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Cartesian products of lens spaces and the Kunneth formula Verster, Jan Frans 1976

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CARTESIAN PRODUCTS OF LENS SPACES AND THE KUNNETH FORMULA by JAN FRANS VERSTER B. Math., U n i v e r s i t y of W a t e r l o o , 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE DEPARTMENT OF MATHEMATICS We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA May, 1976 @ J a n F r a n s V e r s t e r , 1976 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I. f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e H e a d o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mgrtonat-ifg  The U n i v e r s i t y o f B r i t i s h C o l u m b i a 20 75 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 D a t e A p r i l 30, 1976 i i A b s t r a c t The graded cohomology groups of a c a r t e s i a n product of two c e l l u l a r spaces are e x p r e s s i b l e i n terms of the cohomology groups of the f a c t o r s . This r e l a t i o n s h i p i s given by the ( s p l i t ) short exact Runneth sequence. However the m u l t i p l i c a t i v e s t r u c t u r e on the cohomology of a c a r t e s i a n product can i n general not be derived by s o l e l y r e f e r r i n g to the Runneth formula. In t h i s t h e s i s we e x p l i c i t l y e x h i b i t the cup product s t r u c t u r e on a c a r t e s i a n product of two (standard) lens spaces. This r e s u l t i s obtained by an a l y z i n g the Runneth sequence and by making use of the p a r t i c u l a r geometry of the spaces i n v o l v e d . i i i Contents Chapter 1 I n t r o d u c t i o n 1 2n*~l Chapter 2 The Cohomology Ring of the Lens Space L^ 3 2.1 D e f i n i t i o n s 3 o _ -i 2.2 The A d d i t i v e S t r u c t u r e of H*(L ) . 4 P 9n —1 2.3 The M u l t i p l i c a t i v e S t r u c t u r e of H*(L p ) 7 Chapter 3 The Cohomology Ring of C a r t e s i a n Products 12 of Standard Lens Spaces. Chapter 4 The Main Res u l t 20 4.1 The Reduced Cohomology of L 2 n _ 1 x L 2™" 1 20 p q 4.2 C o r o l l a r y H*(RP n x Rp m) 21 Bi b l i o g r a p h y 23 iv Acknowledgements The author would l i k e to thank Dr. Rene* P. Held and Dr. Denis Sjerve f o r t h e i r many h e l p f u l suggestions and t h e i r p a t i e n t readings of t h i s t h e s i s . I would a l s o l i k e to thank the N a t i o n a l Research C o u n c i l and the U n i v e r s i t y of B r i t i s h Columbia f o r t h e i r generous f i n a n c i a l support over the previous two years. 1. I n t r o d u c t i o n The Kiinneth formula r e l a t e s the cohomology groups of a c a r t e s i a n product of t o p o l o g i c a l spaces to the cohomology groups of i t s f a c t o r s . I f X and Y are ( f i n i t e ) CW-complexes, then the Kiinneth formula can be w r i t t e n i n the form of a short exact sequence of c e l l u l a r cohomology groups (with i n t e g e r c o e f f i c i e n t s ) as f o l l o w s : 0 -» (H*(X) ®H*(Y)) H q(XxY) Tor (H*(X) ,H*(Y)) ,. —> 0 q q+1 where m denotes the cross product f o r c e l l u l a r cohomology. I f X and Y are spaces w i t h d i s t i n g u i s h e d base p o i n t s , then we get an analogous sequence f o r reduced cohomology: 0 (H*(X) ®H*(Y)) - 2 * H q(X AY) • Tor(H(X),H(Y)) -»• 0 q q+1 where XAYV==X><Y/XV.Y ^'denotes the smash product and m i s the cross product f o r reduced c e l l u l a r cohomology. S i m i l a r IKunneth). formulas have a l s o been obtained f o r e x t r a -o r d i n a r y cohomology t h e o r i e s h* ; see e.g. [ATIYAH], [ADAMS]. In general the cohomology of XAY and the cohomology of the f a c t o r s X and Y are r e l a t e d by the so c a l l e d .Kiinneth s p e c t r a l sequence i n v o l v i n g higher Tor-terms: E V ' q = Tor A(h*X,h*Y) J E P ' q = Gr h q(X*Y) 2 p ' p+q » p where A = h*S° . - 2 -Although the a b e l i a n group s t r u c t u r e of h*(XxY) seems to be (at l e a s t t h e o r e t i c a l l y ) computable, there s t i l l remains the problem of f i n d i n g the m u l t i p l i c a t i v e s t r u c t u r e of h*(XxY) . The main purpose of t h i s t h e s i s i s to a c t u a l l y compute the cohomology r i n g of a c a r t e s i a n product of lens spaces, i . e . H * ( L 2 n _ 1 x L 2" 1" 1) . p q According to the .Kiinneth formula, H.*(l?n~^~ x L 2 m _ 1 ) contains p q the "tensor product p a r t " H * ( L 2 n _ 1 ) ® H * ( L 2 m _ 1 ) as a subring. The "Tor p a r t " , however, can only be i d e n t i f i e d w i t h a subgroup of H * ( L 2 n _ 1 x L 2" 1" 1) . p q For a p r e c i s e statement of the r i n g s t r u c t u r e of H * ( L 2 n """ x L 2 m ^) , we r e f e r to s e c t i o n 4. We would l i k e to point out some i n t e r e s t i n g f e a t u r e s of the m u l t i -p l i c a t i v e s t r u c t u r e of the r i n g H * ( L 2 n x L 2 m "*") . There are gener-2n.~l 2 m 1 a t o r s a ,3,x e H*(L x L ) (see s e c t i o n 4 ) , where a and 3 P q are the generators c a r r i e d by the f a c t o r s of the tensor product H * ( L 2 n _ 1 ) ® H * ( L 2 m - 1 ) , and x generates R 3 ^ - 1 x L 2*- 1) * T o r C H 2 ^ " - 1 ) ^ 2 ! ^ 1 ) ) , - Z ( p > q ) . 2 I t turns out that x h i t s the "tensor product p a r t " , w h i l e ax and gx l i v e i n the "Tor p a r t " . Hence the "tensor product p a r t " ~ 2n 1 2m 1 can only be imbedded i n t o H*(L x L ) as a s u b r i n g , not as an p q i d e a l , and the "Tor p a r t " i s not closed under m u l t i p l i c a t i o n . These two observations probably i l l u s t r a t e a t y p i c a l f e a t u r e of the cup product s t r u c t u r e on a c a r t e s i a n product of spaces. - 3 -2. The Cohomology Ring of the Lens Space L 2 n  2.1 D e f i n i t i o n s 2n —1 The standard lens space, L , i s defined to be the o r b i t P space of the usual f r e e a c t i o n of the c y c l i c group of order p, Z^, on the standard sphere S 2 n = {z e C n: | |z| | = 1} . R e c a l l that i f i s presented by { t e C : t P = l } , then the a c t i o n i s given by: (2.1) t • ( z Q , . . . , z n _ 1 ) = ( t z Q , . . . , t z n ^ 1 ) where (ZQ,...,Z ^) i s an n-tuple of complex numbers re p r e s e n t i n g a p o i n t of S 2 n . Note that i n the case p = 2, the a c t i o n on S n (n may be even i n t h i s case) i s the a n t i p o d a l a c t i o n and the r e s u l t i n g o r b i t space i s the r e a