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Finite subset spaces of the circle Rose, Simon 2007

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Finite subset spaces of the circle A hyperbolic approach by Simon Rose  B.Sc, The University of Alberta, 2005.  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF Master of Science in The Faculty of Graduate Studies (Mathematics)  The University Of British Columbia April, 2007 © Simon Rose 2007  11  Abstract In this thesis we investigate a new and highly geometric approach to studying finite subset spaces of the circle. By considering the circle as the boundary of the hyperbolic plane, we are able to use the full force of hyperbolic geometry—in particular, its wellunderstood group of isometries—to determine explicitely the structure of the first few configuration spaces of the circle S . Once these are understood we then move onto 1  studying their union—that is, exp (5 ,)—and in particular, we re-prove both an old 1  3  theorem of Bott and a newer (unpublished) result of E . UJenHH (E. Shchepin) about this space.  '  Ill  Contents Abstract  ii  Contents  iii  List of Figures  v  Acknowledgements  vi  1 Introduction and Preliminaries  2  1  1.1  Introduction  1  1.2  Preliminaries  5  1.2.1  A rapid introduction to P 5 L ( R )  1.2.2  Seifert-Fibred spaces  .  2  9  The analysis of e x p ( 5 )  15  1  3  2.1  T h e topology and geometry of C^S )  15  2.1.1  Coordinate-based description of topology  18  2.1.2  Coordinate-independent descriptions of topology  22  Topology of the union e x p ^ S )  25  1  •2.1.3 2.2  5  1  e x p ( 5 ) and the inclusion exp^S ) 1  3  3 Further Research  1  <^-> exp (S' ) 1  3  28  35  iv  Bibliography  39  V  List of Figures 1.1  Exceptional fibres of the S  action  13  1.2  T h e inclusions of C^S )  C (5 )  14  2.1  arg(x + iy)  2.2  T h e homeomorphism between C2(5 ) and M.  20  2.3  T h e action of 7 on the S  21  2.4  Geomety of C (S )  2  2.5  Choice of Fundamental D o m a i n  27  2.6  A neighbourhood of {p, q}  29  2.7  T h e covering used for the Seifert-Van K a m p e n theorem  30  2.8  T h e generator b of ni(A f l B)  31  2.9  T h e homotopy. class of i*(b)  1  1  X  6  : . . .•  19 1  1  coordinate of PSL (R) 2  1  4  2  . . .  .  31  2.10 T h e homotopy class of  32  2.11 Covering of B'  33  3.1  38  Isomorphisms due to exact sequences  vi  Acknowledgements I would like to thank pretty much a l l of the people that I have worked w i t h throughout b o t h my undergraduate and fledgeling graduate careers, and i n particular my supervisor, D r . Denis Sjerve, who has provided me w i t h many ideas and insights, and corrected my typos. Beyond that, I would like to thank a l l of the instructors for the courses I have taken b o t h here and at the University of A l b e r t a , since without their tutelage I would not have been able to understand any of the text contained herein. I would also like to thank D r . Volker Runde, who helped encourage me into higher studies, and without w h o m I never would have appreciated the true nature of that elusive e. I would also like to dedicate this work to my parents, who have been unfailingly supportive of me throughout my education.  1  Chapter 1 Introduction and Preliminaries 1.1  Introduction  Since the m i d 1900s, the study of finite subsets of topological spaces has cropped up i n one guise or another—initially by Borsuk [1], later corrected by B o t t [2] and more recently by the likes of Handel [3], Tuffley [11] and Mostovoy [9]. A few different definitions have been given i n the literature (which are a l l easily seen to be equivalent), but the one that we w i l l use for this thesis is the following.  Definition 1.1.1  L e t X be a topological space, a n d let e x p ( X ) denote the set of fc  all non-empty subsets of X of cardinality at most k. T h a t is exp (X)  := {S C X | 0 < | 5 | < k]  k  There is a natural surjective map 7r : X  k  {x\,...  ,Xk}; we thus endow exp (X) k  -» exp (X)  which takes (xi,...,x )  k  k  i—>  w i t h the quotient topology induced by this  map. It is of course clear that under this topology, e x p ( X ) = X. x  There are a number of results that can be proven about this space and its relation to X. First of all, the map X — i > exp (X) k  is a homotopy functor from U, the category  of topological spaces and continuous maps, to itself. More precisely, we have the following theorem.  2 T h e o r e m 1.1.2  Let X,Y  be topological spaces, let f,g : X —> Y. If F : X x / —>• Y  is a homotopy from f to g, then the map F : exp (X)  x Iexp (Y)  k  ({x  k  u  ... ,x },t) k  t-+{F(x t),...  ,F(x ,t)}  u  k  is a homotopy from exp (/) to exp (g). fc  fe  From which it immediately follows that C o r o l l a r y 1.1.3  The homotopy type o/exp (X) depends only on that of X. fc  There are also natural inclusions exp (X) k  subset {x\,...,  Xk} i—* {xi,..., X  <—> exp (X)  for k < m, taking the  m  Xk}\ thus we have  e x ( X ) ^ exp (X) P l  2  exp (X) ^ fc  •• •  with each exp .(X) embedded as a closed subset of e x p ( X ) . A  m  The proofs of these results (as well as many other topological and homotopytheoretic results) are in [3], which provides a good introduction to many of the basic properties of exp (X) (which he denotes as Sub(X, fc  k)).  It should also be noted that much of this is in marked contrast to the notion of a configuration space, which is not functorial (unless we suitably restrict our category); nor are there nice embeddings as above. The most famous result in the study of these spaces, however, comes from Bott's correction to Borsuk's initial paper. That is: T h e o r e m 1.1.4  The space exp (S' ) is homeomorphic to S , 1  3  3  the three-sphere.  This has been proven so far in a number of different ways—Bott uses an elaborate "cut-and-paste" style argument (in effect showing that it the union of tori together with a calculation of its fundamental group to show that it is a simply connected lens  3 space), Tuffley finds a decomposition of it as a A-complex (in the sense of [5]) and shows that it is a simply connected Seifert-fibered space, while Mostovoy ties it in with a result of Quillen about lattices in the plane. There are of course many other results about similar such spaces, primarily in [11] and [7], most notably the following result. T h e o r e m 1.1.5  The homotopy type o/exp (5 ) is that of an odd dimensional sphere; 1  fc  specifically,  e x p ^ - e x p ^ S  1  ) - ^ -  1  This is proven in [7] using the notion of truncated product spaces—these can be considered to be, in a certain sense, the free Zy^ set on points in our space. More specifically, these are given by TP (X)  :=  n  where SP (X) n  n  is the n-th symmetric product of X defined to be the collection of  unordered n-tuples in X—that that xi H  SP (X)/^  is, SP (X) n  h Xk-2 + 2x ~ Xi -\  = {xi + • • • + x \xi n  G X}—and  we say  x -2 (where we use the convention of writing an k  unordered collection of points as a sum), topologized accordingly. The reason that this is useful is that TP (S ) l  n  = K P (see [8]) and so if we then note the homeomorphism n  ^W/exp^PO =  ^ )/TP _ (X)  TP  X  k  2  this then becomes  We are then able to use the long exact sequence of homology groups to show that exp (5 ) is a homology sphere—since it is simply connected it is then, homotopic to 1  /c  a sphere (by a standard argument).  