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The moore spectral sequence for principal fibrations Donmez, Dogan 1979

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THE MOORE SPECTRAL SEQUENCE FOR PRINCIPAL FIBRATIONS by DOGAN DONMEZ B . S c , Middle East Technical U n i v e r s i t y , 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1979 @ Dogan Donmez, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department nf Mathematics The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date Sept 14, 1979 D E - 6 B P 7 5 - 5 1 1 E i i ABSTRACT A proof of the Moore theorem which i n the case of a p r i n c i p a l f i b r a t i o n gives a s p e c t r a l sequence converging to the homology of the base space i s given. A l s o computed i s the algebra s t r u c t u r e of the homology of the Grassmannians, using Hopf algebra techniques and the cohomology of Grassmanians. F i n a l l y , i t i s shown that a s p e c t r a l sequence f o r re g u l a r covering which was constructed e a r l i e r i s a s p e c i a l case of the Moore Theorem. i i i ACKNOWLEDGEMENTS I wish to express my g r a t i t u d e to my advisor E. L u f t f o r h i s very u s e f u l remarks and suggestions. i v TABLE OF CONTENTS Section Page 1. D i f f e r e n t i a l Graded Algebras, Modules and Tor 1 2. C u b i c a l Singular Homology 10 3. L o c a l C o e f f i c i e n t s 12 4. P r i n c i p a l F i b r a t i o n s 13 5. The A c t i o n of T T 1 (B) 15 6. Serre's S p e c t r a l Sequence 18 7. The Moore Theorem 20 8. The Comparison Theorem ......... .m 24 9. Examples 28 Bi b l i o g r a p h y 34 - 1 -1. D i f f e r e n t i a l Graded A l g e b r a s , Modules and Tor [ 5 ] , [11] A graded module i s a c o l l e c t i o n o f modules {A } n o v e r a n n >_ 0 commutative r i n g K w i t h u n i t . A d i f f e r e n t i a l graded module or a c h a i n complex (DG module f o r s h o r t ) {A ,d } „ i s a graded module {A } n  r n n n >_ 0 n n >_ 0 t o g e t h e r w i t h morphisms o f modules d : A A , such t h a t d n d = 0 . ° n n n-1 n-1 n A morphism f : {A } _ -> {B } „ o f graded modules i s a c o l l e c t i o n n n >_ 0 n n >_ 0 { f } „ o f morphisms of modules f : A -> B . A morphism n n >_ 0 n n n f : {A ,d } -* {B ,d'} o f DG modules i s a c o l l e c t i o n { f } ' of .morphisms o f in n n n n n >^  0 modules such t h a t d'f = f .d . n n n-1 n I f A = {A ,d } and B = {B , d'} a r e DG modules th e n a new DG module n n n n A<£> B i s d e f i n e d by ( A ® B) = £ A. ®) B. and d (a ® b) = n i+j=n 1 K 2 n d .a ® b + ( - 1 ) 1 a « d'.b f o r a e A., b e B.. The r i n g K i t s e l f can be c o n s i d e r e d as a DG module by K = K and K = 0 f o r n > 0 and d = 0 f o r o n n a l l n. Then we have A ® K = K(g>A = A f o r any DG module A. A DG module A i s c a l l e d a DG a l g e b r a i f t h e r e a r e morphisms <j) : A ® A -»--A and n : K + A o f DG modules, such t h a t t h e f o l l o w i n g diagrams commute. A ® A ® A / A ® A ® A A(g)A K ®A n ® A A®A -A®K_^?^.A«A L e t A be a DG a l g e b r a , a DG module M i s c a l l e d a DG l e f t A^-module i f - 2 -there i s a morphism <|> : A ® M -> M of DG modules such that the f o l l o w i n g diagrams commute: A DG r i g h t A-module i s defined s i m i l a r l y . I f M, N are two DG l e f t A-modules, a morphism f : M -* N of DG modules i s c a l l e d a morphism of DG l e f t A-modules i f the f o l l o w i n g diagram commutes. A ® M A ® f A ® W M A morphism of DG r i g h t A-modules i s defined s i m i l a r l y . f g Let M,M',M" be graded modules. A diagram M' -> M -* M" of morphisms of ; f n 8n graded modules i s c a l l e d exact i f each M'' M M" i s exact as morphisms n n n r of modules ( i . e . ker g = Imf ). n n - 3 -A diagram M' •»• M I M" of DG modules and t h e i r morphisms i s c a l l e d proper exact i f M' I M I M", Z(M') Z ( £ > Z (M)-H^. z (M" ) and H(M') H ( £?> H(M) H ( g ) > H(M") are a l l exact as graded modules, where Z denotes the graded module of c y c l e s and H the graded module of the homology of a DG module. Exactness f o r sequences of DG modules i s defined s i m i l a r l y . A graded module P i s c a l l e d p r o j e c t i v e i f any diagram P f IT M y N y 0 TT of morphisms of graded modules, where M N 0 i s exact, can be completed ( i . e . there e x i s t s a morphism g : P.->• M such that jt o g = f ) . A DG module P i s c a l l e d a proper p r o j e c t i v e DG module i f any diagram of DG modules and DG morphisms w i t h a proper exact row P f M 77 , N y 0 can be completed. A DG l e f t A-module P i s c a l l e d a proper p r o j e c t i v e l e f t A-module i f any diagram P f M ^ . N y 0 TT of l e f t A-modules and t h e i r morphisms, where M ->• N •> 0 i s proper exact can - 4 -be completed. For any graded module (DG module or DG l e f t A-module) M there i s a pro j e c t i v e (proper p r o j e c t i v e DG or. proper pro j e c t i v e DG l e f t A) module P and a morphism f : P + M of graded modules (DG modules or DG l e f t A-modules f r espectively) such that P -*• M 0 i s exact i n case of graded modules and proper exact i n case of DG or DG l e f t A-modules. [5], [11]. Let M be a DG l e f t . A-module for some DG algebra A. A proper p r o j e c t i v e r e s o l u t i o n X = {X n,e n} for M i s a c o l l e c t i o n {Xn} n of proper p r o j e c t i v e n >_ 0 DG l e f t A-modules, together with morphisms of DG l e f t A-modules n „n „n— 1 - o „o , , , e : X -* X n > 0 , e : X -»• M such that the sequence n e n n-1 1 e 1 o e -»• X -»• X -> ...X X + M + 0 i s proper exact. The statement above insures the existence of a proper p r o j e c t i v e DG l e f t A-module X° and a morphism e°: X° -> M of DG l e f t A-modules such that X° -> M 0 i s proper exact. Since the kernel of e° : X° -> M i s also a DG l e f t A-module, i t follows by using the homology exact sequence of the exact sequence 0 •> Ker e° -»• X° M 0 of DG modules that • 0 -> Ker e° -»- X° i s a proper exact sequence of DG l e f t A-modules. Let X^ " be a proper p r o j e c t i v e DG l e f t A-module and l e t e~ : X"*- -* ker e° be a morphism of i e 1 1 o DG l e f t A-modules such that X -> ker e -> 0 i s proper exact. Then l e t 1 1 o o