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The Eilenberg-Moore spectral sequence Yagi, Toshiyuki 1973

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THE EILENBERG-MOORE SPECTRAL SEQUENCE by TOSHIYUKI YAGI B. Sc., University of B r i t i s h Columbia, 1970 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Mathematics We accept t h i s thesis as conforming to the required standard. THE UNIVERSITY OF BRITISH COLUMBIA November, 1973 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make It f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by h i s representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department The University of B r i t i s h Columbia Vancouver 8, Canada S u p e r v i s o r : U. S u t e r A B S T R A C T For any two d i f f e r e n t i a l modules M and N over a graded d i f f e r e n t i a l k - a l g e b r a / V ( k a commutative r i n g ) , there I s a s p e c t r a l sequence E r , c a l l e d the Eilenberg-Moore s p e c t r a l sequence, having the f o l l o w i n g p r o p e r t i e s : E r converges to Tor A(M fN) and E 2 = T o r H ^ (H(M),H(N)). T h i s a l g e b r a i c set-up g i v e s r i s e to a "geometric" s p e c t r a l sequence i n a l g e b r a i c topology. S t a r t i n g w i t h a commutative diagram o f t o p o l o g i c a l spaces X x B Y — Y I I X — B where B Is simply connected, one gets a s p e c t r a l sequence E* converging to the cohomology H^(X XgY) o f the space X XgY, and f o r which E 2 = T o r H * ( B ) (H*(X),H*(Y)). In t h i s t h e s i s we o u t l i n e a g e n e r a l i z a t i o n o f the above geometric s p e c t r a l sequence obt a i n e d , by f i r s t e x t e n d i n g the category o f t o p o l o g i c a l spaces and then, extending the coho-rt mology theory to t h i s l a r g e r category. The convergence o f the extended s p e c t r a l sequence does not depend, on any t o -p o l o g i c a l c o n d i t i o n s o f the spaces i n v o l v e d . I t f o l l o w s a l -g e b r a i c a l l y from the way the exact couple (from which the s p e c t r a l sequence Is d e r i v e d ) I s s e t up and from the Suspen-s i o n Axiom o f the extended cohomology theory. C O N T E N T S Chapter I ~ The Algebraic Construction of Eilenberg-Moore Spectral Sequence 1 1. The Categories LDk and LDA 1 2. The D e f i n i t i o n of Proper Projective Resolutions and the Functor Tor 3 3. The Kunneth Spectral Theorem 11 Chapter II - The Geometric Spectral Sequence 13 1. More About Tor 13 2. The Geometric Eilenberg-Moore Spectral Sequence 16 Chapter I I I - The Extension of the Geometric Spectral Sequence 23 1. The Category of Fibered Spaces Over B 23 2. Topological Constructions i n (Top/B)# x 24 3. The Products i n (Top/B)^ 33 4. The Forgetful Functors on (Top/B)* 34 5. The Extended Cohomology on (Top/B)& 35 6. The M u l t i p l i c a t i v e Cohomology 39 7. The Kunneth Spectral Sequence 42 8. The Convergence o f the Spectral Sequence 52 9. The Kunneth Spectral Theorem 60 B i b l i o g r a p h y 62 A C K N O W L E D G E M E N T S I would, l i k e to thank Dr. U. Suter f o r his generous patience and guidance i n writing of t h i s thesis. My thanks also goes to U.B.C. f o r t h e i r f i n a n c i a l support. (1) Chapter I The Algebraic Construction of Blleriberg-Moore Spectral We w i l l give an account of how the Ellenberg-Moore spectral sequence arises i n homologlcal algebra. Throughout this f i r s t chapter, k w i l l denote a fixed commutative ring. 1. The Cateogrles LDk and LDA. The object of the category of graded, differential k~modules, denoted LDk, are the families of k-modules, 16Z, together with a k-module homomorphisms d*:M^—M 1 i+l 1 such that d •dA=0. Such objects i n LDk w i l l be denoted by Ms^M^ieZ. The morphism of degree p between two objects M and N l n Lik i s a family of k-module homomorphism ^ f 1 sM1 ~N i + p l of degree p such that f^-d-d ' f 1 . Note that LDk i s an Abellan category. The tensor product of two objects M and N l n LDk Is defined as the object M ® N i n LDk, where (M ® N ) 1 = ] ^l = 1M m®N n. The differential i n the tensor product i s the one induced by the differential i n M and N. We have that k ® M=M ® k=M, of degree 0 such that the following diagrams commutes: Sequence ' ~ l+m+n=i The graded, differential k-algebra i s an object A i n LDk with two structure morphlsms l : k : — ~ A and ^ : A ® A A (2) q>®i A ® A® A - — - A ® A and l f c f l (f ! d> } A ® A -A The graded k-algebra i s said to be connected i f A°=k and A i = 0 for i«=0. We are now ready to describe the other category of differential A-modules, denoted LDA. Let A be a graded, differential, connected k-algebra. Then, by a l e f t differential A-module, we w i l l mean an object M l n LDk with morphism H^A® M «-M of degree 0 such that the following diagrams are commutative: k <3> M=M A® A ® M -^-^-^A® M m jy and v|>( A « M A ® M — - — — M The right differential A-module i s similarly defined. A morphism between two objects M and N i n LDA i s a morphism i n LDk which i s compatible with the structure morphisms and I mentioned above. From this point on, we w i l l omit the word differential and c a l l objects i n LDA (or i n RDA, the category of right differential A-module.) as l e f t (or right) A-modules. Let M and N be l e f t and right A-module, respectively. Then, the tensor product of M and N, denoted N <S^ M, i s defined as the cokernel of the morphism 1 ; = ^ 1.-13 f M : N <8>A® M — — N M, where f N : N ® A ->N and ^ M ! A ® M -M. (3) The t e n s o r product s i g n without s u b s c r i p t s w i l l always mean t h a t i t I s taken over k. 2. TJig P e f l n l v l o n of. Proper P r o j e c ^ e R e s o l u t i o n and the Functor l o r . L e t M be an o b j e c t l n LDk. Then, denote by Z(M) and B(M), the graded k-modules Ker d=jKer d 1 } i t Z and Im 4={lm d ^ i f e Z , r e s p e c t i v e l y , where d ^ d ^ l f c Z i s the d i f f e r e n t i a l l n M. The homology o f M i s d e f i n e d i n the u s u a l way by H(M)=Z(M)/B(M). D e f i n i t i o n 2 . 1 : A s h o r t sequence 0—*-K —** L ——M —*—0 o f o b j e c t s and morphlsms I n L D A i s s a i d to be pr o p e r exact i f the two sequences, 0—*-K — " - L —«-M —•»» 0 and 0 — « - Z(M) — — Z ( L ) — » Z(M) —<~ 0 a r e exact i n LDk. The s h o r t exact sequences 0 — Z(K) —*-K —«-B(K) — — 0 and 0 — — B ( K ) — Z(K) — ^ H(K) — 0 l n LDk shows t h a t sequences 0 -—-B(K) — » B ( L ) — - B ( M ) —*-0 and 0 —-H(K) — H(L) — H(M) — 0 are a l s o exact i n LDk. From above d e f i n i t i o n , we w i l l d e f i n e the n o t i o n o f l o n g exact sequence i n LDA. L e t . . . — — X n ~ - — • » X n — — X * 1 ^ — « • > , , . be a sequence o f o b j e c t s and morphisms i n LDA. Then, as o b j e c t s and morphlsms I n LDk, the above sequence can be f a c t o r l z e d to a sequence o f s h o r t exact sequences as f o l l o w s : (4) A /\ Each K n i s a ( l e f t ) A-module by the morphism ^ : A ® K n «~K n induced i n the diagram y V ® X n " i — A S> X n — A ® K ^ 0 , tn - l tn P r i i - 1 ^ x n ^ ^n _ where the rows are exact and ^ «, >P are the s t r u c t u r e 1n -1* 1 n morphlsms. Therefore, a long sequence i n LDAcan be f a c t o r i z e d i n t o a sequence o f short sequences. D e f i n i t i o n 2.2: A long sequence o f ob j e c t s and morphlsms, _ _ Y n - l Y n Yn+1 i n LDA i s proper exact i f i t s f a c t o r i z e d short sequences, 0 K n - ^ X n - ^ - K n " ^ — G, i s proper exact i n LDA, f o r a l l n. The proper exactness i n LDA a l l o w s us to d e f i n e the prop ernes s o f the morphlsms i n LDA. (5) D e f i n i t i o n 2.3: A morphism f l n LDA i s p r o p e r epimorphism i f t h e r e i s a morphism g i n LDA such t h a t the s h o r t sequence, 0 — « — K — £ — L - £ » - M — — 0 i s proper exact i n LD/i. D e f i n i t i o n 2.4: A A-module P i s proper p r o j e c t i v e i f f o r every morphism h:P «-N and. every p r o p e r epimorphism f:M — N i n LDA, there i s a morphism K*:P <—M i n L D A s u c h t h a t the t r i a n g l e P commutes. D e f i n i t i o n 2. 6: A proper p r o j e c t i v e r e s o l u t i o n o f A-module^M i s a l o n g sequence o f A - m o d u l e s , ...