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On Hopf multiplications on the wedge of two spaces Chao, Frances Yen-Yen 1973

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ON HOPF MULTIPLICATIONS ON THE WEDGE OF TWO SPACES by FRANCES YEN-YEN CHAO B.Sc, University of B r i t i s h Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1973 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada - i i -Supervisor : Dr. U. Suter ABSTRACT In t h i s thesis we show that i t i s impossible to define a Hopf m u l t i p l i c a t i o n on the wedge X V Y of a connected f i n i t e CW-complex X with an arbitrary connected CW-complex Y, provided neither X nor Y homotopy equivalent to a point. We also show by an example, that the above i s no longer true, i f one does not require that X i s a f i n i t e CW-complex. - i i i -ACKNOWLEDGEMENTS I would l i k e to acknowledge my gratitude to Dr. U. Suter, for suggesting the topic of thi s thesis and his generous guidence and encouragement throughout the work. I would also l i k e to thank Dr. R. Douglas for reading the thesis and the University of B r i t i s h Columbia for f i n a n c i a l assistance. - i v -TABLE OF CONTENTS §1. Preliminaries 1 §2. H-Spaces 5 §3. Cohomology of H-Spaces 16 §4. The Main Theorem 24 §5. Examples of I n f i n i t e CW-Complexes X and Y, such that X V Y i s an H-Space 29 - v -Introduction. A based space (X, X q) i s called an H-space or Hopf-space, i f we can define a m u l t i p l i c a t i o n m : X x X —> X with a two sided unit X q £ X, up to homotopy. Given any topological space, i t i s always of interest to know whether or not i t i s possible to define such a m u l t i p l i c a t i o n on this space; i . e . i f the given space i s an H-space. Here we s h a l l show : If_ (X, X q) is_ & connected f i n i t e  CW-complex with basepoint , and i f (Y, y ) IJB an arbitrary connected CW-complex with basepoint , then their one-point union (X v Y, x v y ) i O ' Q i s not an H-space; provided neither space i s homotopic to a. point. This theorem becomes f a l s e , i f we do not require that X i s a f i n i t e CW-complex. We w i l l give examples of connected i n f i n i t e CW-complexes X and Y , where X v Y carries a Hopf-multiplication. The above theorem i s the main result of this thesis; to prove i t , we use, besides Borel's structure theorem on Hopf-algebras, only c l a s s i c a l theorems of algebraic topology. In §1 , we f i r s t introduce the necessary terminology and state some theorems, which w i l l be needed l a t e r on. Next, i n §2, we prove some basic lemmas about H-spaces. We w i l l exploit the fact, that i f X i s an H-space, then the cohomology H*(X; K) with coefficients i n the f i e l d K i s a Hopf algebra. Therefore, i n §3, we l i s t Borel's structure theorem - v i -on Hopf-algebras . and then give an elementary proof of the following fact ( [ 2 ] ) : If X i s a 1-connected H-space with f i n i t e l y generated integer cohomology, then the highest dimensional non-vanishing cohomology group H n(X; 2) i s isomorphic to 2. After these preparations, we w i l l prove our main resu l t i n §4. In the f i n a l section, we w i l l present an example to show that our main theorem does not hold for arbitrary CW-complexes X and Y . In f a c t , we w i l l prove that the wedge of Eilenberg-MacLane spaces K(Z , n) v K(Z , m) , where p and q are different primes, and n, m >_ 2, P Q i s an H-space. - 1 -§1. Preliminaries. The following are the notations, terminology and conventions which w i l l be used throughout . 