J. o ^ e : X - -»• ker £ c X . Thus, A - •> X •+ M i s proper exact and r e p e t i t i o n of t h i s process shows the existence of proper pro j e c t i v e resolutions for any DG l e f t A-module M. Also f o r any proper p r o j e c t i v e DG module P, and a DG algebra A, A ® P i s a proper p r o j e c t i v e DG l e f t A-module, and for any DG l e f t A-module M, there exists a proper pro j e c t i v e DG module P and a morphism IT = A (g) P -> M of DG l e f t A-modules, such that A ® P •+• M -> D i s proper exact. This, combined with the previous construction of a proper p r o j e c t i v e r e s o l u t i o n , shows we can - 5 -construct, for any DG l e f t A-module M, a proper projective r e s o l u t i o n X = {X n, cn} such that X n = A ® ~PU for some proper p r o j e c t i v e DG module P n. For a DG l e f t A-module N,and a DG r i g h t A-module M we can construct a DG module M ® N as the cokernel of M ® <t> - <}> <g)N : M ® A <& N + M <2> N. A N M A Let A be an algebra with t r i v i a l d i f f e r e n t i a l s . Let M be a DG r i g h t A module also with t r i v i a l d i f f e r e n t i a l s . Then we can construct a proper p r o j e c t i v e r e s o l u t i o n X = {X n, e11} such that each X n has t r i v i a l n _^ U d i f f e r e n t i a l s [9]. Let N be a DG l e f t A-module also with t r i v i a l d i f f e r e n t i a l s . Let Y = {Y n, en} ^ „ be such a proper p r o j e c t i v e r e s o l u t i o n . n 2 U Define ( X ® Y ) n = { ( X ® Y ) , n n ) b y ( X ® Y ) n = I X 1 <g) Y j, n n ( x ® y) = 1 1 - i+j A e 1(x) ® y + (-1)1 x <g> e^y f o r x e X 1, y e Y^ . Then n n - 1 n n = 0, and each ( X ® Y ) n has t r i v i a l d i f f e r e n t i a l s . Then define tor (M,N) to be H ( X ® Y) as a bigraded module. The same l i n e of proof as i n the case of ordinary modules shows t h i s d e f i n i t i o n i s independent of resolutions chosen and also tor (M,N) = H(X <2) N) = H(M ® Y). A A A Let X = {X n,e n} be a proper p r o j e c t i v e r e s o l u t i o n for a DG l e f t A-module M. Define T(X) by T(X) = 7 X* , d (x) = E ^ X ) + (-D V C X) for x e X^ . n . ,L. j ' n i i x+j=n J J J Then d ,d = 0 and T(X) becomes a DG l e f t A-module. The maps e 1 1 induce n-1 n r a morphism of DG l e f t A-modules e : T(X) M which induces an isomorphism : H(T(X)) + H(M). Let M be a DG right A-module, N a DG l e f t A-module, X a proper proje c t i v e r e s o l u t i o n for M, Y a proper proj e c t i v e r e s o l u t i o n for N. Define Tor (M,N) = H(T(X) ® T(Y)). The same l i n e of argument as i n the - 6 -case of ordinary modules shows that t h i s d e f i n i t i o n i s independent of the resolutions involved and also: Tor (M,N) = H(T(X) ® T(Y)) = H(T(X) ® N) = H(M ® T(Y)) . A A A A Also i f A,M,N a l l have t r i v i a l d i f f e r e n t i a l s we obtain T o r A (M,N) n = I t o r A (M,N)g s+t=n ' Example: Let A = E [x ,x 9,...] be the exterior algebra on generators x. L 2 1 l 1 of degree i , with t r i v i a l d i f f e r e n t i a l s . E [x ,x ,. . . ] = E [x.] Z2 1 1 i > l Z2 1 as algebras. Let 1 X n = E „ [x.l ® Z„ 1y deg Xy = n i , ±y =1, & Z 2 x J 2 'n 6 ; n ' ; o ' i n,_ i i r , i o, ,. „ m 1 r i , r n i n, e (1 ® y ) = x . ® y for n > 1. e (y ) = 0. Then { X , e } n x ^n-1 — o n >_ 0 i s a proper p r o j e c t i v e r e s o l u t i o n for the DG l e f t E [x.] module Z . Therefore {X n, e n } n > where X n = ® xx 1 , e n ( x . ® ® x. ) = "'"ex. ® x. ® . ..® x. + . .. r . X. . X X.. X „ X / J ^ n 1 s 1 2 s i . g + x. ® ... ® e x . , i s a proper p r o j e c t i v e 1 s i , i r e s o l u t i o n for the DG l e f t A-module Then T(X) f c = A®4<£(Z2 y . . ..Z' S y ) 1 n s where k = \ ( n . i . +n.), and H(Z (g) T(X)) = Tor (Z , Z ) . The j J 1 J - ^ A elements z. = l ® l ® 1 y , n > 0 are cycles but not boundaries and x,n •'n J dimension z. = (i+l)n. They generate Tor (Z„,Z ) as an algebra and 1 y XI A . ^ T o r A ( Z 2 , Z 2 ) = r Z 2 [ z 1 > r z 2 j l . z 3 f l , . . . ] = r Z 2 [ z r z 2 , . . . ] the divided - 7 -polynomial algebra on the generators z. of degree i + 1 [11], x Theorem [9],[11] Let A be a DG algebra, M a DG ri g h t A-module, N a DG r i g h t module. Then there i s a f i r s t quadrant.spectral sequence {E ,d }^  > ^ S U C n that a ) El,t " t 0 r H ( A ) (H(M),H(N)) S j t where H(A) i s a DG algebra, H(M), H(N) are DG r i g h t and l e f t H(A) modules re s p e c t i v e l y with t r i v i a l d i f f e r e n t i a l s ; o o b) E = G Tor (M,N) the associated graded module f o r sc some f i l t r a t i o n of Tor (M,N). Proof: Let X = {X n,e n} . be a proper p r o j e c t i v e r e s o l u t i o n f o r the DG n ^_ 0 ri g h t A-module N, such that f o r each n, X n = A ® P R for some proper pr o j e c t i v e DG module P n. Then.Tor (M,N) = H(M ® T(X)). F i l t e r A A T(X) by r e s o l u t i o n degree: F T(X) J X^ i < 8 2 This f i l t r a t i o n i s increasing, bounded below (F_£C(X) = 0) and convergent ([F T(X)] = T(X) ), also each F T(X) i s a DG l e f t A-submodule. n n n s F i l t e r M ® T(X) by: A F (M ® T ( X ) ) = M ® F T(X). s . . s A A This f i l t r a t i o n i s also increasing, bounded below and convergent. Therefore we obtain a convergent 0-spectral sequence {E ,d }^  > ^ of K modules, where: - E° = [F s(M ® T(X))] s + t / [ F s _ l ( M <* T ( X ) ] s + t A A and d° i s the induced d i f f e r e n t i a l and d"*" = H , (F /F ,) -> s,t s+t s s-1 H (F /F „) i s the boundary morphism i n the homology long exact S i t - J . S — -i. S — sequence of the t r i p l e ( ^ s _ 2 ' F s _ i ' F s ^ [12]. But E° = [F (M <g> T(X))] / [F (M®T(X))]- = (M ® X S) ' A ' A since F gT(X) = F g ^T(X) $ X as A-modules, and since the r e s o l u t i o n degree also contributes to the gradation of T(X). Also d° coincides with the d i f f e r e n t i a l on M ® X S . Hence E 1 = H ( M ® X S ) . . But H ( M ® X S ) = A S't 1 A A j r t v s s H(M) Qy H(X ) by [9], since X i s a proper p r o j e c t i v e DG l e f t A-module. H(A) Tracing the d e f i n i t i o n of the boundary homomorphism shows.that d * : E 1 = [H(M) <g> H(X 8).l + [H(M) ® H ( X S _ 1 ) ] . = s,t s,t H ( A ) t R ( A ) c s ±,z coincides with H(M) ® el : H(M) <g> H(X S) + H(M) (& H(X S ) . H(A) ' H(A) H(A) F