—^X"*- — ... — ^ X " 1 — X ° «— M — — 0 , which i s proper exact and each X~ n, n^O i s p r o p e r p r o j e c t i v e . The p r o p e r p r o j e c t i v e r e s o l u t i o n o f A - m o d u l e M w i l l be denoted by X — — * - 0 . P r o p o s i t i o n 2.6: Every o b j e c t i n LDA has a p r o p e r p r o j e c t i v e r e s o l u t i o n . P r o o f : Let M be a A-module. Then as o b j e c t l n LDk, t h e r e i s (6) a V In LDk such that V i s projective and. V M — 0. Consider the tensor product A ® V with the morphism dp® 1:A®A® V - A ® V, where <ftA©A—A. The morphism ® 1 makes A® V a yN-module. We w i l l show that j\J& V i s proper projective. Let h : A ® V — — N and f:K — N be morphlsms i n LDA with f a proper epimorphism. Define a morphism, p:V — A ® V by p(v)=l®v f o r every v£V. Then, since V i s projective, we have the l i f t i n g h':V ~ k of h»p:V — N such that h«p=f.h'. The composition of morphlsms, A ® V - ^ C - A® K ^ K - K gives the l i f t i n g of h :A® V — N such that the t r i a n g l e , A ® V /h commutes. f * K — — • N — 0 The composition of morphlsms, A ® V - ^ A ® M-^-M - 0 i s an epimorphism. Hence, i n the usual way, we can construct the proper projective resolution of M. H The proper projective resolution X—*-M — 0 has two d i f f e r e n t l a l s j the i n t e r n a l d i f f e r e n t i a l d j of each TV-module X" n and the external d i f f e r e n t i a l of the resolution, dp"" 1iX""" 1 - X" n. The d i f f e r e n t i a l s are displayed l n the St following diagram: ( 7 ) d I -n,p v-n+l,p-l A » -E X I -n+1, p Denote by Z I(X- n)=[Ker d ^ 1 iX"n^ - X ~ n * i f c Z , and BjCX-nj-llm d J ^ X " " ^ X - n * i + 1 } i f c Z . Then the homology of X with respect to the Internal d i f f e r e n t i a l dj i s H I(X*" n)=Z I(X* n)/B I(X- n). Note that HjtX" 1 1) Is bigraded with gradlngs on n and. 1. Proposition 2.7: If X i s a proper projective resolution of A-module M , then the sequence, ... — i - H j t t - n ) ' -H^X- 1 1* 1) . i s a projective resolution of H(M) as H(A)-mod.ule. Proof: From the proper exactness of the resolution X, the above sequence i s also proper exact. The proJectiveness of H T(X~ n) follows from the projectiveness of X"? • (8) Remark: We need to comment about the a l g e b r a i c s t r u c t u r e o f H(M) and H(A) i n P r o p o s i t i o n 2.?. I f A i s a graded, d i f f e r e n t i a l k - a l g ebra, then H(A) i s a l s o a graded, d i f f e r e n t i a l k - a l g e b r a with the two s t r u c t u r e morphlsms, ( f H ( A ) : H ( A ) ® H ( A ) — H ( A < D A ) H ( y ) ~ H ( A ) , and I H ( A ) s k = H ( k ) BTlT~ h ( A ) , where p i s the e x t e r n a l homology product, and ^ f:A®A — A and I: k —.A> I f M i s a A-module, then H(M) i s a H(A)-module by the morphism, % ( M ) ! H ( A ) ® H(M) ^ — " H(A® M) E{yf H(M) where p i s the e x t e r n a l homology product and S^A® M — M, The d i f f e r e n t i a l i n H(M) i s the one induced i n the homology o f the d i f f e r e n t i a l i n M. D e f i n i t i o n 2.8; A sequence o f o b j e c t s and morphlsms l n LDA, Y n ? l d Y n • • • A A . . . , such t h a t d n , d N ~ 1 = 0 f o r a l l n, w i l l be c a l l e d a complex o f A-modules. A complex o f A-modules w i l l be denoted by X. A morphism between two complexes o f A - m o d u l e s X and Y i s a f a m i l y o f A-module morphlsms f n : X n — Y n , such t h a t f n » d n " " 1 s s d n " ' 1 « f n~ 1%  f o r a 1 1 n» (9) D e f i n i t i o n 2,9: Le t f,g:X *-Y be morphlsms o f complexes o f A-modules X and Y. Then, f and g a r e c h a i n homotopic i f there i s a f a m i l y o f morphlsms £ s n : X ^ -^y 1 1" 1} such t h a t d«s+s.d.=g-f. We w i l l denote the c h a i n homotopy by s : f ^ g . A morphism f:X —Y o f complexes o f A-modules i s c h a i n e q u i v a l e n t i f there i s another morphism h : Y — » ~ X such t h a t h » f - l x and f»h=?ly. I f X i s a complex o f A-modules, then we denote by D(X) the graded o b j e c t , D n(X)= X I = nX p' q. D(X) i s a A-module and has the d i f f e r e n t i a l d^dj+dg, where d^X 1**^ _ x p , q + 1 and d_:XP.^ - X p + 1 ' q . E We s t a t e the f o l l o w i n g p r o p o s i t i o n and theorem without p r o o f . P r o p o s i t i o n 2.10: A - c h a i n homotopy s:f-g:X — Y o f complexes o f A-modules induces a c h a i n homotopy s*:f*sg*:D(X) —D(Y) o f A-modules. Theorem 2.11: (The Comparison Theorem) t L e t f:M •» N be a morphism i n LDA. I f X — — M —«-0 and Y — * - N —"~0 a r e p r o p e r p r o j e c t i v e r e s o l u t i o n s o f M and N, r e s p e c t i v e l y , then there i s a morphism g:X »-Y such t h a t £»g=f»6. Moreover, I f g*:X »-Y i s another morphism such t h a t t«g=f.£, then g and g* a r e c h a i n homotopic. (10) Let M be a r i g h t A-module and N a l e f t A-module. Let X — — M —0 and Y — N — 0 be t h e i r r e s p e c t i v e r e s o l u t i o n s . Then, we s e t L p * q = ( X p (S>^N)q and denote t h i s b i g r a d e d o b j e c t by L. L has the d i f f e r e n t i a l d « A l « l P . * ~ L P ^ ' q , E A * where d E:X p » S « ~ x p + 1 , q i s the e x t e r n a l d i f f e r e n t i a l o f X. D e f i n i t i o n 2.12: With the homology b e i n g taken w i t h r e s p e c t to d P ® i a A g i v e n above, d e f i n e T o r p # q f M , N ) = H p t q ( L ) . Since T o r n ( M , N ) = p ^ = n H p ^ ( L ) = H n ( D ( L ) ) = H n ( D ( X ) ® ^ N ) , we w i l l a l s o denote by Tor(M,N)=H(D(X) & AN). A A The two morphlsms, gqj^l:D(X) C8^D(Y) — M ®^D(Y) and 1 ® A * : D ( X ) O^D(Y) - D(X) <^N, induces the isomorphisms H(D(X) <a D(Y))SH(M <g> D(Y) )=H(D(X) ® A N ) . Th e r e f o r e , T o r c o u l d a l s o have been d e f i n e d u s i n g the r e s o l u t i o n o f N o r u s i n g both r e s o l u t i o n s o f M and N, The Comparison Theorem makes the d e f i n i t i o n o f Tor independent o f the c h o i c e o f the r e s o l u t i o n s . (11) Lemma 2.13: Let P be a proper protectiveA-module, Then f o r any A-module M, H(P <^M)=H(P) ® H ( A ) H ( M ) , Proof: As k-modules, the external homology product, p:H(P (S> M)=H(P) ® H(M), i s an isomorphism. (See Theorem 1 0 . 1 , Chapter V i n ( 1 ) . ) This isomorphism preserves the structure of H(A)-mod.ule on both side, i . e . , the diagram H(A) 63 (H(P) ® H(M)) -±®L-K(A) © H(P ® M) H(P) ® H(M) P - H(P ® M) commutes, where ^ and H"are the modular structure morphlsms fo r H(P) <g> H(M) and H(P ® M), respectively. Hence, p i s an isomorphism of H(A)-modules, • 3. The Kunneth Spectral Theorem. Let X — - * - M — — 0 and Y — — N — — 0 be the proper projective resolutions of A-modules M and. N. We have already mentioned l n Sec.2 that the bigraded object L p , q = ( X p <5^N)q has an (external) d i f f e r e n t i a l d.E g ^ l . There Is also an (Internal) d i f f e r e n t i a l d j : (X p <^N) q — (X p G^N)q+1," where d j i s the i n t e r n a l d i f f e r e n t i a l of X. Since dj.dE+dg'd^O, we have (dj ® ^ ) ' ( d E <^l) + ( d E (2^1)' (dj <^ 1 )=0. (12) We f i l t e r the graded o b j e c t D(L) by ( F * D ( L ) ) % ^ n L P » q . p-a T h i s f i l t r a t i o n determines a s p e c t r a l sequence ^ E R , d r ^ such t h a t , E P « q = H p + q ( F P D ( L ) / P P + 1 D ( L ) ) . (See Sec.6 , Chapter XI o f (1), where t h i s Is shown f o r k-modules.) But, ( F P D ( L ) / F P + 1 D ( L ) ) p + q S L p » q . The r e f o r e , E ^ ^ s R p ^ d ) , where the homology i s taken w i t h r e s p e c t to the d i f f e r e n t i a l d j (8^ 1 • We s i n g l y grade E ^ » Q i n the u s u a l way by ^ p ^ = n E p » q = p ^ = n H p + q ( X p ® A N ) = p ^ = n ( H ( X p ) % ( A ) H ( N ) ) Q f r o m Lemma 2.13. Hence, we get t h a t E 1=D(H(X)) ® H ( A ) H ( N ) . S i n c e H(X) i s a p r o p e r p r o j e c t i v e r e s o l u t i o n o f H(M) as H(A)-module, E 2 = H ( E 1 ) = T o r H ( A ) ( H ( M ) , H ( N ) ) . We have shown the f i r s t p a r t o f the f o l l o w i n g theorem. Theorem 3.1: Le t M and N be a r i g h t and l e f t A-modules, r e s p e c t i v e l y . Then, there i s a s p e c t r a l sequence \E f, d r ^ such t h a t 1 ) E 2 = T o r H ( A ) ( H ( M ) , H ( N ) ) , and i i ) E r converges to T o r ^ M , N ) , For the p r o o f o f the second p a r t , r e f e r to P a r t I o f ( 4 ) , (13) Chapter I I The Geometric S p e c t r a l Sequence We w i l l l o o k a t an example o f the Ellenberg-Moore s p e c t r a l Theorem 3»1 o f the l a s t c h a p t e r to modules o b t a i n e d from the o r d i n a r y cohomology o f t o p o l o g i c a l spaces. Before we begin, we need to mention a few f a c t s about Tor, 1, More About Tor. Let A and P be a graded, d i f f e r e n t i a l k -algebras w i t h a morphism f : A — — P i n LDk. Let g:M »»K and h:N *-L be morphlsms i n LDk, where M and N are A-modules, and K and L are T-modules. Then, the morphlsms g and h a r e s a i d to be f - s e m i l i n e a r I f the f o l l o w i n g diagrams ©ommute: sequence i n geometry. T h i s w i l l be done by a p p l y i n g M ® A g®f J K ®r ?