2 : The ring of integers. : The ring of integers modulo m. <Q : The f i e l d of r a t i o n a l numbers. A(a^, a^) : The exterior algebra over K with generators 1 n th As usual, we denote the i -dimensional homology and cohomology groups of a pair (X, A) with coefficients i n the ring R by EL(X, A; R) and H (X, A; R) respectively, and we set : H^(X; R) = O H ±(X; R) i H*(X; R) = © iT^X; R) . For a space X with base point X q , we write H ^ ( X ; R) for the reduced i - t h homology group, i . e . H _ ^ ( X ; R) = H ^ ( X , X q ; R) . Note that H N ( X ; R) = H N ( X ; R), n >_ 1, and for a connected space X , one has H Q ( X ; R) =0 . The i - t h dimensional homotopy group of a based space ( X , X q ) i s denoted by TT^(X ) . (We drop the base point i n our notation, since we s h a l l only be interested i n path connected spaces.) - 2 -We s h a l l now r e c a l l some of the well-known theorems i n algebraic topology, which w i l l be used, without proofs. As a general reference see Spanier Alg. Top. [ 5 ] . (1.1) the Kiinneth Rule : Let K be a f i e l d . One then has the following isomorphisms : A * 9 H.(X; K) 0 H.(Y;;K) £ H (X x Y; K) i+j=n 1 J — > n A * $ H^X; K) 0 H J(Y; K) _£=_> H n(X x Y; K) . i+j=n (For cohomology, assume that H n(X; K) i s f i n i t e l y generated for a l l n.) The isomorphism X* i s given by : X*(a 0 b) = Tr*(a) U 7r*(b), a e H^X; K) , b e H j(Y; K), where ir , TT are the natural projections of X x Y onto X and Y A I respectively, and U denotes the cup product . We w i l l also need the Kunneth Rule for the pair (X x Y; X v Y), with X v Y being the one point union of the pointed spaces (X, X q ) and (Y, y o) : n (1.2) H (X x Y, X V Y; K) = ® H. (X, x ; K) 0 H . (Y, y ; K) © n . „ i o n - i o i=0 n » Tor(H.(X, x ); H . ,(Y, y )) . n l o n-i-1 o 1=0 (1.3) the University Coefficient Theorems Let G be an abelian group. Then - .3 -(i ) the sequence 0 —> H ±(X; 1) 8 G —> H ±(X; G) —> Tor(H ±_ 1(X; I); G) —> 0 i s s p l i t exact. ( i i ) Assume that H n(X; Z) i s f i n i t e l y generated for a l l n, then the sequence 0 —> ff^X; 2) 8 G —> H^X; G) —> Tor(H 1 + 1(X; Z) ; G) —> 0 i s s p l i t exact. ( i i i ) Let K be a f i e l d . Then H n(X; K) = HomK(Hn(X; K); K) . (The isomorphism i s induced by the Kronecker product) (1.4) Whitehead's Theorem Let X, Y be connected CW-complexes. (i) I f f : X —> Y induces isomorphisms f ^ : -n. (X) —> n\. (Y) for a l l homotopy groups, then f i s a homotopy equivalence. ( i i ) Let X and Y be 1-connected spaces. I f f : X —> Y induces isomorphisms f. : H (X; 2) —> H (Y; Z) for a l l homology groups, * n n then f i s a homotopy equivalence. F i n a l l y , we mention the following consequence of the fundamental theorem on f i n i t e l y generated abelian groups. - 4 -Let G be an abelian group and p be a prime. The elements of G having order a power of p form a subgroup denoted by T . (1.5) I f G i s a f i n i t e l y generated abelian group, then we have an isomorphism G = Z # . . . # Z # T #...•© T •J P i P„ , "V T" '1 " m k summands where p^, p^ are d i s t i n c t primes. The integer k i s called the rank of G . - 5 -§2. H-spaces (Definitions and Lemmas) (2.1) Throughout this section and i n the forthcoming sections, we w i l l assume that a l l spaces considered are (connected) CW-complexes. (We do this for convenience. A l l our statements, with an appropriate modification, carry over to the case of spaces having the homotopy type of CW-complexes.) . Observe that for CW-complexes, the notions "connected" and "path-connected" are equivalent. (2.2) A pointed CW-complex (X, X q) i s called an H-space with unit element X q , i f there exists a map m : X * X —> X , such that m°i^ - moi^ - 1^ . , where i ^ , ±^ ' X —> X x X are the maps defined by i^(x) = (x, X q) and i 2 ( x ) = ( X Q> x). respectively j x e X . (2.3) Lemma. Let (X, x ) be an H-space. Then there exists a m u l t i p l i -cation m with x as an exact unit element ( i . e . moi = 1 = m°i 0). O i A / Proof : Since X i s an H-space, we have a map m : X x X —> X such that the following diagram i s commutative up to homotopy. Here i i s the canonical inclusion and V i s the id e n t i t y on both compo-nents. Let F = (X V X) x i —> X be a homotopy between moi and V . - 6 -The pair (X x X, X V X) has the homotopy extension property with respect to a l l spaces. (This also holds i n the case where X x X i s not a CW-complex any more, see [3]). Hence there i s a homotopy G : X x x x I —> X extending the homotopy F. We set now m(x^, x^) = G(x^, 1), x^ £ X and x_ e X . One checks readily that moi = V . i . e . moi = 1 = moi . (2.4) We say that (X, X q) dominates (Y, y^) i f there are maps (Y, y ) -^J—> (X, x ) ——> (Y, y ) such that r°j = 1 r e l . y . o o o Y o (2.5) Lemma. I f (X, x ) i s an H-space, and (X, x ) dominates o o (Y, y ), then (Y, y ) i s also an H-space. o o Proof : Let m be the m u l t i p l i c a t i o n on (X, X q ) . One has the following homotopy commutative diagram. = m°i We want to construct m' , such that i s commutative. - 7 -Since (Y, y ) i s dominated by (X, X q) , we have (Y, y Q) —J-> (X, X q) -^-> (Y, y Q ) , with roj = Ly r e l . y Q We define now m' to be the composition i x i m r Y x Y —*"—"*—> X x X > X b—> Y m' i. e . m' = romo(jxj) . Then we get : m'oi = romo(jxj)oi = romoiojvj - roVojvj But r°V°jvj = V' and therefore (2.5) i s proved. (2.6) Lemma. I f (X, X q) and (Y, y ) are homotopy equivalent, then (X, X q) being an H-space implies that (Y, y ) i s also an H-space. f g Proof : By assumption, we have maps (Yj y ) > (X, x ) —°—> (Y, y ), _ o' " ' o' ' '• " o' such that g°f - 1„ r e l . y . i . e . (Y, y ) i s dominated by (X, x ) o Y •'o J o o The lemma follows now from (2.5). (2.7) Lemma. Let (X, X q) be a connected H-space with unit X q, and l e t x^ be any point i n X. Then there i s a m u l t i p l i c a t i o n on X with unit x^. Proof : Let m^  : (X, X q) —> (X, x^) be the right translation by x^, i. e . m (x) = m(x, x.), x e X. I t i s clear that IL,, i s a homotopy X-L 1 x l equivalence X - X (choose a path from X q to x^), which gives r i s e to a homotopy equivalence (X, X q) - (X, x^). By lemma (2.5), we conclude that (X, x^) i s also an H-space . - 8 -(2.8) Lemma. (X, X q) and (Y, y ) are both H-spaces i f and only i f (X x Y, X q x y ) i s an H-space. Proof : Observe that (X x Y, X Q X y^) dominates both i t s components (X, X q) and (Y, y ). By (2.5), i f (X x Y, X Q X y ) i s an H-space, then (X, X q) and (Y, y ) are both H-spaces. Now l e t (X, X q) and (Y, y ) be H-spaces with multiplications m and n respectively. Then X x Y admits the m u l t i p l i c a t i o n (m x n ) (1 x T x 1) 5 here 1 x T x 1 : (X x Y) x (X x Y) —> X x X x Y x Y i s the obvious map. (2.9) Lemma. I f X i s an H-space, then TT^(X) i s abelian Proof : Let m : X x X — > X be a Hopf-multiplication and l e t m' : X x X — > X be the m u l t i p l i c a t i o n defined by m'(a, b) = m(b, a) . The c i r c l e i s a co-H-space; i . e . there i s a (obvious) map y : S 1 —> S 1^ s 1 with p^°y - l g i - ?2°y » where p^, p 2 are the canonical projections S^V S"*" —> S^. On the set of based homotopy classes [S^, X ] = TT^(X) , we can now define three products : The f i r s t ; giving the usual group structure on TT^(X), i s - 9 -defined as follows : I f [ f ] , [g] E TT^(X) , then [f]'[g] = [<f, g> ° y] • (Here <f, g> i s the map V S-L > X ; which i s f on the f i r s t and g on the second component.) We define two other products * and *' on TT^(X) by means of m, m' : X x X — > X . [f]*[g] = [m o {f, g}] [f]*'[g] = [m' o {f, g}] ( {f, g} i s the canonical map S 1 > X x X , with t i—> ( f ( t ) , g ( t ) ) ; t E S 1 ) Note : (2.10) [f]*'[g] = [g]*[f] • Now look at : S 1 - Y - > S 1 V S 