4 Moreover, the fact that the exp functor is a homotopy functor has also been used in [12] and [13] to begin the determination of the homotopy type of exp (S) for E /c  a closed surface. The idea is that if you consider a punctured surface, then this is homotopic to a wedge of copies of S —and from the understanding of exp (S' ) Tuffley 1  1  fc  is then able to prove results about  exp (r>), where fc  Te is a graph consisting of wedges  of £ circles. From this, he'uses a Mayer-Vietoris argument, covering a surface with k + 1 copies of Xi = E \ {pi} (for distinct points pi), to determine information about the homology of the space exp (E) (this works since fc  Xi = E \ {pi,... , p/c+i},  and so these cover exp (E)). fc  In this thesis we intend to provide yet another proof Theorem  1.1.4,  beginning  with certain properties of P S L ( M ) . This method will generalize and provide methods ,  2  of calculation not only of the homotopy type of higher dimensional analogues, but possibly even their homeomorphism type. The main difference in this approach as compared to previous ones lies in its inherent geometric nature. This heavy emphasis on the underlying geometry of the circle (or later, of higher dimensional spheres) and its connection to hyperbolic space affords us a much more specific understanding of the topology of the resulting space. In contrast, previous methods (for higher dimensional analogue's) were restricted to knowledge of the homology and homotopy groups of these spaces which permitted a knowledge of the homotopy type of these spaces and no more. N o t a t i o n For the duration of this thesis (unless otherwise noted) we will consider S  to be both the quotiont of [0, n] in the usual manner, as well as the quotient  1  ^ " / l R o ' d P d m g on context. The two views are simply matters of convenience X  e  e n  >  due to our working with PSL (R),  depending on our choice of how we describe the  homeomorphism type of PSL (M.),  as will be seen later.  2  2  We will denote the upper half-plane, R x M  > 0  by H , considered as the subset of  5 complex numbers whose imaginary parts are strictly positive. We w i l l also use the notation C (X)  :={SCX  k  \ \S\ = k}  to denote the k-th unordered configuration space of X. It is of course clear that k  (1.1)  exp (X) = U C i ( X ) fc  Not so clear, however, is just how this union behaves topologically. If we consider {xi,...,  x} k  £ Ck{X) and let Xi —> Xj, then we w i l l end up w i t h a point i n  To properly understand the topology of exp (X)  Ck-i(X).  i n this context, we then need to  k  study exactly how this gluing takes place—which will be the subject of most of this thesis.  1.2 1.2.1  Preliminaries A rapid introduction to  PSL {R) 2  We begin w i t h the most basic definition of PSL (R),  the one most people w i l l have  2  learned of i n a basic linear algebra setting. Definition 1.2.1  Let SL (R) 2  denote the set of 2x2 real matrices of determinant 1.  T h e n we define PSL (R) 2  :=  SL (R)/ 2  ±I  Note that this is equivalant to choosing all matrices of positive determinant, and dividing by all scalar multiples of the identity; that is,  PSL {R) 2  :  a, b, c, d, G K, ad — bc > 0 > j  6 where \c  d)  \c  dj  for all A e R*. This characterization tends to be more useful, as it no longer forces us to ensure that our matrices have determinant 1 (which makes for neater formulae, if nothing else). There is yet another way of considering P S T ^ R ) which is, at its heart, the most useful of the views for our purposes (and follows fairly neatly from the previous comments)—that is, as fractional linear transformations of P and in particular as 1  the group of orientation-preserving isometries of H , the upper half-plane. Purely as sets this is because we have Xaz + Xb  A / a z + b\ _ az + b  Xcz + Xd  X\cz  + dj  cz + d  and so we have a clear bijection between the elements of PSX (]R) and the fractional 2  linear transformations with real coefficients.  However, this correspondance is yet  stronger; a quick calculation shows that faz  + b\  f az + 0\  _ {aa + by)z + (a/3 + b5)  \cz + d) ° V 7 2 + S J ~ (ca + di)z + (c/3 + dS)  and so our bijection is in fact a group isomorphism. This of course means that we can consider P S X ( M ) as a subgroup of the full group of isometries of P . However, 1  2  what we find useful is the fact that since we insist on using real coefficients, it follows that P 5 ' L ( R ) also acts on S = K U 00, the boundary o f i c P . Even stronger, we l  1  2  have the following: L e m m a 1.2.2  P5X (]R)  three points p,q,r  2  in S  1  a c  ^  5  doubly transitively  on S  l  = K U 00.  Moreover  then there exists an element 7 6 P 5 L ( R ) such that they 2  map under 7 (preserving their cyclic ordering) to {0,1, 00}, or any cyclic . thereof.  given  permutation  7 Proof  W i t h o u t loss of generality, we assume that our two distinct points (p, q) are i n  R . Consider the element '7( , )U)-±7— P  9  ^  q  where the sign is chosen such that ± ( p — q) > 0. T h i s maps p to 0 and q to oo. A s p, q were arbitrary, it follows that 7(~\) °7(p,g) takes the pair (p, q) to (a, b) as claimed. For the second part, note that the fractional linear transformation  e _(£zi)£z£ W  \q — pjz  (1  .  2)  — r  takes the triplet (p, q, r) \—> (0,1, oo), and that post-composing this w i t h  z-1 1 =  z  cyclically permutes (0,1, oo) as claimed.  Q.E.D. A s it turns out, this last fact w i l l be the key argument i n our proof of Theorem 1.1.4. First we need to know a bit more about the topology of  Lemma 1.2.3 ,4s a topological space, PSL (R) 2  PSL (M). 