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On the fixed point properties of Grassman manifolds Hurthig, Stephen Peter 1976

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ON THE FIXED POINT PROPERTIES OF GRASSMAN MANIFOLDS by STEPHEN PETER HURTHIG B.Sc, University of British Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the department of MATHEMATICS We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1976 0 Stephen Peter Hurthig, 1976 In p resent ing t h i s t he s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha t permiss ion fo r ex tens i ve copying o f t h i s t he s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r ep re sen ta t i ve s . It i s understood that copying or p u b l i c a t i o n of t h i s t he s i s f o r f i n a n c i a l ga in s h a l l not be a l lowed without my w r i t t e n permi s s ion . Department o f Mathematics The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date November 1, 1976 - i -Supervisor Dr. R. Held Abstractt In this thesis we show that for n even, the Grassman manifolds CG have the fixed point property and that CG_ . has Lusternik - Schnirelmann Category 13» ACKNOWLEDGEMENTS I would l i k e to acknowledge my gratitude to Dr. R. Held, f o r suggesting the t o p i c o f t h i s t h e s i s and f o r h i s generous guidance and encouragement throughout the work. I would also l i k e to thank Dr. R. Douglas f o r reading the t h e s i s and the U n i v e r s i t y o f B r i t i s h Columbia f o r f i n a n c i a l assistance. - i i i -TABLE OF CONTENTS Page Introduction . . . • • . iv Chapter 1« The Lefschetz - Hopf Fixed Point Theorem 1 Chapter 2: Manifolds with the Fixed Point Property 9 Chapter 3* Cohomology of the Grassman Manifolds 13 Chapter L,i Fixed Point Properties of the Grassman Manifolds . . . . 17 Chapter 5 s Properties of some of the CG^ . . . . . . . . . . . 20 Bibliography • • . . . . . . Al Appendix . . . . . . . . . . 42 INTRODUCTION In the f i r s t chapter we give a proof of the famous Lefschetz - Hopf Fixed Point Theorem. In the second chapter we use this theorem to show that various examples of familiar manifolds have the fixed point property. We also give examples of manifolds which do not have the fixed point property and mention how manifolds with this property may be constructed. In the third chapter we find the cohomology of the Grassman manifolds. Due to the nature of these cohomology rings, i t does not appear likely at fi r s t glance that one can prove using only the Lefschetz - Hopf Fixed Point Theorem, as i s done with the projective spaces, that certain of these manifolds have the fixed point property. Bachmann, Glover and O'Neill, however, have shown that for some Grassman manifolds ( See Theorem 4.1 ), the endomorphism of the cohomology ring induced by any self map of such a manifold must take the f i r s t characteristic class to a time3 the f i r s t characteristic class, the second characteristic class to ar the second characteristic class and so on. Such a self map i s known as an Adams type mapping. If a self map of an appropriate Grassman manifold i s an Adams type mapping, i t i s easy to show that it's Lefschetz - Hopf number i s nom - zero through the use of well known theorems concerning the Euler number of these manifolds. We know from Theorem 4«1 that i f n i s even and greater than or equal to ten that CG^ n has the fixed point property. O'Neill, in his doctoral thesis, proved that CG has the fixed point property, and so 3,2 to prove that CG has the fixed point property for a l l even n we only have to show that the remaining three manifolds have the fixed point - V -property. In proving that <CG, . has the fixed point property, we also prove that i t has maximal cuplength and thus we are able to determine i t ' s Lusternik - Schnirelmann Category. ( See Page 39 ). I t should be noted that since H*( CG ;2ZL) has no torsion and there i s the natural injection Tiff K of the integers into the rationals, the Lefschetz - Hopf number w i l l be the same whether calculated with integral or rational coefficients and thus i n a l l the calculations of Chapter 5 we may assume that the variables involved take their values only i n the integers. - 1 -1. The Lefschetz - Hopf Fixed Point Theorem In the following chapter, let X be a connected n - dimensional manifold and R a commutative ring. For more details see (l)» Locally Constant Lemma: Let £ : %(X,X-0;R)—• ^(X^-xjR) be the homomorphism induced by the inclusion X-U —*• X-x for an open neighbourhood U of a point x in X. Every neighbourhood W of x in X then contains a V neighbourhood V of x such that for every y in V, j y i s an isomorphism. Hence a generator \ of Hn(X,X-x;R)= R has a unique continuation in U. Definitions: Given a subspace UCX, an element ^ e ^(XJX-XJR) such that jy(^ ) generates H^XjX-yjR) for each y€U i s called a local R - orientation of X along U. An R - orientation system i s a family of open subspaces which cover X such that for each 1 there i s a local R - orientation ^ H^XjX-U^;R) of X along U and i f x€U i^U i,, then j^M'S ±) = i ^ ' C ^ i . ) . X i s said to be R - orientable (orientable) i f an R - orientation (ZL - orientation) system for X exists. Let X° - ^(x, " S ^ l x C X ^ e ^(XjX-x^R)} and p: X ° — * X be defined by p(x , \ ) = x. The sets <U,\> = [(x , ' S x)|xGU , v S x = f o r m a b a s e for the topology on X°. X° i s known as the R - orientation sheaf of X. For any subspace ACX, a map s: A — X such that ps = inclusion i s called a section over A. Let s(x) = (x,s'(x)) define a section over A. Then x|— (x, Xs'(x)) for some X in R also defines a section over A. Also, i f s^ and s 2 are sections over A, then x|—• (x, s^x) + s^(xj) i s a section over A. - 2 -Therefore the set of a l l sections over A i s an R - module P A. There i s a canonical homomorphism j A : ^(XJX-AJR) • P i defined by j )(*) = U>J X(^)) for x in A. Theorem 1.1: Suppose ACX i s closed. Then i) Hq(X,X-AjR) = 0 for q>n i i ) i s a monomorphism and i t s image i s the submodule P A of sections over A with compact support. A section has compact support i f i t agrees with the zero section outside some compact subset of A. In particular, i f A i s compact, P^A = PA, and i f X i s compacttthen j ^ : BpCX^)•-— »» X i s an isomorphism. Corollary 1.2: Let X be a compact connected manifold and R an integral domain. Then Hn(XjR) i s isomorphic to R i f X i s R - orientable and 0 i f not. An R — orientation of a compact connected manifold X i s therefore determined by a generator of pX or a generator (3 of Hn(X;R) which i s called the fundamental class of the R- orientation of X. The local R - orientation at each point x in X i s then ^(/6). Given an R - orientation of X, that i s , given a global section s: X — • X 3 such that for each x, s'(x) generates ^(XJX-XJR), there exists the dual sheaf 2P »^ X whose fibre over a point x in X i s the local cohomology module ifHXjX-xjR) and a global section s : X — 2 r which i s characterized by (s*(x),s(x)) = 1 for x i n X. Suppose D i s open in X. Denote by P*U the module of a l l sections over U of the dual sheaf. If & i s the diagonal of - 3 -XXX, define U ± X: (X,X-x) >• (XXU,XXU - A ) by U i x(x') = (x',xj, x1 in X, x in U. Theorem 1.3: Let X be an R - oriented n - dimensional manifold, U an open subspace. Then rfHxxu/XXU - A ) = 0 for a l l q<n, and there i s a unique isomorphism $ : ^ (XX^XXU - A ) *• P*U such that 0(<x)(x) = ^ ( U ^ ) ^ ) for a l l oc £H n(XXU,XXU - A ) , x in U. Corollary 1.4: There i s a unique cohomology classJ~> - y in rf^XXXjXXX - A) such that for a l l x i n X, s*(x) = H^X X ) ( ^ ). This class jj i s called the Thorn class of the given orientation. Corollary 1.5: Suppose X i s compact. Let /3 e Hn(X;R) be the fundamental class of the R - orientation. Let fiPCj). H^XXX^XX - A ) vPr^XXX) be the homomorphism induced by inclusion and let yu 1 = H^jHyj ). Then JJ%/^> - 1» where / i s the homology slant product. Proof: For any x in X consider the commutative diagram: U,X-x) (XXX,XXX - A ) j i X X XXX Then 1 = ( r f K l^.sU)) = { ^ ( X ^ . H ( j ^ ) = <fU\)f ,/3> = (H'XjiJ^/S) = ^ S H ^ i ^ ) . However, HXO^S = ^ X x is a property of the exterior homology product and <(yu»',|5X = ^'^3 »x)> a property of the homology slant product. (See appendix) Q.E.D. nm mm Before proceeding to the proof of the Lefschetz - Hopf