i n a l l y H(X S) = H ( A ® P n ) = H(A) (g) H(P n) by [9], since P n i s proper g p r o j e c t i v e DG module. Hence H(X ) i s a proper p r o j e c t i v e DG l e f t H(A)-module both with t r i v i a l d i f f e r e n t i a l s '([5], [9]). Since s s s~~l e A : H(X ) -> H(X ) i s a morphism of l e f t H(A)-modules and since s o •* H(X S) H(X ) ... + H(X°) ->•' H(M) -»• 0 i s exact, H(X) = {H(X n), c*} n > Q i s a proper projective r e s o l u t i o n f o r DG l e f t H(A)-module H(M), H(M) with t r i v i a l d i f f e r e n t i a l s . Therefore - 9 -E 2 = H (H(M) H(X)) = t o r (H(M),H(N)) . s , t s>t H(A) n v A / s > c Also as observed before: [Tor (H(M),H(N))] n= j t o r (H(M),H(N)) = J E 2 . s+t=n s+t=n o s F i n a l l y E = [M®X ] a l s o shows that t h i s s p e c t r a l sequence i s a c t u a l l y S , t A t a f i r s t quadrant s p e c t r a l sequence. Let M be a DG l e f t A-module f o r some DG algebra A. Let X = { X 1 1 ^ 1 1 ^ > o b e a proper p r o j e c t i v e r e s o l u t i o n f o r M. On T(X) consider the f o l l o w i n g f i l t r a t i o n , c a l l e d the s k e l e t a l f i l t r a t i o n : F T(X) = I <f>(A®T(X).) = I A - T ( X ) . i<s i<s The s k e l e t a l f i l t r a t i o n i s i n c r e a s i n g , convergent (F T(X) = T(X) ) n n n and bounded below since F_ 1T(X) = 0. Also each F gT(X) i s a DG l e f t A submodule of T(X). I f each X n i s of the form X n = A ® P n f o r some proper p r o j e c t i v e DG module P n, i n the s p e c t r a l sequence corresponding to the skeletal f i l t r a t i o n we have: E s , t = F s T ( X ) s + t / F s - l T ( X ) s + t - A t ® I P 1 and d° = d.c^y P^. Since P^ " are p r o j e c t i v e modules we see ^ = s,t A L j j J s,t H (A) (X) V T^. Since each F T(X) i s a DG l e f t A-module each E° i s a t . . . j s s,t l e f t A module and the d i f f e r e n t i a l s are morphisms of l e f t A modules, o r o Hence each E^" i s a l e f t H (A) module and d"'" are morphisms of l e f t s,t o s,t H (A) modules. A l s o : o - 10 -H (A) <g> = H (A) ® { H (A) ® £ P* } fc H (A) S'° t H (A) ° l+j=s 3 o o H (A) ® I P* l+j=s = E s,t as bigraded modules. Further H t(A) ® EJ H (A) S'° o Ht(A) <g) " C H (A) S X'° o E 1 -> s ,t s,t J s - ' l , t commutes. 2. C u b i c a l Singular Homology [4], [10J: Let I = [0,1] the n-dimensional standard cube, I = one p o i n t . For a t o p o l o g i c a l space X, l e t Q n Q 0 denote the f r e e a b e l i a n group generated by continuous maps c : I N •> X f o r n >_ 0. On the graded group {Q^CX) ) N > Q define maps: : Q (X) -> Q j^X) by (X^c) ( x ^ . . . ,x n_ 1) = c(x ^ , . . . . x ^ ^ ,x ±,. . . x ^ ) where e = 0 or 1, and 1 < i < n. Define d : Q (X) -> Q 1 (X) f o r — — n n . n-± n . 1 n > 0, by d (c) = ).. (-l)1(X°c - ATc), d = 0. Then d .d =0. Let ' J n . U i i o n-1 n i=0 - 11 -Q ( X ) = {Q ( X ) , d } rt be the r e s u l t i n g chain complex. This c o n s t r u c t i o n n n n >_ 0 i s n a t u r a l w i t h respect to continuous maps, i . e . any continuous map f : X -> Y gives r i s e to a morphism Q(f) : Q ( X ) Q(Y) of chain complexes. We a l s o have s^ : Q N ( X ) -»• 0^ ( X ) . defined by (s^c) ( x ^ , . . . > x n + 1 ) = c(x.,,...,x. . , x . x ,..) f o r 0 < i < n+1. Let D ( X ) c Q ( X ) be the 1 i - i l + l n+1 — — n n subgroup generated by the elements {s^c}, c : I n ^ •> X , 1 <_ i <_ n. We have d(D ( X ) ) c D , ( X ) . For any continuous map f : X -* Y, Q(F)(D ( X ) ) c D (Y) n n-1 n n f o r a l l n > 0. Hence we can d e f i n e a fu n c t o r C ( X ) from the category of t o p o l o g i c a l spaces and continuous maps to the category of chain complexes and t h e i r morphisms by C ( X ) = Q ( X ) / D ( X ) . I f X = p o i n t , then C Q ( X ) = Z C Q , c : 1° -»• X the only map and C ( X ) = 0 n > 0, i . e . as DG modules o n C ( p t ) = Z . Let G be an a b e l i a n group. Define C (X,G) = G ® C ( X ) and n n d = G ® d : C (X,G) C , (X,G) , then { C (X,G) ,d } i s a chain complex, n n n n-1 n n Also f o r a continuous map f : X -> Y, G<S)C(f) defines a morphism between C ( X,G) and C(Y,G), t u r n i n g C(G,-) i n t o a n a t u r a l f u n c t o r . Let X,Y be t o p o l o g i c a l spaces., there i s a n a t u r a l morphism p of chain complexes: P : Q ( X ) ® Q ( Y ) -»• Q ( X x Y) Z defined by p(c P® c q ) = c P x c q where c P : I P X , c q : i q ->• Y and C P x c q . xp+q = iP x i q ^  X x Y ( c P x c q ) ( t r . . . , t ) = ( c P ( t 1 5 . . . , t ) , c q ( t l l 9 . . . , t , ) ) . Also p i s a s s o c i a t i v e , i . e . 1 P p+1 p+q - 12 -Q < X ) ® Q ( Y ) ® Q ( Z ) p g > Q ( Z ) , Q ( X x Y ) ( g ) Q ( Z ) Q ( X ) < g ) P Q ( X ) ® Q ( Y x Z ) ^ Q ( X x Y x Z ) commutes. For 1 £ i £ p+1 , p ( s i c P ® c q ) = s i p ( c P ® c q ) and f o r 1 <_ j < q+1 P ( c P ®s.c q) = s , . P ( c P C 3 c q ) . J P+J Hence P defines P : C(X) ® C(Y) ^  C(X x Y ) , which i s a l s o a s s o c i a t i v e . Z For a commutative r i n g K wi t h u n i t , we can define P : C(X,K) ® C(Y,K) + C(X x Y,K) by p ( k c P ® k ' C q ) = kk 'p(c Pcg) c q ) , and K a s i m i l a r diagram f o r a s s o c i a t i v i t y commutes. Since point x point = point C(point, K) becomes a DG algebra and we have C(point, K) = K as DG algebras. 3. L o c a l C o e f f i c i e n t s [ 4 ] , [10] Let X be a t o p o l o g i c a l space. A l o c a l system on X i s a c o l l e c t i o n of groups G^ f o r each x e X, such that f o r every path to : I—>X w i t h w(0) = a, w(l) = b there i s a given isomorphism T : G —*G which only CO c i D depends on the homotopy c l a s s of co and al s o T . , = T • T f o r another path t o ' w i t h to'(0) = <o(l) where: ( tO*U)')(t) = co(2t) f o r 0 ^  t <_ 1/2 OJ' ( 2 t - l ) f o r 1/2 < t < 1 - 13 I f X i s path connected a l l the groups G are isomorphic. We can X choose a base point b, and f o r any other p o i n t x, a f i x e d path to w i t h to (0) = b, to (1) = x. This allows us to i d e n t i f y each G w i t h G = G, X X X D v i a T . to We form a chain complex by s e t t i n g Cn(X,G) = G® Cn(X) and by d e f i n i n g d (g * c n ) = I (g 8 X°c n - T ( c n , i ) g * X 1c 1 1) n i=o 1 where T ( c n , i ) i s the automorphism of G = G^  defined by the path t •> c n(0,0,. . . ,0,t,0,. . .0) and t appears at the i * " * 1 place. Using the f a c t T only depends on the homotopy c l a s s of the path, i t f o l l o w s that d ..d = o. n-1 n 4. P r i n c i p a l F i b r a t i o n s [9] A continuous map p ->r.