2  = Hx S. 1  We w i l l provide two proofs of this lemma which w i l l b o t h be used later as we examine two parallel proofs of the main theorem. Proof  For any  = ( ° ^ ) G PSL (R),  we can decompose it as the composition  2  of a rotation about i (a fractional linear transformation of the form j|fij_) and a translation (one of the form az + (3). T h a t is, ad—be ac+bd c +d c +d | | 2  2  2  2  |  'T 3)  T h i s yeilds us an explicit map 0 : PSL (R)  - ^ 1 x 5 ' given by  2  a  p\  ( cos 9  sin#\  0  1J \--sm9  cos9J  Similarly, we have the map HI x S —> PSL (R)  by  l  2  cos 9  (x + iy,9)  sin 9  sin 9 cos 9  and these maps can be checked to be inverses of each other; T h e y are clearly b o t h continuous, and so the conclusion follows.  Q.E.D. Second proof  Given T G PSL (R),  we note that as T maps EI to itself that T(i) G EL  2  az+bthen Also, note that if T(z) = 2*±J cz+d dT . .  ad — be  3rW = (ci + d ) dz w  2  which up to scaling is an element of 5 ; to see this, note that we have the liberty to 1  choose c, d above such that ci + d is i n the upper half-plane together w i t h the (strictly) positive real axis, and so squaring it we see that we can indeed get any element i n S . 1  Combining a l l of the above, we define the map ^ : PSL (M) 2  r ~ ( r  W  —> EI x S  1  , f ( < )  which' is then easily seen to be a homeomorphism.  Q.E.D.  9  1.2.2  Seifert-Fibred spaces  For the purposes of this thesis, the simplest definition that we need of a Seifert-Fibred space is that of a space with a nice ^ - a c t i o n on it. Equivalently, we can consider it as a space that can be foliated by circles. More precisely, we can define a Seifert-fibred space in the following manner (We mostly follow [6]). A model Seifert fibring of the solid torus S x D is a decomposition of S x D 1  2  1  2  into disjoint copies of S (called fibres) in the following way. If we consider S x D 1  as a quotient of  [0,7r]  1  2  x D , then instead of the usual identification of {0} x D 2  2  and {IT} X D we identify them with a twist of 2np/q (that is, we instead identify 2  (0,a;) ~ (-7T, e / x)). 2mp q  We then retain an S action in the obvious way (simply 1  moving a point along its respective fibre), but due to the twist we end up with an extra detail—if we take a point in S x D (other than one in the central fibre) and 1  2  move it along the fibre than it returns to its starting point after q times the number of passes it would take a point in the central fibre to return to its starting point. W i t h that in mind, we come to the following central definition. D e f i n i t i o n 1.2.4 A Seifert-Manifold is a triple (M,F,ir)  where M is a 3-manifold,  F is a 2-dimensional surface (oriented or not, closed or not) and 7r : M —> F is a map such that: (i) For each x G F, the preimage of x under n is homeomorphic to S . l  (ii) For each x G F there is a D neighbourhood of x such that ir~ (D ) 2  l  2  is  fibre-preserving diffeomorphic to a model Seifert fibring as described above. That is, M is an "almost" locally trivial S bundle—if we stay away from certain 1  exceptional fibres, then the resulting space is a locally trivial (in fact, trivial) S  1  bundle.  10 Remark  Note that there is a more general notion of a Seifert-Manifold i n which  F is allowed to have a boundary—this corresponds to allowing a more general sort of model Seifert-fibring i n which reflections as well as rotaions are allowed i n the gluing process. W i t h the above definition an S  1  action becomes immediately apparent; simply  move a point along its fibre. Moreover, we can consider the space. F to be a quotient of M determined by collapsing each fibre to a point. Conversely, if we are given a sufficiently nice S action on a space M, then we obtain a map M —• M/gi 1  which is  a Seifert fibreing; M is clearly foliated by circles as well, the circles being the orbits of S . Thus we can be justified i n saying that Seifert Manifolds are i n 1-1 correspondence 1  w i t h S actions on 3-manifolds. 1  Now, there is no a priori reason why this definition should permit us to accurately classify 3-manifolds—however, it turns out that the imposed structure is sufficient to provide quite a thorough classification of a great many manifolds, as follows. T h e first thing to do is to identify the types of fibres that occur i n our manifold. If we consider the model Seifert fibring containing x on its central fibre, we have two cases. T h e first is that this tubular neighbourhood was obtained w i t h a 2-xp/q twist (with q ^ ± 1 ) , and the second was that it wasn't.  In the former case we call it  an exceptional fibre, and i n the latter, a regular fibre. Under the assumption that F be compact (if and only if M is also compact), it can be shown that there are only finitely many exceptional fibres i n a space. Now, if we remove a l l of these tubular neighbourhoods then we obtain a t r i v i a l S bundle over F; \ ( D i U • • • U D )—this 1  k  is since principal S bundles over a space X 1  11 are in one-to-one correspondence with the elements of [X,  BS  1  ' ^ [x, CP°°; [X, K(Z, 2)] H (X,Z) 2  which in our case can be easily seen to be zero. Thus to construct our manifold M , we consider the following information. Let g be the genus of our surface F, and let • • • -.Vkllk) be a collection of integers with gcd(p;, qf) = 1. The idea then is to glue in solid tori Tj along each boundary according to the homology relation Hi ~ piAi + qiBi  (1.4)  where Hi is the meridian of T; and Ai and Bi are the respective generating homology classes of the i-th boundary component of F \ (D\ U • • • U D ) x S . l  k  We then call (g;pi/qi,  • • • ,Pk/Qk) the Seifert invariants of M. Conversely, given a  collection of Seifert invariants (g;pi/qi,  • • • ,Pk/qk), we denote by M(g;pi/q ,...  ,Pk/qk)  1  the Seifert manifold constructed from these invariants as described above. W i t h this in mind, we have the following classification theorem for Seifert manifolds (cf [6]). Theorem 1.2.5  Let (M,F,n)  be a Seifert manifold with F closed and orientable.  Then 1. M is fibre-preserving diffeomorphic  to some M(g;pi/qi,...  ,Pk/qk) f  or  some  9,Pi,Qi-  2. M(g;pi/q ,... 