Fixed Point Theorem, we need some additional facts about compact manifolds* i) A compact manifold can be embedded in a Euclidean space. i i ) A space i s called an Absolute Neighbourhood Retract. (ANR), i f for any normal space Y and a map f: B »»X of a closed subspace B of Y into the - space X, f extends to a map of an open neighbourhood of B into X. It i s known (see for instance (2)) that every compact manifold i s an ANR and consequently i f a compact manifold i s embedded in Borne Euclidean space, i t is the retract of some open neighbourhood (just apply the universal property to B = the manifold and f = the identity). Lemma 1.6: If X i s a compact manifold and A i s the diagonal in XXX, then there i s an open neighbourhood V of A such that the identity map of V i s homotopic in XXX to a retraction of V onto & • N Proof: Embed X in R , and let U be an open neighbourhood having a retraction r: U — X . Let £ = the distance from X to RN - U, and let V be the £ -neighbourhood of A in XXX. Define F: XXXXI >-RN by F(x,x',t) - (1 - t)x + tx'. Then F maps VXI into U. Let G = r(F|Vxl): VXI >»X so that G(x,x',0) = r(x) = x, G(x,x',l) = x». Define H : VXI >XXX by H(x,x*,t) = (x,G(x,x',t)) which i s the required homotopy. Q.E.D. Lemma 1.7: If ^ eHP(XXX,XXX - A ), r^GH^X), then H P ( J ) ( ^ ) W (ryx l) -HP(j)(^ ) W (lXry) where j : XXX —>- (XXX,XXX - A.) is the inclusion. Proof: By Lemma 1,6, there i s an open neighbourhood V of A in XXX and a retraction r: V—*~ A such that i r c ^ k , where i : A— > • XXX and k: V >• XXX are inclusion maps. Denote the inclusion (V,V-A)—>(XXX,XXX - A ) ' by k1, and note that k' i s an excision. The following diagram i s commutative: H*(X X) rfl(A) #(k) HP + (*(X X , X X - A ) K?+q(k«) I?+<1(V,V-A) Let p ±: XXX—>• X, i = 1,2; be the projections. Then l x r ^ = H°(Pl)(l) W H q(p 2)(r^) =r H q(p 2)(r^), r\X 1 = I?(p^Cry). Let p: A ^X be the common restriction of p^ and p 2 to the diagonal. From the diagram we have * KJ &{T>±Kry) = K^VfVuOa ) U H q ( P i i r ) ( ^ > = rf^k'^(k-)(X ) H°»(pr)(»"^ ) for both i = 1,2. By the properties of the cup product we have ffCjJU ) W rfkp^) = #*q( ^XJ iftp^CryH which proves the lemma. Q.E.D. Lemma 1.8: The basic formula relating a l l the products i s : for ^  C 'lfUxY),^ €. H5(X), r^eH^Y^oc e H g(x). Proof: SupposeY e«p + q + r_ 8(Y). Then [ryw| ^ r\^/c<j,V] = [^ rOu/oC, ry^Y] - a property of r\ = \t ,(^r\OC)X(ry^y)] - a property of/ , W « ) [ ^ ) ( ^ X r v ) ^ ( 0 C X > ) ] = (-D = ( _ 1 ) q ^ p + q + r " s ^ ' - a property of r\ [|(^xr^) \ j^/bc/y/] Q.E.D. - 6 -Theorem 1.9: Let X be a compact R - oriented n - dimensional manifold with fundamental class (3£H n(X). Then for any p^n, the inverse to the Poincare Duality isomorphism H^X) —>- H (X) i s given by <X —>(-l) Pyu«/c< for n-p / cxeH (X). n-p Proof: If r^LH^X), then /S,r\ry i s i t s image in Hn_p.(X) and y j ' / p ^ r ^ = l ^ y u ' / p x r ^ j = (_i)P(n+p+0-n) | ( ^ x !) w y / ^ ^ ^ Lemma 1 # 8 = (-l)p 2^(lxr^) \JjJ*/£>^ by Lemma 1.7 - {-l)W{-l)°i\y {fj'/irsfi} by Lemma 1.8 .= (-l) Pr\ W {jj'/p ) = (-l)pr\^» 1 by Corollary 1.5 = (-l) pr\ Q.E.D. Suppose we have a map f: X **Y, where Y i s another compact R - oriented manifold of dimension m. We define the cohomology class jj^ of the graph of f by jjf ~ H ^ f X i d X y u ' K l A x X Y ) where jj' £ Hm(Y Y) i s the image of the Thorn class of Y. The class JJ ^  completely determines the homomorphism induced by f on the cohomology. Lemma 1.10: For any <\<Ll?{l), lP(t){<(\) = i-lfjJf / f i j ^ ^ where fd^C ^(Y) is the fundamental class of Y. PE22£» jJf / r l ^ N ^ Y = K P + m - p ( f X i d ) ^ y P y ^ / C i = ifCfJC^J/H^ ( i d ) ( r y ^ | Q Y ) by naturality of / = HP(f)( / J.)/r{^ i6 i = ( - l W C f ) ^ by Theorem 1.9 Q.E.D. Theorem 1.111 Let f» X »-X where X i s a connected compact R - oriented manifold. If yu f / 0 then f has a fixed point. Prooft If f has no fixed point, then we have the factorization XXX »- XXX fX id where i i s the inclusion. Since lfti)Hn(jj = 0 and JJ^ = AjK^), JJ = i f C f X i d ) ^ ^ ) = 0. Q.E.D. Next, define the Lefschetz - Kopf class L£ = lT(f,id)(}J %) and the Lefschetz - Hopf number f\r =^Lj..ft). Theorem 1.121 The Lefschetz - Hopf Fixed Point Theorem Let X be a compact connected R - oriented manifold, where R i s a field. If fx X >-X i s any map, then the Lefschetz - Hopf number of f i s given by A p = H (-l)qTrace H q(f). If lA. f / 0, then f has a fixed point. q Proof? Let^c< A be a basis for H*(X), where i runs through a finite set; q i / let be the integer such that H (X). By the Kunneth formula (since R is a field), the o^x (Xj form a basis for H*(XXX), so that / " = S ° i J ^ X U h e r e C U = ° l f ^ + q J * n ) 'Let H*(f)(oci) = ^ (where a ^ = 0 i f q k / q±) Let y k j = ( o C j , ^ ^ ) =(^\J 0Cyp>) so that y j k = ( - l ) ^ ^ when q k + q^ = n, and v k j = 0 when q k + qj ^  n. - 8 -* iScijaki<ockw °VP> = i ? j c i j a k i y k j = E \ i ( 9 V k j ^ Howeveri q, (-1) KOC k = yu'/oC k/^p by Lemma 1.10 = pi0^"1^ ^ ( ^ j * ^ ^ z5) b y L e m m a 1 4 8 ' ? C E c j i y J k ) o c i 5 0 t h a t ? c j i y j k = ( - l ) q ^ i k b e c a u s e t h e ° Y 8 f o r m a basis. Since the right inverse of a matrix i s also its left inverse, ? C i / k j = ( - l ) q k q S i k -Thus = C (-1) k« . Q.E.D.. - 9 -2. Manifolds with the Fixed Point Property A topological space, X , i s said to have the fixed point property i f every self map of X has a fixed point. That i s , i f f: X >-X i s a map then f(x) = x for some x in X . There are many examples of topological spaces that have the fixed point property, (see for example (3))> howevdr relatively few of these are manifolds. The following is a l i s t of some of these manifolds. 1. A compact manifold X i s said to be Q - acyclic i f FPUJQ) = 0 for a l l p ^  0, where Q denotes the field of rational numbers. Such a manifold has the fixed point property since-A-^ = 1 for a l l maps f» X — » - X . In particular, the n - discs D have the fixed point property. This result i s known as the Brouwer Fixed Point Theorem. 2. Real protective 2n - space. jR F^n, n ^ l has the fixed point property _2n"*"l since i t i s Q - acyclic. The manifolds, iR r , however, do not have the fixed point property. Since IRr = the set of equivalence classes of vectors ( r Q , . . . , r 2 n + 1 ) e IR 2 n + 2 - (0,0,...,0) where (r Q,..•»r 2 n + 1) i s equivalent to ( r j , . . ) i f a n d ovl7 i f ( V , , r 2 n + 1 ^ = C* r0'' * * ,r2n+l^ for some non - zero real number c, the function f s R2n+2 - (0,0,...,0) —>(R 2n+2 - (0,0,...,0) defined by f ( r Q , . . . f r 2 n + 1 ) = (-VV ,- -'~ r2n+l , I2n ) induces a self map of (R p211*1- which fixes no point. 3» Complex pro.lective 2n - space. CP211, n>l, has the fixed point property. The cohomology ring of CP 2 1 1 with rational coefficients i s the truncated polynomial algebra generated by a generator c of IT( CP 2 1 jQ) where c = 0 Therefore i f f i s a self map of CP 2 1 1 and f*fc = ac thenJV^ = 1 + a + a 2 +.. + a 2 n . If a / 1 then_/Vf = ( l - a 2 n + 1 ) / ( l - a ) / 0. If a = 1 thenA_ f = 2n + 1 and hence CP 2 1 1 has the fixed point property. However, i f f: C P 2 n ^ C P 2 n + 1 is defined by f [ ( c 0 , . . . , c 2 n + 1 ) ] = [(-c1,^),...,-«ijnfl,ft2B)], we see that the CP 2 1 1*^^ do not have the fixed point property. U» Quaternionic projective space. NP n, has the fixed point property for a l l n greater than 1. (IHF^^S^ does not) Proof i B*(lHPnjQ) = Q [ h J / h n + 1 , the truncated polynomial algebra generated by h-^ a generator of H^HF^jQ). Therefore the Lefschetz - Hopf number,-A-f for a self map f of IHP i s 5^ (-l^a* where f * ^ = ah^. JV, = 0 only i f a = -1. We shall show however that a cannot assume the value -1 Let P1: tfKXj'Z. )• ) be the reduced power operation. (See 3 3 There exists a map g which makes the following diagram of Hopf bundles commute and i s such that g*hj= c^ where c^ i s the generator of S 1 S3 S 4n-l _ c4n-l CP 2 1 1 >. HP11 - l i -l t is sufficient to show this for the case n = 1. Here the fibre of g is 1 2 just CP ^ S and we have the result from the Gysin cohomology sequence, Since dim c^ = 2, P^ c-^  - c^ and the Cartan formula gives P^(c^) = 2c^. Since P 1 is natural F L(h ) = 2h^ and by the Cartan formula PL(h^) = 4h^. Therefore P 1 f* P 1 H^OHP 1 1;^) — — • H L 2(lHP nj2^) •->• a 3 2 implies that Ubt=~. 4a (mod 3) which implies that a ^  -1. 