l p r o j e c t i v e space RP n . In t h i s s e c t i o n we g i v e an e x p l i c i t c a l c u l a t i o n of the cohomology groups of L 2 n ^ . We f i r s t d e s cribe a c e l l u l a r s t r u c t u r e on S^ n ^ which i s e q u i v a r i a n t w i t h regard to the a c t i o n of Z^ . The c a n o n i c a l p r o j e c t i o n onto the o r b i t space induces a c e l l u l a r s t r u c t u r e on L 2 n From the corresponding c e l l u l a r chain and cochain complex we w i l l d e r i v e the a d d i t i v e s t r u c t u r e of H * ( L 2 n "S . To determine the m u l t i -P p l i c a t i v e s t r u c t u r e we use a b a r y c e n t r i c s u b d i v i s i o n of the o r i g i n a l c e l l u l a r s t r u c t u r e , which i s again compatible w i t h the a c t i o n of Z^ c 2 n - l on S _ 4 -2.2 The A d d i t i v e S t r u c t u r e of H * ( L 2 n 1 ) P The f o l l o w i n g i s standard and f o l l o w s the treatment i n [DOLD] 2x1 1 Give S the f o l l o w i n g c e l l u l a r s t r u c t u r e : Let e 2 k = { (z„,. . . ,z ) e S 2 n "*": z . = 0 f o r j > k and r U n-1 j arg(z, ) = r — or z, = 0} k p k ( 2 ' 2 ) and e 2 k + 1 = { ( z Q , . . . , z ^ ) e S 2 n _ 1 : z = 0 f o r j > k and 2TT , s / i -i \ 2Tr _ ! r — < arg(z, ) < ( r + 1 ) — or z =0} • ... • p .,- 6 k - p r f o r r = 0,...,p - 1 ; k = 0,...,n - 1 be the 2k (resp. 2k+l) dimensional c e l l s of S 2 n . I f t e Z , k k t = exp(2Tri/p), then t ' e r = e r + i • Hence t h i s c e l l u l a r s t r u c t u r e 2n—l i s compatible w i t h the a c t i o n of on S I f D 2 k = {z = ( z 0 , . .. , z n _ 1 ) e C n: | |z| | <_ 1 and z = 0, j >_ k} and D 2 k + 1 = {z = ( Z Q , . . . ^ ^ ) e C n: ||z|| <_ 1 and z = 0, j > k and z 1 e R} are the standard 2k (resp. 2k+l) dimensional closed d i s c s , then we can give the c e l l s (2.2) o r i e n t a t i o n s by homeomorphisms ' k ^k k f : D —y e : r r f 2 k ( z ) = (z ,. .. ,zk_±, g(z)exp(2irir/p) ,0,.. . ,0) ^ • 3 ) 2k+] f ^ + 1 ( z ) = ( z 0 , . . . , z k _ 1 , g ( z ) e x p ( ( ( z k / g ( z ) ) + 2r+l)Tri/p),0,...,0) 2 2 2k+l where g(z) = / l - ( z Q + ... + z f c _ 1 ) • Note t h a t can be w e l l defined f o r those z f o r which g(z) = 0 . - 5 -With respect to these o r i e n t a t i o n s the c e l l u l a r boundaries are given by: 9 ( e 2 k ) r 2k = { ( z n , . . . , z ^ ,) e e : z = 0} i=0 2k-1 (2.4) 2k+l N • \ 2k+l , . 3 ( e r ) = { ( z 0 , . . . , z n _ 1 ) e e r : a r g ( z k ) or a r g ( z k ) = ( r + 1 ) ^ } = r 2TT 2k 2k e . - e r+1 r 2k 'P 2k „ „2n-l 2 n - l , . i » . Let IT : S *• L be the canonxcal pro-0 p k k k i v j e c t i o n . Denote the image of by T r(e r) = e'v, where r = 0,...,p - 1 . Passing to the c e l l u l a r c hain complex C ^ ( L 2 n 1 ) of L 2 n _ 1 we have one generator i n dim k denoted by e k f o r P k = 0,1,...,2n - 1 - 6 -2n X The chain complex C.