-E : B i s c a l l e d a f i b r a t i o n i f i t has homotopy l i f t i n g property f o r f i n i t e s i m p l i c i a l complexes, i . e . f o r any f i n i t e s i m p l i c i a l complex A and any commutative diagram of continuous maps: AxO Axl admits H : A x I E such that H = f and p • H = h. o Let X be a t o p o l o g i c a l space and u : X x x X a continuous map, e e E, - 14 -then (X,u,e) i s c a l l e d a top o l o g i c a l monoid i f the following diagrams commute: Xx X xx yxx X x X where p^ and p^ are projections onto the f i r s t and second factors r e s p e c t i v e l y . A f i b r a t i o n p : E + B i s c a l l e d a p r i n c i p a l f i b r a t i o n i f for some point b e B, G = p "*"(b) e e G, there i s a map u : G x E -»• E such that E py E(g,x) = p(x) and (G,u G = ^ EIQ X Q> e) i s a t o p o l o g i c a l monoid and the following diagrams commute: where p„(e,x) = x. - 15 -Therefore when we pass, to the s i n g u l a r c u b i c a l chains over a commutative r i n g K w i t h u n i t we o b t a i n , f o r a p r i n c i p a l f i b r a t i o n w i t h f i b e r G, morphisms of DG modules: cb : C(G,K) ® C(G,K) C(G x G,K) ^—» C(G,K) an a s s o c i a t i v e m u l t i p l i c a t i o n w i t h u n i t : e* n : K = C ( p o i n t , K) + C(G,K) so that C(G,K) becomes a DG algebra and p (w E) A cb : C(G,K) ® C(E,K) -> C(G x E,K) =—»C(E,K) E turns C(E,K) i n t o a DG l e f t C(G,K) module. 5. The a c t i o n of TT (B) [8] Let p : E -*• B be a f i b r a t i o n . Let to : I->B a path from a to b. We can construct T : C ( p _ 1 ( a ) , K ) •> C ( p _ 1 ( b ) , K ) as f o l l o w s : to Let c : 1° p ''"(a) be a O-dimensional cube, l e t be a l i f t i n g of to which s t a r t s at c ( 0 ) . Proceeding i n d u c t i v e l y , when S c i s defined on a l l n-dimensional cubes, f o r an (n+1)-dimensional cube c consider the f o l l o w i n g commutative diagram: ^-r^+1 ^n+1 -dl x i u I xQ -> E I n + 1 x l B - 16 -where f| = S and f| ,., = c X. ( i d , . ) ( I ) x I A.c I x 0 l n+1 x f (x^,. . . > x n +2» t) = co(t). Since g l n + " ' ' x I u i n + - ' - x 0 i s a deformation n+2 r e t r a c t of I , there i s a l i f t i n g S^ of f, using the f a c t that p i s a f i b r a t i o n . Define ( J c)(x ) = S ( x , l ) f o r any s i n g u l a r cube c. By to c c o n s t r u c t i o n dT = T d. Also any other l i f t i n g gives r i s e to a map to CO which i s chain homotopic to f and i f co,co' have the same endpoints and are homotopic to each other by a homotopy l e a v i n g the endpoints f i x e d , the r e s u l t i n g maps and T i are chain homotopic. Hence f o r each homotopy c l a s s of paths j o i n i n g a to b and any^abelian group G : HCp-^.G) + H(p _ 1(b),G) ,^ " which s a t i s f i e s T . , = T ,T . Al s o T = 1 f o r the constant path c o . C0*CO CO CO co Hence T i s an isomorphism f o r any path t o , s i n c e c o * c o and co ^ * c o are.homo-co t o p i c to the constant path, where co "^(t) = c o ( l - t ) . Thus H^(F,G) i s a l o c a l system on B f o r any q > 0. A l o c a l system defines homomorphisms c o ^ : TT^(B) •*• Aut (H n(F,G)). P r o p o s i t i o n 1: Let p : E + B be p r i n c i p a l f i b r a t i o n w i t h f i b e r G, which i s path connected. Then TT^(B) acts t r i v i a l l y on H(F) ,CO^TT^(B)) = 1 f o r a l l n >_ 0. Proof: Let co : S"*" B be a loop. I t w i l l be shown that we can choose T^ : C(G) C(G) to be the i d e n t i t y map. Let p 1 : E' S 1 be the p u l l b a c k f i b r a t i o n , which i s a l s o a p r i n c i p a l f i b r a t i o n w i t h f i b e r G. In the homotopy exact sequence of t h i s f i b r a t i o n p.. : TT (E') TT^  (S"*") i s onto s i n c e G i s connected. Hence there e x i s t s s' : S"*" E' such that p^s' i s homotopic to the i d e n t i t y map on s \ Using the homotopy l i f t i n g - 17 -property we can l i f t t h i s homotopy to E', o b t a i n i n g a s e c t i o n s of p : E' + S 1 w i t h s ( l ) = e. Now de f i n e F : G x S 1 + E' by F(g,x) = (x,y (g, s (x) )) . For a s i n g u l a r cube c : l n -> G, de f i n e S^c : l n x I -* E - - 1 <3 by S c ( x , t ) = u(c(x) ,p„s(t)), where s : I ->• S E' and p : E' E. to z . 4 These maps s a t i s f y a l l the p r o p e r t i e s r e q u i r e d i n the d e f i n i t i o n of the a c t i o n . But then T c(x) = S c ( x , l ) = S c(x,0) = c(x) since p„s(l) = to to to z P2s(0) = e. Therefore TT^(B) acts t r i v i a l l y on H(G). P r o p o s i t i o n 2: Let p : E . + B be a p r i n c i p a l f i b r a t i o n w i t h f i b e r G. Then 3 : TT^(B) -> T r 0(G) i n the homotopy exact sequence of the f i b r a t i o n preserves the m u l t i p l i c a t i o n . Proof: Let to and to^  be two loops i n B s t a r t i n g and ending at b = p ( e ) . Let to^ , to^  be l i f t i n g s of to^  and to^  s t a r t i n g at e. Then to : I -»• E defined by <o(.t) = tOQ(2t) f o r 0 «_ t <_ 1/2 and to(t) = u C ^ C l ) , to^(2t-l)) f o r 1/2 <_ t <_ 1, i s a l i f t i n g of UQ*W-|. s t a r t l n g a t e > Since u ( l ) = y ( f f l o ( l ) , f f l 1 ( l ) ) , we have 9 [to^to^ = [p (2^(1) ,(^(1) ) ] = y( [u ( D l j i . C l ) ] ) = y(9[to ],3[to1]). o 1 o i C o r o l l a r y 1: Under the same conditons of the p r o p o s i t i o n 2, ker 9 aets t r i v i a l l y i n H(G). Proof: Let to be a loop i n B such that 9 [to] = 0. Let p : E' •> S 1 be the p u l l b a c k f i b r a t i o n as i n p r o p o s i t i o n 1. Then the homotopy exact sequences of p and p^ give r i s e to the f o l l o w i n g commutative diagram: TTxCE') — • T T 1 ( S 1 ) — y TT o(G) P 2 * to TT (E) P* TT-i (B) 9 TT (G) l >• ± >- o - 18 -Hence for [ S 1 ] e ir (S 1) we have ^[S1] =9wJS 1] = 3 [ to] = 0. Therefore 1 1 [S ] e TT^(S ) i s i n the image of P-^. Then the same construction as i n proposition 1 shows that [ to] acts t r i v i a l l y on H(G). Corollary 2: If i n addition to the conditions i n proposition 2, E i s path connected, then ^(G) i s a group. Proof: Since E i s path connected 9 : TT (B) •+ TT (G) i s onto and 3 1 o preserves the m u l t i p l i c a t i o n by proposition 2. Proposition 3: Let p : E B be a p r i n c i p a l f i b r a t i o n with f i b e r G, then the action of T r^Bjb) on H(G) i s given by [ to ] • x = (w(l)) ) V(x) where to i s a loop i n B, x £ H(G) , to a l i f t i n g of to starting at e and ( ^ ( l ) ) y . i s the map induced by g 4 u ( g , t o ( l ) ) . Proof: Define S^c : i " x I •> E for c : i n + G by S wc(x,t) = u(c(x),&(t)), which s a t i s f i e s a l l the conditions required i n the d e f i n i t i o n of the action-? But then T^cCx) = S uc(x,l) = u(c(x) , t o ( l ) ) which i s exactly the morphism C(G) •+ C(G) induced by the map G + G defined by g + u ( g , t o ( l ) ) . 