5. The Cayley Plane The Cayley numbers are ordered pairs of quaternions. They are added by adding coordinates and multiplication i s defined by (q-^ qgHq^ fq^ ) ~ ^ q l q l " ^ 2 q 2 , q 2 q l + q2^P* d>°) i s a t w 0 sided unit. Also i f the conjugate of c = (q^,q2) i s defined to be c~ = ("q^ ,-^ ) t h e n c ^ = * s r e a l a n d n o n ~ negative and equals zero i f and only i f c = (0,0) = 0. It can also be shown that multiplication is distributive with respect to addition and cd = 0 implies that either c = 0 or d = 0. Thus the set of a l l Cayley numbers forms a division algebra. The associative law does not hold in general and so we cannot use the equivalence relation used for the real, complex and quaternionic spaces to define projective spaces based on the Cayley numbers - 12 -Using the fact that any two Cayley numbers generate an associative algebra isomorphic to a subalgebra of the quaternions, we can construct a fibering of S t o v e r S3 with S 7 for a fibre to define the notion of a Cayley plane. (See (5))» It i s , in fact, the homogeneous space F /Spin(9) where F i s the Lie group which i s the quotient group SO(8)/U(4.). Its rational cohomology ring is the truncated polynomial ring generated by a generator c in dimension 8 such that c^= Oj and hence by the Lefschetz -Hopf Fixed Point Theorem i t has the fixed point property. Since a l l that i s used to show that these manifolds have the fixed point property i s the Lefschetz - Hopf Fixed Point Theorem, any manifold with the same rational cohomology algebras as these will have the fixed point property. Such manifolds may be obtained for instance by mixing homotopy types. (See (6)). Using the Runneth formula we see that cartesian products of the above manifolds also have the fixed point property, however, in (7) Husseini constructs manifolds with the fixed point property such that certain cartesian products of them do not. In (2), Brown shows that i f X is a compact manifold and M i s an n -manifold with the fixed point property and n>3 then the mapping cylinder of any map f: X—>• M has the fixed point property. In particular i f I i s the unit interval then MX.I has the fixed point property. - 13 -3. Cohomology of the Grassman Manifolds Let K be fi^ , (TJ. or |H , the real, complex, and quaternionic fields respectively, and let G denote the Grassman manifold of n - dimensional subspaces of K11*^. Let ^  £ denote the bundle (and total space of the bundle) of G . which i s the set whose points are the pairs: (n - plane, point in the n - plane) and ^ £ the set of pairs: (n - plane, point in the complementary k - plane). For a vector bundle ^ = (E,B,tr ) let the projectivization, P(^ ), be the space of one - dimensional subspaces of the fibres of E, and let l ( ^ ) be the bundle over P(^ ) consisting of the pairs (cr,e) where o i s a one - dimensional subspace of a fibre and e i s a point in that subspace. Proposition 3.1: (See (8) for further details of Chapter 3) Let ^ be a locally tr i v i a l vector bundle over a compact Hausdorff space B. There exists a finite dimensional vector space V over K and a surjective bundle map e: VXB —>-B and a commutative diagram: . J , ' 7Z • P(E) ^P(VXB) = P(V)xB »- P(V) I B B For example, i f ^ = X£ then V = Knand P(V) i s K- projective (n-l) -space, - H -Now let A = 7L i f K = <C or IH and 7L i f K = 1R . Let o<v be the class in ^(POOjA) SO that H*(P(V);A) i s the free A - nodule on 1, dimV - 1 Theorem 3.2: Let ceH k(P(E);A) be the class j*(o<v). Then H*(P(E)JA) i s the free H*(B;A) module (viaTT*) on the classes l,c,...,<P~^  and there exist unique classes ) € ^ ( B ^ ) , (where k = the real dimension of K) and a0(\ ) = 1 such that c n - c * " 1 * * * ^ ^ ) ) + . . . • ( - l ^ c w M c r )) • (-i)\r*(cr (*)) = 0. n The class c r ( ^ ) = 1 + CT ) + ... + a ) i B called the total characteristic class o f ^ • These classes are also called: i) K = R ; Stiefel - Whitney class to ) i i ) K = C : Chern class c(^) i i i ) K = IH : (Symplectic) Pontrjagin class -f>C^) Theorem 3.3: The total characteristic class cr ) has the following properties: 1. CT (^) = 0 i f i>dim^. 2. ) is natural:, i f f; B' >B then CT(f*^ ) = f * c r ( ^ ), 3. If ^ and are two vector bundles over B then c r ( ^ ©f|) = C J ^ ) v ^ c r ( r ^ ) 4. If 1 i s the canonical line bundle over P(V), then CT^l) = Cxr^ . Now, let O" = CCCK?), O, = o t t f ) . Property 3. of the previous X X K X K theorem implies that CV^O = 1. - 15 -Proposition 3.4: H*(G , jA) i s the quotient of the polynomial algebra over A on the CT^ , i ^ n , by the relations imposed by = 0 for j>k. (Note: O" i s the polynomial of degree j i n a, given by the formal inversion of O .) Proof: This polynomial algebra i s certainly mapped into H*(Gn ^;A) and to prove that this i s indeed an isomorphism we induct on n. For n = 1, G l k = K j k + 1 s o t h a t H k>A) i s g e n e r a t e d by c< = with the one v+i 2 \k k relationoc =0 but since CJ = 1 - oc + oc - ... + (-1) oc. + ..., the condition O. = 0 for j>k gives this and only this relation. If the result holds for a l l G . with n<s. then consider G . . A point n,k 7 s,t r in P ( ^ ) i s a line a in an s - planejj • The orthogonal complement of a in u i s an s-1 - plane and hence a point of G , . ,. Therefore ?(% f) / _ s-l,t+l t t+i e.g. s=2, t = 1 2 2. a€PU ) and aeP (K 2 ) We have the diagram: t i t+x TT Tf s-l,t+l *s,t - 1 6 -Letting 1 = 1(% I) = 1(1 *"*) one has that T T * ( ^ J) = ^® 1, T f * U ^ ) = ry® 1 with ^ © ffi 1 being the tri v i a l bundle. Since c = ^ ( l ) we have 0 = ^ t - l j V ' V ^ U *)) = C(- l )cf ^ D ^ a ^ ^ f l )) by naturality = E C - l ) 1 © ^ ! ) 8 " 1 ^ ^ ) + 0 i. 1(^)0 1 ( D ) = CSaC%) (the other terms canceling in pairs). Therefore regarding P( ^  **) as a bundle over G the above property and the induction hypothesis say that H*(p(^ ^ )»A) i s generated by the characteristic classes of ^ , 1, and subject only to relations imposed by the conditions: 1. dim cr(r^e 1 ) = t 2. O (r^® 1 ) = 1 Since C7UC = 1 we already have that H*(G jA) i s generated by CT , S , t J . i-^s with one of the relations being that dim(CS !*)~^) = t. Now looking at P(^ f) as a bundle over G ^ we see that this must be the only relation, t s,t Q.E.D. V 17 4. Fixed Point Properties of the Grassman Manifolds In (9), F. Bachmann, H.H. Glover and L.S. O'Neill prove the following: Theorem 4.3,: i) For q>p 2,H G has the fixed point property. i i ) For q>p2and pq even, CG^ ^  has the fixed point property. i i i ) For q>p 2 ( P* q) o d d a n d P ^  & " 1> ^ G p q the fixed point property. This theorem partially satisfies their conjecture: Conjecture 4.2: IH G has the fixed point property i f and only i f p / q. p,q For F = iR or € , FG has the fixed point property i f and only i f p ^  q and pq even. If this conjecture i s true then the question of which Grassman manifolds have the fixed point property i s completely settled. This i s because the self map of KG , where K = R ,C, orlH, induced by orthogonal complementation P*P is fixed point free and also because of the following theorem: Theorem 4.3: For p + q even let fx F p + q *-F p + q be defined by f(u_,...,u ) = (-TJL/U, ,. ..,-u ,H ) and f be the induced map of 1 p+q 2 1 p+q p+q-1 V Stiefel manifolds f : FV > FV which is seen to be well defined i f V P,q p,q F = ift or € . If p and q are both odd and F = iR or € , then the - 18 -induced map f Q : F Gp~^ *"FGp,q i s f i x e d P o i n - t free, Pgoofj In the case that F = ^  , f is given by the direct sum of matrices (0 -1\ of the form I j • It follows then that the characteristic equation of f i s (z*2 + l) P + 0 ^ / 2 = o, which is seen to have no real roots and thus f has no eigenvectors. Now i f W i s an odd dimensional invariant subspace of f then f|w must have at least one eigenvector, a contradiction. For the case F = C, we refer the reader to (10), page 304 for a proof using differential geometry. Q.E.D. Theorem 4»1 follows from an application of the Lefscheta - Hopf Fixed Point Theorem and the following theorems also proved by Bachmann, Glover and O'Neill in (9). Theorem 4.4: Let R be ZZ i f F = C or IH, 7Z^ i f F = (R . Let f be a self map of FG • Then clearly f*o, = key. for some k in R , C5, being P,q 1 1 1 the f i r s t characteristic class. Suppose q^ .p (and p ^ 2 - 1 i f F =1R ), Then f * ^ = k 1 ^ for i = l,...,p. A self map f with the above property i s called an Adams type mapping. Corollary 4.5: For F = IR , C or IH and q^p 2 (and p £ 2 r - 1 i f F = IR ) every self map f of FG has Lefschetz - Hopf number J\, = P»q 1 £ qdim H (FG jR) k 1 where d = 1,2,4 i f F = IR, <C or \H respectively. o p>q Corollary 4.6: i) For F = C or VA and q^.p2 and f a self map of FG , _\ = 1 (mod k) p,q r - 19 -i i ) For q>p2, ( P + q^ odd and p / 2 r - 1 and f is a self map of IRG j \ P / P»Q -Ay = 1 (mod 2). Theorem 4.1 now follows from the Lefscheta - Hopf Fixed Point Theorem. If F = C or IH ,_/Vj, = 0 only i f k = -1 perhaps. But then 7 ^ = IK-l^dim H ^ F G ^ j Z ) = D - l J ^ C t R G ) ="X^Gp,q» t h e number, since dim Hdi(FG ;Z) i s the number of cells in dimension di in the p»q canonical CW structure for FG where F = {R, C or|H ( d = 1, 2, or A ) P>°i ( See ( l l ) ) It is well known, however, that"XfRG for p £ q, pq even, p»q i s greater than zero, ( See (10) page 303 ). In the case where F = IH and pq i s odd we may use P*" in the Steenrod Algebra mod 3 to eliminate the case k = -1 in the same way i t was eliminated for IHG^ 2r&\ ~ 'HP20*** In the case F = IR , (P**) odd, J V j = dim KP^IRG ; Z 2 ) (mod 2) == ^Ppq^(mod 2) = 1 (mod 2) and thus Theorem 4.1 i s proved. 