(L ) comes w i t h the f o l l o w i n g boundary homomorphisms: (2.5) 8 ( e 2 k ) = 3 ( T r ( e 2 k ) ) = T T ( 9 ( e 2 k ) ) p-1 = pe 1=0 2k-1 2k-1, e i } 0 < k < n - 1 2k+l Moreover we get 3(e ) = 0 f o r 0 _< k <_ n - 1, and 0 3(e w) = 0 We are now able to read o f f the homology groups: (2.6) H . ( L 2 n = \ Z i = 0 , i = 2n - 1 Z i odd, 1 < i < 2n - 1 P -0 otherwise In passing to cohomology we choose a c o l l e c t i o n of generators 2x1 X f o r the cochain complex C*(L ) dual to the ones used f o r the P chain complex. Moreover by abuse of n o t a t i o n we w i l l use the same k symbols f o r the duals of e ; k = 0,1,...,2n - 1 . The coboundaries of C * ( L 2 n 1 ) are given by P (2.7) 2 k w 2k+l x 2k / ; c 2k+l N <5(e ) (e ) = e (6e ) = e 2 k ( 0 ) = 0 2k and t h e r e f o r e 6(e ) = 0 , 0 £ k £ n We f u r t h e r evaluate - 7 --, 2k+l N 2k+2 _ , , j r/ 2 n - l N „ 6(e ) = p e , 0 <_ k < n - 1, and 6(e ) = 0 . From t h i s cochain complex we compute the cohomology groups of i T 2 n - l the lens space L : (2.8) H V 1 1 " 1 ) P Z = <e > i Z = <e > P i = 0, 1 = 2 n - l i even, 0 < i < 2n - 1 otherwise 2n —1 2.3 The M u l t i p l i c a t i v e S t r u c t u r e of H * ( L ;Z) E In the previous s e c t i o n we computed the cohomology groups of L from a f r e e cochain complex which i s of rank one i n dimension P i , where i = 0,1,...,2n - 1 . To determine the m u l t i p l i c a t i v e s t r u c t u r e we w i l l c a l c u l a t e cup products on the cochain l e v e l . To c a r r y out t h i s c a l c u l a t i o n we r e f i n e e q u i v a r i a n t l y ( i . e . e s s e n t i a l l y b a r y c e n t r i c a l l y subdivide) the c e l l u l a r s t r u c t u r e i n the previous 2n—1 s e c t i o n to get a s i m p l i c i a l s t r u c t u r e f o r S . C a l c u l a t i n g cup products i n the corresponding cochain complex and r e l a t i n g t h i s cochain complex w i t h the one i n the previous s e c t i o n , w i l l give us the required cup products. The a d d i t i o n a l v e r t i c e s we introduce are given by the barycenters of the o r i g i n a l c e l l s , i . e . the images of the o r i g i n under the maps k f i n the previous s e c t i o n : r - 8 -(2.9) x k = f k ( 0 ) = 2x1 1 ( 0 , . . . , 0,exp ( 2 rT r i/p) , 0 , . . . , 0 ) e S k even [ (0,...,0 , e x p((2r+l)iri / p),0,...,0) e S 2 n 1 k odd f o r r = 0 ,.. . ,p - 1; k = 0 , . . . , 2 n - 1 and the non-zero entry occurs i n the ([-^]+l)th component. Let K n be the s i m p l i c i a l complex w i t h the above v e r t i c e s and a l l s i m p l i c e s of the form (x^ x^ ... K1 ) where 0 < i < j < ... < h < 2 n - l r . r . r. — — 1 J h 2 k 2 k+l and i f x x occurs, then = or ^ i + 1 M p) . 2 k 2 k+l We c l a i m that K i s homeomorphic to S n 2n - l We show t h i s by i n d u c t i o n on n . When n = 1, S 1 c o n s i s t s of.'.the c e l l s e° and e 1 f o r r r r = 0,...,p - 1 . The b a r y c e n t r i c s u b d i v i s i o n of the c e l l e"*" i s r Oe^)*xJ = (x°,x°+1}*xj = (xV)u(x° + 1xJ) . Hence | K l | i s homeo-morphic to S"^  . I l l u s t r a t i o n n = 1, p = 3 - 9 -2xi 3 Assume i n d u c t i v e l y that l ^ n _ ] J -*-s homeomorphic to S Let Kj^ be the s i m p l i c i a l complex obtained from K^ by r e p l a c i n g 0 . ^ , 2n-2 , 1 _ u 2 n - l , x w i t h x and x w i t h x . Then r r r r IK I = IK , |*|K,'I = S 2 n ~ 3 * S 1 = S 2 n _ 1 as r e q u i r e d . Since the homeo-. . I k Z k+-c+l phism |K^| = S and the homeomorphism S *S = S (assuming mor s u i t a b l e