1  ,p /qk) and M(g';pf /q[,... k  phic if and only if (a)  9^9'  1  are fibre-preserving diffeomor-  12 (b) Ignoring  any of the Pi/qi andp\jq\  which are integers (and not equal to  cx), the remaining Pi/qi (mod 1) are a permutation  of the remaining p'j'q[  (mod 1). • k I • (c) ^^Pi/qi  — ^^p'i/q'i,  i=l  where we use the convention  that I/O = —1/0 = oo  i=l  and a + oo = co for any a G l U {oo}Note that this is not a complete classification; there is still the case where F is either not closed or not orientable to deal with. There are also a number of manifolds (see [4], [6], and the remark below) which have multiple non-isomorphic fibrations. However, for the purposes of this thesis this case is sufficient. In particular though, we have the following. T h e o r e m 1.2.6  The 3-sphere S is a Seifert-fibred 3  manifold  with Seifert  invariants  (0; 1/2,-1/3). Remark  This is not the only fibreing of S —in  fact, any p, q relatively prime will  3  provide a fibreing with invariants (0; 1/p, ±l/<?) in a similar manner. Proof  We consider S as the unit sphere in C ; Then we have the action of S (seen 3  2  1  as a subset of C) via \-{z ,z ) 1  = {\ z ,\- z ) 2  2  .  3  l  2  which has exceptional fibres given by the orbits of (1,0) and (0,1) with multiplicity 2 and —3 as claimed. Q.E.D. The reason this interests us is that exp (S' ) has a natural S action on it given 1  1  fc  by C • {Xi,  . . . ,X } = {(Xi, .. ,(x } k  k  13 T h a t is, rotation of our collection of points i n a counter-clockwise direction about the circle. If one w i l l take for granted for a moment that exp (5' ) is a 3-manifold, we see :L  3  that it is i n particular a Seifert manifold and that the above classification theorem can be used to describe its homeomorphism type. Now, i f we have an S  1  action, then we should examine the exceptional fibres.  It should be fairly easy to see that the collection of points of the form {z, —z)— antipodal points—will form an exceptional fibre of multiplicity 2, while the points of the form {z, £32, £|z)—those that form the vertices of an equilateral triangle—will form one of multiplicity 3, as shown i n figure 1.1.  Figure 1.1: Exceptional fibres of the 5" action 1  Now, there is no real reason to stop here—if we have exceptional fibres i n e x p ( S ' ) , 1  3  we should have analagous objects i n e x p ( 5 ) for higher values of k (although e x p ( 5 ) 1  1  fc  fc  is no longer a manifold past k = 3). If nothing else, consider k equally spaced points; then this is certainly analagous to the two situations above. However, the situation is more interesting than that. In fact, we have the following. Proposition 1.2.7 For each d which divides k, C^iS ) contains a homeomorphic 1  copy of Cd{S ) such that rotation about S l  1  by ^ fixes this subspace. Moreover,  14 This can alternately be worded as saying that C ^ S ) contains exceptionally fibred 11  subspaces C ^ S ) of multiplicity _ for every d dividing k. 1  How can we see this? Well, the simplest non-trivial examples are C^S ) 1  and  CeiS ); we will examine the latter, being the more interesting of the two. 1  For the copy of C^S ), 1  consider figure 1.2.  If we let T  3  be the collection of  pairwise antipodal points, then we have a map T —> C ([0, TT/2] /Q ^ 3  3  n  ^  which is  easily seen to be a homeomorphism.  Figure 1.2: The inclusions of C^S ) ^ C (S^) 1  6  For the copy of C (S' ), we instead consider T 1  2  2  to be the subset of points that  make up the vertices of equilateral triangles. Yet again there is a clear map T —> 2  C ([0,7r/3]/g ^ 2  which is a homeomorphism.  Having seen these two cases, it is easy to see how to generalize this idea for arbitrary inclusions Cd(S ) l  C^S" ) whenever d divides k. Note also that this 1  means that we can learn quite a bit about these spaces by studying C (S ) for p 1  P  . prime.  15  Chapter 2 The analysis of  exp3(5 ) 1  The topology and geometry of C^S )  2.1  1  The main structure of the proof of Theorem 1.1.4 is as follows. For any three points of S  1  = E U o o there is an element of PSL (M) 2  (by lemma 1.2.2) which takes the points  to { 0 , 1 , oo}, preserving cyclic ordering. T h i s element is not unique; post-composing it w i t h any element which cyclically permutes (0,1, oo) yields another possible choice. If we let T be the subgroup of P S L ( R ) generated by a l l elements cyclically permuting 2  (0, l , o o ) , then this subgroup acts on P S X ^ R ) by left multiplication; if we quotient out by this action (choosing, i n effect, the space of left cosets of T i n P S X ( R ) ) then 2  we obtain the following result:  Lemma 2.1.1  The subgroup o / P S I / ( M ) which cyclically permutes (0, l , o o ) is sim,  2  ply the cyclic group T =  generated by 7 =  thus  C (5 )^P5L (R)/ 1  3  2  r  B y similar reasoning we have that for any two points of S  l  there is an element of  P S X ( R ) taking those points to (0, 00); if we then let E be the subgroup of P S X ( R ) 2  2  which fixes the set {0, 00} we similarly have the following:  Lemma 2.1.2  The subgroup o / P S X ( R ) which setwise fixes {0, 00} is generated by 2  the elements r = — \ and o\ = Xz (for X G R>o ). In matrix form these are the /  16 elements of the form (? V ) and ( o ? ) > d  they generate the subgroup  an  and so we have C^S ) 1  A n d of course, Ci(S )  ^PSL (R)/  is simply the circle  l  2  E  S. l  T h i s tells us less than half of the story, however. We also need to understand how these quotients behave topologically, as well as determine how these three pieces glue together to form exp (5' ) (cf. eq (1.1)). 1  3  We first need a quick lemma, however. Lemma 2.1.3 If a E PSL (M) 2  fixes three points, then a(z) —;z for all z.  Proof T h i s follows since if we have , az + b a(z) = := z cz + d N  w  then this is equivalent to cz + (d- a)z - b = 0 2  which of course has only two (or less) solutions unless c = b = 0,a = d. Q.E.D. Proof of lemmas 2.1.1,2.1.2 L e t a ^ 1 be any element permuting the set {0, l , o o } (and of course preserving the cyclic ordering); note that as a  3  must be the identity by the previous lemma (if a  2  has to fix this set, it  were the identity, then it would  reverse orientations). Choose an appropriate power fc of a; such that (0,1, oo) maps under a to (1, oo, 0)—thus a o 7 fixes (0,1, 0 0 ) a n d so by the lemma above, it must k  k  17 be the identity. T h a t is, either a = 7  r = (7)  or a  2  _1  = 7  -1  = 7. 2  It then follows that  = z/ . 3  For the case of {0, 00}, it is clear that any element permuting these two points either fixes or swaps them. If it fixes them, then it has to be of the form Xz; otherwise it has to be of the form — j (in b o t h cases, A > 0), which proves the lemmas.  Q.E.D. Before we move onto a topological description of these quotients, a. quick word about orientation should be mentioned. It may appear i n the above like we are only quotienting out cyclic permutations of { 0 , 1 , 00}, and so we shouldn't get the full configuration space, which is i n fact ((S ) 1  3  — A ) w h e r e A is the fat diagonal.  