5. Properties of some of the <^ Gj > k'a Theorem 5.1: Any self map of CQ. i s an Adams type mapping and hence 3>2 CG has the fixed point property. Proof; Since CG i s homeomorphic to CG , i t s cohomology ring is 3»*- 2,3 generated by a generator x in dimension 2 and a generator y in dimension A subject to the relations imposed by the sums of elements of the same dimension in the formal inverse of the total class, 1 + x + y. ( l + x + y)""^ = 1 - x - y + x 2 + 2xy + y 2 - x 3 - 3x2y - 3XJT2 - y 3 + x^ + Ax?7 + bx2?2* Axj3 + y 4 - x 5 - 5xAy - 10xV - lOxV - 5xyA - y 5 + x 6 +... Thus dimension 8 i s completely determined by three generators, and 2 Z. 2 2 2 y with the one relation : x* - 3x y + y = 0. We can therefore choose x y 2 and y as a basis for dimension 8 with 1. xA = 3x2y L y 2 Dimension 10 i s generated by x-5, x^y, and xy 2 subject to the two relations: - xy (equation 1. multiplied by x) 3. -x5 + Ax?y - 3xy2 = 0 Solving this system of equations gives us the single element xy as a basis with A* x^y = 2xy2 and 5xy2 5. x5 - — 2 Let f: CG > CG with f*(x) = ax and f*(y) = bx2 + cy. Using 3 , 2 3 ; / equation 1. and the fact that f* i s a ring homomorphism we have a x = f*(x 4) = f*(3x2y - y 2) = 3a2x2(bx2 + cy) - ( b 2 / + 2bcx2y + c 2y 2) = 3a2bx^ + 3a2cx2y - b2x*-- 2bc x 2y - c 2y 2 = 3a2b(3x2y - y 2) + 3a2cx2y - b 2(3x 2y - y 2) - 2bcx2y - cV = (9a2b • 3a 2c - 3b2 - 2bc)x2y • (-3a2b + b 2 - c2)f. Since a^x^ = 3a^x2y - a^y2 and ^y, y 2 form a basiB we have* 6. 3&A = 9a2b + 3a2c - 3b2 - 2bc 7. -a^ = -3a2b + b 2 - c 2 Adding three times 7. to 6. gives us; 8. 3a2c - 2bc - 3c 2 = 0 Using equations A* and 5. in a similar manner gives us» 5a5xy2 = a V = f*(x 5) = f*(5xf) - 5ax(&y£ + 2bcx2y + c 2 / ) = 5a.b2x? * lOabcj^y + 5ac2xy2 = (25ab2 + 20abc + Sacf^Jxy2. Therefore 9. a^ = 5ab2 + 4abc + ac 2. Case l i a = 0. If b = 0 as well then equation 8. becomes = 2bc - 0 which means that c = 0 as well as a and b. If b ^  0 then after dividing b into equation 6. we have 3b = -2c or b = (-2/3)c Substituting this into equation 8. gives us 3c 2 = (A/l)<? which implies that c = 0. But i f c =0 then according tb equation 6. b must also = 0, contradicting the assumption. Therefore i f a =0 then b = c = 0. Case 2t a / 0 If a / 0 then dividing equation 9. by a gives us the equation: 10. a^ = 5b2 + Ate + c 2 Now i f b f 0 then c / 0 for i f c = 0 then equation 10. becomes a^ = 5b2 to which there are no solutions consisting of non - zero integers. - 22 -Adding equation 7. to equation 10. produces 11. 6b2 + 4-bc - 3a2b = 0 and dividing this by b gives 12. 3a2 = 6b + 4c Substituting for 3a in equation 8. and then dividing equation 8. by c* 13. c = -4b Substituting for c in equation 12. gives 3a2 = -10b and therefore 9a^ = lOOb2" A 2 A 2 2 Substituting for c in equation 10. gives ar - 5b and thus 9eT = 45b = 100b This implies that b = 0 contradicting the assumption. Thus i f a / 0 then b = 0 and by equation 9. we see that c / 0 and then by equation 8. we see that 2 c = a and thus f is an Adams type mapping. Q.E.D. Theorem 5.2t Any self map of CG i s an Adam's type mapping and hence 3»4 CG. has the fixed point property. 3>4 Proof t The ideal of zeros of H^dLG^ jjTD are generated by the following relations in dimensions 10, 12, 14, 16, 18, 20, 22, 24 respectively! 1. -x5 + 4x^ y - 3X3T2 - 3x2z + 2yz = 0 (z being the third Chern class) 2. x 6 - 5x^y + 6x 2y 2 - y 3 + 4x3z - 6xyz + z 2 = 0 3. -x7 + 6x5y - lO^y 2 + 4x3^  - 5x^ z + 12x2yz - 3y2z - 3xz2 = 0 4. x 8 - 7x6y + X^J2 - lOx 2^ +. + 6x5z - 20x3yz + 12x3^ + 6x 2z 2 - 3yz2 = 0 5. -x9 + 8x7y - 21x5^ + 20x3y3 - 5xy^ - 7x6z * 30x^yz - 30x2y2z + Ay3* -lOxV + 12xyz2 - z 3 = 0 6. x 1 0 - 9x8y + 28x6y2 - 35xV* + ^x 2/^ - y 5 + 8x7z - 42x5yz + 60x3y2z -20xy3z + 15x^z2 -•30ac2yz2 + oy 2z 2 + 4xz3 = 0 7. -x 1 1 + 10x9y - 36x7y2 + 56xV - 35X3/ + 6x3^  - 9x8z + 56x6yz - 105xVz + 60x2y3z - 5/z - 21x 5z 2 + 60x3yz2 - 30xy2z2 - 10x 2z 3 + 4yz3 = 0 - 23 -8. x 1 2 - llx^-Oy • ^ ^ y 2 - S ^ y 3 + 70xV" - 21x2y5 + y 6 + 10x9z - 72x7yz + IbSxtyz - U O x ^ z + 30x/z + 28x 6z 2 - 105x^yz2 + 90x 2y 2z 2 -lCy^z 2 + 20x3z^ - 20xyz3 + z^ = 0 3 2 2 Dimension 10 has a basis of four elements: x^y, xy , x z and yz with 9. x5 = 4x3y - 3XJT2 - 3x2z + 2yz Dimension 12 has two relations: equation 2 and 10. -x^ > + 4xV - 3x 2y 2 - 3x3z + 2xyz = 0 (equation 1. multiplied by x) By row reducing the associated 2 by 7 matrix we get a basis of 5 elements: x^ 2, x^z, y^, xyz, and z 2 with 11. - 3x 2y 2 + x 3z - y 3 - 4xyz + z 2 12. x 6 = 9xV+ x 3z - 4y 3 - 14xyz 4z 2 Dimension 14 has four relations: equation 3 and 13. -x 7 + 4x5y - 3x^y2 - 3x^z + 2x^z = 0 (equation 1. multiplied by x ) 14. -x5y + Ax^y2 - 3xy3 - 3x^z + 2y2z = 0 (equation 1. multiplied by y) 15. x 7 - 5x-*y + 6x 3y 2 - xy 3 + 4x^z - 6x2yz + xz 2 = 0 (equation 2. mult, by x) 3 2 2 2 By linear algebra we have a basis of 4 elements: xy , x yz, y z, and xz with 16. x 7 = 14xy3 + 7x2yz - 28y2z - 7xz 2 (-14(13) - (14) - 23(15) - 10(3)) 17. x^r = 5xy3 + 5x2yz - 10y2z - 4xz 2 (-4(13) - (14) - 8(15.) - 4(3)) 18. x V = 2xy3 + 2x2yz - 3y2z - xz 2 ((13) + 2(15) + (3)) 19. x4s = 3x2yz - y 2z - 2xz 2 - ((H) + (15) + (3)) By.Poincare Duality we know that Dimension 16 i s spanned by 4 elements. To find 4 such elements we multiply equations 16, 17, 18, and 19 by x to get: - 24 -20. x 8 = Ux2^ + Ix^yz - 2Sxy2z - 7x 2z 2 21. x 6y = S x V + 5x3yz - lOxy^z - 4x 2z 2 22. x 4y 2 = Sx 2^ + 2x3yz - 3xy2z - x 2z 2 23. x 5z = 3x^yz - xy 2z - 2x 2z 2 Multiplying equation 1. by z and substituting for x^z with equation 23. gives: 2A. x-*yz = -2yz2 + 2xy2z + x 2z 2 Substituting x^yz with equation 24. in equations 20,21,22, and 23 and collecting terms gives: 25. x 8 = U x V 3 - Uxy 2z - Myz 2 26. x 6y = Sx 2^ + x 2z 2 - 10yz2 27. x V = 2x 2y 3 + xj^z + ^ z 2 - 4yz 2 28. x 5z = Sxj^z + x 2 z 2 - 6yz2 Multiplying equation 11. by y and substituting gives: 29. y 4 = x 2 ^ - Jsq^z * 3yz2 Since CG~ . i s an orientable 24 - dimensional manifold, we know that -?»4 can be spanned by a single element. To find such an element 3,4 we solve the following system of 22 equations (equation 8. plus equations 3© to 50 inclusive) in 19 variables. Equation 30. is equation 24. multiplied by xz, 31 i s 25. multiplied by xz, 32. is 26. by xz, 33. i s 27. by xz, 34. i s 28. by xz, 35. i s 29. by xz, 36. i s 11. by z 2, 37. is 12 by z 2, 38. is 16. by yz, 39. i s 17. by yz, 4O. i s 18. by yz, 4I. is 19. by yz, 42. is 20. by x4-, 43. i s 21. by x 4, 44. i s 22. by x 4, 45. is 4. by x 4, 46. i s 5. by xy 47. is 5. by x 3, 48. i s 6. by y, 49. i s 6. by x 2, and 50. i s 7. by x. We find that z 4 generates dimension 24 with xyz 3 = z 4, x 3 z 3 = z 4, jr^z 2 = z 4, x ^ z 2 = 2z^ , x^yz2 = 3zA, x 6z 2 = 5z4, xy^z = 3^, x ^ z = 6z^ , x 5 ^ = l l z ^ , x 7yz = 21zA, x 9z = 42Z4, y 6 = 5Z4, x 2 ^ = llzA, x^/ = 23 z^, x6y3 = A7z^, ' x 8 ^ = 98zA, x*°y = 2IO2A, x 1 2 = 462z^ . which of course can be checked by substituting these values into equations 8. and 30 to 50, Let ht CG — >- CG where a,b, c,e,f, g are integers such that h*(x) = ax h*(y) = bx2 + cy h*(z) = ex3 + fxy + gz Using equation 1. and the fact that h* is a ring homomorphism we have: hHx5) = hM^y - 3xy2 - 3x2z + 2yz) = Aa^Cbx2 + cy) - 3ax(b2xA + 2bcx2y + c 2^) - 3a2x2(ex3 + fxy + gz) + 2(bx2 + cy)(ex3 + fxy + gz) = Aa3bx5 + 4a3cx3y - 3ab2x5 - 6abcx3y - 3ac2xy2 - 3a2ex5 - 3a2fx3y - 3a2gx2z + 2bex5 + 2bfx3y + 2bgx2z + 2cex3y + 2cfxy2 + 2cgyz = (l6a3b - 12ab2 - 12a2e + 8be + Aa3c - 6abc - 3a2f + 2bf + 200)3^ + (-12a3b + 9ab2 + 9a2e - 6be - 3ac2 + 2cf)xy2 + (-12a3b + gab2 + 9a2e - 6be - 3a2g + 2bg)x2z + (8a3b - 6ab2 -6a2 e + Abe + 2cg)yz Since h*(x5) also equals a 5x 5 = 4a5x3y - 3a5xy2 - 3a5x2z + 2a5yz and x3y, xy 2, x 2z and yz form a basis, we have the following equations: 51. 