~M A ® N g and f®h r ® L where ^ K ©nd H^ L a r e the r e s p e c t i v e modular s t r u c t u r e morphlsms f o r M> N, K and L. (14) The f-semilinear morphlsms g and h induces the morphism T o r f (g,h):Tor A(M,N) — Torp(K, L), Moreover, with { E R , d r^ and ( E ^ , being the respective spectral sequences associated with (M, N) and (K,L) from Theorem 3*1 of Chapter I, we have the following lemma. Lemma 1.1: The f-semilinear morphlsms g and h induces the morphism T o r f ( g , h ) r : E r - E * . Furthermore, i f g and h induces the isomorphisms, H(g) :H(M)=H(K) and H(h) :H(N)SH(L),. i n the homology, then T o r f ( g , h ) r i s an isomorphism f o r a l l r ^ 2 . Proof: Let X and Y be the projective resolutions of H(M) and H(K), respectively. Then, by the Comparison Theorem, g induces the morphism g':X —Y i n the resolutions. In the same way, i f R and S are the respective resolutions of H(N) and H(L), then h induces the morphism h':R ~-S. Consider the diagram, D(X) ® H(A) ® D(R) g #3H< f ) g > n l D ( Y ) «s, H(P) ® D(S) D(X) ® D(E) ~ D ( Y ) ® D ( S ) D(X) ®i u )D (B ) . ~-D(Y) ® | ( P ) D ( S ) (15) The top square o f the above diagram commutes by the two diagrams l n the d e f i n i t i o n o f f - s e m i l l n e a r i t y . Hence, g'<3^h* i s induced on the c o k e r n e l s . Then, g*® Ah* induces the morphism, T o r f ( g , h ) 2 : T o r H ( A ) ( H ( M ) , H ( N ) ) — T o r H ( A ) (H(K), H(L)).' I f g and h induces the isomorphisms i n the homology, then g'fii^h' i s an isomorphism. T h e r e f o r e , T o r f ( g , h ) 2 : E 2 S E | . • Let M be a l e f t A-module. Then the t e n s o r product o f k-modules, M ® M, i s a l e f t A®A-module by the morphism ( A ® A ) ® (M ® M) — M ® M, d e f i n e d by f o r X&A and x, x'eM, where A® M *-M. S i m i l a r l y , I f N i s a r i g h t A-module, then we have a r i g h t A®A-module, N ® N. Let X — — M — 0 be the pr o p e r p r o j e c t i v e r e s o l u t i o n o f theA-module M« Then the sequence ... — ( X ® X)"* _ ... — (X ® X)° — M ® M i s a complex o f A®A"" m o d u l et where (X ® X) ' "s l^ Z^^xP® X q i s a p r o J e c t i v e A®A-module f o r a l l n=sO. I f P i s a pro p e r p r o j e c t i v e r e s o l u t i o n o f M ® M as A®A"?aodule, then the i d e n t i t y morphism induces the morphism I:X ® X *-P. Hence, w i t h N b e i n g the l e f t A-module, t h e r e Is a morphism iE:TorA(M,N) ® Tor ^ M , N) — T o r ^ ^ M ® M, N ® N). (16) E x p l i c i t l y , the morphism ii i s the composition H(D(X) ® A N ) ® H(D(X) ® A N ) IP' H(D(X ® X) ® ^ A ( N ® N)) |H(I®1) H(D(P) ^ ^ ( N ® N)) where p* i s the morphism induced, by the e x t e r n a l homology product. The two s t r u c t u r e morphlsms, ^ s ( A ® A ) ® ® M ) M ® M and t':(N ® N) ® (A®A> — N ® N, d e f i n e d above, i s <?-semilinear, where : A ® A *~A« T h e r e f o r e , by Lemma 1.1, t h e r e i s a morphism T o r ^ ( t , t ' ) : T o r ^ A ( M ® M, N ® N) — Tor A(M,N). The composition o f morphlsms Tor^S-^'J ' i E makes Tor A(M, N) a k - a l g e b r a . 2 . The Geometric Bllenberg-Moore S p e c t r a l Sequence. In t h i s s e c t i o n , we w i l l assume t h a t a l l t o p o l o g i c a l spaces have the homotopy type o f a countable CW-complex and t h a t t h e i r o r d i n a r y cohomology i s f i n i t e . We w i l l a l s o assume t h a t k i s a f i e l d . F or any t o p o l o g i c a l space X, l e t C^(X) denote the normalized s i n g u l a r c h a i n complex o f X ( I . e . , the s i n g u l a r c h a i n complex o f X w i t h a l l i t s degenerate elements taken o u t ) . And denote by C*(X)=sHom(C^(X), k ) . (17) Then, the graded o b j e c t C*(X) i s a d i f f e r e n t i a l k - a l g e b r a by the two morphlsms, ^iC*{X) ® C*(X) ~ C * ( X ) and I:k ~ C * ( X ) , d e f i n e d as follows: The morphism SPis the composition C*(X) ® C*(X)—P_H^Hom (C^(X) ® C*(X), k) w - C*(X), where p I s the e x t e r n a l product and w i s the morphism induced by a map o f E l l e n b e r g - Z i l b e r Theorem and the d i a g o n a l map A:C*(X) — C*(X) x C*(X). The morphism I i s g i v e n by I a J C (X) — k , I a ( x ) = a f o r a l l x £ C n ( X ) . Furthermore, the k - a l g e b r a C*(X) i s connected s i n c e C G(X)=Hom(C 0(X), k)£k and G n(X)=0 f o r a l l n^O. We w i l l be i n a s i t u a t i o n where we have a commutative diagram o f t o p o l o g i c a l spaces X Xp, Y - Y I' f ' * — B where g i s a Se r r e f i b r a t i o n and X x g Y= [(x,y)eXxY| f (x)=g(y)}. The space B i s assumed to be simply connected. The maps f and g above g i v e a modular s t r u c t u r e to C*(X) and C*(Y) i n the f o l l o w i n g way: Define , ^ X : C * ( B ) <g> C*(X) - C * ( X ) by (18) ^ x ( a ® x ) = ( a » f )x f o r a®xtC*(B) ® C*(X), where f:C±(X) -C*(B) i s induced by f . T h i s morphism makes C*(X) a l e f t d i f f e r e n t i a l C*(B)-module. The r i g h t C*(B)-module s t r u c t u r e morphism i s a n a l o g o u s l y d e f i n e d . S i m i l a r l y , C*(Y) i s a l e f t and r i g h t C*(B)-module, Remark: We remark t h a t the modular s t r u c t u r e o f C*(X) i s pr e s e r v e d i n the homology. Hence, H*(X;k)=H(C*(X)) i s a l e f t and r i g h t H*(B;k)-module. Let 4>:C*(X) ® C*(B) ® C*(Y) -C*(X) ® C*(Y) be the morphism g i v e n by 4>(x®aay)^(a. f )®y - x®(a.g)y f o r x®a®ytC*(X) ® C*(B) ® C*(Y), I.e., Coker 4> =C*(X) ® < ^ B ) C * ( Y ) . Then, d e f i n e the morphism Cp:C*(X) ® C*(Y) ~C*(X x B Y) as f o l l o w s : We have the morphism v:C*(X x B Y) ® C*(X x B Y) ~Hom(C*(X x B Y) ® C*(X x B Y), k) g i v e n by v ( a ® b ) ( x ® y ) * ( - l ) d e g a * d e g x a ( x ) ® b ( y ) , f o r a£>b£C*(X x B Y) ® C*(X x B Y) and x®y£C^(X x B Y) ® C^(X x B Y). There i s a l s o a morphism (19) w:Hom(C,,,(X x B Y) ® C*(X x f i Y), k) ~C*(X x f i Y), which i s induced by the composition of the map of the Ellenberg-Zllber Theorem and the diagonal map A:X x B Y «~(X x B Y) x (X x f i Y). We set the composition of morphlsms by U=w. v. Let f':X x B Y -Y and g':X x f i Y —X be the induced maps of f and g, respectively, i n the commutative diagram given above. Then, f ' and g # induces the morphlsms f*:C*(X x BY) — C*(Y) and g*:C*(X xfiY) — C^(X). Define V:C*(X) <f> C*(Y) — C*(X x B Y) <® C*(X x B Y) by V(a®b)=(-l) d eS f* d e g a(a.g*)x(b.f*) f o r a®b£C*(X) ® C*(Y). The morphism i s defined as the composition of morphlsms Lemma 2 .1 : The composition of morphlsms f»<|>=0. Proof; Let x®a®y C*(X) ® C*(B) ® C*tY). Then, (p.4>(xQa®y) =w((x (a. f ) ) . g*®y• f *-x. g*®( (a. g)y)• f * ) . Considering C*(X x B Y) as a ri g h t and l e f t k-module, then (x(a.f)).g*®yf*-x.g*®((a.£)y).f* i s 0 i n C*(X xfiY) ® C*(X x f iY). Hence, <£.<j>(x®a®y)=0. • By the above lemma, there i s an induced morphism tP*:C*(X) fc<ftB)c*(Y) -C*(X x B Y). (20) Let P C*(X) — 0 be the proper p r o j e c t i v e r e s o l u t i o n o f C*(X) as C*(B)-module. Define the morphism © as the composition o f morphlsms, D(P) • ( f t B ) C * ( Y ) -£®Uc*(X) & ^ ( B ) C * ( Y ) — £ ~ C * ( X x f iY). Lemma 2.2: The morphism © Induces an Isomorphism 3*:Tor c* ( Bj(C*(X),C*(Y))SH*(X x f iY). P r o o f : L e t \ l r t d ^ be the Serre s p e c t r a l sequence f o r the f i b r a t i o n F *-X x f iY —X, i . e . , there i s a f i l t r a t i o n F^C^X x BY) o f H*(X x B Y j k ) which determines a s p e c t r a l sequence E such t h a t E£' q=H p(X;H Q(F;k)) and E converges r — r to H*(X x B Y ; k ) . S i m i l a r l y , l e t $E£, d^) be the Serre s p e c t r a l sequence f o r the f i b r a t i o n F " - Y — — B w i t h the f i l t r a t i o n FPC*(Y) o f H^Yjk). Now, s i n c e B i s simply connected and k i s a f i e l d , the Serre Theorem s t a t e t h a t E|feH*(B;k) ® H*(F;k). S i n c e the f i b r a t i o n F — — X x B Y —X i s Induced from the f i b r a t i o n F — Y «~B, we a l s o have t h a t I 2^H*(X;k) ® H*(F;k). The f i l t r a t i o n F 1 ( D ( P ) ) ^ = 2 Z P m ' n o f the graded o b j e c t m+n=j D(P) determines a s p e c t r a l sequence ^E r, d") such t h a t E 1=D(H(P)), where the homology Is taken w i t h r e s p e c t the e x t e r n a l d i f f e r e n t i a l d E o f the r e s o l u t i o n P . The d i f f e r e n t i a l d i : E 4 *-E 1 corresponds w i t h the i n t e r n a l d i f f e r e n t i a l d j (21) o f P. Note t h a t both E* and. E R a r e H*(B;k)-modules, and f u r t h e r t h a t E_ i s p r o j e c t i v e s i n c e H(P) i s p r o j e c t i v e , r fo i f PA E?' =H p q(P)=] 1 p ' q Lcq(x) i f p=( Prom the d e f i n i t i o n o f the r e s o l u t i o n s , f p^O 0 Hence, 2 \.Hq(X;k) i f p=0 Define the s p e c t r a l sequence [E^*, d*'^ by E r ' = E r ®Ar Er» wi t h the d i f f e r e n t i a l d* ' : E £ - — E £ ' g i v e n by d£'=d rfi> A rl - 1 ®A rd*. The d i f f e r e n t i a l k - a l g e b r a A r Is defined. I n d u c t i v e l y by A T = C * ( B ) and A ^ ^ A ^ - H ^ B ; ^ f o r a l l r2*2. Prom Lemma 2,13, we have the isomorphism E ; ; i = E r + i ® A r + 1 E ; + i = H ^ r ) ® H ( A r ) H < E r ) S H ( E r V ^ 3 ^ ' T h e r e f o r e , E** i s a s p e c t r a l sequence. Since E R converges to H(D(P)) and E£ converges to H (Yj k) =H(C*(Y)), the s p e c t r a l sequence E** converges to the o b j e c t H(D(P)) ® H * ( B ; k ) H ( C * ( Y ) ) = H ( D ( P ) ® Q * ( B ) C * ( Y ) ) by Lemma 2.13. Hence, E*' converges to T o r c * ( B ) (C*(X), C * ( Y ) ) . We have the Isomorphism 0 2 ! E 2 * £ E 2 S i v e n D V E 2 ' = E 2 ® H * ( B ; k ) E 2 = H * ( X » k ) ® H * ( B ; k ) H * < B ; H * < F * k > > =H*(XJk) ® H * ( B j k ) H * < B ! k > ® H*(F;k)SH*(X,k) ® H*(F?k) =H*(X;H*(F;k))=E 2 . • ( 2 2 ) We are now ready to a p p l y Theorem 3.1 o f the l a s t c h a p t e r to the f o l l o w i n g theorem, Theorem 2 . 