T h i s is, however, not the case. There are two ways to see this. T h e first is that there is an implicit ordering of points around a circle, and that we simply choose those isometries which preserve that ordering. T h e second view is somewhat more explicit, however.  In principle we should  begin w i t h the collection of all isometries of H — t h a t is, the group G generated by PSL {M.) 2  and• f(z)  = 1/z—and quotient out as before the subgroup of isometries  which permute the set { 0 , 1 , 00}. to S  3  = (a, b I a  3  = b  2  T h i s subgroup can be shown to be isomorphic  = baba = 1), the permutations on three letters.  Now,  this group G is topologically the disjoint u n i o n o f two copies of P S ! ^ (M)-one copy 1  representing those elements which reverse orientation (ie those involving z), and the other those that preserve it. Thus when we quotient out by the action of b = it simply identifies the two copies of PSL (R), 2  1/z,  and we are left w i t h quotienting out  the resultant by a = 7 (as above). Thus our initial choice of PSL (R), 2  restriction to orientation preserving isometries, is i n fact the correct one.  despite its  18 We will now begin with a topological description of these configuration spaces. However, we will present two different methods of looking at them, one involving the natural coordinates obtained from PSL (R)  and the second more coordinate  2  independent, involving the second proof of lemma 1.2.3.  2.1.1  Coordinate-based description of topology  We begin with C ^ S ) , the simplest of the two. 11  P r o p o s i t i o n 2.1.4  C2(S ), 1  described as the quotient P S L ( M ) / — above, is homeo-< I  2  morphic to the open Mobius band M..  Proof  Recall that S is the group generated by r and the o\ (for A > 0), subject to  the relations TO T X  = o  CA<7ji =  1 x  = CTA-I (2.1)  CAM  and so in particular from the first of those it suffices to consider the actions of o\ and r separately. Writing both of these as matrices we have  and so examining the action of the former on an element of P 5 L ( M ) we have that 2  cos 9 0"A •  sin 9  — sin 9 cos 9  Xy  Xx  0  1  cos 9  sin 9  — sin 9 cos 9  19 and thus v i a our action we have that, as elements of M x S , (x + iy, 9) ~ (\(x + iy), 6) 1  for any A > 0. Thus if we quotient out by this action we find that (0,7r) x S  1  v i a the  map  (see  figure  PSL  2  2.1)  [x + iy, 9] ^  (0,9)  where the square brackets denote the equivalence class, and 4> = axg(x + iy).  x + iy  Figure 2.1: arg(x + iy) A s for the action of r , we first note that by (1.3) we have  y x 0  | x +y 2  x  x +y  2  2  2  -y  1 / ) ( _5°?n # cose) corresponds  and so combining these two actions, and noting that  to the angle 9 — <fi we find that the whole quotient is simply  (°' ) 7 r  x 5 l  / ( 0 , ^ ~ ( r - <M-0) 7  which can be represented pictorially as i n figure 2.2 (where the numbered triangles are identified w i t h one another), showing that PSL (R)/ 2  = M P \ {*} ^ M 2  E  as  claimed.  Q.E.D.  20  Figure 2.2: T h e homeomorphism between C2{S ) l  and A4  Proposition 2.1.4 can also be proven i n any number of ways; there is a quite simple pictoral proof of the matter i n [11]. We now move onto the more complicated case (which, unfortunately, does not have such a simple proof as the one cited above). T o aid i n its description we w i l l use the machinery of Seifert-fibred spaces described i n section 1.2.2—this turns out ;  to be a very natural sort of mechanism to describe exp (S ) l  k  Proposition 2.1.5 C^S ), 1  omorphic Proof  for k < 3.  described as a the quotient PSL2(M.)/-p  above, is home-  to the (open) model Seifert fibreing with twist 2TX/.2>.  Recall that T = (7) where 7 =  = (\ ~Q ). We begin by computing the  action of 7, and note that  7' and thus if we let <fi be as before (cf. figure 2.1) our isomorphism yields that  < *»)~( -?f^ ra»-*) I+  1  +<  21 We end up then with the following description of the action of 7 on PSL (M). 2  In  the H coordinate, it is the composition of two reflections—first around the geodesic x + y = 1, and then across the line x = \—and thus is a rotation about the point 2  2  \ + i^- by an angle of ^ . In the S coordinate, 7 shifts a point (x + iy, 9) back by the angle between x + iy l  and the positive x-axis, as seen in figure 2.3 (recall that S = [0,7r]/^). 1  „ --/ 1  \  ^—• \  /  \  Figure 2.3: The action of 7 on the S coordinate of PSL (W) 1  2  To see that this is then homeomorphic to the model Seifert-fibreing stated above, we first choose an explicit description of that model, which will be  Q:=Mx [0,1]/^  (z,l)~ ( l - ' i  0)  That is, a model Seifert-fibreing with twist Now, I first claim that the set F = {{z,6) e PSL (R) 2  | 0 < 9 < arg(z)} forms  a fundamental domain for the action of the group V. To see this we begin by noting that under the action of the group V we have the identifications (z,0)~ ( ^ , 0 - a r g ( * ) ) - (~,0  - arg(z - 1))  (2.2a) (2.2b)  Thus noting that 0 < 9 < n we have two cases to deal with, assuming that a pair (z,  B)iF.  22 (i) arg(z) < 9 < arg(z — 1) In this case we look at (2.2a). Since arg(z) < 9 we have that 9 — arg(z) > 0. Next, we see that  > 9 — arg(z) and so this element here is i n F as claimed. (ii) axg(z — 1) < 9 < ix In this case we look at (2.2b). Similar to the first case, we have that 9 — arg(z — 1) > 0. Moreover, we have that  > 9 - arg(z - 1) and thus the claim follows. W i t h F identified as a fundamental domain, it is easy to see that this space is homeomorphic to Q above. appropriate (z, 9) G F to (z,  a r  Simply define a map ^ ( S * ) —> Q by mapping the 1  \ , 9); this is clearly a homeomorphism. Q.E.D.  2.1.2  Coordinate-independent descriptions of topology  In this section we simply re-prove propositions 2.1.4 and 2.1.5 using more geometric rather t h a n coordinate based methods. Proof 2 of proposition 2.1.4 A s before, we have that E is generated by r and the o\, subject to the relations given i n (2.1), and so it suffices to consider the actions of o  x  and r separately.  23 We again consider the action of o~\ first. Now we have both ( a o T ) ( 0 = A(T(t)) A  d(o~\ oT) ^z—  {%)  dT = ~dz  and so the action of o\ is given by o\ • (z, 9) = (Xz, 9). If we then quotient out by the action of the subgroup of E generated by the o\'s we find  where the angle <fi £ (0,7r) is once again as infigure2.1. Moving on to the action of r, we note that (roT)(z) =  dz  K  1 T(«) T*(i)dz  }  W  and so since up to scaling (as an element of (0,ir)) —xfi)  =  ~  n  a  n  d similarly  —Ly = —20, we find that PSL (R) 2  /E  - (0, vr) x 5 7  ( 0 | d ) r  ^  ( ? r  _ ^  _  2  0  )  which can be pictorially represented as in figure 2.2 (subject to suitable rescaling), whence follows the conclusion. Q.E.D. We also have the following succinct proof. Proof 3 of proposition 2.1.4 Given two ordered points (p, q) in S , let z be the point 1  halfway between the two as shown in figure 2.4. Let 9 be the angle from p to q. This yields a homeomorphism between the pairs of distinct ordered points in S to 1  24 S  1  x (0,27r).  