4a5 = l6a3b - 12ab2 - 12a2e + 8be + Aa3c - 6abc -3a2 f + 2bf + 2ce 52. -3a5 = -12a3b + 9ab2 + 9a2e - 6be - 3ac2 + 2cf 53. -3a5 = -12a3b + 9ab2 + 9a2e - 6be -3a2g + 2bg 54. 2a5 = 8a3b - 6ab2 - 6a2e + Abe + 2cg Subtracting 53. from 52. gives: 55. -3ac2 + 2cf + 3a2g - 2bg = 0 - 26 -Adding three times 54. to twice 53. we have: 56. 6cg - 6a g + 4bg = 0 Adding four times 52. to three times 51. we have: 57. -12ac2 + 8cf + 12a3c - 18abc - 9a 2f + 6bf + 6ce = 0 Using equations 11. and 12. and the given basis of dimension 1Z we have: h*(x6) = h*(9x2y2 + y?z - Aj3 - Hxyz + A*2) giving us the equations: 58. 9a6 = 81a2b2 + 9a3e - 36b3 - 126abe + 36e2 + 54a2be + 3a 3f - 36b2c -42abf -42ace + 24ef + 9a2 c 2 - l^bc 2 - 14acf + Af2 59. a 6 = 9a2b2 + a 3e - 4b3 - 14abe + 4e2 • 18a2bc + a 3f - 12b2c - 14abf -14ace + 8ef + a 3g - 14abg + 8eg 60. -4a6 = -36a2b2 - 4a3e + 16b3 + 56abe -I6e2 - 18a2bc - a 3f + 12\£c + 14abf + 14ace - 8ef - 4c3 61. -14a6 = -126a2b2 - 14a3e + 56b3 + 196abe - 56e2 - 72a2be - 4a 3f + 48b2c + 56abf + 56ace - 32ef - 14acg + 8fg 62. 4a 6 = 36a2b2 + 4a3e - 16b3 - 56abe + l6e 2 • 18a2bc + a 3f - 121^0 - Uabf - 14ace + 8ef + 4g2 Subtracting three times equation 59. from 58. gives: 63. 6a6 = 54 a 2b 2 + 6a3e - 24b3 - 84abe + 24e2 + 9a 2c 2 - 12bc2 - Uacf + 4^ - 3a3g + 42abg - 24eg Adding equation 59. to 60. gives: 64. -27a2b2 - 3a3e + 12b3 + 42abe - 12e2 + a 3g - 14abg + 8eg - 4c3 = -3a6 Subtracting four times equation 60. from 61. yields: 65. 18a2b2 + 2a3e - 8b3 - 28abe + 8e2 + 16c3 - 14acg + 8fg = 2a 6 Adding equation 60. to equation 62. provides: - 27 -66. g 2 = c 3 Since x^y = 21C-5S4 and x 1 2 = 4622A, l l x ^ y = 5X12. h ^ l l x ^ y ) = l l a ^ x ^ b x 2 + cy) = l l a ^ b x 1 2 + 11a 1 0 cx^y = (5082a10b + 2310a10 c ) ^ = h*{5*12) = 5a 1 2x 1 2 = 2310a 1 2 A Therefore: 67. 5a 1 2 = l l a 1 0 b + 5a 1 0c Since x V = 98zA and x 1 2 = 462zS 33x8y2 = 7X 1 2. h*(33x8y2) = 33a8x8(b2x^ + 2bcx2y + c V ) = 33a 8b 2x L 2 + 66a8bcx10y + 33a8 c ^ y 2 = (l5,246a8b2 + 13,860a8bc + 3234a8c2)z^ = h*(7x12)" = 7a 1 2x 1 2 = 3234a 1 2 A So: 68. 7a 1 2 = 33a8b2 + 30a8bc + 7a 8c 2 Since x 9z = 42z^  and x 1 2 = 462z^ , l l x 9 z = x 1 2. h*(llx 9z) = lla 9x 9(ex 3 + fxy + gz) = 11a 9ex 1 2 + 11a 9fx^y + lla 9gx 9z - (5082a9e + 2310a9f + 462a9g)zA = h H x 1 2 ) = a^x 1 2 = 4 6 2 a 1 2 A Therefore: 69. a 1 2 = 11a9e + 5a9f + a 9g Since y6= 5^ and x 2y 5 = l l z ^ , l l y 6 = $x 2y 5. h*(lly 6) = l K b x 2 + cy) 6 = l K ^ x 1 2 + 6b 5cx 1 0y + I S D V X V + 20b 3c 3x 6y 3 + 15^</^hU + 6bc5x2y5+ c 6y 6) = h*(5x2y5) = 5a 2x 2(b 5x L O + 5tAcx8y + 10b3 c ^ y 2 + lOt rVxAy 3 + 5bcAxV" + c^y5) Using the basis in dimension 24 gives us:: 70. ll(462b 6 + 1260b5c + 1470tAc2 + 940b3 c 3 + 345b2c^ + 66bc5 + 5c6) = 5(2310a2b5 + 5250a2lAc + 4900a2b3c2 + 2350a2b2c3 + 575a2bc^ + 55a2c5) Case 1: a =0, g = 0. If g = 0 then according to equation 66, c = 0 - 28 -If c = 0 and a = 0 then according to equation 70, b = 0 If a = b = c = g = 0 then according to equation 64, e = 0 and then i t follows from equation 63. that f = 0 as well. Case 2: a = 0, g ^  0: I f g ^ O then according to equation 66, c ^  0. If a = 0 then equation 56. becomes 6cg + 4bg = 0 and i f we are assuming that g 0 then we have that c = (-2/3)b. Substituting a = 0 and c = (-2/3)b into equation 70. gives us (4I,342/729)b^ = 0 which means that b = 0 and c and g are both zero contradicting the assumption. Therefore i f a = 0 then b = c = e = f = g = 0. Case 3; a ^  0: If a £ 0 then dividing equations 67. and 68. by a 8 gives: 71. 5a2 = l i b + 5c 72. 7a^ = 33b2 + 30bc + 7c2 Squaring 71. and multiplying by 7 produces: 73. 175a4 = 847b2 + 175c2 + 770bc Multiplying equation 72. by 25 produces: 74. 175a^ = 825b2 + 175c2 + 750bc Subtracting 74. from 73. yields: 75. 0 = 22b2 + 20bc = 2b(llb + 10c) Therefore i f a ^  0 then either b = 0 and c = a 2 (by equation 71.) or b = (-lO/ll)c and c = -a 2. Since by equation 66. g 2 = c 3, i f c = -a 2 , then g 2 = -a^ which means that both a and g are equal to zero, contradicting the assumption. Therefore i f a / 0 then b = 0 and c = a 2 and g = a 3. Equation 55. becomes 2a f = 0 which means that f = 0 and after dividing equation 69. by a 9 and substituting we have that e = 0 as well, hence h i s an Adams type mapping. Q.E.D. - 29 -Theorem 5.3: Any self map of CG^ ^  is an Adams type mapping and hence <CG has the fixed point property. 3,o Proof: Let x,y, and z be the f i r s t , second, and third universal Chern classes as usual. Dimension 14- i s spanned by seven elements with 1. x 7 = 6x5y - lOx^y2 + Axp - 5xAz + l ^ y z - - 3XZ2 If h i s a self map of G , such that h*(x) = ax, h*(y) = bx2 + cy, and h*(z) = ex3 + fxy + gz, then taking h* of both sides of equation 1. and comparing the coefficients of the x^y2, x4z, xy3, ^ yz, j^z, and xz 2 terms gives us the equations: 2. -10M - 10a3c2 + 12abc2 + 12a2cf - 6bcf - 3^6 - 3a^ = -10a7 3. -5M - 5a^g + 12a2bg - 3b2g - 6aeg = -5a7 4. W + Aac3 -3<?£ = 4a 7 5. 12M + 12a2cg - 6bcg - 6afg = 12a7 6. -3M - 3c2g = -3a7 7. -3M - 3ag2 = -3a7 where M = 6a5b - 10a3b2 +. Aab3 - 5a^e + 12a2be - 3b2e - 3ae2 From these equations we get: 8. c 2g = ag 2 ( (6) - (7) ) 9. 12a2cg - 6bcg - 6afg = 12c2g ( (5) + 4(6) ) 10. 15a^g - 36a2bg + 9b2g + 18aeg = 15c2 g (5(6) - 3(3) ) 11. - ^ a c 3 + 9 ^ = -12c2g ( 3(4) + 4(6) ) 12. 30a3c2 - 36abc2 - 36a2cf + 18bcf + 9c 2e + gaf 2 = 30c2g ( -3(2) + 10(6) ) Dimension 16 i s generated by 10 elements and 2 relations: 13. x 8 - 7x6y + lSx^y 2 - lOx 2 ^ + y^ + 6x5z - 20x3yz + 12x3^ + 6x 2z 2 - 3yz2 = 6 as well as equation 1. multiplied by x. - 30 -By row reduction, we find that dimension 16 i s spanned by 8 elements with: 14. x 6y = Siry2 - 6x 2y 3 + y 4 + x 5z - 8x 3yz + 9xy2z + l^z2 - 3yz2 15. x 8 = 202TJ2 - 32x 2y 3 + 6y* • x 5z - 36x 3yz + 51x7^ + l S x 2 ^ - Wyz2 Taking h* of equation 14. and comparing the coefficients of y 4, yz 2, and x^z of both sides gives us the equations: 16. 6M + N + c 4 = 6a6b + a 6c 17. -IBM - 3N - 3cg2 = - l S a S - 3a6c 18. M + N + a 5g - 8a3bg + 9ab2g + 6a2eg - 6beg = a 6b + a 6c where M = 5a 4b 2 - 6a 2b 3 + b 4 + a 5e - 8a3be + 9ab2e + 3a2 e 2 - 3be2 and N = 10a4bc - 18a2 b^c + 4 ^ 0 • a 5f - 8a3bf - 8a3ce + 9ab2t * 18abce + 6a2 ef - 6bef - 3ce • From these equations we get: 19. c 4 = eg2 ( 3(16) + (17) ) Dimension 18 i s generated by 12 elements and 4 relations: 20. -x9 + 8x7y - 21x5y2 + 20x3y3 - 5XV 4 - 7x6z • 30x£yz - 30x2y2z + 4y3z - lOx^z2 + 12xyz2 - z 3 = 0. 21. equation 13. multiplied by x 22. equation 1. multiplied by x 23. equation 1. multiplied by y Solving the associated matrix, we find that dimension 18 i s spanned by 8 elements with: 24. x 9 = 48x3y3 - 54X3T4 + 9x^2 - 135x2y2z + 81^% - 9x 3z 2 • 108xyz2 -21z3 ( -47(21) - 21(20) - 27(22) - (23) ) 25. x 7y = U x 3 ^ - 14xy4 + 7x4yz - 42x 2y 2z + 21y3z - 6x 3z 2 + 33xyz2 - 6z 3 ( -6(20) - 12(21) - 6(22) - (23) ) 26. x^y2 = 4x 3y 3 - 3XV 4 + 2xVz - ^ j^z + 4y3z - x 3z 2 + 6xyz2 - z 3 - 31 -( (20) + 2(21) + (22) ) 27. x 6z = 5x^yz - oxVz + y*z - 4x 3z 2 + 6xyz2 - z 3 ( (20) + (21) + (23) ) Taking h* of equation 27. and comparing the coefficients of z 3, x 3y 3, and xy^ we have: 28. -21M - 6 N - P - Q - g 3 = -21a6e - 6a 6f - a 6g 29. 48M + U N + 4P - 6a 2c 2f + 3bc2f + c^e + oacf2 - f 3 = 48a6e + H a 6 f 30. -54M - U N - 3P + c*t = -54a6e - U a 6 f where M = 5a^be - 6a2b2e + b^e - 4a3e2 .+ 6abe2 - e 3 and N = 5a^bf + 5a^ce - 6a 2b 2f - 12a2bce + t^f + 3 ^ 0 6 - 8a 3ef + 12abef + 6ace2 - 3e 2f and P = 5a^cf - 12a2bcf - 6a 2c 2e + 3b 2cf • 3bc2e - ^ f 2 + oabf2 + 12acef - 3ef 2 and Q = 5a^bg - 6a2b2g + b 3g - 8a3eg + 12abeg - 3e2g Case l a : g ^  0, c = 0 If g 0 and c = 0 then according to equation 8, a = 0 and consequently by equation 10. we have that 'b = 0 as well. However, i f a = b= c= 0 and g ^ 0 then equations 28., 29., and 30. become; 31. 