6. Serre's Spectral Sequence [10], [12] In his paper "Homologie Singulaire Espaces Fibre" (Ann of Math., 1951) Serre proved that for a f i b r a t i o n p : E -> B with f i b e r F where E i s path connected, and for any commutative ring K, there i s a f i r s t quadrant 1-spectral sequence {E ,d } ^ ^ of K modules, such that: 1) E ^ = H t(F,K)® C s(B) such that d ^ : E ^ + ^g-1,t xs defined by d_, on C(B) and the fact that H (F,K) i s a system of l o c a l a t 2 coefficients as i n section 3. Hence E ^ = H (B ; H (F,K)) with l o c a l s, t s t c o e f f i c i e n t s . CO 2) E = GH(E,K) the associated graded module corresponding to the f i l t r a t i o n of H(E,K). - 19 -More e x p l i c i t l y t h i s s p e c t r a l sequence i s obtained by f i l t e r i n g C(E,K) = K ® C ( E ) where C(E) = the normalized c u b i c a l s i n g u l a r chains of E whose v e r t i c e s l i e i n F = p "*"(b), b e B the base p o i n t . This f i l t r a t i o n i s defined by: F C (E.K) = the submodule generated by cubes c : i n -*• E, such s n that pc : i n -> B i s independent of the f i s t n-s coordinates. Let p : E -*• B be a p r i n c i p a l f i b r a t i o n w i t h f i b e r the t o p o l o g i c a l monoid G. By p r o s p o s i t i o n 3 i n s e c t i o n 5 the a c t i o n of TT^(B) i s given by m u l t i p l i c a t i o n by an element of H q(G). Hence : H ( G , K ) ® C (B) •* H (G,K) ® C q (B) i s given by 1 i o 1 d (x ® c) = £ (-1) (x ® X.c - T.x ® X.c) 1=1 x i x where T^ : H^(G,K) ->• H^_ (G,K) i s the map induced by m u l t i p l i c a t i o n by 6^(1) where ffi^ i s a l i f t i n g of the loop to^ : I -> B ai (t) = c(0,. .. , t , . .. ,0) , where t appears at the i * " * 1 p l a c e , s t a r t i n g at e. In case of a p r i n c i p a l f i b r a t i o n w i t h f i b r e G, the m u l t i p l i c a t i o n H_(G,K) (E> E 1 -> E 1 t v ~ s,o s,t induces an isomorphism H (G,K) (g) E 1 E 1 H (G,K) S ' ° S , T such that - 20 -H (G,K) <S> E 1  C H Q(G,K) S ' ° -> E s,t s,t (-I)1 Ht(G,K)®dJ q H (G,K) <g) Ej -> E Ho(G,K) s - l , t commutes by. inspecting d using proposition 3 i n section 5. s, t 7. The Moore Theorem [6], [9] Let p : E B be a p r i n c i p a l f i b r a t i o n with path connected t o t a l space, and the f i b e r a topological monoid G. Let K be a commutative ring with unit. Let also C(E,K) be the cubical singular chains of B with a l l vertices mapped onto b = p(G). As mentioned i n section 4 C(G,K) i s a DG algebra and C(E,K) i s a DG l e f t C(E,K) module. Let X = {X n,e n} n ^ be a proper projective resolution of C(E,K) as a DG l e f t C(G,K) module such that X n = C(G,K) % P n, where p n i s a proper projective DG module. Then T(X) = C(G,K) (g) T(P). The augmentation e° : X° -> C(E,K) induces a morphism e : T(X) C(E,K) of DG l e f t C(G,K) modules such that : H(T(X)) H(E,K) i s an isomorphism. Consider the sk e l e t a l f i l t r a t i o n on T(X) and the Serre f i l t r a t i o n on C(E,K). e preserves these f i l t r a t i o n s since for any c : I P ->- E and c' : I q -> G, p(c' x c) : I P + q •> B i s independent of the f i r s t q coordinates, therefore c' x c l i e s i n F C(E,K). Therefore e induces q a morphism {e r> r > ±: {E rT(X),d r} -»•' {ErC(E,K) ,dr} of the corresponding - 21 -f i r s t quadrant s p e c t r a l sequences of K-modules. The isomorphism e : H(T(X)) -> H(E,K) induces a l s o an isomorphism of the corresponding f i l t r a t i o n s e.I: F H(T(X)) -> F H(E,K), and hence an isomorphism *"' s s £°°: E°°(T(X)) E°°(C(E,K)) . . Also E : E T(X) E C(E,K) i s a morphism of bigraded l e f t H(G,K) modules and the f o l l o w i n g diagram commutes s i n c e i n both f i l t r a t i o n s the submodules are submodules of C(G,K) modules: H.(G,K) €5 E T(X) C H o(G,K) S'° -> E T(X) s,t H (G,K)<2 E t s, o ~s,t H ( G , K ) $> E 1 C ( E , K ) > E 1 C ( E , K ) H ( G , K ) S ' ° S , T o Since T(X) and C ( E , K ) are l e f t C ( G , K ) modules f o r a l l n and the d i f f e r e n -n n o t i a l s are morphisms of C Q ( G , K ) modules, t h e r e f o r e {E T(X),d }^  > ^ and {E C ( E , K ) , d ^ r > i a r e s p e c t r a l sequences of H Q ( G , K ) modules and {e }^  > ^ i s a morphism of H q ( G , K ) s p e c t r a l sequences. 2 P r o p o s i t i o n 1: Under the c o n d i t i o n s above i f E G i s an isomorphism 2 f o r s < n and an epimorphism f o r s = n, then the .same i s true f o r e f o r s, t any t . Proof: E C(E,K) = H (G,K) <8> C (B) i s a f r e e H (G,K) module sin c e s, o o s o C (B) i s a f r e e a b e l i a n group. E 1 T(X) = H (G,K) ® £ P^ i s a p r o j e c -s ,o i+j=s J t i v e H O ( G , K ) module sin c e ^P^ i s a p r o j e c t i v e K-module. (Since £p^ i s a p r o j e c t i v e i t i s a d i r e c t summand of a f r e e K-module F, then H Q ( G , K ) # £p^ i s a d i r e c t summand of the f r e e H ( T , K ) module H ( G , K ) < £ > F.) o o - 22 -Therefore the mapping c y l i n d e r ( [ 5 ] , [8]) M of : E* T(.X) E* C(E,K) i s a complex of p r o j e c t i v e l e f t H (G,K) modules (M = E 1 C(E,K)© E 1 ., T( X ) ) , and the long exact sequence f o r n n,o n-l,o ° n 1 2 e A ( [ 5 ] , [8]) the hypothesis f o r e correspond to H (M) = 0 f o r y O S j O S s <^  n. Since are a l l p r o j e c t i v e , we can then construct a ' p a r t i a l ' c o n t r a c t i n g homotopy ([12]), i . e . morphisms : >L •> M ^ + 2 ~ x £. 