Now, when we quotient out by the action of E , then we find that  (z,9) ~ (-z,2ir-9).  2  Thus  which is easily seen to be the Mobius band.  Figure 2.4: Geomety of  C (S ) l  2  Q.E.D. Lastly, we move onto the case of  Cs(S ). l  Proof 2 of proposition 2.1.5 T h e proof of this theorem is identical i n method to the second proof of proposition 2.1.4. Since 7 = 1 - \ , we have immediately that  A(«)  d(7 o T) dz and so 7 does the same to the S  1  f\  v  v  2  1  c\  dT  T (i) 2  dz  coordinate as r . Now, i n the H coordinate, it turns  out that I — I can be written as i?i o R R (z)  1  = l-z  where R {z) 2  =  ^  25 and so it is the composition of two reflections, first about the circle \z\ = 1 and then about the line tftz = | — i t is thus a rotation of the angle  A s w i t h the previous  proof, it follows that this is indeed homeomorphic to our model Seifert fibreing as claimed.  Q.E.D.  2.1.3  Topology of the union exp (S' ) 1  fc  A t this point we have a solid grasp (via a variety of methods) as to exactly what the three pieces Ci(S ), i = 1, 2, 3 are. Remaining still is to describe the topology of their l  union—that is, what happens when points begin to coalesce. We begin by looking at e x p ( 5 ) . First, i n Ai = C2(S ), we find that a pair {p, r} 1  1  2  maps to the element c f  \  ~P .z — r Z  in P S L ( K ) , and so using the homeomorphism ^ from lemma 1.2.3 we see first that ,  2  cf-\ and so that as p —> t,.£(i)  ~P i —r  $r\ dz .  l  ~P (i + r )  d  r  —> 1, while ^ ( z ) = d^+r) ' 2  s  2  m  c  e  w  e  c  a  n  ig  n o r e  scaling, stays  constant. We can use this to prove the first part of our gluing process. Proposition 2.1.6 The space exp (<5' ) = C ^ S ) U C ^ S * ) is homeomorphic to the 1  1  1  2  closed Mobius band Ai*. Proof W e consider the closed Mobius band to be the quotient of [0, IT] x S  1  the action of Z v i a  modulo  (cj), 9) ~ (ir — (j), 9 — 20) (cf. the second proof of proposition  2.1.4). We then have a clear embedding of C ( 5 ) <—»• Ai*. :  2  Now, from the comments above it is easy to see that as p —> r , 0 —> 0 (see figure 2.1) and 9 stays constant—and thus that Ci(S ) U C (S ) = Ai* as claimed. l  1  2  26 Q.E.D. We now need to move on to the situation with three points. Recalling back to lemma 1.2.2 (and in particular, equation (1.2)) our triple {p, q, r} (with p < q < r, without loss of generality) maps to the element 'q — r\ z — p \q-pj  z-r  which, described in terms of equation (1.3) becomes  (  ( q - r ) ( p - r )  ( g - r ) ( l + p r ) \  ( , - p ) ( l + r 2 )  (  9  /  A  - p ) ( l + r 2 )  0  (2  1  3)  r)  We note again that as in the case of two points, the position in the S coordinate 1  only depends on a single one of these points—in this case, r. Now, the gluing together of exp (S ) and C^S ) 1  1  2  is somewhat of a surprise and is  strongly reminiscent of the notion of a blowup from Algebraic Geometry. Recall that the blowup of a point p in an n-dimensional variety V is a space BL (V) P  with a map ir : BL (V) P  together  —> V which separates the lines through V; that is, away from  7 r ( p ) this map is a homeomorphism, while 7 r ( p ) is a copy of (n — l)-dimensional _1  _1  projective space. The key idea, of course, is that the distinct lines through a point become separated in the resulting space, and this is what we will see when we glue e x p ^ S ) onto C ( 5 ) . 1  1  3  Before we move on though, we should first once and for all select a region of PS'L (]R) to use as our fundamental domain for the action of T; that region will be 2  the one shown in figure 2.5, crossed with S and with top and bottom edges identified 1  as per the identifications given in proposition 2.1.5. Note also that the round corners are exaggerated for reasons which will momentarily become clear. To aid in our description, we will assume that it is the point q which tends towards either p or r. That being the case, it should be noted that as q varies, the quantity  27  i Figure 2.5: Choice of Fundamental D o m a i n ^  varies between —oo and 0. A s such (see equation (2.3)), by fixing b o t h p and r  we trace out a path i n a particular H slice which happens to be a straight line from the point 0 + i0 of slope ^ j ^ - L e t t i n g q —> r (for simplicity's sake) we approach either 0 + iQ or 1 + zO, depending on the sign of  (Consider our choice of fundamental  domain). T h e n the Mobius band is glued on to C (S ) l  3  entirely at the points 0 - M O  and 1 + iO—however, the point which we end up at on the M o b i u s band depends on the slope of the path we take towards these points, and so much like, a blowup we find that these points are i n fact separated by the set of lines passing through them, which are then glued to the corresponding point on the open Mobius band. B u t what about C^S ) 1  = d(M*), and what of the edge running from 0 + iO to  1 + iO? T h e first thing to note is that as the union exp (<S ) must be compact, any 1  3  sequence {pi, qi, r^} which tends to that edge must necessarily converge to an element of C^S ) 1  i n the union. T h e only question remaining is what that element is.  Let {pi,qi,ri}  be a sequence i n exp (S' ) which converges (regarded as a point 1  3  28 i n the quotient H x S /^) 1  to some point P =  (A + iO, 9) where A € (0,1), 9 €  [0,7r). Choose a neighbrouhood N oi P such that TV is completely contained i n the fundamental domain of figure 2.5. A s such, each  q^, r;} is eventually contained i n  this neighbourhood,'and thus i n our particular choice of fundamental domain.  '  However, by our choice of fundamental domain there is an explicit ordering now on these points, and thus the HI slice that any {p  i}  qi,  = [(pi,  r,)] finds itself i n  is determined by a particular one of these points, say rj. A s this converges towards (A + zO, 9), we must have that  —» 9 (as well as pi, qi). T h a t is, the point i n  C^S ) 1  that we converge to is simply the point that a l l of Pi,ri, qi are converging to. Thus i n e x p ( S ' ) , the edge (0 + iO, 1 + iO) x {9} simply collapses to {*} x {#}. 