21e3 + 18e2f + 3ef 2 + 3e2g - g 3 = 0 32. -48e3 - 42e2f - Uef2 - f 3 = 0 33. 54B3 + 42e2f + 9ef2 =0 Equation 31. implies that e ^ 0 and therefore equation 32. implies that f ^ 0. Adding 9 times equation 32. to 8 times equation 33. and then dividing this sum by f gives: 34. -42e2 - 36ef - 91*2 = 0 Dividing equation 33. by e and then adding to equation 34* produces: 35. 12e2 + 6ef = 0 or f = -2e - 32 -But i f f = -2e then according to equation 33. 6e3 = 0 which implies that e = 0. Therefore i f g ^  0 we only have to consider the case: Case lb: g ^  0, c ^  0 Equation 19 when divided by c becomes: 36. c 3 = g 2 Equations 36. and 8. then imply that c = a 2 and g = a 3. Equation 11. then implies that f = 0. Equation 9. then implies that b = 0 and then equation 10 implies that e = 0. Thus i f g / 0 then h i s an Adams type mapping. Case 2a: g = 0, a = 0 If g = 0 then according to equation 18. c = 0 and i f a = 0 as well then equations 16. and 18. become: 37. 6b4 - lBbe2 - 6bef = 0 38. b 4 - 3be2 - 6bef =0 Subtracting 6 times equation 38. from equation 37. yields 39. 30bef =0 Subtracting equation 38. from equation 37. makes 40. 5b( b 3 - 3e2 ) = 0 so that either b = 0 or 0 b 3 = 3e2 Now i f a = b = c = g = 0 then equation 28. becomes: 41. 21s3 + 18e2f + 3e^ = 0 and we also have in this case equations 32. and 33. Equations 4l. and 32. imply that e = 0 i f f f = 0, however, as we have shown before with equations 34. and 35. e and f must equal zeros in this case. On the other hand, i f 0 / b 3 = 3B2 then by equation 39. we have that f = 0. But i f a = c = f = g = - 33 -then equation 28* becomes: 42. -21b3e + 21e3 = 0 , \" which contradicts 0 ^  b 3 = 3e2. By Proposition 3.4, the universal Chern classes x, y, and z are subject to the following equations: 43. x 1 0 - 9*^ + 28xV - 35x4y3 + 15x2/ - y 5 + 8x7z - 42x5yz + bOxtyz - 20x^2 + lSx^z 2 - 30x2yz2 + d^z2 + 4xz3 = 0 44. -x 1 1 + 10x9y - 36x7y2 + 56x5y3 - 35x3y4- + 6xy5 - 9x*z + 56x6yz - 105x4y2z + 60x2v3z - 5 ^ - 21ac?22> 60x3yz2 - 30xy2z2 - lOx^ 3 + 4yz3 = o 45. x 1 2 - l l x 1 0 y + / ^ y 2 - 84x6v3 + 70x^y^ - 21x^5 + y 6 + 10x9z - 72x7yz + I68x5y2z - l40x 3y3 z + 30xjr4z • 28x6z2 - lOSx^yz2 + qOx2?2^ - 10y3z2 + 20x3z3 - 20xyz3 + z 4 = 0 We now continue with the final case: Case 2b: g = 0, a ^  0 If c = g = 0 and a ^  0 then we have by equation 12. that f = 0. We further subdivide this case into three subcases: i) b = 0, i i ) e = 0, and i i i ) b ^  0, e ^  0. Case 2bi: I f b = c = f = g = 0 then h* of equations 20', 43, and 44 • become: 46. -a 9 - 7a6e - 10a3e2 - e 3 = 0 47. a 1 0 + 8a7e + 15a4e2 + A&e3 = 0 48. - a 1 1 - 9a8e - 21a5e2 - 10a2e3 = 0 Equation 46. implies that e ^  0. Dividing the sum of 4a times equation 46 - 34 -plus equation 47. by a gives us the quadratic equation: 49. 25e2 + 20a3e + 3a6 = 0 Equation 49. has two solutions: e = (-3/5)a3 or e = (-l/5)a3 both of which when substituted into equation 48. imply that a = 0. Case 2bii; l f c = e = f = g= 0 then h* of equations 20, 43, and 44. become: 50. -a 9 + 8a7b - 21a5b2 + 20a3b3 - 5ab* = 0 51. a 1 0 - 9a8b + 28a6b2 - 35a^b3 • 15a2tA - b 5 = 0 52. - a 1 1 + 10a9b - 36a7b2 + 56a5b3 - 35a3tA + 6ab5 =0 Equation 50. implies that b ^  0. The sum of a 2 times equation 50. plus 2a p times equation 51. plus equation 52. when divided by ab i s : 53. -a 6 + 6a^b - 10a2b2 + 4b3 = 0 Letting c = e = f = g=0in equation 16. and then dividing by 6b results in: 54. -a 6 + 5a^ b - 6a2b2 + b 3 = 0 Subtracting 4a times equation 54. from equation 53* gives us the quadratic 55. 3a 6 - 14a^b + 14a2b2 = 0 which i s readily seen to have no non-zero integer solutions. Case 2 b i i i : c = f = g = 0, a ^ 0 , bj*0, e ^ 0 . According to Proposition 3.4 we have 56. 1 = ( 1 + x + y + z ) ( + "©2 +... ) where the are the polynomials of degree i in the formal inversion of 1 + x + y + z and a l l but a finite number of the £5^ are equal to zero. If we have c = f = g = 0 then h* of equation 56. gives us the equation: 57. 1 = ( 1 + a + b + e ) ( O x + 0 2 +... ) - 35 -where the are now the polynomials of degree i in the formal inversion o f l + a + b+ e and since a l l but a finite number of these are equal to zero, we have that any integral solution of this equation must satisfy: 58. a + b = -e Also i f c = f = g = 0 then equations 20, 43, and 45. become: 59. -a 9 + 8a7b - 21a5b2 + 20a3b3 - 5atA - 7a6e + 30a^be - 30a2b2e + 4b3e - l O a V + 12abe2 - e 3 = 0 60. a 1 0 - 9a8b + 28a6b2 - 35a^b3 + 15a2tr> - b 5 + 8a7e - 42a5be + oOa^e - 20ab3e + ^ a^e 2 - 30a2be2 + 6b 2e 2 + 4ae^ = 0 61. a 1 2 - l l a 1 0 b + 45a8b2 - 84a6b3 + 70a^tA - 21a2b5 + b 6 • 10a9e - 72a7be + I68a5b2e - 140a3b3e + 30atAe + 28a6e2 - lOSa^be2 + 90&2\?£ - lOlPe2 + 20a3b3 - 20abe3 + «A = 0 Equations 58. and 59. imply that a, b, and e must a l l be even. If 2^\a, 2k|b, and 2 ^ 6 for some k^-1 ( m\n means m divides n ), then equation 59. implies that 24k|e3 and therefore 2 k + 1| e. If 2 k + 1/ b then according to equation 60. we have 25k|6b2e^because ^ l e 3 , 25k|4ae3and clearly 25k is a factor of the other terms) and thus we also have 2d|e where d = 3k/2 i f k is even and d = (3k - l)/2 i f k i s odd. If k is even then according to equation 60. 2 5 k + 1| -b5 ( since 2 5 k + 1 now divides a l l the other terms of equation 60. in particular 25 k +l|6b 2e 2) and therefore 2 k + 1|b which contradicts our assumption. If k is odd then according to equation 61. 2 6 kje^ ( in particular 26k|-10b3e2 since 23k-l| e2 ) a n d t h u g 2 d | e w h e r e d = (3k + l)/2 ( since k is odd ) and therefore again by equation 60. we have that 2 k + 1|b, a contradiction. Therefore, i f 2k|a and 2 k + 1| e then 2 k + 1|b. However, by equation 58. i f 2 k + 1|b and 2 k + 1| e then 2 k + 1|a and we have thus shown that 2 kU, 2k|b, and kl 2 |e for a l l positive integers k which is impossible. 2 Therefore i f g = 0 then a = b = c = e = f = 0 and i f g ^  0 then c = a , g = a and b = e = f = 0. Therefore any self map of CG^ ^ i s an Adams type mapping and hence CG^ ^ has the fixed point property. Q.E.D. Theorem 5.4: Any self map of CG ft i s an Adams type mapping and hence CG has the fixed point property. 3,8 Proof: Let x, y, and z be the fir s t , second and third Chem classes. As for H*(CGo .j7Z) we also have for H*( CG„ i~H) the following: 3,o 3,8 1. x 9 = 8x7y - 21x5y2 + 20x3^ - 5XT4 - 7x6z + 30x4yz - 30x2y2z + 4y3z - l O ^ z 2 + 12xyz2 - z 3 2. x 1 0 - 9xBy + 28x6y2 - 35x4y3 + IS*2?^ - y 5 + 8x7z - 42x5yz + oOx^z - 20xy3z + 15xAz2 - 30x2yz2 + 6y2z2 + 4xz3 = 0 3. -x 1 1 + 10x9y - 36x7y2 + SoxSy3 - 35x3y4 + 6xy5 - 9x8z + 56x6yz - lOSx^y2 + 60x2y3z - Sy^z - 21x5^ + 60x3yz2 - 30xy2z2 - 10x 2z 3 + 4yz3 = 0 Dimension 18 is spanned by 11 elements with one dependent element, x , as shown by equation 1. Dimension 20 has two relations: equation 2. and equationl. multiplied by x. It is therefore spanned by 12 elements with: 4. x 1 0 = 35x6y2 - lOOxV3 + 75X2/ - 8y5 + x 7z - 66x5yz + 210x3y2z - 124xy3 + 30x4z2 - 132x2yz2 + 483^  z 2 + 23xz3 5. x8? = ly^y2 - 15x4y3 + l O ^ y 4 - y 5 + x 7z - 12x5yz + 30x3y2z - l6xy 3z + 5yrz2 - 18x2yz2 + oj^z 2 + 3xz3 Let h: CGj g — C G ^ g be a map with h*(x) = ax, h*(y) = bx2 + cy, and -37 -h*(z) = ex3 + fxy + gz. Taking h* of equation 1, and comparing the coefficients of the terms x 3y 3, xy^, x^yz, y^z, x 3z 2, xyz 2, and z 3, we get the following equations: 6. 20M + 20a3 c 3 - 20abc3 - 30a 2c^f + l^bc^f + ^ e + ^ a c f 2 - f 3 = 20a9 7. -5M - 5ac^ + Ac?t = -5a9 8. 30M + 30aAcg - 60a2bcg + 12b2cg - 20a3fg + 24abfg + 24aceg - 6efg = 30a 9 9. 4M + Ac?g = 4a 9 10. -10M - 10a3g2 + liabg 2 - 3eg2 = -10a9 11. 12M + l^acg 2 - 3fg 2 = 12a9 12. -M - g 3 = -a 9 where M = 8a7b - 21&>}? + 20a3b3 - 5atA - 7a6e + 30a^be - 30a2 t^e + 4b3 e - 10a3e2 + 12abe2 - e 3. Taking h* of equation 5» and comparing the coefficients of the y 5, xy 3z, j ^ z 2 , and xz 3 terms gives us the following equations: 13. -8N - P - c 5 = -8a8b - a 8c 14. -124N - 16P - loac^g + 12c2fg = -124a8b - l6a 8c 15. 43N + 6P + 6crV = 48a8b + 6a8c 16. 23N + 3P + 3ag3 = 23a8b + 3a8c where N = 7a 6b 2 - lSa^b 3 + 10a2lA - b 5 + a 7e - 12a5be + 30a3b2e - l6ab3e + Sa^e2 - 18a2be2 + 6b 2e 2 + 3B6 3 and P = 14a6bc - 45a4D2c + 40a2b*c - 5b^c + a 7f - 12a5bf - 12a5ce + 30a 3b 2f + 60a3bce ,- l6ab3f - 48ab2ce + lOa^ef - 36a2bef - 18a2ce2 + 12b2ef + 12bce 2 + 9ae 2f. From these equations we get the following: 17. c?g = g 3 ( 4(12) + (9) ) 18. -5ac^ + ^ <j3f = -5g3 ( (7) - 5(12) ) 19. 