1 1. n d.,..T. + T . ,.d. = M. e x a c t l y as i n the case of a c y c l i c complexes [12]. l + l l i - i l l J Now consider M, the mapping c y l i n d e r of H (G,K) ® el : H (G,K) ® ^ - > , H ' . ( G , K ) ® E * . H O ( G , K ) H Q ( G , K ) H Q(G,K) Then M = (G,K)® M, th e r e f o r e H (G,K)® x ± gives a ' p a r t i a l ' c o n t r a c t i n g homotopy f o r M, therefore H s(M) = 0 f o r s <_ n, th e r e f o r e H (G,K)02) e ^ induces isomorphism up to dimension n, and an epimorphism t , o on dimension n. And the f o l l o w i n g commutative diagram H ( G , K ) <g> E * T ( X ) Z H ( G , K ) ' ° H ( G , K ) « e. t *, o y H (G,K) ® E ! C(E,K) E * , t T ( X ) '*,t H Q ( G , K ) .C(E,K) shows that e i s an isomorphism f o r s < n and an epimorphism f o r s, t s = n f o r any t . This r e s u l t using the Comparison Theorem, which i s proved i n - 23 -2 s e c t i o n 8, guarantees that e i s an isomorphism f o r a l l s and t . In s, t 2 p a r t i c u l a r e i s an isomorphism f o r a l l s, i . e . s,o { e 1 } : { E 1 T C X ) ^ 1 } •> { E 1 CCE.lQ.d 1 } s,o s,o s,o s,o s,o induces isomorphisms between t h e i r homologies. Therefore the same proof as i n p r o p o s i t i o n 1 proves t h a t : K <g) el : K ® E^ T(X) -> K ® 1 H Q(G,K) *° H Q(G,K) *° H o ( G , K ) E * , o C ( E ' K ) induces an isomorphism between t h e i r homologies. But K Ig) T(X) = K ® ' {H (G,K) <g> = K ® T(X) H (G,K) *° H (G,K) ° J C(G,K) o o and K <g) EJ C(E,K) = C(B,K) ^ * o H CG.K) ' as DG modules, hence Tor ' (K,C(E,K)) = H(B,K). C(G,K) Theorem: (Moore [ 6 ] , [9]) Let p : E + B b e a p r i n c i p a l f i b r a t i o n w i t h f i b r e the t o p o l o g i c a l monoid G. Then f o r any commutative r i n g K with u n i t , there i s a f i r s t quadrant O - s p e c t r a l sequence {E ,d } > Q of K modules such t h a t : 1 } E s , t " t 0 r H ( G , K ) ( K ' H ( E ' K ) ) s , t o o 2) E = GH(B,K) the associated graded module corresponding to some f i l t r a t i o n of H(B,K). Proof: The theorem i s j u s t the l a s t statement combined w i t h the s p e c t r a l sequence f o r Tor, i n s e c t i o n 1. - 24 -C o r o l l a r y : Let p : EG + BG be a c l a s s i f y i n g f i b r a t i o n , i . e . a p r i n c i p a l f i b r a t i o n w i t h f i b e r G, such that EG i s c o n t r a c t i b l e . Then H(BG,K) = T o r c ^ G ^ ( K , K ) f o r any commutative r i n g K w i t h u n i t . Proof: Consider the augmentation e : C(EG,K) K which i s a morphism of DG l e f t C(G,K)-modules and induces an isomorphism between t h e i r homologies, since EG i s c o n t r a c t i b l e . Then e induces a morphism 10 10 TO -v" — ~£ {e } n of s p e c t r a l sequences {E ,d } _ and {E ,d } _ which r >_ 0 ^ r > 0 r > 0 converges to T o r ^ g R^(K,C(EG,K)) and T o r c ^ G R^ (K,K) r e s p e c t i v e l y . But E s , t " t 0 r H ( G , K ) ( K ' H ( E G ' K ) ) s , t a n d E s , t = t 0 r H ( G , K ) ( K ' K ) s , t 2 since z i s an isomorphism, e i s there f o r e an isomorphism. Therefore oo e i s an isomorphism. This proves T° rC(G,K) ( K> £ ) : T o r C ( G , K ) ( K ' C ( E G ' K ) ) ^ T ° r C ( G , K ) ( K ' K ) i s an isomorphism using the F i v e Lemma. Hence T° rC(G K ) ( K ' K ) = H ( B G ' K > -8. The Comparison Theorem [ 3 ] , [15] Theorem: Let {E ,d } „, {E ,d } „ be two f i r s t quadrant s p e c t r a l r > 2 ' r > 2 TO TO —IT ~ r sequences, f : {E ,d } -> {E ,d } a morphism of s p e c t r a l sequences which 0 0 0 0 — 0 0 induces an isomorphism f : E ->• E . Suppose al s o that f has the 2 2 f o l l o w i n g property P: I f f i s an isomorphism f o r 0 < s < n, and f i s s j o n y o 2 2 an epimorphism then f g ^ i s an isomorphism f o r 0 < s < n and f ^ ^  i s an epimorphism f o r any t . 2 Then f i s an isomorphism f o r any s and t . s, t We f i r s t prove the f o l l o w i n g Lemmas [3], [15]: Lemma 1: Let 0 = K cLc'.. .c'R = K and 0 = L c L . c . c L = L be o 1 n o 1 n two f i l t e r e d modules. Let f : K -> L be a morphism of modules pre s e r v i n g - 25 -these f i l t r a t i o n s . a) I f f : K + L i s an epimorphism and i f f ^ : K j < . / K j c _ ^ ^y/^\-± ^s not an-epimorphism, then f. : K./K. . L./L. . i s not a monomorphism J J J - l J J - l f o r some j > k. b) I f f : K ->• L i s a monomorphism and i f f. : K./K. ,-»- L./L. , i s J J J - l J J - l not a monomorphism, then f ^ : K^ /K^ ..-^  L k / L k - 1 ^ S n 0 t a n eP-'- m o rP' l l s i n f o r some k < j . Proof: a) Let j the the l a r g e s t number such that f i s not an isomorphism, then j > k. By the F i v e Lemma and i n d u c t i o n f. : K. -»• L. ~ J J J i s an epimorphism. Therefore f i s an epimorphism, t h i s proves j > k. Since f i s not an isomorphism i t can not be a monomorphism. b) Let k be the smallest number such that f. i s not an isomorphism, then k <_ j by the Fiv e Lemma. The Fiv e Lemma a l s o proves that f k i s a monomorphism, hence k < j . f ^ can not be an epimorphism because t h i s would by the F i v e Lemma force f to be an isomorphism, K. Notation: Let { E r , d r ) r ^ Q be a f i r s t quadrant s p e c t r a l sequence. Define [n,p,s] = d P _ S ( E P ~ S ) c E P _ S , f o r p-s > 2. Then [n,p,s] = 0 p,n-p s,n-l-s — unless s >_ 0 and n > s. Let [n,p,s] = 0 f o r p-s < 2. Let a l s o r r 2 B ^ and Z ^ be the submodules of E which correspond to the s,t s,t s,t r r boundaries and cycles, of E g t r e s p e c t i v e l y f o r r >_ 2 (since E g fc f o r 2 r >_ 2 i s a subquotient of E ). Then we get a f i l t r a t i o n of s, t E 2 : 0 c B 2 c B 3 c ... c B t + 1 c Z S c ... c Z 3 c Z 2 c E 2 s,t s,t s,t s,t s,t s,t s,t s,t wi t h the associated graded modules [s+t+1, s+2, s ] , [s+t+1, s+3, s]. [s+t+1, s+t+1, s ] , E°° , [s+t, s , 0 ] , [s+t, s , , . l ] , , [s+t, s, s-2]. s J t . Let f : E -> E be a morphism of s p e c t r a l sequences where E and E are f i r s t quadrant s p e c t r a l sequences. Suppose f : E ->• E i s an isomorphism. - 26 -Lemma 2: [3], [15] In a d d i t i o n to the conditons above suppose a l s o 2 2 f i s an isomorphism for: s <c p and t < n - l - s and. f i s an epimorphism s, t P > t f o r t < n - l - s . Then f : [n,p,s] -y [n,p,s] i s an isomorphism. Proof: The c l a i m i s t r i v i a l l y v a l i d unless 2 < s+2 < p < n. Suppose f : [n,p,s] •+ [n,p,s] i s not an epimorphism f o r 2 < s+2 < p < n. Consider: 0 c B 2 c B 3 ... B n _ P + 1 c Z P c . . . c Z 2 c E 2 p,n-p w i t h the as s o c i a t e d graded modules [n+1, p+2, p ] , [n+1, n+1, p ] , oo E _ , [n,p,0], ...