1  3  2.2  e x p ( 5 ) and the inclusion e x p ( 5 ) 1  1  3  exp (5 ) 1  ^  1  3  We w i l l now prove (after a quick lemma) the m a i n result of this thesis, a stronger version of Theorem 1.1.4 (proven also i n [11]). Lemma 2.2.1 The space exp (S' ) is a compact 3-manifold without boundary. 1  3  Proof Compactness is immediate as exp (S' ) i s ' a quotient of (S ) . 1  1  3  3  A s for it being  a manifold, the only place where this might fail is on e x p ( 5 ' ) C e x p ( S ' ) , so we 1  2  1  3  simply need to verify that each point therein has a euclidean neighbourhood. For points i n C ^ S ) this is rather easy. 1 1  forp^q  A point i n a neighbourhood of {p, q}  w i l l be i n one of the configurations shown i n figure 2.6, and so it is fairly  easy to see that there is a neighbourhood of {p, q) which is homeomorphic to two copies of (—e, e) x C ((—e, e)) glued along their common boundary—which is then 2  homeomorphic to (—e, e) as required. 3  For a point i n Ci(S ), 1  it is similarly easy to see that it has a neighbourhood  homeomorphic to e x p ((—e, e ) ) which considered as a quotient of the space X 3  =  29  Figure 2.6: A neighbourhood of {p, q] {(x,y,z)\ — e < x < y < z < e} is also readily seen to be homeomorphic to a euclidean ball.  Q.E.D. Theorem 2.2.2  The space e x p ( S ' ) is homeomorphic to S , 1  3  3  e x p ( 5 ) <—> e x p ( 5 ) = S 1  1  1  3  3  and the inclusion S  l  is the trefoil knot.  Proof T h e majority of this proof w i l l rely on calculating the fundamental groups of e x p ( S ' ) and e x p ( 5 ' ) \ e x p ( S ' ) , relying on classification theorems to show the above 1  3  1  1  3  1  result. T h e m a i n tool w i l l be the Seifert-Van K a m p e n theorem. F r o m the covering shown i n figure 2.7, the Seifert-Van K a m p e n theorem yields the following pushout of groups.  7Ti(A n B)  — ^  7Ti(B)  3*  TTI(A) —  7T (exp (5 )) 1  1  3  where A deformation retracts onto a circle (and so it\(A) = (s)), and B deformation retracts onto M* ^ S  1  (hence Tt\(B) = •(£)); thus we have that Tt\(exp (S' ))  R). It remains to determine what the relations R are.  1  3  = {s,t \  =  30 2  T  t  2  Figure 2.7: T h e covering used for the Seifert-Van K a m p e n theorem Now, A f l B is simply the "boundary" of C (S ); L  3  simply a torus T . 2  in the S  1  that is, up to homotopy it is  Thus its fundamental group is (a) © (b) where a is the generator  direction, and b is the meridional generator.  We can explicitely describe this homotopy torus i n terms of points on the underlying circle S  1  i n the following manner.  T h e longitudinal direction (ie the one  corresponding to a above) is given, as expected, simply by rotation of points along S; 1  as we are avoiding the exceptional fibre, there is nothing unusual here and we  need a full rotation at any given point to return to the starting point. Now, we obtain the other generator b (demonstrated i n figure 2.8) simply by rotating each point i n a counter-clockwise manner to the next point along. It is easy to see that this commutes w i t h a, and that together these two paths make a torus. So to determine the relations R, we first examine i*(o) and j*(a), the simplest of the two to deal with. Now, as the generator of TTX(A) is the p a t h along the exceptional fibre from 8 = 0 to 8 = | , it is easy to see that j*(a) = s . T h e exact same reasoning 3  shows that i*(a) = t , and so we have the relation that s = 2  3  t. 2  So for the meridional generator, b, we have the situation shown i n figure 2.9 (cf.  31  Figure 2.8: T h e generator b of n^A  f l B)  figure 2.2) which can easily be seen to be homotopic to the generator t of TTI(B);  thus  it only remains to see what happens to j*(6) (shown i n figure 2.10) to fully understand what 7 r i ( e x p ( 5 ' ) ) is. W h i l e it would be tempting to suggest that it simply collapses 1  3  to a homotopically t r i v i a l path, this is not indeed the case.  i*{b)  L  t Figure 2.9: T h e homotopy class of i*(6)  If we restrict the front end of this path to have S  1  coordinate of 0, then recalling  the proof of P r o p o s i t i o n 2.1.5 we find that as we contract our p a th towards the center (that is, the exceptional fibre). of A the S  1  coordinate of our back end begins to  recede-until finally, when our path has been deformed completely to the center we find that it has become a path from — | to 0; that is, the generator s of TX\(A). A s an aside, it is worth noting that figure 2.8 gives an alternate geometric idea as  32 2 ^  2  6  \  / \  /  L  _  I  Figure 2.10: T h e homotopy class of j*(b) to why j*(b) is the generator of it\(A). If we perturb that diagram so that our three points are equally spaced about the circle, then it is easy to see that the description of the path b is exactly the path which generates ~K\(A). Combining a l l of this together, we find that 7Ti ( e x p ( 5 ) ) = (s,t \ s = t , s = t) = 1 1  3  2  3  and so e x p ( 5 ) is simply connected.  F r o m [10] it follows immediately that as a  1  3  simply connected Seifert fibred space, this must be homeomorphic to S . 3  For brevity we will now define X := e x p ( S ' ) \ e x p ( S ' ) . Now, for the calculation 1  3  of iri(X)  1  1  the majority of the details above still hold through—the only difference  is that what was labelled as B above (now to be denoted B') has become slightly different. I n the calculation of 7 T i ( e x p ( 5 ' ) ) , we claimed that B deformation retracts 1  3  onto M*; this fails for B'. C l a i m : TVI(B') = (t,u | [t ,u] = 1), where t is the generator of ni(Ai) and u is 2  simply the image of the meridional generator i n B'.  33 F r o m this claim it follows that •Ki(X) = {s,t,u | [t ,u] = l , s = t ,s = u) 2  2  3  = (s,t\ [t ,s] = l,s 2  3  = t) 2  = (s,t\ s . = t ) 3  Now,  2  to see that this implies that the inclusion e x p ( 5 ) <—> e x p ( S ' ) is the trefoil 1  1  1  3  knot, we proceed as follows. T h e first thing is t o note that the center of the M o b i u s band—its exceptional fibre—is unkotted i n exp (S' ). T h i s is due t o the fact that :L  3  7 r i ( C 3 ( 5 ) ) = Z . Thus i f we consider a tubular neighbourhood of this subset we ,  1  obtain a torus i n e x p ( S ' ) — a n d the intersection of the boundary of this torus w i t h 1  3  the Mobius band is thus a torus knot which is isotopic t o e x p ( 5 ' ) . 1  1  We can now  use the fundamental group to say that it is a (2, 3) torus knot, or a trefoil knot as claimed. Proof of claim: Let us examine B' a little more closely. Figure 2.11 shows a Slice of B', separated into open sets U,V.  