12acg2 - 3fg2 = 12g3 ( (11) + 12(12) ) 20. -10a3g2 + 12abg2 - 3eg2 = -10g3 ( (10) - 10(12) ) 21. 30aAcg - 60a2bcg • 12b2cg - 20a3fg + 24abfg + 24aceg - 6efg = 30g3 22. 20a 3c 3 - 20abc3 - 30a2 c^f + l^bc^f + U<?e + ^ a c f 2 - f 3 = 20g3 23. c 5 = c 2g 2 24. 16c5 - loacPg + ^ c ^ f g - l ^ g 2 + 12ag3 = 0 (The linear combinations for equations 21. to 24. being ( (8) + 30(12) ), ( (6) + 20(12) ), ( 6(13) • (15)), and ( -16(13) + (H) - 2(15) + 4(16) ))-Case 1: g ^  0 If g / 0 then equation 17. tells us that c 3 = g 2 and thus c ^  0. Dividing equation 18. by c 3 gives 4f = 5(ac - g) and dividing equation 19. by g 2 gives 3f = 12(ac - g) and therefore 15(ac - g) = 48(ac - g) which implies that ac = g and f = 0. From this we have by equation 24. that c 5 = ag3. Thus - a 2g^ = a 2c 9and therefore c = a 2. Also g^° = (p-5 = a 3g 9 and therefore g = a 3 and a j4 0. We then have from equation 20. that e = 4ab and thus b = 0 i f f e = 0. Equation 21. reduces to -60a2b + 12b2 + 96a2b = 0. If b ^  0 then b = -3a2 and e = -12a3 but these values along with equation 22. imply that 12a9 = 0 and thus i f g £ 0 then c = a 2, g = a 3, and b = e = f = 0. Case 2t g = 0 If g = 0 then according to equation 23. c - 0 and then by equation 22. f = 0. We have 4 subcases to consider: Case 2a: b = c = f = g = 0 - 39 -Case 2b; c = e = f = g = 0 Case 2c: a^O, b^O, e^O Case 2d: a = 0 Cases 2a, 2b, and 2c have a l l ready been covered in the proof of the previous theorem because the only equations used there were equations 1. 2. and 3. We need only consider the remaining case: Case 2d: If a = c = f = g = 0 then equations 1. and 2. become: 26. 4b3e - e 3 = 0 27. -b 5 + 6b 2e 2 =0 Equations 26. and 27. imply that b = 0 i f f e = 0. The sum of 6b2 times equation 26. plus e times equation 27. i s 28. 23b5e = 0 which implies that b = e = 0. Therefore we have shown that i f g = 0 then a = b = c = f = e = 0 and thus any self map of C.G^  g i s an Adams type mapping. Q.E.D. Combining Theorems 5.1, 5.2, 5*3, 5.4, and 4.1 gives us the following: Theorem 5.5: For n even, (DG has the fixed point property. 3,n The Lusternik - Schnirelmann category of a topological space X is the smallest integer k 1 such that X may be covered by k open subsets which are contractible in X. The category of the sphere, S2, is then 2 and the category of the torus, S1"* S1, i s 3. It i s well known that the Lusternik - Schnirelmann category i s greater or equal to the cuplength of the space plus one. The cuplength of a space X over a ring R i s the largest number - AO -m such that there exist elements x^,..., x^ €. H*(X;R) with the cup product XjX 2«..x m / 0. It i s also well known that the category of a manifold i s less than or equal to i t 1 s dimension plus one. Thus we have from these upper and lower bounds, for instance, that the category of the real projective space, tRP^ = IRG^ ^  i s k • 1. In the complex case, we observe that since there exists a cell decomposition of C G , which contains only cells of even dimension, ( See (l l ) ), cat C.G- , = k + 1 as well. Heinz X,K and Singhof ( See (12) ) have shown that the cuplength ( with TL coefficients) of C G 2 i s 2k and thus cat C G 2 K = 2k + 1 for any k 1. In the proof of Theorem 5*2 we found that the cuplength of <LG~ , i s 12 ( since x = i.*A 4.62) and so we have: Theorem 5.6: The Lusternik - Schnirelmann category of C G is 13» - Al -BIBLIOGRAPHY (1) M. Greenberg, "Lectures on Algebraic Topology", W. A. Benjamin, Inc. 1971. (2) R. F. Brown, "The Lefschetz Fixed Point Theorem", Scott, Foresman and Company, 1971. (3) R. Bing, "The elusive fixed point property", American Mathematical Monthly 76 (1969) 119 - 131. (4) N. E. Steenrod and D. B. A. Epstein, "Cohomology Operations", Annals of Mathematics Studies Number 50, Princeton University Press, 1962. (5) N. E. Steenrod, "The Topology of Fibre Bundles", Princeton University Press, 1957. (6) P. J. Kahn, "Mixing homotopy types of manifolds", (to appear). (7) S. Y. Husseini, "The products of manifolds with the fixed point property need not have the fixed point property", BAMS 81 (1975). (8) R. Stong, "Notes on Cobordism Theory", Princeton University Press, 1968. (9) F. Bachmann, H. H. Glover and L. S. O'Neill, "On the fixed point property for Grassman manifolds" (to appear). (10) J. A. Wolf, "Spaces of Constant Curvature", Second Edition, Publish or Perish, 1972. (11) J. W. Milnor and J. D. Stasheff, "Characteristic Classes", Annals of Mathematics Studies Number 76, Princeton University Press, 1974. (12) H. Heinz and W. Singhof, "On Cuplength and Lusternik - Schnirelmann Category of Grassman manifolds", (to appear). (13) A. Dold, "Lectures on Algebraic Topology", Springer - Verlag, 1972. - 42 -toffldlx - Products Ellenberg - Zilber Theorem: The functors S(x) ® S(Y) from Top X Top to the category of chain complexes are homotopy equivalent. More precisely, there are unique ( up to homotopy ) natural chain maps 1 : S(x) ® S(Y) —> S(XXY) and "V: S(XXY) ^S(X) ® S(Y) such that $0(a<2>T) = (a,X) and Y 0 ( a , t r ) = a®TT for zero simplices O :AQ- X, ~ £ : AQ Any such chain map is a homotopy equivalence; in fact, there are natural homotopies ~ id, I V ~ id. Any such chain map will be called an Eilenberg - Zilber map and will be denoted EZ. For more details and proofs see (13)» Corollary: For arbitrary Eilenberg - Zilber maps the following diagrams are homotopy commutative: SX ® SY SY® SX EZ EZ -J-S(XXY) set) S(YXX) SX ® SY 03 SZ id ® EZ SX® S(YXZ) ^ ~ EZ ® id EZ SX ® SP EZ S(XXP) jid®f^ |proj. SX® (7LyQ) -^-sx ^ S(XXY)® SZ EZ S(XXYXZ) where t(x,y) = (y,x), t(u®v) = (-1) * u n v * v«>u ( I \ denotes gradation), P i s a point and i s augmentation Proof: In each case the two ways of going from corner to corner diagonally induce the identity in dimension 0 (or on HQ), hence are (naturally) homotopic. Q.E.D. Corollary: For arbitrary EZ map6 jE , Y and arbitrary pairs of spaces (X,A) and (Y,B) we have commutative diagrams with exact rows: SA <8> SY + SX ® SB ^SI ® SY —v SX/SA ® SY/SB — ^ 0 Y' V I" 0 - > S{AX Y,Xx B* .S(XX l ) - ^ ^ g I x B | — 0 The vertical maps are induced by I , Y and S{AXY,XXB} = im S(AXY) 8 S(XXB) >S(XXY). Moreover, there are natural homotopies 1 '"Y1 ~ id, Y ' i d , Y ' ' <P'~id. Proof: Natural!ty of I applied to j : A-^ =-^ X and id shows 1 (SA® SY) C S(AXY) similarly I ( S X ® S B ) C S(AXY) and analagously f o r Y . This gives the desired maps. Since the homotopy id i s natural i t maps S{AXY,XXB} into itself and hence induces I ' Y ~ i d , ^ ' Y ' ^ i d . The other homotopies are similar. Q.E.D. The Exterior Homology Product Consider the composite chain map (SX®LA © r S(XXY) & (L©RM)_4,S(XXY) toU®* \SA y \ SB ) S\AXY,XXB) K S(AX YUXXB) K where (X,A),(Y,B) are arbirary pairs of spaced and L,M are R - modules* Passage to homology and composition with the unique map o C . H ( g ® L ) < X > R H ( § ® M ) : — H ( | ® L ® R § ® M ) such that ©c(rx]<8>ly]) = [x<8>y] for x C z ( g ) and %y£ z(||) gives jft(EZ),c<: Ht(X,AjL) <X>R H^Y^jM) H i + k(XX Y,AX YOXXBjL ® R M) This map i s called the exterior homology product and we write ^ X r ^ = J*(EZ)#OCi^® ). In terms of representative cycles this reads as Ca] X Lb] = [EZ(a®Rb)] where a £ SX ® L, <^  a £ SA <8> L, b £ SY ® M, & b £ SB <S> M. Properties : Naturalitvt If f: (X,A) ^(X',A') and g: (Y,B) >.