1 — - — > X x X — - — > X where F = { <f, c>, <c, g> } = <{f, c}, {c, g}> , and c : S 1 —> {x Q} i s the constant map . One checks readily <f, g > ° y - m ° F o y - m o { f } g} Hence [f]«[g] = [f]*-[g-] . Si m i l a r l y , [f]«[g] = [f]*'[g] . Now with (2.10), [f]»[g] = [f]*'[g] = [g]*[f] = [g]-[f] • Q-E.D. - 10 -(2.11) Lemma. Let X be a connected H-space. Then H ^ ( X ; Z) = 0 i f and only i f X - pt. Proof : Suppose H . ( X ; Z) = 0, then i n par t i c u l a r H n ( X ; Z) = 0. Since * 1 the fundamental group i r ^ ( X ) i s abelian (2.10), we conclude with [5; p.394], TT 1(X) = H ^ X ; 2) = 0; i . e . X i s 1-connected. Now, consider the map X —^—> pt; t r i v i a l l y p induces an isomorphism i n homology. Whitehead's theorem (1.4) then implies that p i s a homotopy equivalence, i . e . X - pt . The converse i s t r i v i a l l y true. (2.12) Lemma. I f X i s a connected H-space, with f i n i t e l y generated fundamental group such that r a n k ( i T ^ ( X ) ) = k >_ 1, then there exists a connected H-space Y , with ir (Y) f i n i t e , such that X - Y x s 1 x s 1 x . . . x s 1 ( S 1 i s the c i r c l e . ) — v k factors Proof : According to (1.5) the f i n i t e l y generated abelian group TT-^(X) has a subgroup C = Z , such that TT;l(X) = C + G = 2 $ G ; rank G = k - 1 The covering - 11 -which corresponds to the subgroup G c TT-^(X) ( i . e . P^TT^(Y) = G ) i s a p r i n c i p a l Z-bundle, (see [5; I I , §5]), and by the c l a s s i f i c a t i o n theorem [6; 19.3] we have a commutative diagram : c being a c l a s s i f i c a t i o n map of the bundle Y — X . Passing to the homotopy groups, we get : 0 — > IT. ( Y ) - — > TT1 ( X ) > IT (2) = 2 —> 0 -> TY, (S 1) >TT (2) = 2 > 0 1 o and obtain a s p l i t exact sequence of abelian groups 0 — > TT ( Y ) —> T T 1 ( X ) —> T\±(Sl) > 0 Since i s onto, there i s a map g : S — > X with c°g = 1 ^, and s hence ^ ( X ) = P # ^ 1 ( Y ) + g # T r 1 ( S i ) . We define now F : Y x S — > X by F = m ° (p x g) where m i s a m u l t i p l i c a t i o n on X . I t i s easy to show that - 12 -F„ : u (Y x s1) = TI (X) ; n > 2 u n n — Furthermore, F^ : i r ^ ( Y x S"*-) •—> TT^(X) i s onto. (Both p; and g factor over F.) Since the f i n i t e l y generated abelian groups TT^(Y X S^) = 7T^ ( Y ) <& TT^ (S"'") and TT^(X) are (algebraically) isomorphic, an epimorphism between them must have kernel zero. Hence F^ i s an isomorphism for a l l homotopy groups, and Whitehead's theorem (1.4(i)) implies that F i s a homotopy equivalence. By (2.6) and (2.8) Y i s an H-space, and our lemma follows by induction on the rank k . Now, l e t X denote the Universal Covering space of X , and l e t p : X — > X be the covering projection. We have the following lemmas : (2.13) Lemma. If X i s a connected H-space, then X i s also an H-space. Proof : Let m be a m u l t i p l i c a t i o n on X with exact unit x . We f i r s t o f i x a point X q e p (^x- ), and consider the following diagram : X X --> X x X pxp -> X x X m •> X m -> X where i ^ and 1^ are given by i^(x) = (x, X q ) ; i 2 ( x ) = (X Q, X) Si m i l a r l y for i ^ and 1^ • Since X i s simply connected, X x X i s also simply connected. We have then (m ° pxp)^(-rr^(X x X)) = 0 cz P^TT^(X) , and hence the map - 13 -m o (pxp) has a l i f t i n g m : X x x —> X, with m ( x 0 > X Q ) = X Q (see [5; p. 85]). Since m°i.. = 1 , moi = 1 , we have a commutative triangle J. X L X moi. X -> X P \ ^ / P " X We check that m©i (x ) = x , hence moi = 1~ . S i m i l a r l y , m°i0 = 1~ J. O O X X 2. X (see [5; 2.2.4], un i q u e - l i f t i n g property) Q.E.D. (2.14) Remark : Let y e p ^ ( X Q ) an<* l e t m~ : x translation with y ; i . e . m~(x) = m(y, x ) , x e X -> X be the l e f t The triangle m~ y X -> X P \ /P i s commutative. Hence, m~ i s a covering transformation and G = { m~ y e p 1 ( X Q ) > i s the group of covering transformations of p : X —> X (see [5; 2.5]). The map m~ i s homotopic to 1~ ( j o i n y and x by a path), and hence y x o (m~) A : H^(X; Z) — > H^(X; Z) i s the i d e n t i t y homomorphism . - 14 -(2.15) Lemma. I f X i s a connected H-space with f i n i t e fundamental group TT^(X), and such that H^  ( X ; 2) i s f i n i t e l y generated for a l l j , then H^ ( X ; Q) = H ^ X ; Q) . Proof : The projection p : X — > X induces p^ : H A ( X ; 2) —> H ^ X ; 2). If the covering i s f i n i t e , then there i s a homomorphism x : H^(X; 2) —> H^ ( X ; 2), the socalled transfer homomorphism [7; V, §7]. The group of covering transformations of our universal covering i s naturally isomorphic to TT^(X) [5; I I , 6.4]. Let Y E ^ ( X ) , we write : : H^(X; 2) —> H^ ( X ; 2) for the isomorphism induced by the covering transformation that corresponds to Y • By (2.14) we have Y * = ^ ( X ; 2) ' Let r be the order of TT^(X) . The homomorphisms p^ and x are then related as follows [7; V, §7] : P* ° x(a) = r • a , a e H^ ( X ; 2) T o p^( a) = I Y*(a) > a s H A ( X ; 2) , Y E T T ^ X ) and hence x o p A(a) = r • a . - 15 -It follows that (X; 2) and H^. (X; 2) have the same rank for a l l j , and our lemma i s proved . (2.16) Lemma. Let X be a connected H-space with f i n i t e fundamental group TT^(X), and such that H A ( X ; 2) i s f i n i t e l y generated. Then X i s not homotopy equivalent to a point . Proof : I f we had ir ( X ) = 0 for a l l n , then TT ( X ) = 0, n > 1 ; n n — hence X would be an Eilenberg-MacLane space K(TT^(X) , 1). But H ^ C K C ^ C X ) , 1); 2) i s according to [5; 9.5.7 and 9.5.9] not f i n i t e l y generated. This proves (2.16) . - 16 -§3. Cohomology of H-spaces. (3.1) A Hopf algebra over the f i e l d K i s a p o s i t i v e l y graded K-algebra oo A = © A , together with a homomorphism • i|> : A —> A ® A of graded O n K K-algebras; such that : (i) A has an unit element 1, that spans A ; i . e . A q = K . ( i i ) If a i s a homogeneous element with degree (a) >_ 1 ( i . e . a e A , n >_ 1) , then i K a ) = a 0 1 + 1 8 a + I u . 8 v. L l l 00 where u. , v. e A = €> A . i i i n n=l The homomorphism i s called comultiplication of the Hopf algebra A . The Hopf algebra i s said to be n o n - t r i v i a l , i f A = § A n i s non-zero. n>l (3.2) Assume now that X i s a connected H-space, with m u l t i p l i c a t i o n m : X x X —> X . Using the Kunneth formula for X x X , we see that . m induces a homomorphism of graded algebras : m* : H*(X; K) > H*(X; K) 0 H*(X; K) . I t i s e a s i l y seen that H*(X; K) i s a Hopf algebra with comulti-p l i c a t i o n m* (Property ( i i ) of (3.1) follows from the fact that X has 2-sided unit X q . Let i ^ , 1^ : X —> X x X be the canonical inclusions, i . e . i..(x) = (x, x ), i 0 ( x ) = (x , x); with x e X . Since moi - 1 -i o z o i X moi^ , one checks readily with (1.1) that m*(a) = a ® l + 1 8 a + ||u. 8 v^, where u±, v ± e H*(X; K); deg(u i) +'deg(vi> = deg(a ±)). - 17 -Next, we introduce some terminology, and l i s t some properties of Hopf Algebras. (3.3) The height of an element a e A i s the integer h, such that a*1 ^ ^  0, a*1 = 0. If no such integer exists ( i . e . i f a^ / 0 for a l l k) , then we define the height of a to be 0 0 . (3.4) A Hopf Algebra i s called monogenic, i f i t i s generated by 1 and a homogeneous element a with degree (a) > 0. We denote such a mono-genic Hopf Algebra by M(a). The following theorem of Borel [1; Thm 3.1] gives a complete description of monogenic Hopf Algebras. (3.5) Theorem. Let M(a) be a monogenic Hopf Algebra over a f i e l d K of ch a r a c t e r i s t i c p, and l e t h be the height of a. (a) If p 2, and degree (a) i s