3 ; L e t X x B Y — Y g x £ — — B be a commutative diagram o f t o p o l o g i c a l spaces, where B i s a simply connected space and g i s a Serre f i b r a t i o n . Then, there i s a s p e c t r a l sequence o f k- a l g e b r a s , ^E r, d p}, such t h a t i ) E r converges to the k - a l g e b r a H*(X x B Y ; k ) , and i i ) E ^ T o r H * ( k j (H*(X; k ) , H*( Y; k ) ) . P r o o f : We know th a t C*(X) and C*(Y) are C*(B)-modules. T h e r e f o r e , by Theorem 3.1, there i s a s p e c t r a l sequence |E r, d ^ such t h a t E 2 = T o r H # ( B . k ) ( H * ( X ; k ) , H * ( Y ; k ) ) and t h a t E r converges to T o r c # ( B ) ( C * ( X ) , C * ( Y ) ) which i s isomorphic to H*(X x f iY;k). The f a c t t h a t \E r, d i s a sequence o f k-algebras f o l l o w s from the d i s c u s s i o n i n S e c . l . D (23) Chapter I I I The E x t e n s i o n o f the Geometric S p e c t r a l Sequence We have seen i n Chapter I I how the Eilenberg-Moore s p e c t r a l sequence came about l n geometry. Such s p e c t r a l sequence e x i s t e d because there were c e r t a i n a l g e b r a i c s t r u c t u r e s i n the o r d i n a r y cohomology on the c a t e g o r y o f • t o p o l o g i c a l spaces. Our aim l n t h i s c h a p t e r i s to g e n e r a l i z e the c o n d i t i o n s under which t h i s s p e c t r a l sequence e x i s t s . The f i r s t step i s to g e n e r a l i z e the c a t e g o r y o f t o p o l o g i c a l spaces. And then, a f t e r d e f i n i n g a cohomology on the extended category we w i l l show t h a t we o b t a i n an extended v e r s i o n o f the Eilenberg-Moore s p e c t r a l sequence. In t h i s chapter, a l l t o p o l o g i c a l spaces w i l l be assumed to have the homotopy type o f a countable CW-complex. 1. The Category o f F i b e r e d Spaces Over B. We w i l l denote the ca t e g o r y o f (pointed) t o p o l o g i c a l spaces by (Top)*. Then, w i t h some space B i i n (Top)* f i x e d , we d e f i n e the f o l l o w i n g c a t e g o r y : The o b j e c t i n the category i s the sequence o f maps X • P x ' B s j T x i n (Top)^. such t h a t p x » s x = l g . We w i l l denote the o b j e c t I n the c a t e g o r y by X:X — B — — X . (24) The morphism <P:X —Y between two o b j e c t s i s a map Cp:X- —Y i n (Top)-ft such t h a t the diagram | | commutes. M:—— B — = — — Y Py By T h i s category w i l l be c a l l e d the c a t e g o r y o f f i b e r e d spaces o v e r B, and w i l l be denoted by (Top/B)&. Note t h a t I f B i s a one-point space, then (Top/B)* i s j u s t (Top)&, Remark: Fo r o b j e c t X i n (Top/B)*, each p o i n t beB determines a subspace F D(X)=(xtX| p x(x)=b} o f X, which has the baseppint, s x ( b ) . The subspace F f e(X) i s c a l l e d the f i b e r o f X a t b. Thus, the name " f i b e r e d spaces". The i n i t i a l and t e r m i n a l o b j e c t l n the c a t e g o r y ( T o p / B ) * i s B:B — B — B. T h i s i s e a s i l y seen from the f a c t t h a t , g i v e n an o b j e c t X, p x : X — — — B and s x : B — — X a r e the o n l y morphlsms between X and B. 2. T o p o l o g i c a l C o n s t r u c t i o n s i n (Top/B)*. We have extended the c a t e g o r y (Top)* feo (Top/B)*. We w i l l d e f i n e a cohomology on (Top/B)*.. But, whenever t h e r e i s a cohomology (or homology), we need to have such t o p o l o g i c a l n o t i o n s as homotoples, mapping c y l i n d e r s , cones, c o f i b r a t i o n s , (25) e t c . , and s i n c e (Top/B)^ i s an e x t e n s i o n o f the c a t e g o r y (Top)*, the t o p o l o g i c a l c o n s t r u c t i o n s i n (Top/B)* s h o u l d be an e x t e n s i o n o f the ones i n (Top)^. We, t h e r e f o r e , mimic the t o p o l o g i c a l c o n s t r u c t i o n s i n (Top)*. D e f i n i t i o n 2.1: Let X be an o b j e c t i n (Top/B)*. Then, X' i s a f i b e r e d subspace o f X o v e r B i f i ) X* i s a t o p o l o g i c a l subspace o f X, and i i ) the i n c l u s i o n map i : X - —*X i s a morphism i n (Top/B)*. The p a i r (X,X #) w i l l be c a l l e d the p a i r o f f i b e r e d spaces o v e r B. From t h i s , one can d e f i n e the c a t e g o r y o f p a i r s o f (Top/B)*. I f (X,X*) i s a p a i r o f f i b e r e d spaces over B, then we d e f i n e the q u o t i e n t space X/X' o v e r B to be the o b j e c t i n (Top/B)fc, where the sequence o f maps X/X* B — g * — X/X' X X i s g i v e n by P x ( ( x ) ) = p x ( x ) f o r (x)eX/X' and s x ( b ) = ( s x ( b ) ) f o r b£B, D e f i n i t i o n 2.2: A continuous morphism F:X x I — Y, where I=DO,ll, i s a continuous map F:X x I — Y wi t h F(«,t) a morphism i n (Top/B)* f o r a l l t f c l . (26) Two morphlsms <fQ, <p^  :X Y i n (Top/B)# a r e homotopic i f t h e r e i s a continuous morphism F:X x I *-Y such t h a t F(x,G) = ( P 0(x) and F(x, 1 (x) f o r a l l xeX. The f a c t o f the homotopy between (f>Q and <Pt w i l l be denoted by <P0-CP 1 . As i n (Top)^, the homotopy i n (Top/B)^ i s an eq u i v a l e n c e r e l a t i o n , and the composite o f homotopic morphlsms a r e homotopic. Thus, we can form the homotopy category o f f i b e r e d spaces o v e r B, The morphism <P:X <*-Y i s a homotopy eq u i v a l e n c e i f there i s a morphism <j>:Y — X such t h a t <t>'^slx and cp.^-ly. Remark: The equivalence c l a s s o f homotopic morphlsms from X to Y, denoted (X, Y), has a base p o i n t (# B)=the homotopy c l a s s o f the t r i v i a l morphism 0fi:X p-_— B — Y . D e f i n i t i o n 2.3: A p a i r o f f i b e r e d spaces (X,X*) i s s a i d to have the homotopy e x t e n s i o n p r o p e r t y w i t h r e s p e c t to the f i b e r e d space Y i f , g i v e n morphism C P:X- — Y and a continuous morphism G:X*x I —Y such t h a t G(x*,0)=<P(x*) f o r x'eX', then t h e r e i s a continuous morphism F:X x I — Y such t h a t F(x,0)=f(x) f o r xfcX and F *G. X x l (27) D e f i n i t i o n 2.4: A morphism <[>:X *-Y I s a c o f i b r a t i o n i f , g i v e n a morphism W i Y — a n d a continuous morphism G:X x I — « — Z such t h a t G(« ,0)=TP»4>, t h e r e i s a continuous morphism F:Y x I ; — » - Z such t h a t F(*,0)=SP and F(<|>(x),t)=G(x,t) f o r a l l xeX and t e l . C o n s i d e r i n g the diagram X'x 0 — X'x I the i n c l u s i o n map i : X ' — X i s a c o f i b r a t i o n i f and o n l y i f (X,X') has the homotopy e x t e n s i o n p r o p e r t y w i t h r e s p e c t to any o b j e c t i n (Top/B)^. There a r e many c o f i b r a t i o n s i n (Top/B)^. We make heavy use o f the one we d e f i n e now. Let X be an o b j e c t i n (Top/B)^. We denote by C(X) the t o p o l o g i c a l space X x I w i t h the i d e n t i f i c a t i o n o f the p o i n t s ^ ( x , l )=(x*l) i f and o n l y i f p x ( x ) = p x ( x ' ) f o r x,x'sX, and V s Y ( b ) x I,, the d i s j o i n t u n i o n o f s Y ( b ) x i j b&B A Note t h a t i f B i s a one-point space, C(X) i s Just the o r d i n a r y reduced cone o v e r X. The space C(X) i s an o b j e c t i n (Top/B)^ w i t h the two maps P c ^ x j ( ( x , t ) ) = p x ( x ) f o r (x,t)GC(X) and s c ( x j ( b ) = ( s x ( b ) , 0 ) f o r b&B. The f a c t t h a t C(X) i s a f i b e r e d space o v e r B w i l l be shown by C(X). (28) I f qp!X—— Y i s a morphlsm i n (Top/B)fc, then the mapping cone o f <p, denoted O f f ) , i s d e f i n e d as the space C(<?)=C(X)yY w i t h the i d e n t i f i c a t i o n o f the p o i n t s {(x,0)=cP(x) f o r a l l xeX, and s Y ( b ) = ( s x ( b ) , r ) f o r a l l beB and. r e l } . The two maps P C ^ ! C ^ ) — * " B a n d s c (CP) : B ~ ^ C W' which makes C(fl?) an! o b j e c t l n (Top/B)^, are g i v e n by [pn(vAx) f o r (x)ec(qp), p c ( C W ) ( ( x ) ) = p y P Y ( ( ^ ) ) = ° X MMTJ C(X) Y ip (x) f o r (x)eY, and s c ^ j ( b ) = ( s y ( b ) ) f o r bGB. There i s a n a t u r a l i n c l u s i o n morphlsm —C(<?) g i v e n byf 0(y)=(y)£C(qp) f o r a l l yeY. P r o p o s i t i o n 2,5s The n a t u r a l I n c l u s i o n morphlsm 9? 0 g i v e n above i s a co f i b r a t i o n i n (Top/B)*. Proof: C o n s i d e r the diagram Y x 0 Y x I C(<P) x 0 Given any morphlsm4s*C(<P) • — Z and a continuous morphlsm G:Y x l — w e need to f i n d the continuous morphlsm F which makes the above diagram commutative. We e x p l i c i t l y d e f i n e the morphlsm P as f o l l o w s : F ( ( y ) , r ) = G ( y , r ) f o r (y)ec(<0 and r e l , and (29) F((x,r),r'H f ( ( x , ( r - r ' ) / ( l - r ' ) ) f o r (x,r) C(q?), O ^ r ' ^ 1 , ( ( x , ( r ' - r ) / ( l - r * ) ) f o r (x,r) C(<0, O^r-r'^1, U ( x f O ) ) f o r r=r'. Then, F ( ( x , r ) , 0 ) = ( ( x , r ) ) f o r a l l (x,r) C ( f ) , and F ( ( y ) , 0 ) = G(yt0)=t(tp0(y))-4>((y)) f o r a l l (y)6C((p). Hence, P f ' , 0 ) ^ . Now, F(^ 0 ( y ) , r ) = F | ' ( y ) , r ) = G ( y , r ) f o r a l l y€Y and r&I. There f o r e , the f i b e r e d p a i r (CC*?),Y) has the homotopy e x t e n s i o n p r o p e r t y . a D e f i n i t i o n 2.6: Any sequence o f morphlsms isomorphic to the sequence X — ^ - Y °^ Q — C($) i s c a l l e d a c o f i b r a t i o n sequence. Lemma 2.7: _ tf> _ <P0 -L e t X — Y — be a c o f i b r a t i o n sequence. Then, C ( ^ ) - Y 0 / Y i s a homotopy equivalence i n (Top/B)*. Proof; We may assume t h a t YQ==C(<?). Then, we have t h a t Y Q/Y= C(C?)/Y=C(f 0)/C(Y). The c a n o n i c a l map C(<?) ^ C ( f Q ) / C ( Y ) I s a homotopy eq u i v a l e n c e . (An e x p l i c i t d e s c r i p t i o n o f the homotopy i s made i n C o r o l l a r y 6, S e c . l o f Chapter 7 i n (2). The same d e s c r i p t i o n a p p l i e s to the above c a n o n i c a l map wit h care taken on C ( Y ) , the e x t e n s i o n o f the o r d i n a r y cone o v e r (30) Lemma 2 . 