ff 60 b3 60 s 62 s x 62 K*L 67 / G / K * / H / L 67 VK 71 VK{N) 72 Mf 89 Qf 89 CX 90 B barycenter 60 c carrier 38 category 7 Clifford algebra 12 even 16 Clifford group 25 compactly generated 88 cone 90 coproduct 4 F flag manifold 10, 58 complete 10 functor 7 96 diagonal 9 functor category 8 G group action free 11 proper 11 I identification 4 J Jahn-Teller effect 1 join of lattices 67 of simplicial complexes 67 of spaces 52 L lattice construction 31 link : 62 locally compact 6, 7 locally finite 87 M map 4 mapping cylinder 89 stunted 89 matrix antisymmetric 3 Hermitean 3 orthogonal 3 symmetric 3 symplectic 3 unitary 3 N natural transformation 8 neighbourhood 4 o open star 46 P preorder 7 Principal Axis Theorem 81 product S simplex closed 35 open 38 simplicial approximation 49 simplicial complex 35 locally finite 36 realization of a 35 simplicial map 49 simplicial neighbourhood 62 subdivision 49 symplectic group 81 T topology compactly generated 88 generated 86 identification 4 induced 86 weak 86 u universal arrow 8, 9 universal element 8 97 

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