odd, then M(a) = A^C3) • (Here ^ ( a ) i s the exterior algebra with one generator a) (b) If p £ 2, and degree (a) i s even, then h = p^ or 0 0 . (c) I f p = 2, then h = 2 k or «= . In order to state the next theorem, we need two d e f i n i t i o n s . Let K be a f i e l d of chara c t e r i s t i c p, then K i s called i f p = 0 or i f K contains a p-th root of each of i t s elements (3.6) perfect, - 18 -( i . e . x° = k has a solution i n K for a l l k e K.) Note i f p i s a prime number ( i . e . p ^ 0), then 2^  i s perfect. oo (3.7) The graded K-algebra A = # A i s said to be of f i n i t e type, i=0 i f for each i , the K-module A 1 i s f i n i t e l y generated. The following theorem i s called Borel's structure theorem for Hopf Algebras. (3.8) Theorem (see [1; Thm 3.2]) If A i s a n o n - t r i v i a l Hopf Algebra of f i n i t e type over a perfect f i e l d , then i t i s isomorphic (as an algebra) to the tensor product of monogenic Hopf Algebras. We now mention the following two c o r o l l a r i e s of the above theorem. (3.9) Corollary (Hopf's theorem) Let X be a connected H-space, such that i t s r a t i o n a l cohomology H*(X; <Q) i s n o n - t r i v i a l , and of f i n i t e dimension as vector space over Q (e.g. i f X i s a f i n i t e CW-complex.). Then H*(X; Q) i s isomorphic to an exterior algebra with generators of odd degrees, i . e . H*(X; <Q) = A Q ( a 1 5 • .., a m) ; degree (a^) odd for a l l i . Proof : The Hopf Algebra H*(X; Q), by (3.8) and (3.9),is a tensor product of monogenic Hopf Algebras M(a^) , where the height of a_^  i s 2, i f degree (a.) i s odd, and 0 0 , i f degree (a.) i s even.. - 19 -Since H*(X; <Q) i s a f i n i t e dimensional Q-vector space, we have no elements x. of i n f i n i t e height. Hence m H*(X; Q) = 0 A (a.) = A (a , a ), degree (a ) odd . . - Q l Q 1 m l 1=1 (3.10) Corollary. Let X be a connected H-space, such that i t s 2 ^ -cohomology H*(X; 2^) i s of f i n i t e dimension as a 2^-vector space (e.g. If X i s a f i n i t e CW-complex.). Then the heighest dimensional non-vanishing cohomology group H n(X; 2^) i s isomorphic to 2^ . Proof : As for (3.9), one gets that there are no elements i n H* of i n f i n i t e height. Hence by (3.9) : H*(X; 2 ) = M(aJ 0 ... 0 M(a ) , p i m where h_^  = height (a^) < 0 0 ; i = 1, m . I t follows then, that the element h , - l h -1 1 „ a m a = a, 0 ... 0 a 1 m generates the highest nonvanishing cohomology group. Note m (3.11) n = degree (a) = £ (h. - 1)'degree(a.) i = l 1 X The following theorem and corollary w i l l play an important role i n the next section. Their proof depends on Borel's structure theorem and the Universal Coefficient theorem. - 20 -(3.12) Theorem. Let X be a 1-connected H-space, such that H*(X; 2) i s f i n i t e l y generated as an abelian group (e.g. I f X i s a f i n i t e CW-complex.). Then the highest dimensional non-vanishing cohomology group H n(X; 2) i s isomorphic to 2 . Proof : I f X has the homotopy type of a one-point space, the theorem i s t r i v i a l . So we assume for the remainder of this proof that X i s not homotopy equivalent to a point. The CW-complex. X then has n o n - t r i v i a l cohomology; i . e . H^X; 2) 0 for some j > 1 . (see (2.11)) Let n be the largest integer with H n(X; 2) ^ 0 (since H*(X; 2) i s f i n i t e l y generated, there exists such an integer ), and l e t £ be a prime, such that H n(X; 2^) ^ 0 . I t follows from the Universal Coefficient Theorem ( 1 . 