8 : _ Cp _ q?0 _ Let X — — Y — * - Y Q be a c o f i b r a t i o n sequence. Then, the sequence o f homotopy c l a s s e s 05* CP* (Y 0,Z) T ° - (Y,Z) ~ • (X,Z) I s exact f o r any o b j e c t Z i n (Top/B)fc. P r o o f : See Theorem 3 , S e c . l o f Chapter 7 i n (2).ffl The suspension o f an o b j e c t X l n (Top/B)^ i s , as expected, a f u n c t o r on (Top/B)^. The suspension o f X w i l l be denoted, by lEx. The o b j e c t lEIx i s the t o p o l o g i c a l space Xx=X x I w i t h the i d e n t i f i c a t i o n o f the p o i n t s ( x , 0)=(y , 0 ) l f and. o n l y i f P x ( x ) = p x ( y ) f o r x,yeX, ( x , l ) = ( y , l ) f o r x,y6X, and. J s x ( b ) , r ) = ( s x ( b ) , r ' ) f o r b B and r,r*£l, t o g e t h e r with the two maps P j - x s Z x — B and Sjp x:B— — l E x g i v e n by pj- x( (x, t ) )=p x(x) f o r a l l xfeX and t£l, and S j - X ( b ) = ( s x ( b ) , t ) f o r a l l b£B and. t£l. I f QiX — Y i s a morphlsm i n (Top/B)*, d e f i n e the morphlsm 3>:Zx—ZY by Z 5 ? ( ( x , r ) ) = ( ^ ( x ) , r ) f o r a l l (x,r)eS. Note t h a t Z? i s a morphlsm l n (Top/B)* s i n c e the diagram Zx S5X I commutes. -Zx^r— B-=-TZY £Y (3D Furthermore, whenever makes sense i n (Top/B)#, then Z^+J-Z^Z* and Z l X^^Zx* H e n c e t Z ] i s a co variant functor from (Top/B)* to i t s e l f . Remark: If ^sX — Y i s a monomorphism, i . e . , <P:X — Y i s a monic map commuting with ^ p ^ s ^ a » 4(p , 8 ^ ] , then also a monomorphism. This i s because we have the i d e n t i f i c a t i o n of the points f (<?(x),0)=(<P(x'),0) i f and only l f PY*<?(x)=py'q?(x'), < (<?(x),l)=(<e(x'),l), and Ux),r)«(^'I,r#) f o r q?(x)=<?(x')=sy(bh i n Y from the I d e n t i f i c a t i o n of the points f(x,0) = (x',0) i f and only l f p x(x)=p x(x'), < (x,l)=(x*,l), and l ( s x ( b ) , r ) = ( s x ( b ) , r ' ) , i n X. The suspension of the fibered. p a i r (X,X*) i s defined, as Z(X,X') = (Xx,Xx*). The nth suspension of X i s defined, i t e r a t i v e l y as, ^ X=X and. ^=ZJ^L X). The functor^-" has the expected properties as exemplified i n the following two lemmas. (32) Lemma 2 .9: - <? - <?0 -Let X — Y Y Q b e a c o f i b r a t i o n sequence. Then, 2>c(^0). Moreover, there i s a morphism A : Y Q —^~X which i s homotopically equivalent to an inc l u s i o n . Proof: For the f i r s t part, a l l we need to do i s to observe that ZS=Y0/Y=C(^)/Y. Then, by Lemma 2.7, ZJ-Z^Q)* Now, there i s a canonical i n c l u s i o n ^ :YQ "~C(<^0) into the mapping cone. The morphism A i s the composition where i s the homotopy equivalence. IS Lemma 2.10: Let <jp:X — Y be a morphism i n (Top/B)^. Then, Zc<?)=c(Z<P). Proof: An e x p l i c i t r e a l i z a t i o n of the objects Tc{<%) and CiT^) shows that the topological spaces ^"c(f)=Cif) x I (with c e r t a i n points i d e n t i f i e d ) , and C i f t l s C ^ X l v I l (with c e r t a i n points i d e n t i f i e d ) are Isomorphic. B Corollary 2.10: I f X— ^ - Y — - * Y A i s a c o f i b r a t i o n sequence, then j > ~ X — Y 0 | " ^ Y o l s a l s o a c o f i b r a t i o n sequence. Proof: We may assume that Y0=C(^f). Then, "^c(tf)=Ct^ ~ff), i . e . , (33) i s the mapping cone of the morphism 3. The Products i n (Top/B)*. We w i l l be interested i n two products of objects i n (Top/B)^. The one i s the so-called pullback of the maps PX:X ~ B and py:Y —B o f objects X and. Y i n (Top/B)*. The other i s the smash product, X AfiY, of objects X and Y. There are others, such as the cross-product of X and Y, but they are of no i n t e r e s t to us here. We define, f o r objects X and Y i n (Top/B)$, X XgY to be the topological space X XgY={(x,y)£X x Yl p x(x)=p"y(y)^ together with the two maps P xXgp y:X XgY »~B and. s x X g 8 y : B «~X XgY given by P xXgP v(x,y)=p x(x)=p Y(y) f o r (x,y)6X xfiY and s x x B p v ( b ) = ( s x ( b ) , s y ( b ) ) f o r b&B. Note that i f B i s a one-point space, then X xfiY i s the usual cross-product i n (Top)*. The smash product of X and Y i s the quotient space of X XgY with the points ^(x,s v(b))=(s x(b),y) f o r (x,y)6X xfiY and bGB} i d e n t i f i e d . We denote t h i s quotient space by X AgY. The two maps P XA BP Y:X AgY — - B and s x A ^ y J B — - x A B Y are given by (34) PX A BPy ( ( x » y ) ) = p X ( x ) = P Y ( y ) f o r (** v> e xA BY, and. s x A f i S y ( b ) = ( s x ( b ) , s Y ( b ) ) f o r b£B. The above maps make the smash product, X A Y, an object i n B (Top/B)*. For the fibered p a i r (X,X'), we define YA, (x,x') to be the fibered p a i r (Y A ^ , Y AgX'). The following lemma gives a relationship between the suspension functor J~ and the smash product. Lemma 3»1 : (X /\BY)*X Ag^Y i s a homotopy equivalence. Proof: The two maps ^:Z (XA B Y ) —~ XA^ty and $:XAB£Y -Z(X A B ?) defined by <£(<<x,y),t))=(x, (y,t)) f o r ((x,y),t)£Z(X gY) ^ ( ( x , (y,t)))=((x,y),t) f o r (x, (y, t) )&X A BZ5 gives the homotopy equivalence Between Z ^ X A B Y ) and X AgZJY. • 4. The Forgetful Functors on (Top/B)%. We w i l l be Interested i n two c e r t a i n f o r g e t f u l functors on (Top/B)^ and. t h e i r respective adjolnts. The f i r s t functor Jr£ forgets the fibered. structure of the object X, and. the other functor F forgets the basepoints of each f i b e r space of X. (35) For describing the second functor F, we need another category c a l l e d the category of spaces over B, denoted Top/B. The objects of Top/B are the maps PX*X =- B. The morphism between two objects p x:X «*B and. p^:Y «-B i s a map <?:X «-Y commuting with p x and. p y . The f o r g e t f u l functor F i s from (Top/B)* to Top/B. The functor 5 from (Top/B)* to (Top)* Is defined by JE(X)=X/s x(B), f o r object X i n (Top/B)*. The functor P, given by P(X)=X x B f o r the topological space X, i s adjoint to 5JT. Note that P(X) i s an object i n (Top/B)^ by two maps "Thx x B — — B and. i : B — — X x B. The second functor F i s the functor which drops the map s x:B — X i n the object X. Its adjoint functor G i s defined by G(X)=X y B f o r the object p :X — B i n Top/B. The maps p x \ / l B : X V B — B and. i : B — — X V B make G(X) an object i n (Top/B)^. 5 . The Extended Cohomology on (Top/B We w i l l use the notation of Chapter II f o r the ordinary cohomology H"*"(';k) with c o e f f i c i e n t k. Throughout the following sections, k w i l l be assumed a f i e l d . We extend H * ( » ; k ) to a cohomology on (Top/B).*. by f i r s t d efining the cohomology theory on (Top/B)*-and then show that the extended, cohomology s a t i s f i e s the axioms of the theory. (36) D e f i n i t i o n 5.1: A cohomology theory on (Top/B)* i s a f a m i l y o f c o n t r a -v a r i a n t f u n c t o r s $h n;neZ^ from (Top/B)* to a category o f Z-graded o g j e c t s from an A b e l i a n category, such t h a t i ) (The Homotopy Axiom) I f morphlsms ty and $ a r e homotopic, then hn(<?)=hn(<i>) f o r all n£Z i i ) (The Suspension Axiom) For a l l neZ, t h e r e i s a n a t u r a l e q u i v a l e n c e cr:hn Vnz, i i i ) (The Exactness Axiom) Given a c o f i b r a t i o n sequence X — - £ ~ Y ^~^0' there i s a l o n g exact sequence .. . - h " ' 1 (X ) l h n(Y 0) J^SLh n(Y) h n(X)i.. . where S i s the connecting homomorphism. Remarks: (1) We d e f i n e the cohomology o f the p a i r (X,X*) as the cohomology o f the mapping cone o f the map i : X ' — — X , h n ( X , X ' ) = h n ( C ( i ) ) f o r a l l neZ. From the Exactness Axiom, we have the l o n g exact sequence f o r the p a i r (X,X*). (2) In the Exactness Axiom, i t i s enough to know feteat h n ( Y Q ) — — h n ( Y ) — h n ( X ) i s exact f o r a l l nfcZ. S i n c e Y0=C(<?) and 21x^C(cpQ) by Lemma 2.9, the composition o f the morphlsms h I 1 ( £ f 1 ) # ( T i s the c o n n e c t i n g homomorphism <f, where <P (37) I s the morphlsm from the Suspension Axiom a n d . ^ r Y g — — C(f°0) I s the ca n o n i c a l i n c l u s i o n i n t o the mapping cone. I n o t h e r words, ... - h n"^(X) h n ( Y Q ) - h n ( Y ) — h n(X) ~... h n ( C ( % ) ) i s an exact sequence. Using the f o r g e t f u l functorjfT, we extend, the o r d i n a r y cohomology on (Top)^ to a cohomology theory Hg(«) on (Top/B)^ by H|(X)=H*(5(X);k)=H*(X,s x(B);k) f o r o b j e c t X. Theorem 5.2: H*(» ) i s a cohomology theory on (Top/B)^, w i t h the A b e l i a n category being the category o f k-modules. Proof: I f ^/^JX *-Y are two nomotopic morphlsms l n (Top/B)^, then, «\<p:X * and. ^ | S x ( B ) ^ | s x ( B ) s s X ( B ) ~ ^ S Y ( B ) a r e a l s o homotopic i n the ©ategory (Top)^. Therefore, the maps H*(q>)=H*(4>) a n d H*(<?js ( B J J ^ H ^ + L ( B ) ) are i d e n t i c a l . Prom X * X the long exact sequences f o r the t o p o l o g i c a l p a i r s ( X , s x ( B ) ) and ( Y , s y ( B ) ) i n the category (Top)^, we have that H*(<£)=H*(4>). Let X — Y — Y Q be a c o f i b r a t l o n sequence I n (Top/B)^ and c o n s i d e r the diagram (.38) Hn+1(sc(<p) (B) ;k) H n + 1(C(<p);k) H n(c(f), S 5- ( £ ? > )(B);k) H h + 1 ( s Y ( B ) f k ) H n + 1 ( Y ; k ) H n ( Y , s Y ( B ) ; k ) H n + 1 ( s x ( B ) ; k ) H n + 1 ( X - k ) H n ( X , s x ( B ) ; k ) I The f i r s t two rows and a l l the columns are exact f o r a l l n i n the above diagram. Then, the t h i r d row i s a l s o exact f o r a l l n. Hence, Hg(«) s a t i s f i e s the Exactness Axiom. The Suspension Axiom f o l l o w s from the f i v e lemma a p p l i e d on the diagram . . HgC Y Q ) — — Hg(Y) — H^X) - H^ + 1 ( Y 0) •Hfi1+1 ( Y ) — 7 . . ' • ^ H B C V - - H B ( Y ) J H B + 1 ( C ( % ) ) - H ^ 1 < Y G ) — H ™ (*>--».. n+ Tn+i ffl. Hf^Zx) where the rows are exact by the Exactness A x i o m . 