V  Figure 2.11: Covering of B' Now,  U ~ T , and V ~ M. Lastly, U C\V ~ 5 , and so we end u p w i t h the 2  following pushout t o calculate  1  7Ti (_£?'):  34  (*>  © (b)  — ( A )  Tn(B') from which it follows that  TTI(B')  2* (a, b, c | [a, b] =  1, c  2  =  a)  ^(6,c|[c 6] = l ) 2  )  as claimed; Clearly, c is the generator of ir-i(M), and b is the meridional generator and so the full claim follows. Q.E.D.  35  Chapter 3 Further Research Having determined the homeomorphism type of exp .(S' ) for k = 1,2,3, the next 1  A  obvious goal is either to determine the homeomorphism type of these spaces for higher and higher values of k, or alternately, to move on to see if the method generalizes for higher dimensional spheres. T h e former generalization would conceivably involve considering certain bundles over P S X ( R ) — r e c a l l that i n P S X ^ R ) we are limited to choosing fractional linear 2  transformations which take at most three arbitrary points to a select group of points, and as such we cannot directly continue the process used above. However, if we let {p < q < r < s} € C^S ),  then there remains still, a fractional  1  linear transformation which takes three of these points to { 0 , 1 , 00} as considered before. W h a t we can then do is consider what happens to that fourth point. If we let T be this given transformation, then T(s) geodesic between (say) 0 and 00.  G HI and i n particular, it lies on the  Thus for fixed p < q < r there is an R degree  of freedom i n choosing s. If we consider these points as ordered, then we have an R bundle over the ordered configuration spacs F (S ), l  3  and so we simply have to consider  what happens when we quotient out by the action of the group of isometries which permute these points. T h i s works i n general; i n fact for sufficiently nice spaces X, if we let F (X) k  denote  the fc-th ordered configuration space of X, then we obtain a sequence of fibrations  36 given by  ,p _!} <-> F (X) A  X \ { ,...  fc  Pl  k  F ^(X) k  where the projection map n is given by (xi,..., x ) >—• ( x i , . . . , z^-i)- It should be k  noted that for X = S , this is in fact a fibre bundle. n  This being said, the generalization that we intend to follow .is that of considering higher dimensional spheres first.  If we consider S in particular, then S is the 2  2  boundary of H and more importantly, the group of isometries of this spaces is then 3  going to be given by PSL (C)—as 2  such the general methodology implemented above,  as well as a number of the calculations used, will still be valid for this case, making it a more natural generalization to approach first. This leads us to examine yet again the first step.  What is the topology of  PSL {C)1 2  We push this question aside for the moment and see what we can determine about the spaces Ci(S ) = F^S )/^.2  2  We can begin with at least obtaining a rough idea  using the fibrations given above. From the first one S \{*}^>W (S )^F (S ) 2  2  we obtain that F (S )  = S  2  2  2  1  ~ 5 ; the second fibration is then (up to homotopy)  2  2  2  S -> F (S ) 1  .  S  2  2  3  and so we can use either the Serre spectral sequence to compute the homology or the long exact sequence of homotopy groups to help understand this space. In the latter approach, since ^(S* ) = 0 for i > 1, we immediately find that 1  ^(S )) 2  ^  MS ) 2  for a l H > 2, as well as a (somewhat short) exact sequence 0 - 7T (F (5 )) - 7T (5 ) 2  2  3  2  2  -> TTxtS ) 1  TTl(F (S )) 2  3  - 0  37 or 0 -  n (F (S )) 2  2  3  -  Z ^ Z — 7n(F (5 )) 2  3  0  and so ^ ( ^ ( S ) ) = ker(fc) and 7Ti ( ^ ( S ) ) = coker(fc). So determining the nature 2  2  of this map k tells us a l l of the homotopy groups of  F (S ). 2  3  So what is kl Well, to track it down we have to examing a chain of isomorphisms; namely n (F (S ),  S \ {0, oo}) a n (F (S ),  2  2  2  3  (0, oo))  2  2  2  ^TT (F (5 ),5 \{0}) 2  2  =  2  2  7T (5 ) 2  2  all of which come from the homotopy exact sequences and isomorphisms i n the diagram i n figure 3.1, where the vertical column comes from the exact sequence of relative homotopy groups, and the horizontal isomorphisms come from the isomorphsism 7 r ( X , A) = Tr (B) for a fibration, and from the homeomorphism Fi(X) n  n  = X.  If we then trace the generator of n (S ) back through a l l of these isomorphisms, 2  2  we find that k must simply be multiplication by 0. ^ ( F ^ ^ Z ^ n ^ S Remark  2  T h i s of course implies that  ) ) .  T h i s can also be seen by noting that it is necessary that 7Ti (F^S )) 2  = Z.  In fact, we can describe a generator explicitly. Let (0,1, oo) G /^(IP ) be the 1  chosen basepoint; then the path t H-» ( 0 , e cyclic subgroup of 7Ti (F (S )). 2  3  27nt  , o o ) is a generator of an infinite  A s this must also be a quotient of Z , it follows  that the map k i n the exact sequence above must be 0.  38 .  ^(5 \{0» 2  TT (F (S ),S \{Q,OO}) 2  2  3  2  TT (F (S ),(0,OO)) 2  2  2  at  n (F (S ),S \{0}) 2  2  2  2  n (Fi(S ),(y) 2  2  *i(S \{0}) 2  Figure 3.1: Isomorphisms due to exact sequences  =  7r (S ) 2  2  39  Bibliography [1] K . Borsuk. O n the t h i r d symmetric potency of the circumference. Fund. Math., 36:236-244, 1949. R . B o t t . O n the t h i r d symmetric potency of Si. Fund. Math., 39:264-268, 1952. D . Handel. Some homotopy properties of spaces of finite subsets of topological spaces. Houston J. of Math, 26:747-764, 2000. A . Hatcher. Basic topology of 3-manifolds. In preparation, available on the web at  http://www.math. C o r n e l l . e d u / ~ h a t c h e r / .  A . Hatcher. Algebraic Topology. Cambridge University Press, 2002. M . Jankins and W . D . Neumann. Lectures on Seifert Manifolds. Brandeis Lecture Notes, 1983. S. K a l l e l and D . Sjerve. F i n i t e subset spaces and a spectral sequence of biro. J . Mostovoy. Geometry of truncated symmetric products and real roots of real polynomials. Bull. London Math. Soc, 30:159-165, 1998. J . Mostovoy. Lattices i n C and finite subsets of a circle. Amer. Math. Monthly, 111:357-360, 2004. H . Seifert and W . Threlfall. A Textbook of Topology, pages 359-422. Harcourt Brace Jovanovich, 1980. Translated from the 1934 G e r m a n edition.  40 [11] C . Tuffley.  Finite subset spaces of S . 1  Alg. & Geom. Topology, 2:1119-1145,  2002. [12] C . Tuffley. Finite subset spaces of graphs and punctured surfaces. Alg. & Geom. Topology, 3:873-904, 2003. [13] Christopher Tuffley. F i n i t e subset spaces of closed surfaces, 2003.  

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