3 ( i i ) ) , that there i s such a prime and that (3.13) H n(X; 2 ) = H n(X; 2) 0 2 P P HJ (X; 2 p) = 0 ; j > n . By Borel's structure theorem, we have m (3.14) H*(X; 2 ) = 0 M(a.) P 1=1 1 where M(a^) i s a monogenic Hopf Algebra generated by , i = 1, m, the elements a^ a l l having f i n i t e height (see proof of (3.11)). Let d^ = degree (a^) , = height (a^) - 1 ; i = 1, m . - 21 -According to (3.10) and (3.11), we know (3.15) * - k l d l + k 2 d 2 + + km dm • Furthermore, H n(X; 2 ) = 2 . Hence by (3.13), we get P P (3.16) H n(X; 2) 0 J = 2 . P P We claim that the simply connectedness of X implies (3.17) H n - 1(X; 2 ) = 0 . P This i s obtained as follows : Referring to (3.14), we see that the homoge-neous elements r l r2 r a. 0 a„ 0 ... 0 a , 0 < r. < k. , i = 1, .... m 1 2 m — x — l form a basis of the graded 2^-vector space H*(X; 2" ). We calculate : r l rm degree (a, 0 ... 8 a ) = r,d, + ... + r d 1 m i l mm For a simply connected space X, one has H"*"(X; 2 ) = 0 ; which implies d^ >^  2, i = 1, m. Therefore, since k^d^ + ... + k m d m = n , by (3.15), there are no integer r,, r (0 < r. < h.), such that r,d, + ... + r d = n - l . Hence 1 m — I — I 1 1 mm there are no n o n - t r i v i a l elements of degree n - 1 , which proves (3.17). The Universal Coefficient Theorem ( 1 . 3 ( i i ) ) gives H n - 1(X; 2 ) = H n 1(X; 2) % Ti « Tor(H n(X; 2) ; 2 ) , P P P - 22 -and we i n f e r from (3.17) , Tor(H N ( X ; 2) ; 2 ) =0 . P Hence (3.18) H N ( X ; 2) i s p-torsion free. I t follows then from (3.16) and the fundamental theorem on f i n i t e l y generated abelian groups (1.5) that H N ( X ; 2) contains precisely one 2-summand. This i n turn implies v i a (1.3) : H N ( X ; 2 q) + 0, for a l l primes q . And we get with (3.18), that H N ( X ; 2) i s q-torsion free for a l l q, and hence torsion free. The proof of theorem (3.12) i s now completed. Remark : Theorem (3.12) has been proved among other things i n [2], The main tool used i n there i s the Bockstein spectral sequence associated to an H-space. Next, we w i l l have an important consequence of the above theorem. (3.19) Corollary. Let X be a connected f i n i t e CW-complex which i s an H-space, and l e t K be any f i e l d . Then the reduced homology H A ( X ; K) i s n o n - t r i v i a l ; i . e . H n(^> K) ^  0 for some n >_ 1, unless X i s homotopy equivalent to a point . Proof : Suppose f i r s t that TT,(X) i s f i n i t e . The universal covering - 23 -space X i s then a 1-connected f i n i t e CW-complex, which i s not homotopic to a point (see 2.16), and i t carries a m u l t i p l i c a t i o n (2.13). From (3.12), (1.3) and (2.15) we deduce : Q = H n(X; <Q) = H n(X; Q) = H n(X; <Q) , where n >_ 1 i s the highest dimension, for which X has n o n - t r i v i a l integer cohomology. Hence : H ( X ; 2) = 2 © T , n where T i s some f i n i t e group. Therefore by (1.3) : H N ( X ; K) ^ 0 . In case TT^(X) i s not f i n i t e , we have according to (2.12), that X - Y x (S 1) 1^. One knows that ^ ( ( S 1 ) 1 ^ ; K) + 0 , and the Runneth formula (1.1) gives : H X(X; K) = H (Y x (S 1) 1*; K) = K 8 ^ ( ( S 1 ) 1 * ; K) <S> H^Y; K) 8 K i 0. Hence H.(X; K) ^ 0 . Q.E.D. - 24 -§4. The Main Theorem. In this section, we investigate the question,- whether the wedge X v Y of two pointed spaces can carry a m u l t i p l i c a t i o n or-not. Before giving the proof of our main theorem stated i n the introduction, we s h a l l f i r s t provide the following two lemmas . (4.1) Lemma. Let (X, X q) and (Y, y Q) be two connected pointed CW-complexes. Then X V Y an H-space implies that X and Y are both H-spaces. Proof : I f X v Y i s an H-space, we may choose a m u l t i p l i c a t i o n with unit x Q v y Q (2.7). The lemma i s now immediate from (2.5), i f one observes that (X v Y, x v V ) dominates both (X, x ) and (Y, y ) . o o o o (4.2) Lemma. Let (X, X q) and (Y, y Q) be as i n (4.1) and l e t X v Y be an H-space. Then the canonical inclusion j : X V Y —> X x Y i s a homotopy equivalence . Proof : We i d e n t i f y the wedge X v Y with the subspace X x y^ I