(Y',B') are maps then the naturality of EZ implies ( f X g ) t ( ^ X n^) = (f*^ )X (g*r^) Commutatlvitv: t * ( ^ X r^) = ( - l ) ' ^ ^ lyX ^ Imciatiyity.. ( ^ X f^)x^ = ^x ( r ^ X ^ ) Unit element; If Y = P i s a point, B = 0, and 1 P = 1 P£ R = H0(Y}R) then (XXY,AXYUXXB) = (X,A) and 1 PX^ = ^ x i P = y . These properties follow from the corresponding ones for EZ maps. We also have that the following diagram is commutative: (coefficients ommitted) H(X,A>® H(Y,B) — H(XXY,AX YUXXB) , \ , vdim \ \ H(AXYUXX B,AXB) ( c ) * ® id^-lj^id® d # ) | ' X © X • l [ il * , 1 2 » ) (HA <S) H(Y,B)) $ (H(X,A)<S) HB) >H(AXY,AXB) 6 H(XXB,AXB) - 45 -where i p i g a*"e inclusions. That i s C**(^X Cy) = i ^ U c ^ )x r\) + i 2 < ) ( ( - l ) ^ X When B = 0 we have i ^ = id, i 2 # = 0 and c \ ( ^ X r^) = ( c ^ )Xir^ where ^  €. H(X,A), r^ e HY. Proof: Let a €1 SX, b£l SY be representatives of ^  , ^  ; in particular ^  a 6SA, c>b€ SB. EZ(c^a®b)eS(AXYUXXB) represents )x f | ) , EZ(a®^b)€ S(XXB)C S(AXYUXXB) represents i 2«(^Xc}*r|) and c)(EZ)(a®b) = (EZ)cMa®b) = EZ(c^a®b) + ( - l / ^ EZ(a®^b) represents ; 4 ( ^ X r ^ ) . Q.E.D. The Exterior Cohomology Product: Let (X,A),(Y,B) be pairs of spaces such that (XX Y;AX Y,XXB) i s an excisive triad and let L,M be R - modules. Consider the composite chain map: Bom ( g , L) <S>r Horn ( § , M ) Horn ( § f ® § , L ® R M ) ^ ( S ^ X I I I X B ^ ' L ® R M ) j > ^ ( S I & U I X B I ' L ® R M ) where the chain map V i s defined by U (^®y))(c®d) = (-1) | c U d *p(c)®"y/(d) and j i s induced by the inclusion s(AXY,XXB}c S(AXYUXXB), which like EZ, is a homotopy equivalence since (XX YjAX Y,XX B) is an excisive triad. Passage to homology and composition with CC: H*(X,A;L) ® R Hom*(Y,B;M) —H(S*(X,AjL) <&R S*(Y,BJM)) gives (j*)" 1^)*^*;. ^ ( X J A J L ) ® R H^YJBJM) >-H^UXY,AX YUXXBjL ® M) This map is called the exterior cohomology product and i t is written: xXy = (jar^EZ^^Ufcy) or in terms of cocycles ( ^ ) X ( y ) = Properties! Naturalitvi If f; (X,A) >- (X',A«), g» (Y,B) ^(Y',B») are maps of pairs such that (X'X Y«;A«X I',X«X B») is also an excisive triad then (fx g)*(x'® y') = (f*x' )x (g*y' )• Commutativitvi t*(xxy) = ( - l ) l x M y l y X x where t: X X X — » - X X X commutes factors. Associativity! (x xy)x z = X X (yx z) Units; If Y = P i s a point, B = f6> and l p € is the cohomology class of the augmentation ry: S^ P — R , Pl—»-1, then lpX x = x = x x l p (where PX(X,A) = (X,A) = (X,A)XP). If Y is an arbitrary space again, andIT : Y >P then 1^ = Tr*(l p) € H°(Y;R) is the class of the augmentation SQY y-R and naturality gives x X l v = (idXTr)*(xXl p) = p*(x) where p: (X,A)X Y —»-(X,A)XP i s the projection. The following diagram (coefficients ommitted) i s comutative: or &*(i*) (axy) = ( b*a)xy for a e H*A, y €. H*(Y,B). In the case where B = 0, i = id and &*(axy) = ( o*a)x y. H»A <& H»(Y,B) X >- H»(A^Y,AXB)S H»(AXYUXXB,XXB) H»(X,A) ® H*(Y,B) X H*(XX Y,AX YUXXB) - Ul -Duality: If ^ £ H(X',A;R), rye H(Y,B;R), x £ H*(X,A;L), y£H*(Y,B;M) then { xxy^Kry) = ( - l ) , y l ^ <x,^>® <y,ry) . The Interior Cohomology Product (The Cup Product): This product is equivalent to the exterior cohomology product because Eilenberg - Zilber maps EZ J S(XXY)—>- SX ® SY and natural diagonals DJ SX —>• SX ® SX are formally equivalent notions and we get the definition of the interior cohomology product by replacing the EZ map which occurs in the definition of the exterior cohomology product with a diagonal map. Consider the composite chain map: * > * ( i i v M i ) ® R * > » ( f x / M 2 ) * ' * » ( § ® 12> M i ® R * a ) * • ( l i A ^ A j ^ "x ® R * — J M i ® A ) where, as before, U t ^ ® <^ 2))(a 1®a 2) = (-1) ± (f^)® (^ 2 a 2) and j is the homotopy equivalence induced by inclusion. Passage to homology and composition with OC as before gives (j*r1D»,K #0C : HL(X,A1|RL) ® Hk(X,A2;M2) Hi*k(X,A1 ® R *t>) We write the cup - product as x^\j x^ - (j*)~*D*^ #°c (x-j® x 2) and in terms of representative cocycles V ^ V ^ ' (y^) ^  (y? 2) = ^ D^ where y? G S*(X;Mi), y?^ !^ SA± = 0, yi ±o S = 0. Propert3.es> Naturalitvt If ft ( X ^ ^ ) >-(X1 ;A^,A£) is a map of excisive triads then tfHyjOyg) = (f*y x) W (f*y 2) for j± € H*( Y,B;M^ ). Uj) \x2\ Assgc^Uvity: x^Kj^KJxJ - {x^x^k^x^ Units t 1^w x = x = x \ j l x , where ± x £. H°(XjR) is the augmentation class. The following diagram i s communist: H*A^  63 H*(X,A2) i d g > i * H * ^ ® H»(A 1,A 1AA 2) — —> H*(A^ ,A^  A A^ &*®id H*(A1\JA2,A2) j * H*(X,A1) <S> ffKX.Ag) . i ^ ^ H*(X,A1UA2) That i s : V l j * ) " ^ ^ ^ ) = ("b*a)wx for a t H*^, x £ H*(X,A2) and i f A 2 = fbt $>*(a^i*x) = ( W ) u x for a £ HfA^ x £ H*X. x 1 v j x 2 = A*(x 1X.x 2), x ± e H*(X,Ai), where A : '(X^UAj) =9-(XXX,A.jX XuXxA^ is the diagonal map. xxy = (p*x)w(q*y), i f x € H*(X,A), y €_ H*( Y,B) and p: (XXY,AXY) (X,A), q: (XXY,XXB) (Y,B) are projections and (XXYjAXY,XXB) i s excisive. Therefore we have; \y1\ \x2\ M j a i i p l i c ^ y i i y ; (x^y^\yU2 X y 2) = (-1) U xwx 2) x (y±w y 2) i f X i £ H^X^), H»(Y,BA) and ( X ; ^ ^ ) , (YjB1,B2) are triads such that the products above are defined. The Cohomology SJlant Product Consider the chain map: E: Hom(D,M)® (C®D®L) S: (C<X>L) ® (Hom(D,M)®D)— i d® e^ C<SL(S>M where C,D are R - complexes, and L,M are R - modules,(JO permutes factors and e i s the evaluation map. E("y®c®d® 1) = (-1) i>Ucl 0®l®]/(d). Passage to homology and composition with o<gives E*c*; H*(D,M) ® Hn(C®D<&L) »- H (C<»L«>M) This map is called the cohomology slant product (for complexes) and is written y \ ^ = E ^ ( y ® ^ ) G H^C&LfcM) for y e ^ ^ M ) , ^ e H n(C®D®L). The cohomology slant product for spaces (X,A),(Y,B) i s obtained by taking C = S(X,AjR), D = S(Y,B;R) and replacing S(X,A;R)® S(Y,B;R) by the homotopy equivalent complex S(XXY,AXYUXXB) — s{AXY,?XXB;R^ ^ S(X,A;R)® S(Y,B;R) assuming that (Xx Y;AX Y,XX B) i s an excisive triad. In terms of representative cocycles Lz3 = (-1) ^ ( 1 z t y l ^ i® V < b i >] where EZ(z) = E a ^ b ^ & ± € S(XjL), b ± e S(YjR) for Si(Y;M), z e S(XxY;L). The representative z must be such that c^ze s{AXY,XXB;L} as well as being in S(AX Y\JXX BjL). Properties: (ommitting coefficients) Naturalitv: If f: (X,A) >(X',A«), g» (Y,B) ^(Y»,B') are maps of pairs then f*(g*y'\X> ) = 7 ! \ ( f X g ) ^ , for y'e H*(Y',B«), ^ € H(XX Y,AX YUXX B). Associativity: ( x X y M = x\(y\>0, for x<LH*(X,A), y&H*(Y,B), ^<L H((W,U)x(X,A)x(Y,B)). In particular i f W is a point and U = 0 we have (xxy,^) =<x,y\^) , for xtH»(X,A), y£H*(l,B), H(Xy Y,AXYUXXB). The following diagrams are commutative: H*(Y,B) ® H(XXY,AXYUXXB) — H(X,A) (-l) d i mid®^ H*(Y,B)<& H(AXYUXXB,Xy,B) i d S J * H*(Y,B) H(AXY,AXB) — HA That i s : \iy\\) = (-l) ^ VXJ*1^*\, i f y€_H*(Y,B), ^ £. H(X XY, Ax YOXXB). H*B <$) H(XX Y,AXYUXX.B) . ^ H*(Y,B) & H(XX Y,AX YUX X B) v - ( - l ) d i m i d ® ^ I \ H»B(g)H(AXYuXXB,AXY) i d < ^ » > H » B H(XXB, AXB) *-H(X,A) That i s : ( VbjX^ + (-1)'^bNj^c^ = 0, i f bCH*B, H(XXY, AX YUXXB). Multiplicativitv: y\ujx^ = (-l) 1 7 U C P > O J X (y\^), i f y & H*(Y,B), UJ £ H(W,U), \eH(XXY,AXYOXXB), and (W,U),(X,A),(Y,B) are pairs of spaces such that the products above are defined. Units: iyV^ = P « ^ > where 1 Y e H°(YjR), ^£H(XX Y,AXY), and p: (XXY,AXY) (X,A) i s the projection. T^ e Cap Prod,u,cv: Let (XjApA 2) be an excisive triad, and let M^ , M2 be R - modules. Consider - 51 -E ^ SX where D i s a natural diagonal and E is the same as in the slant product. SX Passing to homology ( using H( S^ A ^ A ^  ) HCXjA^U k^) ) and composing with <X we obtain ^(id®D) #eU I^U.A^Mg) ® H ( X , ^ A^!^) *n k ^ , A l * M l ® R ^2^* ^ J l i s 1115115 i s c a l l e a - t l i e c a D product. W e write x /"\ ^ = E^(id®D)#oC(x ® ^ ) , i f xeR>(X,A2jM2), HU.AjU A 2JM 1). In terms of representatives this reads C^}/^ £cl = ( - l ) ^ ^ c I " ^ I ^ c}®f ( c 2)] where Dc =E<^ X>c2 . a s s u m i n g ^ £ S » X , y?|SA2 = 0, = 0, c € SX, Properties. Naturalitvt f*((f*x')r\*\ ) = 'x* r\(f*\ )» i f f is a map of excisive triads and x'£ H»(X',A2), ^ e HU.AjUA^. Aas9ctattYa,iar,» ( X 1 V J . X 2 ) ' " ^ = xir\U2r\^) i f x t e H » ( X , A 1 + 1 ) , ^ e • H ( X , A 1 U A 2 - U A ) . Duality: x ^ ) = ( x ^ / ^ ) , i f x A £ H»(X,A ), ^ £ H U ^ U A ^ . In particular, ( l . x ^ ) = <x,^) , for x e Hj(X,A), ^  e Hj(X,A). Units: lr\\ = ^  , i f ^eH(X,A), and 1 £ H°(X;R) i s the augmentation class. The following diagrams are commutative; H*(X,A2) ® H(X,A1UA2) ?-H(X,A^ ) ( - D d l V ® \ ^ T id® j * ^ V u H ^ A ^ H A , , ) ® H(A1UA2,A2) ^ ^ ( A j ^ ^ r V A g ) ® H(A1,AinA2) HA^  That i s : &*(xr>t%) = (-1) U \ i * x ) / " \ ( j ^ c } ^ ) i f x £ H*(X,A2), ^ £ H(X,A1UA2) H*A2 ® HU^VJ A2) <^*® i d If^XjAg) ® H U.A^Ag) H(X,A;L) -(-l) d i mid®^ i * H»A 2® H(A1UA2,A1) S H*A 2® H U ^ ^ ) : > H l A ^ ^ U ^ ) That i s : ( V a ) r \ ^ + (-1) i * ( a ^ J * 0 * ^ ) = 0 i f a e ^ , ^ £ H ( X , A 1 U A 2 ) xr\\ = *\A>V i f xe'B»(X,A2), ^ £ H(X,A 1VJA 2 ) and A : (X.A^A^ ^ ( X X X , A ^ x X U X X A 2 ) i s the diagonal map and the triads are excisive. = p*(q*yr\V» i f y £ H * ( Y , B ) , H ( X X Y , A X Y U X X B ) and p: ( X X Y , A X Y ) — > ( X , A ) , q: ( X X Y , X X B ) — - ^ ( Y , B ) are projections and (XXYjAXY , X X B ) i s excisive. MultjJDlicatiYJVY* ( x X y ) A ( V ^ ) = ( - l ) * 7 ^ (x r<\ ) X (yr\ ry) i f x £ H * ( X , A 2 ) , y £ H * ( Y , B 2 ) , ^ €. H U ^ V J A ^ , f^e Hd.B^B^ and the triads are such that the products are defined. 

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