placement) can be given by p r o j e c t i n g from the o r i g i n , we 2x1 X deduce that the homeomorphism |K | = S can be given by p r o j e c t i n g from the o r i g i n . k k If t e 2^, t = exp(2Tri/p), the a c t i o n (2.1) gives t * x r = x r + i > where r + 1 i s assumed to be mod p . Hence the s u b d i v i s i o n i s compatible w i t h the a c t i o n of Z^  . Under the p r o j e c t i o n TT: S 2 n > L 2 n \ we denote the image of P each simplex by: (2.10) T T C U 1 ... x j )) = [ X 1 ... x j ] r. r . r . r . i J i J 2x1 1 Let C'(L ) be the chain complex of the a s s o c i a t e d s i m p l i c i a l * p st r u c t u r e on L 2 n and l e t C ' * ( L 2 n be the corresponding cochain P P complex. Let i. : C . ( L 2 n > C ' ( L 2 n "*") be the chain complex » * p P homomorphism induced by the i n c l u s i o n . The map i ^ i s determined by: (2.11) i * ( e ) = 2 . [ x 0 x o x i X i ] + ^ [ x l X 0 X i 2 *•• X i ] where the above two sums are to be taken over a l l i 2 , . . . , i r such that i 2 k = i 2 k + 1 or i 2 k = i 2 k + 1 + 1 (mod p) and 0 < i f c < p - 1 - 10 -Let c S be the f o l l o w i n g cochain i n C ' * ( L 2 n "*") f o r P 0 < s < 2n - 1 : (2.12) c S ( [ x ^ ... x S ] ) = r 0 r s -l+l - Ll 0 otherwise r» - 1 (mod p) 0 <_ I < s 2xi 1 2xi 1 Under the induced s u r j e c t i o n i * : C'*(L n _ ) — * C*(L n ) we c l a i m P P 2 s that i * ( c ) = e . We v e r i f y t h i s by a d i r e c t c a l c u l a t i o n : ( c S ) ( e S ) = c°(i.(e")) = c°([x"x"x 0 1 2 f O p - 1 * s s Hence i * ( c ) = e . To determine the m u l t i p l i c a t i v e s t r u c t u r e of 2n — l H*(Lp ) we proceed as f o l l o w s . We c a l c u l a t e the cup products r s c ^ c using the Alexander-Whitney diagonal approximation. This y i e l d s : (c r~c w)([x"° ... x / + s n '0 'r+s i„ 1 . 1 i . r / r 0 r , . s / r r r + s i \ = c ( [ x . ... x. ])c ([x. ... x. J) J 0 J r J r 3, 'r+s (2.13) 1 = j - 1 (mod p) 0 < _ £ < r o r r < _ £ < r + s 0 otherwise r+s, r 0 c ([x.. 0 r+s,. . x J) Jr+s , , r r s r+s and t h e r e f o r e c c = c 2x1 1 Thus i n C*(L ) we have P e r v - - e S = e r + S , 0 < r + s < 2 n - l In c o n c l u s i o n we have: - 11 -(2.14) H * ( L 2 n 1;Z) = Z[a]/(pa,a n) <© <g> P where the generator a i s i n dimension 2, and g i s i n dimension 2n - 1 . The top dimensional generator g generates an i n f i n i t e c y c l i c group, i . e . <g> = Z . - 12 -3. The Cohomology Ring of C a r t e s i a n Products of Standard Lens Spaces, 2T1-*X R e c a l l that the cohomology r i n g s of the lens spaces L^ and L^ have the f o l l o w i n g s t r u c t u r e s r e s p e c t i v e l y , (see 2.14): H * ( L p n _ 1 ) = Z[a]/(pa,a n) © <g> (3.1) 9 -H*(L ) = Z[b]/(qb,b m) $ <h> q where dim a = dim b = 2, dim g = 2n - 1, dim h = 2m - 1 and <g> = <h> = Z . We f i r s t f i n d the a d d i t i v e cohomology s t r u c t u r e of H * ( L 2 N ^ x L 2 M """) . For that purpose we assume that the graded cohomology groups H * ( L 2 N ^ x L 2 M "*") are derived from cochain complexes which have one generator i n each dimension; see previous s e c t i o n . We l e t c(resp. d) denote the cochain. generator i n 2x1 X 2m X dimension 1 f o r L (resp. L ) . Note that c ( r e s p . d) does 2 2 not represent a cohomology c l a s s (6c = pc , resp. fid = qd ); 2 2 2n-~l 2m_ 1 however [c ] = a (resp. [d ] = b ) , [c ] = g (resp. [d ] = h) where the square brackets denote cohomology c l a s s e s as usual. Using the Kiinneth formula we deduce: H j a 2 n _ 1 >< L 2 1 1 1" 1) * e HV 1 1 - 1 ) ® H j - r a 2 m _ 1 ) P q r = 0 P q (3.2) j+1 TorCH ' a ^ - b.nJ^ - ' a 2 ^ 1 ) ) r=0 P q - 13 -R e f e r r i n g to (3.1) we deduce: H 2 k a 2 n _ 1 - >< L2"1"1) p q 2k © H r ( L 2 n ~ ^ ) ® H ^ a 2 ^ 1 ) r=0 P q = <£ <a? x b r=0 k-r 0 < k < n + m Let a = a x 1, 3 = 1 x b . Then H 2 k ( L 2 n - 1 x L 2 1 1 1" 1) ^  ® <a r 3 k _ r> p q r = 0 0 < k < n + m (3.4) . ..and . <a r 3 k r> = r = k r = 0 r > n - l o r k - r > m Z , N otherwise (p»q) Note that i f (p,q) = 1, then ag = 0 . For 2k = 2n + 2m we have: H2n+2m-2 2 n - l x L2m-1 . H2n-1 2 n - l ^ ^ 1 2 ^ 1 p q p q = <gxh> (3.5) = <yn> where y = g * 1, n = l x h - 14 -When j Is odd, say j = 2k + 1, the term i n the d i r e c t summand f o r ( L 2 N x L 2 M ^ ) not i n v o l v i n g Tor i s : P q 2k+l e H ^ L 2 1 1 " 1 ) ® H ^ - ' a 2 " - 1 ) r=0 P q B H2(k-m) +2 2 n - l H2m-l(I2m-l H2n-l ( L2n-l 0 2(k-n)+2 2», p q p q k-m+l .k-n+1 where each term occurs only i f m - l _ < k < _ m + n - 2 or n - l < k < m + n - 2 r e s p e c t i v e l y and, (3.7) k-m+l _ J <a n> = i Z k = m - 1 Z k > m - 1 ' P 0k-n+l _ J <Y3 > = i Z k = n - 1 Z k > n - 1 ' q To complete the c a l c u l a t i o n of. H 2 K + " ' " ( L 2 N x L 2 M ^) we need P q only f i n d an element i n H ^~+^~ (L^1 x L ^ M "*") generating a subgroup i • m / T T 2 r / T 2 n - l . 2k+2-2r / T 2m-l. s , , „T,.. xsomorphxc to Tor(H ( L ^ ),H ( L ^ )) under the "Kunneth isomorphism" (3.2). Since that isomorphism depends on the s p l i t t i n g chosen f o r the Kunneth exact sequence, we have to go back to a proof of the Kiinneth formula i n order to get a hold on the choice of the s p l i t t i n g i n v o l v e d . - 15 -I f X and Y are f i n i t e CW-complexes, then the f o l l o w i n g i s a s p l i t short exact sequence: (3.8) 0 -+ (H*(X)®H*(Y)) H q(XxY)-* Tor(H*(X) ,H*(Y)) —> 0 q q+1 Proof. (Spanier) Let C = C*(X) and C* = C*(Y) . Let Z' and B' be the complexes defined by ( Z ' ) q = Z q(Y) and ( B ' ) q = B q + 1 ( Y ) , and where the boundary maps are a l l t r i v i a l . From the short exact sequence (3.9) 0 — • Z' • C • B' • 0 we get the f o l l o w i n g short exact sequence (3.10) 0 —* C®Z' — y C<8C' —• COB' —»- 0 since C i s f r e e . From t h i s we ob t a i n an exact cohomology sequence (3.11) ... Hq(C®Z') - y H q(C< a C ) ( ^ ^ Hq(C®B') H q + 1(C®Z !) -> Observe that C8Z' = $ C J, where ( C 3 ) q = Cq~^(X)®Z^(Y) and COB' = ® where (C^ ) q = C ^ (X)®B^ + 1(Y) . Since Z^  (Y) and B J (Y) are f r e e , i t f o l l o w s from the U n i v e r s a l C o e f f i c i e n t Theorem, that Hq(C®Z') = $ H q(C j) = • © H 1(X)®Z j(Y) (3.12) H q(C®B') = 8 H q(C j) = © H1(X)®B:i(Y) j i+j=q+l - 16 -Under these Isomorphisms, the map 8 * corresponds to the homomorphism (-1) 1 © , where v_. i s the i n c l u s i o n map v.: B^(Y) y Z J(Y) . Theref ore there i s a short exact sequence (3.13) 0 -*- @ [coker(-l1)(§v.] Hq(C@C') -> © [ker(-l)1®v. ] i+j=q J i+j=q+l J Tensoring the f o l l o w i n g short exact sequence w i t h H X(X) (-DV (3.14) 0 -> B J(Y) J - »Z J(Y) -* H J(Y) -»• 0 gives (since 7? (Y) i s f r e e ) : (-1)^. (3.15) 0 -*• Tor(H 1(X),H ; l (Y)) -> H.1(X)®B*' (Y) 1 + H1(X)®Z~' (Y) H1(X)®H:)(Y) 0 and hence: (3.16) coker((-l)1®vj) = H 1(X)®H j(Y) and k e r ( ( - l ) 1 0 v j ) = Tor(H 1(X),H^(Y)) Using the E i l e n b e r g - Z i l b e r equivalence we i d e n t i f y Hq(C*(X)®C*(Y)) w i t h H q(XxY) i n (3.13) and thus get the Kiinneth formula. This completes the proof. In the f o l l o w i n g lemma we c h a r a c t e r i z e elements i n H*(X*Y) " l i v i n g i n the Tor-part", i . e . elements i n H*(X*Y) which are pre-images of elements i n Tor(H*(X),H*(Y)) under the map (3.8) - 17 -Lemma Let u(resp. v) be a cocycle representing a c l a s s of order s i n H 1 ( X ) ( r e s p . H J ( Y ) ) , and u'(resp. v') be a cochain such that <5u' = su (resp. 6v' = sv) . Then under the n o t a t i o n of the above theorem (3.17) [u'<8^ - ( - i ) 1 ^ ' ] i + i - 1 i s a c l a s s of order s i n H (XxY) such that (3.18) (l®<5)*[u'<S>v - (-DV J V*] = (-l):1•+1[u]®6v, e H 1(X)®B j (Y) i s a c l a s s of order s i n ker[ (-l)1®^ ] = Tor (H^X) ,HJ (Y) ) . Proof 6(u'g>v - ( - l ^ u ^ v ' ) = 6u'®v + (-l)1_1u'®6v - (-D^uSV* - (-l)2lu®6V = s(u®v - u®v) = 0 Hence u'®v - (-l^uOv* i s a cocycle. ( - l ) 1 ^ . ^ [ ( - l ) i + 1 [ u ] 8 > 6 v ' ] = -[u]«6v' = -[u]®sv = -[su]®v = 0 since v e (Y) . Suppose &([ w.Oz, ) = I 6w ®z = t[u€>6v'] e C k k k R k - 18 -Since 6v' generates a d i r e c t summand of B^+"*"(Y) (v i s a c l a s s of order s ) , and B^+"^(Y) i s f r e e , we can assume, using p r o p e r t i e s of the tensor product, that = 6v' f o r a l l k . Hence I 6w ®z = (6 I w )®<5v' = tu®6v? k k k k k Since C*(X) i s f r e e we can conclude 6/w, = tu and hence that s k k d i v i d e s t . Therefore (-1) :'~+"'"[u]®6v, i s an element of order s i n ker(-1) ^ v ^ . ^ . Observing that 6 ( ( - l ) 1 " 1 u , < a v , ) = s(u'®v - (-1)^8?') we conclude that [u'®v - ( - l ^ u ^ v ' ] i s a l s o a c l a s s of order s . This completes the proof. C o r o l l a r y I f X = L 2 n and Y = L 2 m the generator of the subgroup P q . 2 k + l / T 2 n - l _ 2m-l. of H (L x i , ) corresponding to _ / T T 2 r / T 2 n - l s „2k+2-2r ,T 2m-l s N , , . Tor(H (L ), H (L ))-=.>.Z, . * under the isomorphism p q v.(p,q) (3.2) i s n 2r-L_,2k+2-2r , 2r_,2k+l-2r, [k^ c ®d - k 2 c ©d J = a r _ 1 B k " r t k 1 c®d 2 - k 2 c 2 ® ^ r - l k-r = a 3 x 2 2 where k.. = , q . , k n = , P and T = [k., c®d -.c ®d] . 1 (p,q)' 2 (p,q) 1 - 19 -Remark In g e n e r a l , a complete s e t of g e n e r a t o r s f o r H*(XxY) may be found i n the above manner. To complete the c a l c u l a t i o n of t h e r i n g s t r u c t u r e we need o n l y t o determine the cup p r o d u c t s i n v o l v i n g x . x 2 = [ ( - l ) 2 * 1 ^ 4 - (-D 2* 2 k^cW 3 ( 3 > 2 0 ) - ( - i ) M k l V V + ( -D 1 , 2 k^cV] 2 2 2 2 = k^ aB + k 2 a B 2 2 where k^ and k 2 a r e of c o u r s e t a k e n mod (p,q) . n-1 r 2n-2^ , a x = [ c ®1J x = [ k 1 c 2 n " 1 ® d ' ] = k l Y B -S i m i l a r l y B™ "'"x = k„an and xy = xri = 0 . - 20 -The Main Result , , ~ , i „ i i ^ T 2 n - l T 2m-l 4.1 The Reduced Cohomology of L x L P q In the preceding s e c t i o n we have shown that H*(L, 2 n ^ x L 2 m ^) p q i s a r i n g w i t h 5 generators a , 3 , T , y , r i , where dim a = dim 3 = 2, dim x = 3, dim y = 2n - 1, dim n = 2m - 1, and which s a t i s f y the f o l l o w i n g r e l a t i o n s : a = 3 = ay = 3n = Y = 1 = TY = tri = 0 n ~ i / q \ o a T = TpTqT Qm-1 , p . 