[n, p, p-2]. Since p > s+2, we have n-p < n - l - s . p ,n p -2 Therefore f ^ ^ i s an epimorphism. Then by Lemma l a f : [n,p,s^] -»• [n,p,s^] i s not a monomorphism f o r some s < s^ < p-2. Now consider: 2 n - S l S l 2 2 O c B c B c Z c . . . c Z c E s ^ , n - l - s ^ w i t h the,associated graded modules [n, s^+2, s ^ J , [n, s^+3, s ^ ] , ... 0 0 [n, n, s 1 ] , E g n_i_s > [n-1, s 1 , 0 ] , [n-1, s.^2]. By 2 hypothesis f • _, i s an isomorphism, t h e r e f o r e by Lemma l b there s^, n X—s^ e x i s t s p^ w i t h s^+2 <_p^-< P such that f : [n, p^, s^] [n, p^, s^] i s not an epimorphism. Now we can repeat the same argument o b t a i n i n g i n t e g e r s . . . > > s^ > s and p > p^ > Y>2 > s a t i s f y i n g p_^  >_ s^+2, which i s impossible. This c o n t r a d i c t i o n proves that f : [n,p,s] -* [n,p,s] i s an epimorphism. Suppose f : [n,p,s] -> [n,p,s] i s not a monomorphism. Consider 0 c B 2 c . . . B n _ S c Z S c . . . c Z 2 c E 2 s, n - l - s with the as s o c i a t e d graded modules [n, s+2, s ] , [n, s+3, s ] , ... [n,n,s], - 2 7 -o o 2 E , [ n - 1 , s, 0 ] , [ n - 1 , s, s - 2 ] . Since f . i s an s, n-±-s s, n - l - s isomorphism by hypothesis, Lemma l b provides us w i t h p^, s+2 < p^ < p such that f : [n,p^,s] [n,p ,s] i s not an epimorphism. Now consider 2 n - P ! _ + 1 P i 2 ? O c B c , . . c B c Z X c . . . c Z c E P l ' n _ P l w i t h the associated graded modules [n+ 1 , p 1 + 2 , p ^ ] , [n+ 1 , p + 3 , p ^ , .. [n+1 , n+ 1 , E- ^ n _ p , [n, p ^ O ] , [n, p l 5 s ] , [n, p . ^ p . ^ 2 ] , 2 1 1 Since f i s an isomorphism by hypothesis, Lemma l a provides us P 2 » n~v1 w i t h s^, Vj~2 L si > s s u c h t n a t f : [^.p^s^] [ n , p 1 , s 1 ] i s not a monomorphism. Repeating the argument shows the existence of int e g e r s p., s i such t h a t : l p > p^ > p^ > ... and s < s^ < s^ < ... such that p_^  >_ s^+2 which i s impossible. This c o n t r a d i c t i o n proves that f : [n,p,s] -> [n,p,s] i s a l s o a monomorphism, hence an isomorphism. Proof of the Comparison Theorem: By v i r t u e of the property P i t 2 s u f f i c e s to prove f i s an isomorphism f o r a l l s > 0 . Hence i t s,o — s u f f i c e s to prove f o r a r b i t r a r y n the statement 2 2 I : f i s an isomorphism f o r s < n and f i s an epimorphism. n s,o n,o We proceed by i n d u c t i o n on n. 2 0° Since both E and E are f i r s t quadrant s p e c t r a l sequences E = E n r n 090 0,0 —2 —00 2 0 0 —2 —00 2 2 E = E , E. = E n and E.. = E., , hence f and f_ are 0,0 0,0 l , o l , o l , o l , o 0,0 l , o isomorphisms. Therefore 1^ holds. Suppose I has been proven. To show I ,. i s true i t s u f f i c e s to n n+1 2 2 show f n q i s a monomorphism and f n + ^ 0 a n epimorphism. Consider ^ r,n „2 „ 2 O c Z c . . . c Z c E n,o - 28 -wit h associated graded modules E°° , [n,n,0], [ n , n , l ] , [n,n,n-2]. n, o 2 2 By i n d u c t i o n hypothesis f i s an isomorphism f o r s < n and f i s an J ; r s,o nvo 2 epimorphism. T h i s , using the property P shows that f i s an s, t 2 ismomorphism f o r s < n and f . i s an epimorphism f o r any t . This, by n, t Lemma 2 shows that f : [n,n,s] [n,n,s] i s an isomorphism f o r any s. 2 Therefore using the F i v e Lemma repeatedly shows that f ^ q i s a c t u a l l y an isomorphism. 2 To prove f n + ^ Q 1 S a n epimorphism consider: n 7 n + 1 7 2 p 2 n+1, o wi t h associated graded modules E™ +^ G ' ^ n +^' n + x>0]> ••• [ n + x> n+1, n-1] 2 Again by the Fiv e Lemma i t s u f f i c e s to show f ,n induces epimorphisms & J n+l,o on the associated graded modules. Suppose f : [n+1, n+1, p] ->• [n+1, n+1, p] i s not an epimorphism f o r some 0 < p < n-1, consider: 0 c B 2 c . . . c B n - P + 1 c Z P c . . . Z 2 c E 2 p, n-p wi t h associated graded modules [n+1, p+2, p],...,[n+l, n+1, p ] , oo 2 E , [n,p,0], [n,p,p-2]. Since f i s an isomorphism, p, n-p p, n-p Lemma l a provides us wi t h s, 0 <_ s < p-2 such that f : [n,p,s] ->- [n,p,s] i s not an monomorphism contrary to Lemma 2. This c o n t r a d i c t i o n shows 2 that f , i s an epimorphism. n+1, o 9. Examples Let G = 0 = |^ _J 0(n). 0 can be given a c e l l complex s t r u c t u r e n >_ 0 such that the c e l l u l a r chains C(0,Z ) = K ® E [x ,x_, . . . J where E^ [x^,X2>--.] i s the e x t e r i o r algebra on generators x^ of dimension i 2 and K i s the r i n g Z^CtJ/Cl+t ) considered as a graded algebra over - 29 -Z^, C(0,Z^) has t r i v i a l d i f f e r e n t i a l s [13J. Then by the example i n s e c t i o n 1: Tor^ ( Z 2 , Z 2 ) = T ^  [y^ > y 2 > ' ' - ] t h e d i v i d e d polynomial algebra on y^ dim y. = i+1, where A = E [ x . ] . Also Tor (Z0,Z„) = r„ [y] dim y = 1. i ^2 l K z z ^2 Therefore H(B0,Z ) = r [z.] dim z. = i , i = 1,2,... as graded vector . z 2 i i spaces. S i m i l a r l y f o r SO = U SO(n) we have H(BSO,Z ) = T [z ,z , n >_ 0 Z Z 2 Z j as graded vector spaces, but not as algebras [14]. ft BO, the Grassmannians, i s . a t o p o l o g i c a l monoid and H ( B 0 , Z 2 ) = Z2 [co^,a)2> • • • ] the polynomial algebra generated by the S t i e f el-Whitney c l a s s e s co^  of dimension i [ 8 ] . The c o m u l t i p l i c a t i o n i n H ( B 0 , Z 2 ) , induced by the m u l t i p l i c a t i o n on BO, by Cartan's formula, i s given by n * A(co ) = T co. ® co . , 10 = 1 . These operations turn H (BO.Z.) i n t o n , L n 1 n - i o 2 1=0 Hopf algebra [ 7 ] . Since Z 2 i s a f i e l d H A ( B 0,Z 2) i s the dual Hopf algebra. P r o p o s i t i o n : H A ( B 0 , Z 2 ) as an algebra, i s a polynomial algebra on generators x^ of dimension i , i = 1,2,... . ft Proof: I t s u f f i c e s to prove that i n H ( B 0 , Z 2 ) there i s only one p r i m i t i v e element i n each dimension, since t h i s , using the d u a l i t y of P(H ( B 0,Z 2)) and Q(H^(B0,Z 2$ [ 7 ] , forces H^(B0,Z 2) to have only one indecomposable element i n each dimension and f o r dimensional reasons they cannot have any r e l a t i o n s . Let p , n >_ 1. be defined i n d u c t i v e l y by p - p ..co-. + .. . + p..co 1 i f n i s even n n-1 1 r l n-1 - 30 -Claim: p^, n >_ 1 are the only p r i m i t i v e elements i n H (BO,!