9 Remark: For the o r d i n a r y cohomology H n on the category (Top)^, H n i s zero f o r n*0. Therefore, we make a note that the extended cohomology Hg(« )=0 f o r n«*0. There i s a modular s t r u c t u r e i n Hg(X) induced from the ( 3 9 ) one i n C*(X)=Hom(C*(X),k) of Chapter I I . We have already remarked l n Chapter II that H*(X;k) i s a r i g h t and l e f t H*(BJk)-module l f there Is a map f:X — B. Therefore, f o r any object X i n (Top/B)&, Hg(X)=H*(X,s x(B);k) Is also a H*(B;k)-module. D e f i n i t i o n 5*3-The n-sphere i n (Top/B)^ w i l l be defined by Sn=Rsn), where S n i s the n-sphere i n R n + 1. The c o e f f i c i e n t module f o r the cohomology Hg(«) i s H3(Sg)=H*(S°x B, B;k)£H*(B;k). Hence, H*(X) i s also a H*(S°J-module. a D 6. The M u l t i p l i c a t i v e Cohomology on (Top/B)*. The cross product of the ordinary cohomology y4*H*(X/A;k) ® H*(Y/B;k) — H*(X x Y/(A x Y)v(B x X);k) fo r the topological pairs (X,A) and (Y,B), gives a multl-p l i c a t i v e structure to jfche extended cohomology Hg(* )• D e f i n i t i o n 6.1: A cohomology theory $hn;nez} on (Top/B)* i s m u l t i p l i c a t i v e i f there i s a natural (external) product h*(X) ® h*(Y) h*(X A BY) fo r objects X and Y i n (Top/B)fc. (40) For the extended cohomology Hg(» ) on (Top/B)*, the composition of the morphisms, Hg(X) ® H*(Y) H*(X x Y/(s x(B) x Y)v(s Y(B) x X);k) 4u)| H*(X A BY) where Jl i s given above and i i s the Inclusion map, i s the exte-Ttmi! product. Let X and Y be objects l n (Top/B)^. Consider the p u l l -back of the maps P X*X — B and p y»Y —B, X Xx,Y — X Then, the maps s x » B ' — — X and s y:B »-Y Induces the maps r:Y—<— X XgY and u:X —X XgY. E x p l i c i t l y , r and u are defined by r(y)=(s xp y(y),y) f o r y&Y and u(x)=(x, S y p x ( x ) ) f o r x6X, Lemma 6 . 2 : X x BY/(r(Y) v u ( X ) ) £ X A B Y / s x A B s Y ( B ) Proof: Both of the above spaces are the quotients of the space X x BY. The I d e n t i f i e d points r(Y) v u ( X ) i s same as the i d e n t i f i e d points {_(x,s Y(b)=(s x(b),y) f o r a l l (x,y)€X xfiY (41) and a l l b6B, and (s x(b),s y(b) )=(s x(b'),s y(b')) fo a l l b and b*£ . Hence, the lemma. B Theorem 6.3: Let B be a simply connected space. Let Hg(X) be f i n i t e and projective as Hg(Sg)-module and. l e t p y i n Y be a Serre 4 — fibration with H£(Y) also f i n i t e . Then, 4(X"> ®H|(SB) HB^^ Hf(X A B Y ) . Proof: Let ^E r, d p^ be the Serre spectral sequence for the fibration p y:(Y, s y(B)) "-(B,#). Then, E r converges to H*(Y,sy(B);k) and E2^H*(B;k) ® H*(F, s y(b);k), where (F,s y(b)) i s the fiber of the map p y. The Induced, map p y:(X xgY, r(Y)vu(X)) — (X,s x(B)) Is also a Serre fibration. Let JE^, d^} be the spectral sequence for this map p y. Then, E£ converges to H*(X x BY,r(Y)vu(X);k) and. E|^H*(X,sx(B);k) ® H*(F, s y(b);k), where (P,s y(b)) i s the fiber of the map p y. Let the spectral sequence (E r, dr} be defined by I r=H*(X,s x(B);k) % * ( B , k ) E r w l t h d r=l®d r, where d r:E^—- E r. By the proJectlveness of H*(XjS x(B);k) and the Lemma 2.13 of Chapter I, we have that Hd r)=H(H*(X,s x(B);k) ® H * ( B ; k ) E r ) £ H % , s x ( B ) ; k ) ® H * ( B . k ) H ( E r ) =H*(X, s x (B) • k) B. k ) E r + 1 =E r + 1. Therefore, E r i s a spectral sequence. (42) Since E^ converges to H*(Y,s Y(B);k), the spectral sequence E r converges to H*(X,s x(B);k) ® H * ( B . k ) H * ( * . s v(B)'k). We have the Isomorphism E 2=H*(X , S x(B);k) ® H * ( B ; k ) E 2 =H*(X,s x(B)jk) ® H * ( B # K ) H * ( B ; k ) ® H*(F,s v(b);k) SH*(X,s x(B);k) ® H*(F,s v(b);k)=E|. Hence, E^=E£> o r H*(X,s x(B);k) <^ ( B ; k )H*(Y,s Y(B);k)SH*(X x F IY, r ( Y ) v u ( X ) ; k ) . By Lemma 6.2, we have the isomorphism of t h i s theorem, • Corollary 6.3: With the same conditions as i n the above theorem, we have the isomorphism f o r the p a i r ( X , X * ) , Hg(Y) ( ^ * ( s 0 )H|(X,X5)=H|(Y A B ( x , x ' ) ) . B 7, The Kunneth Spectral Sequence. The following d e f i n i t i o n and theorem form the basis from which we b u i l d the Kunneth spectral sequence. D e f i n i t i o n 7.1: The cohomology theory \hn;n£z} on (Top/B )^ i s said to s a t i s f y the Atiyah property i f f o r a l l object X i n (Top/B).^, there i s an object "Y and a morphism0(^:X — Y such that i ) h n 0 ( x ) i s an epimorphism f o r a l l n£Z, and i i ) h*(Y) i s a projective h*(sg)-module. (43) Theorem 7.2: The extended cohomology Hg(» ) s a t i s f i e s the A t l y a h property. Proof: Let o(:X »-(X/sx(B)) x B be a map defined by 0((x)=((x),p x(x)) f o r a l l xex. Note that o( I s a morphlsm i n (Top/B)£. We set Y = f l ; x / s x ( B ) ) , The n a t u r a l mapv fix ( X / s x (B)) x B — X given by ^c?((x),b)=x f o r a l l ((x),b)£(X/sx(B)) x B, Is w e l l - d e f i n e d and i s a morphlsm l n (Top/B)^. Now, the composition o f the maps ^ •d(=l x-. Therefore, H*(^)«H§(^)«lH*^j.' Hence, Hg(X) i s an eplmorphism f o r a l l n€Z. Since H*(X) i s f i n i t e and k i s a f i e l d , from the Kunneth theorem f o r o r d i n a r y cohomology, H*(Y)=H*((X/s x(B)) x B, B;k)=H*((X/s x(B));k) ® H*(B;k). Therefore, H*(Y) i s a fre e H*(B;k)-module. B The morphlsm fl(x o f Theorem 7.2 i s " n a t u r a l " i n the f o l l o w i n g sense. Theorem 7. 3: Let tfxX — V be a morphlsm i n (Top/B)^. Then, w i t h O^xX—•» Y and &yXV—«—W being the morphisms and o b j e c t s obtained from Theorem 7»2, there i s a Y'in (Top/B)^ and a morphlsmU^xX »-Y'such that i ) Hg^jf) I s an eplmorphism f o r a l l n£Z, i i ) H*(Y #) i s a p r o j e c t i v e Hg(S°j)-module, and (MO i l l ) the diagram X" X — V Y commutes. Proof: Let the t o p o l o g i c a l space Y* be defined by Y'=(X/s x(B)) x (W/s w(B)) x Y. Then, Y* i s an o b j e c t i n (Top/B)^ by the two maps p Y,:Y — — B s v,:B — Y * given by p y , ( ( x ) , (w),y)=p y(y) f o r a l l ( ( x ) , (w),y)6Y*, and s v * ( b ) = ( s x ( B ) , s w ( B ) , s Y ( b ) ) f o r a l l b6B. The above maps are w e l l - d e f i n e d since s x ( B ) and s w ( B ) are the i d e n t i f i e d p o i n t s i n Y'. Now, there i s a morphism fl(x:X —Y* d e f i n e d by o(*(x)«((x) t(0^-^(x)),O^x(x)) f o r a l l x 6X. Then, as i n the proof o f Theorem 7.2, HgOC^ ) i s an epimorphism and H*(Y') i s a p r o j e c t i v e H|(S B )-module. The p r o j e c t i o n map I f J Y - — — W d e f i n e d by 1H(x),(w),y)=w f o r a l l ( ( x ) , (w),y)£Y', and the map O^Y *-Y* given by <Hy)=((s x»p y(y)), (s w«s y(y) ),y) f o r a l l y£Y make the diagram o f t h i s theorem commutative.9 Now, we begin the complicated process o f c o n s t r u c t i n g a f i l t r a t i o n on a c e r t a i n space, which w i l l give us the Kunneth s p e c t r a l sequence. (45) Let the sequence o f c o f i b r a t i o n sequences be defined, as f o l l o w s : Set X Q = X f o r some o b j e c t X i n (Top/B )&. Then, from Theorem 7.2, there i s an o b j e c t Y Q and. a morphism <^ 0:XQ—— Y Q i n (Top/B)# such t h a t HFI(X0) i s an epimorphism and Hg(YQ) i s a p r o j e c t i v e Hg(Sg)-module. The i n d u c t i v e step i s made by s e t t i n g X _ N = C ^ N + 1 f o r n*0 and o b t a i n i n g the ob j e c t s Y_^ and the morphism N :X — Y „ from Theorem 7.2. ^A-n -n -n Then, w i t h ^„ N :Y~^FI—~"X_ N being the n a t u r a l I n c l u s i o n i n t o the mapping cone, we have the sequence o f c o f i b r a t i o n sequences, R e c a l l that f o r each c o f i b r a t i o n sequence X-n+i ~ Y-n+l ~ X - n » f o r n*°» there i s a morphism A n <„ 1 : X J ^ — * 2 ^ . x - n + l *"1,0111 Lemma 2.9 o f t h i s chapter. A p p l y i n g the suspension f u n c t o r on each o f the c o f i b r a t i o n sequences and c o l l a p s i n g them, we get the sequence x - n An-1~ • •' J ? j jr R LX'. P_ 1 n ^ l ^ j f f l - p ~* • ' j ^ i ^ J ^ V f o r O^p^n. Since A n _ i l s homotopically equivalent to an i n c l u s i o n we have the ( p a r t i a l ) f i l t r a t i o n o f the smash product Y A B ^ X Q , by t a k i n g the smash product o f Y w i t h the above sequence. (46) Theorem 7 .4 : There I s an exact sequence where each H * ( t ^ X _ p t n " p ^ X ^ p ^ 1 ) i s a p r o j e c t i v e H^sl )-module f o r a l l O-p^n-1 • Proof: By the Exactness Axiom o f the cohomology and the f a c t that Hg^Y _) i s an eplmorphism, tb©r® i s a short exact sequence of H*(S°j)-modules, 0 -H*(X n . ) ~ H * ( Y ) - H * ( X ) - 0 o -p -1 B -P B -P f o r 0^p-n-l. By the Suspension Axiom o f the cohomology, the sequence I s a l s o exact f o r 0-p^n-l. Now, we s p l i c e the appropriate sequences from above t o -gether and get the long exact sequence, 4<Z*-n+l > — — H B < J E P * - P > ~''' ^ H | ( ^ X ) — 0 . Consider the sequence where k I s the I n c l u s i o n Into the mapping cone. Then, by Lemma 2 . 