3 T = ( 7 i ^ y ) a n i i a B generates a c y c l i c subgroup of order p i f 1 <_ i < n and j = 0, of order q i f i = 0 , 1 < j < m, and of order (p,q) i f 1 <_ i < n and 1 <^  j < n . o^g^T generates a c y c l i c subgroup of order (p,q) i f 0 <^  i < n and 0 ^ j < n . Y> n>Y n each generate an i n f i n i t e c y c l i c subgroup, a 1™ generates a c y c l i c subgroup of order p i f 1 <_ i < n . B 1y generates a c y c l i c subgroup of order q i f 1 <_ j < m . From the above r e l a t i o n s i t f o l l o w s that i f n = 1 (resp. m = 1) x i s not necessary as a generator s i n c e x = ("^—^y)y3 (resp. x = (, P , )ari) . A l s o i f n = m = 1, i n which case (p>q) L' x L' = S 1 x S 1, x = 0 . P q - 21 -4.2 C o r o l l a r y H*(RP n * KP™) I f n and m a r e odd, say n = 2k - 1 and m = 2L - 1, the s t r u c t u r e of H*(RP n x RP m) i s t h e same as i n the p r e v i o u s s e c t i o n . I t i s a r i n g w i t h 5 g e n e r a t o r s a , S , T ; Y , r | where dim a = dim 3 = 2, dim T = 3, dim y = n and dim n = m, and which s a t i s f y t h e f o l l o w i n g r e l a t i o n s : n+1 m+1 2 2 2 2 a = 3 = ay = 3n = Y = r l = T Y = T r i = 0 (4.2) n-1 m-1 2 2 2 2 2 x = a 3 + a g ; a x = y&, 3 T = an and e v e r y non-zero p r o d u c t g e n e r a t e s a subgroup i s o m o r p h i c t o Z^, except Y>n, and yri which g e n e r a t e subgroups i s o m o r p h i c t o Z . I f n i s even and m i s odd, say n = 2k and m = 2Z - 1, H*(RP n x RP m) i s a r i n g w i t h 4 g e n e r a t o r s a,3 ,T,n where dim a = dim 3 = 2 , dim T = 3, and dim n = m, and which s a t i s f y the f o l l o w i n g r e l a t i o n s : T 2 2 a = 3 = 3 n = r i = T r i = 0 (4.3) n m-1 2 2 2 2 2 T =a3 + a 3 , a x = 0 , 3 x = a n and e v e r y non-zero p r o d u c t g e n e r a t e s a subgroup i s o m o r p h i c to except T) which g e n e r a t e s a subgroup i s o m o r p h i c t o Z . I f n and m a r e even, say n = 2k and m = 2£, H*(RP n x RP™) i s a r i n g w i t h 3 g e n e r a t o r s a , 3 , T where dim a = dim 3 = 2 and dim T = 3, and which s a t i s f y the f o l l o w i n g r e l a t i o n s : - 22 -^+1 ™+l a = 3 = 0, 2 Q 2 2. T = a3 + a.f (4.4) a T = 0, m 6 2T and every non-zero product generates a subgroup which i s isomorphic to Z - 23 -B i b l i o g r a p h y Adams, J.F. Lectures on Generalized Cohomology, Category Theory, Homology Theory and t h e i r A p p l i c a t i o n s . I l l , Springer Lecture Notes Volume 99, 1968. A t i y a h , M.F. Vector Bundles and the Kiinneth Formula, Topology, V o l . 1 , pp.245-248, 1962. Dold, A. Lectures on A l g e b r a i c Topology, Sp r i n g e r - V e r l a g , 1972. Spanier, A l g e b r a i c Topology, McGraw-Hill, 1966. 

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