^). Proof: p^ •- OJ^ i s c l e a r l y p r i m i t i v e . Suppose by i n d u c t i o n i t has been proven that p^, PN_-^ are p r i m i t i v e . I f n i s even: p = p + ... + p 0) n n-1 1 1 n-1 A(p n) = M p ^ u ^ ) + . . . + A ( P 1 c o n _ 1 ) = A ( p n _ 1 ) A ( u 1 ) + . . . + A ( p 1 ) A ( t o n _ 1 ) n-1 = (1 ® P ^ I + P J J . ^ ® 1) (1 ® UJ^+OJ-^ ® 1) + . . . + (1® P 1 + P 1 ® 1) ( £ UK ® " N _ 1 _ I ) i=0 n-1 n - i n - i n - i - i = i ®p +p ®i + y y ai. ® P . .+ y y to.p , . ® t o . n n i - l j - l 1 J n - x - J j = l i=o ^ - ^ - J J -1 n- i - 1 But n - i n - i ui. ® to . i odd l n - i y t o . ® p.to . . = t o . ® y p.to . . = i .L± i j n - i - j i .L± * j n - i - j Q l even and n-j-1 n-j-1 y to.p . . ® a i . = c • y to.p . .) ® t o . = •< i - o I . n _ 1 - J J i - o l n _ 1 - J J to . eg t o . j odd j even. Therefore a l l undesired terms cancel proving p^ i s p r i m i t i v e . I f n i s odd: A(p n) = A ( ( V , " P 1 C V ; L + + p n - l W l ^ = h^r? + A ^ P l u n - l ^ + " ' " + A ^ P n - l ^ A ^ l ^ y t o . ® t o .+ (l®p ,+p T ® 1) (1 ® t o , + t o , ®1)+. , L n l n - i n-1 n-1 1 1 i=0 n-1 + (l®p +p ®1)( y t o . ® t o . . ) 1 1 . L n l n - l - i i=0 n-1 n-1 n-i n-1 n-j 1 ® p n + p n ® 1 + I c o . ® u + I I t o . ® P u + J . I co P • to , i = l i=0 j = l J j = l i=0 J - 31 -But n-i n-i ) 10, 8 P-OJ . . = OJ. ® / P.OJ . . = { j = l 1 J n _ : L - J 1 j = l n ~ 1 ' 2 to. ig to . i even l n-i i odd and n-i-1 - 1 - 1 n-j-1 / OJ.p . . ® OJ. = ( / O J . p . . ) <8> OJ. = i = 0 1 ^ " J 3 i=0 1 n - x - J J oj . ® OJ. j even j odd . Therefore a l l undeslred terms cancel proving p^ i s p r i m i t i v e . To prove the uniqueness, consider the exact sequence [7]: 0 -»- P(?H*(B0,Z 2)) + P(H*(B0,Z 2)) -> Q(H*(B0,Z 2)) where £H (B0,Z 2) i s the subalgebra of H (B0,Z 2) generated by the squares. ft ft Hence CH ( B 0 , Z 2 ) n = 0 f o r n odd. Since QH (B0,Z 2) has only one nonzero element [OJ ] i n each degree n and f o r odd n, p -»• [OJ ], there can not n n n be any other p r i m i t i v e element of odd degree. I f p 2 n i s a p r i m i t i v e element of degree 2n, then p 2 n i s decomposable. The reason f o r t h i s i s that i f p ^ i n i t s expansion i n terms of U K ' s contain 0 J 2 n i t can not be p r i m i t i v e , since i n MP2 n) the term l i i n^ u n w°uld appear and i t 2 cannot be cancelled by any other term because A(OJ^) does not contain W n ® W n * Suppose p 2 n i s another p r i m i t i v e element, then both p 2 ^ and ft P 2 n l i e i n CH (B0,Z) because of the above exact sequence, hence they have square roots x^ and y^ r e s p e c t i v e l y . But i n a polynomial algebra over Z 2 the square root of a p r i m i t i v e element i s p r i m i t i v e . I f deg x^ = deg y^ = n i s s t i l l even we can repeat the same argument u n t i l 2 k 2 k we get x k , y k such that deg x f c = deg y f c odd, ^  = p 2 n and y k = p ^ , Xk'^k ^ o t ^ P r i m l t i v e . But the argument i n the beginning of the - 32 -uniqueness proof shows that x^ = y^, hence p^ n = Pzn» proving the uniqueness. Remark: In [1] the following theorem has been proved: Theorem: Let p : E ->• B a regular covering, let TT = T ^ W / P ^ T T ^ C E ) . r 10 Then there i s a f i r s t quadrant 2-spectral sequence { E ,d }^  > ^ s u c n that: 2 ~~ 1) E = H (TT;H ( E ) ) where TT acts on E as covering transformations S 3 t t S turning H ( E ) into a ir-module, and H (TT;'E ( E ) ) is the group homology s s, with coefficients i n IT [2] . OO 2) E = GH(B) the associated graded group for some f i l t r a t i o n of H(B). Theorem: This spectral sequence is the same as the Moore spectral sequence of the principal fibration p : E -> B with fiber the discrete group TT. Proof: The spectral sequence is obtained by f i l t e r i n g C 81 C ( E ) TT by resolution degree, where C = {C ,d } _ and J n n n >_ 0 d d c + 1 c T -»-....- C- C + Z + 0 n n-1 1 o is exact and TT acts on C on the l e f t , d preserves the action of TT and n n each C i s Tr-free [2]. C 8 C ( E ) is the quotient o f C a C ( E ) by the n TT submodule generated by {a ® sb - as ® b} a e C, b e C ( E ) , s e TT. If we identify TT with the fiber G, we see C(G,Z) = Z(TT) where C(G,Z) = C(G) = the normalized cubical chains of G. With this terminology a Tr-free abelian group is nothing but a free Z(TT)-module. Therefore C is a proper projective resolution for C(G,Z) = Z(Tr)-moduie Z. Also C ® C ( E ) = C ® C ( E ) = C. ® C ( E ) . Since the f i l t r a t i o n is obtained 1 1 Z(TT) C(G,Z) - 33 -by the resolution degree this spectral sequence coincides with the Moore spectral sequence. We also obtain: t o r H ( G , Z ) ( Z » H ( B ' Z ) ) s , t = Ht<*;H8<B,Z)) - 34 -BIBLIOGRAPHY 1. H. Cartan: Seminaire Cartan 1950/1951 Exp 11,12. 2. S. E l l e n b e r g : Seminaire Cartan 1950/1951 Exp 1,2. 3. J.P. H i l t o n - J . Roitberg: On the Zeeman Comparison Theorem of Quasi N i l p o t e n t F i b r a t i o n s . The Quarterly Journal of Math. Dec. 1976. 4. J.P. H i l t o n - S. Wylie: Homology Theory, Cambridge U n i v e r i s t y Press. 5. S. MacLane: Homology, Springer-Verlag. 6. J.P. May - V.K.A.M. Guggenheim: On the Theory and A p p l i c a t i o n s of D i f f e r e n t i a l Torsion Product. Memoirs of AMS. V o l . 142. 7. J . Milnor - J.C. Moore: On the Stru c t u r e of Hopf Algebras. Ann. of Math. V o l . 81, 1965. 8. J . M i l n o r - J.D. Stasheff: C h a r a c t e r i s t i c Classes. AMS Studies. P r i n c e t o n U n i v e r s i t y Press. 9. J.C. Moore: Seminaire Cartan 1959/1960 Exp 7 10. J.P. Serre: Homologie S i n g u l a i r e Espaces F i b r e . Ann., of Math. 1951. 11. L. Smith: The Eilenberg-Moore S p e c t r a l Sequence. Transactions of AMS. V o l . 129 1967. 12. E.H. Spanier: A l g e b r a i c Topology, Academic Press. 13. N. Steenrod - D.B.A. Ep s t e i n : Cohomology Operations. Ann. of Math Studies 50. Pri n c e t o n U n i v e r s i t y Press. 14. R.M. Switzer: Homology Comodules. Inventiones Mathematicae 1973. V o l . 20. 15. C.H. Zeeman: A Proof of the Comparison Theorem f o r S p e c t r a l Sequences. Proc. of Camb. P h i l . Soc. V o l . 53, 1957. 

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