9 , we have the Isomorphism f o r 0^p*sh-l. Therefore, by the Suspension Axiom, H B ( ^ P Y . p ) a H B ( ^ ' , X . p , n - ^ X . p . 1 ) . Hence, we have the long exact sequence o f t h i s theorem. (47) 4(c n—p n—p—1 The above isomorphism a l s o shows that H g ( V x _ p , X_ p_^) i s a p r o j e c t i v e Hg(Sg)-module. • We are now ready to descr i b e the Kunneth s p e c t r a l sequence f o r the cohomology Hg(* ). In order to s i m p l i f y the notations s l i g h t l y ( and get more co n f u s e d l ) , we denote by z]J_p=Y A g J^Lp w i t h the i n c l u s i o n morphism l yAg ^ A p * Note that Z n i s o n l y def i n e d f o r O^p^h. Therefore, we l e t n-p f~n * _>^ T n-p f o r p-n, f o r p^O. Then, f o r each n^G, we have the f i l t r a t i o n o f the ob j e c t Z^, Let the exact couple o f blgraded k-modules, D(n) — D ( n ) Y E (n) ' dependent on n, be defined by D U r ^ - H ^ - 1 ^ ) and * ( » > - » ' « - H ? " * < ^ p . £ p - 1 > . f o r a l l p and q. The three maps 0(t ^  and Y come from the long exact sequence o f the f i b e r e d p a i r (z£ , z " ^ ^ ) • We make a note that £>( has the bidegree ( - 1 , 1 ) , ^ has the bigjegree ( 1 , - 1 ) and Y has the bidegree (0 ,1). ( 4 8 ) The above exact couple determines a s p e c t r a l sequence ( E r ( n ) , d r ( n ) } , dependent on ri^O. Theorem 7. 5: There I s a n a t u r a l isomorphism E " p * q ( n ) S E " P , q + 1 ( n + l ) f o r 0 * p * h - l . r r Therefore, d^ p» q(n)=d~ p' q + 1(n+1). Proof: Applying Lemma 3» i » a c a l c u l a t i o n shows th a t f o r o^b^h-l, -p+<I*.l -n+l n+\ ) - H ~ p + q + 1 V f z 1 1 z n ) Bv the H B ^(n+n-p'^n+n-p-i^B Z}Z^Zn-P-1K B y t h e Suspension Axiom, E- p» q(n)=E- p' q + 1(n+l). • The above theorem makes the f o l l o w i n g d e f i n i t i o n o f the Kunneth s p e c t r a l sequence independent o f n. D e f i n i t i o n 7 » 6 : The Kunneth s p e c t r a l sequence i s d e f i n e d by E ; p ' q ( X , Y ) = E ; P ' q + n ( n ) — — -»p, q _ _ f o r o b j e c t s X and Y i n (Top/B)*. I t s d i f f e r e n t i a l d r (X,Y) i s d ; p ' q + n ( n ) . Remark: -p,q+m /%-,-.-p.q+i*V \ I f m=sri*0, then by Theorem 7.5, E r (m)-.. .«E r (n), f o r O^p^h-1. I n oth e r words, f o r p^O, E ; p • q (x, Y) =E;p • q * P ( n ) S E; p • q + p + 1 <P+l), I.e., independent o f n. Note th a t E ^ P , Q + P ( p ) # E £ P , Q + P + 1 (p+1). (49) Lemma 7« 7' The Kunneth s p e c t r a l sequence i s a second, quadrant s p e c t r a l sequence. Proof: Consider the morphlsm o( i n the diagram —n —n —n _/~'P«Q.'*'n w, f o r p^O, Zn-p'Zn-p-i^^' H e n c e » f o r f l x e d an Isomorphism. Prom the long exact sequence, . — - D - p ' q + n ( n ) D-P- 1^*"* 1 (n) E " P * q + n ( n ) D-p,q+n+l ( N ) J 2 C D-P-l,q+n« ( n ) # # f we get that EjPt* ( f , Y)=E~ p» q + n(n)=0, f o r p-^ 0. Consider the c o f i b r a t i o n sequence, Z* „ , J - Z n n _ C ( J ) . n-p -1 n-p Now, E ^ P » Q ( X , Y ) = H g P + q + n ( C ( j ) ) . By Lemma 3 . 1 and the Suspension Axiom, Prom the remark made a f t e r Theorem 5 . 2 , H Q ( Y A B X. p)=0 f o r q^O and H g + 1 ( Y A g X . p ^ )=0 f o r q+i-f). Hence, from the l o n g exact sequence o f the above c o f i b r a t i o n sequence, H ~ p + q + n ( C ( j ) )=0 f o r q^O. • Since we w i l l be u s i n g Theorem 6.3 and. i t s C o r o l l a r y to make the i d e n t i f i c a t i o n on the E 2(n)-terms, we need, to assume, at t h i s p o i n t , the c o n d i t i o n s o f Theorem 6.3, i . e . , t hat B i s simply connected, th a t H | ( X ) i s a f i n i t e , p r o j e c t i v e H ^ S g ) -(50) module, and that p Y : Y ~ B i n Y i s a Serre f i b r a t i o n w i t h H | ( Y ) a l s o f i n i t e . For each s p e c t r a l sequence { E r ( n ) , d r(n)} , E 2 ( n ) i s the homology o f the complex ... E - n + 1 ' * ( n ) -E°' q(n) — 0 . . . where E " n + 1 » ^ ( n ) » H 3 n + 1 + q ( z J , ^ 1 ) , f o r a l l i ^ i ^ n . Then, from C o r o l l a r y 6.3, we have the isomorphisms B ^ ^ H W f * * ^ . ^ ) = H B » + I + Q ( Y AB(Z.x-n+i» Z^-n+i-l > > = (H*(Y) « H * ( S 0 ) H * ( ^ X ^ 1 , ^ X - n + l - l > ) * n + 1 + q , f o r a l l l ^ l ^ h . Hence, the two complex o f Hg(S f i)-modules, . . . 0 fc Hg"(ZN, ZQ ) — — ... ^-Hg(Z^ Z^ _^  ) • . . H B (^n* ^ n-1 * 0 * •' A N D . . . 0 H*(Y) ® H* ( S0)Hf Q T * . n + 1 , X _ N ) ... — -®4(S§) HB ( i: I-n+i'^-n+i-l ) ' " H B ( ^ ) ® H f ( s g ) H B ( ^ 0 » ^ . l ) — °V - . aim c h a i n e q u i v a l e n t . Now, by Theorem 7.4, the above complexes are a ( p a r t i a l ) p r o j e c t i v e r e s o l u t i o n s o f H | ( J ^ ? X ) . Therefore, by the d e f i n i t -i o n o f the Tor f u n c t o r H - ^ W - t o r J I t J o , < H | ( Y > , H * ( ^ X ) ) f o r 0*£p^ h-i. By the Suspension Axiom, " ^ ( n ^ r j ^ O , «*<*>.!£<*>>. Hence, f o r a l l p and q, (51) E2 P' q(X,Y)=E; P» q + P + 1(p +l) = T ° r H i ( S B ) ( H B ( f ) ' H B ( I ) ) -We have shown above that the s p e c t r a l sequence $.E r(X,Y), d r(X,Y)} I s Independent o f the choice o f the morphism 0( x from Theorem 7.2. We w i l l now show that E r(X,Y) i s f u n c t o r i a l w i t h respect to the obj e c t X and Y. Theorem 7.8: Let <?:X — V and4>:Y ~W be morphlsms i n (Top/B)^ w i t h both p a i r s o f o b j e c t s , \_X,Y} and ^ V , l } s a t i s f y i n g the co n d i t i o n s o f Theorem 6.3. Then, there i s a n a t u r a l morphism Er(<f?,<|>) :E r(V, W) E R ( X , Y) w i t h E2(<Pt^)»TorH|(sO)(Hj(q?),H*(4>)). Proof: Consider the two diagrams obtained from Theorem 7.2, X — X ^ V Y — Y ^ J*X J<*X and J<*Y °<Y W^ U — U' — U " f — f * - — T " Now, the morphisms(qp, lf\and{^,"^1 Induces a morphism E'(f,4):E'(V,W) — E'(X, Y) r * r r through the c o n s t r u c t i o n o f the Kunneth s p e c t r a l sequences, E£(?,W) and E^(X,Y), s t a r t i n g w i t h the morphlsms c(£ and D(yJ r e s p e c t i v e l y . Let Er(V,W") and Er(X*, Y) be the Kunneth s p e c t r a l sequence u s i n g the morphlsms 0CX and o( Y , r e s p e c t i v e l y . Then, (52) s i n c e E r(V,W) and E r(X,Y) are independent of the choice o f the morphlsms and 0 v^, we have the morphism E r(^,^):E r(V,W)SE^(V,W) E^(X, Y)=E r(X,Y). On the E 2 - l e v e l , we have the commutative diagram E2(V,W) Tor H | ( s g)(H|(W) fH|(V)) — T o r h % 2 ) - * h B ( Y ~ M B ( X ) ) . B ;E2 ( f , 4 ) E 2(X,Y) 8 . The Convergence o f the S p e c t r a l Sequence. The convergence o f the Kunneth s p e c t r a l sequence w i l l be discussed from the p o i n t o f view o f the convergence o f each s p e c t r a l sequence ^ E r ( n ) , d r(n)} d e f i n e d i n the l a s t s e c t i o n . The convergence o f E r ( n ) i s a consequence o f the f o l l o w i n g lemma. Lemma 8 . 1 : For each s p e c t r a l sequence ^ E r ( n ) , d r ( n ) } , E; p' q(n)«B;5l*(n)fi ... S » 2^(n). Proof: E 2 ( n ) I s the homology o f the complex ...0 - E j n + 1 ' q ( » ) — r . . -E°' q(n) «- 0... Hence, the lemma. • With the notations o f the l a s t s e c t i o n , we have the (53) f o l l o w i n g theorem. Theorem 8.2: For a l l riso, E r ( n ) converges to Hg(Y A B ^ X ) . Proof: The f i l t r a t i o n • • • Y A B x - p ~ l ™~ Y A B ^ X _ p — ... — Y A j | o f Y A B ^ 0 Induces the f i l t r a t i o n F P H b ( Y A f i Z x 0 ) o f H B ( Y / \ B ^ X 0 ) through the s i n g u l a r c h a i n complex o f the t o p o l o g i c a l p a i r ( Y A B / ^ 0 ' W re? ( B ) ) ' ^— _0 ru-Hence, E p ( n ) converges to H B ( Y A B V x o ^ * * We w i l l be t a k i n g the in v e r s e l i m i t s o f the f a m i l i e s o f s p e c t r a l sequence i E r ( n ) } ^ 0 and. o f o b j e c t s \.H(n)}^ 0, both indexed on the p o s i t i v e i n t e g e r s . The d e f i n i t i o n o f i n v e r s e systems and f a c t s used i n t h i s s e c t i o n are the ones l n S e c . l of the I n t r o d u c t i o n i n (2). Although the next theorem i s s l i g h t l y more general than what we w i l l need, we w i l l apply the theorem to show the convergence o f the Kunneth s p e c t r a l sequence. Theorem 8.3: Let l B r ( n ) } r i s » Q D e a f a m i l y o f s p e c t r a l sequence indexed on the non-negative Integers. Let ^H(n)} ^ be a f a m i l y o f graded o b j e c t , indexed on the non-negative i n t e g e r s , w i t h morphisms{g™:H(m) ^ H ^ n ) } m ^ n o f degree r(n,m) such t h a t the Inverse l i m i t , H q=llm,{H q(n)} e x i s t s , f o r each q. (5*0 Then, l f 1) E (n) converges to H(n) f o r each n, and r i i ) each f i l t r a t i o n pPH(n) o f H(n) i n i ) i s such t h a t g ^ p P h ^ d n J J C p P H ^ ^ ^ ^ n ) f o r n 4 , there e x i s t s 1) a f a m i l y o f morphisms (f™:E pj q(m) _ E P ' q + r ( n ' m ) (n)i such t h a t the Inverse l i m i t E P o , q=ii^i{E P e» q + r ( 0» n> (n)) e x i s t s , and 11) a f i l t r a t i o n F p H q o f H q such that E P ^ S F P H V F P ' 1 ^ , f o r every p and q. Remark; The Inverse l i m i t H q=lim?H q(n)} i s not w e l l - d e f i n e d . We make a convention that H q i s the Inverse l i m i t o f the system, { . . . — H q - r ( o » n ) ( n ) — . . . — H q - r ( o » 1 ) ( i ) ^ - H q ( 0 ) } . Proof; Let hg-gg F P H q ( m ) s F P H q ( m ) P P H q * r f e , m ) ( n ) f o p n - & ; Then, we can take the inv e r s e l i m i t o f the system ... p P H q - r ( 0 » n > (n) . . . - J P H * - P ( O - 1 > ( 1) -pPH q(0) and denote by F pH q=UslF PH ( 1~ r* 0 , n ) (n)l. Since by d e f i n i t i o n and & 4 ^ 0 ^ ~ l l 0 ' a ) ^ K < l * J . »«}. pPH q i s a f i l t z a t i o n o f H q, f o r each q. m The morphlsm h n Induces the morphlsm EP» q(m)SF PH q(m)/pP* 1H q(m) I f m n E P , q + r ( n , m ) ( n ) s F P H q + r ( n , m ) ( ^ ) / p p - i H q + r ( n , m ) ( n ) (55) The commutatlvity f n * f k = f n f o r n ~ k - m f o l l o w s from the commutatlvity g j ^ g j ^ g j j . Hence,we can take the Inverse l i m i t o f the system, t # ^ EP,q-r(0,n) ( n ) _f f i E p f q-r(0, 1) ( 1 q ( 0 ) and l e t EP» q=lim{EP; q- r ( 0» n ) (n)}. Let the morphlsm h n : F p H q ^ F p H q " r ( 0 » n ) (n) be defined. by " n ^ ^ ^ s o ^ ^ n f o r ^xri f^O^^^' Then, f o r n=*m,we have the commutatlvity h^1(.{xn)W)J.h;(iB)-xn^ ({xIJfr0). We w i l l show, subsequently, the Isomorphism p P H V ^ ^ H ^ l l m n ) (n)/!?'^'^0'n) (n)> ( = W E ^ q - r ( ° » n ) ( n ) } ) . Then, ^ I q f i P p H q / P p " l H q . The morphlsm h n d e f i n e d above Induces f n : F p H q / P p - 1 H q -FPH q" r ( 0» n>(n)/P p- 1H q- r ( 0' n )(n) as f o l l o w s : f „ ( ^ n } n a o F P " l H q » = h n < ^ „ l * 0 » ? P * l i j q " r < 0 - n ) < ' ' ' f o r a l l i x \ ^„FP- 1H qtpPH ( l/P P" 1H < l. v n J n=0 Then, f o r n^in, we have the commutativity f™»f =f„» ' n m n Now, i f f n d x ^ ^ p P ^ ^ J - ^ d x ^ ^ p P " 1 ^ ) f o r a l l n " ^ K W K W 1 ^ t h e n X n F P - i H q - r ( 0 , n ) ( n ) = x , F p - l H q - r ( 0 , n ) ( n ) f o r a l l n < Q r x ^ m o d p P ^ H ^ ^ ^ ^ n ) f o r a l l n. n n Hence, K W ^ i W ) " 4 * * " 1 * * ' Moreover, i f [ x ^ ' 1 ^ ' ^ 0 * n * (n$ f&O i s l n the i n v e r s e l i m i t , lim[pPH q*" r ( 0» n )(n)/F p" 1H q'" : r ( C )' n )(n)}, then by d e f i n i t i o n xy-V : ?" ( 0'^(n)=h m(x m)FP- 1H q- r ( 0' n )(n) f o r „ * . (56) Therefore, {x ^  ^,n$>~1jfikFpKq/Fp~1Kq, and we have that * 1 n» n^O * f n ( U n ' i h ^ o F P " l H q ) * x n j P " l H q " r ( 0 , n ) ( n ) f o r a 1 1 n* H e n c e ' t h e isomorphism above* B We w i l l apply the above theorem to the f a m i l y o f s p e c t r a l sequence ^ E ^ t n ) , d p(n)} d e f i n e d i n Sec.7. We w i l l denote by H*(n)=H*(Y A B ^ C ) and by FPH*(n), the f i l t r a t i o n o f H*(n) such th a t E P ; q ( n ) = P p H q ( n ) / P p " 1 H q ( n ) . Let g m:H q(m) — H q + n " m ( n ) , f o r .n<n, b a t t e l © isomorphism n a D given by the Suspension Axiom o f the cohomology. Then, we can take the i n v e r s e l i m i t o f the system 4 + n ( n ) ... - H B + 1 < 1 ) <-H q ( 0 ) , and l e t B%«llm[H|*n(n A . Lemma 8.3: With the above n o t a t i o n s , ggfFPrijGa)) c P P H | + n - m ( n ) f o r n*in. Proof: We w i l l prove the lemma u s i n g i n d u c t i o n on n. For n=0, E P * q ( n ) = 0 f o r a l l p and q. Therefore, the f i l t r a t i o n o f H q(n) i s FpHq,(n)=Hq,(n) f o r a l l p. Then, f o r any man, we have g g \ a / — x i g i £(F pH|<m)) C H | + n - m ( n ) = P D H q + n - m ( n ) . Assume tha t g™. t (FPH^im)) £ F p H q + n " * m * 1 ( n - 1 ) f o r a l l m^n-1, and c o n s i d e r the f o l l o w i n g isomorphisms, g™(FPHB(m) J / g ^ F ^ H l ( m ) )^PH|(m)/FP- 1H|(m) (57) g FP Hq+n-m ( n ) / p p - l H | + n - m ( n ) •Cl ( p P4 + n" m ( n ) ) / s£-l ^ ^C'^n)). f o r -n+l^p^O, from Theorem 7 . 5 . We al s o have, from Theorem 7» 5» the isomorphism ^ ^ ^ M / ^ ^ ^ n ) ^ ^ ' 1 ^ 1 <n~l ) / F p - 1 H | + n - m - 1 (n - 1 ) . By i n d u c t i o n hypothesis, g x ( F p H q + n " m ( n ) ) C F P H ^ 1 1 " " 1 1 1 " 1 (n-1) and g^(F pH q(m))Sg^ 1.g^(FP H|(m)) - g ^ (FPH q(m) ) £ ¥ p ^ * n ' m ' 1 (n -1 ) . Hence, f o r -n+l^p^t), we have that g*(F pH q(m) J S F P H ^ " - 1 1 1 - 1 (n-1 )S?V%+™ln). For p=-n, we, again, have the isomorphisms F - n + 1 H B + n - m ( n ) / F - ^ % m ( p - n + l H q ( m ) / g m ( p - n H q ( m ) K and ^(P" n + 1H|(m))Sr" n + 1H| + n"' m(n) from Theorem 7 . 5 . Hence, g m(F- nH B(m) ^F'^l^in). F i n a l l y , f o r p^-n, E p , q ( n ) = 0 , o r tha t F p"' : lH qj(n)=F pH B(n) f o r a l l p ^ n . Hence, g^(F pH|(m)) E^(P"n4<*>*"ir,14+n"m<n> f o r a 1 1 P ^ - H By Theorem 8.2, there e x i s t s an i n v e r s e l i m i t EP,'^ llmlE p» (l - r ( 0 V n)(n)} and a f i l t r a t i o n F pH| o f H|= l ^ m f a q " , r * 0 ' n ) ( n ) ) (In t h i s case, r(0,n)=~n. ) such that E p^=F PH|/F p- 1H|. (58) Lemma 8,kt With the above no t a t i o n s , we have the isomorphisms, E p a ' q S E p > J q ( X , Y ) , where E ( X , Y ) i s the Kiinneth s p e c t r a l sequence, and H | S H | ( Y A B X ) . Proof; Let the mo r p h i sm g n : H q ( Y AgX) — H J * " ( Y ABj>%) be defined as the isomorphism given by the Suspension Axiom. Then, g^Cg")"" 1. The commutatlvity gg«gm=Sn, f o r n^m, f o l l o w s from the commutatlvity SQ'S^~S^» The c o n d i t i o n that g n ( x ) = g n ( x ' ) f o r a l l n Imply x=x* f o r x and. x* i n H g ( Y AgX) l s t r i v i a l s i n c e g n i s an isomorphism. Let ( x ^ ^ e ^ ^ " " ^ . Then, x 0 = g n ( x n ) f o r a l l ri&O, by d e f i n i t i o n o f the i n v e r s e l i m i t . Hence, g n ( x 0 ) = g n ' g n ( x n ) = x n f o r a l l ri*0. Therefore, H B ( Y A g X ) = l ^ { H | + 1 1 ( n ) } . We w i l l show that f o r each p and. q, E ^ * q ( X , Y ) i s the in v e r s e l i m i t o f the system . . E 2 > » q + n ( n ) — - E ^ + P * 1 (p+i ) — . . . Note t h a t , f o r 0*h4>+l, E t P , q + n ( n ) * 0 . Let the morphlsm f n:E^p* q (X, Y ) — - E ^ > q + n ( r i ) be defined. by _ ( ( f p 4 - l ) " 1 f o r "*P + 1» n l o f o r 0*h*p+l. For m^h^p+1, the commutatlvity fJJ« tm**tn f o l l o w s from the commutatlvi t y f n + 1 • f g = f m + 1 . For rn^p+1 and. n^p+1, we have the commutatlvity ( 5 9 ) (n)=0 E~ P' q(X,Y) The f i n a l case, n*m^p+l, i s t r i v i a l s i nce E^P»q+m(m)=0. Hence, fm«f_=f„ f o r a l l 0=^ n^ m. I f f o r a l l riao f n ( x ) = f n ( x ' ) w i t h x and x' i n E^^Oc, Y), then, s i n c e f i s an isomorphism f o r rPsp+l, x=x*. Moreover, i f i ^ ^ ^ E ^ ' ^ t n ) such that * n=f™(* m), then we have y = x p + i e E ^ ' q " f p 4 ' 1 ( p + l ) = E ^ ' q ( X , Y ) such t h a t f (y)=x f o r a l l n*0. Hence, E 2 ' 4 ( X , Y ) S l i m ^ E ^ ' q + n ( n ) } . B n n n I t i s evident from Theorem 8.3 and the previous two lemmas that the f i l t r a t i o n F pHg(n) o f H q(n) induces a f i l t r a t i o n F p H q ( Y A g X ) o f H q(YA f iX) such t h a t E?^ q (X f Y)=FPH| (Y A B X ) / F D " 1 H q (Y AgX ). 9 . The Kunneth S p e c t r a l Theorem. We have shown, i n the l a s t two s e c t i o n s , the f o l l o w i n g theorem. We w i l l use the notations of the previous s e c t i o n s . Theorem 9 . 1 * Let X and Y be the o b j e c t s i n (Top/B)^ such t h a t H^(X) and H | ( Y ) are f i n i t e as Hg(Sg)-modules. Then, i f H | ( X ) i s p r o j e c t i v e and P X : Y B i n Y i s a Serre f i b r a t i o n , there i s a n a t u r a l , second quadrant s p e c t r a l sequence ^ E r ( X , Y ) , d r(X,Y)} such t h a t (60) I ) E r(X,Y) converges to the o b j e c t Hg (Y AgX), and 11) E ^ ' ^ Y ^ T o r * ! ^ ^ f o r a l l p and q. In o r d e r to see how Theorem 9.1 g e n e r a l i z e s the geometric Theorem 2.3 o f Chapter I I , we note two p r o p e r t i e s o f the a d j o i n t f u n c t o r G : T o p / B — — ( T o p / B ) ^ defined, i n Sec. 4, P r o p o s i t i o n 9.2: The a d j o i n t f u n c t o r G has the f o l l o w i n g two p r o p e r t i e s : 1) f o r p x:X —B i n Top/B, H^(G(X) )=H*(X;k), and i i ) f o r X XgY — B i n Top/B w i t h p x:X — B and P Y:Y — B , G(X XgY)SG(X) A f iG(Y). Proof: The f i r s t p roperty f o l l o w s from the f a c t that3I»G(X) = X v B / M , where 3T i s the f u n c t o r used i n Sec. 5 to extend the o r d i n a r y cohomology. For the second property, we have t h a t (X vB)/\g( Y vB)= (X XgY) V £(b,b)| b&B^ from the d e f i n i t i o n o f the smash product. Hence, the isomorphism above. • Consider the commutative diagram o f t o p o l o g i c a l spaces X x^Y — Y where, as i n Chapter I I , B i s simply connected and the n&p p Y i s a Serre f i b r a t i o n . Then, from Theorem 9.1, we have (61) a s p e c t r a l sequence [ E r , such that i ) E r converges to H*(G(X) A f iG(Y) )=H*(X XgY Sk), and 11) E j P ' ^ T o r ^ l O ) (H*(G(X)),H*(G( Y))) =Tor-f»qO,(H*(X),H*(Y)) f o r a l l p and q. (62) B I B L I O G R A P H Y (1) Mac Lane, Saunders: Homology. Springer??Verlag, New York, 1967. (2) Spanler, Edwin H. : A l g e b r a i c Topology. McGraw-Hill, Inc., 1966. (3) Smith, L a r r y : Lectures on the Eilenberg-Moore S p e c t r a l Sequence., Lecture Notes I n Mathematics, V o l . 134, Sp r i n g e r - V e r l a g , New York, 1970. (4) Smith, L a r r y : Ho mo l o g l c a l Algebra and the Ellenberg' Moore S p e c t r a l Sequence. Transactions o f A.M.S. 129 (1967) , PP. 5 8 - 9 3 . (5) Baum, Paul F. : On the Cohomology o f Homogeneous Spaces. Topology V o l . 7, PP.1 5 - 3 8 . 

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