J X Q X Y C X X Y . Let j : X v Y —> X x Y be the inc l u s i o n , and define maps f, g : X x y —> X v Y by f ( x , y) = (X q, y) and g(x, y) = (x, y Q) . - 25 -I f X v Y i s a n H - s p a c e , t h e n a c c o r d i n g t o (2.2) a n d (2.3) t h e r e i s a m u l t i p l i c a t i o n . m : ( X V Y ) x ( X V Y ) — > X V Y o n X v Y , w i t h ( x , y ) a s e x a c t u n i t e l e m e n t , o o N o w , l e t h : X x Y — > X V Y b e t h e p r o d u c t o f f a n d g , i . e . h ( x , y ) = m ( ( x , y ) , y ) ) . W e t h e n h a v e t h e f o l l o w i n g c o m p o s i t i o n : X V Y — J > X x Y > X V Y . I t i s e a s i l y c h e c k e d t h a t h ° i = 1,, 3 X V Y ( F o r a n y ( x , y ) e X v Y , h ° J ( x o , y ) = h ( x Q , y ) = m ( ( x Q , y ) ; ( X Q , V Q ) ) = ( X q , y ) . S i m i l a r l y f o r ( x , y ) e X V Y , w e h a v e h o j ( x , y Q ) = ( x , y Q ) ) • L e t : V Y ) — > ^ ( X x Y ) b e t h e m a p i n d u c e d b y j o n t h e h o m o t o p y g r o u p s . S i n c e 1^ = ( h o j ) ^ = h ^ o j } w e h a v e t h a t i s a n o n e - t o - o n e m a p . O n t h e o t h e r h a n d , i t i s w e l l k n o w n t h a t i s a l s o o n t o [4; p . 42] . ( I n f a c t , o n e h a s : ^ ( X x y ) = i ^ n ( X ) + y n ( Y ) . X Y w h e r e i a n d i a r e t h e c a n o n i c a l i n c l u s i o n s o f ( X , x ) a n d ( Y , y ) o o X Y i n t o ( X x y , ( X q , y ) ) r e s p e c t i v e l y . B u t b o t h i a n d i f a c t o r o v e r i . e . x - ^ X V Y — ^ > x x Y i s c o m m u t a t i v e . ) - 26 -Therefore, i s an isomorphism for a l l n . Since both X and Y are CW-complexes, X V Y and X x Y are of the homotopy type of a CW-complex. By Whitehead's theorem (1.4), one gets that j : X v Y —> X x Y i s a homotopy equivalence. (4.3) Remarks. For connected spaces X and Y, one has according to [4; p. 42], the isomorphisms T T^ X x Y ) = ir^X) 9 ^ ( Y ) , and •n±(X V Y) = ^ ( X ) * ir^Y) where * denotes the free product of groups. Hence i f X v Y - X x y , we must have that TT^(X) or ir^(Y) i s the t r i v i a l group. (4.4) Theorem. Let (X, X q) be a connected f i n i t e CW-complex with base point, and l e t (Y, y ) be an arbitrary connected CW-complex with base point. Then thei r wedge X v Y i s not an H-space, provided neither X nor Y i s homotopy equivalent to a point . Proof : Let (X, X q) and (Y, y ) be as stated above, and assume that X v Y i s an H-space. We w i l l show that this leads to a contradiction. If X v Y i s an H-space, then by (4.1) (X, X q) and (Y, y Q) are H-spaces too. Since Y i s not homotopic to a point, we have that H (Y; Z) ^ 0 for some i .> 1 (2.11). Hence there i s a f i e l d K , such that - 27 -(4.5) \(Y>- K ) ^ 0 > f o r s o m e k 1 !• (In f a c t , K = <Q or 2^ . Using the Universal Coefficient Theorem, th i s i s seen as follows : I f IL(Y; 2) has elements of i n f i n i t e order, i . e . i f i t has a subgroup isomorphic to 2, then H^Y; Q) = H^(Y; 2) 0 Q ^  0 . If H i(Y; 2) has an element of order p , then Tor(H^(Y; 2); 2^) ^ 0, which implies H.,,(Y; 2 ) + 0 ). 1+1 p For the H-space X, we know from (3.19) that : (4.6) H (X; R) + 0, for some n > 1. n We now consider the exact homology sequence of the pair (X x Y, X si Y) over the f i e l d R : . . .—> H ±(X v Y; R) —> H ±(X x Y; R) > H ±(X x Y, X V Y; R) > H (X v Y; R) — — — > ... . By (4.2) j i s a homotopy equivalence, and so j . i s an isomorphism. Hence (4.7) H^(X x Y, X v Y; R) = 0, for a l l j . But the Runneth formula for the pair (X, X q) and (Y, y ) (1.2) gives n+k H n + k ( X x Y, X V Y; R) = • H ( X , X q ; R) 0 H.(Y, y ; R) 1=0 n+k-1 s « H n + k - i ( X ; K ) 8 H i ( Y ; K ) ' i = l and with (4.5) and (4.6), we conclude - 28 -(4.8) H , (X x Y, X V Y; K) / 0 , n+K since H n(X; K) 8 \ ( Y 5 K) ^ 0 . We see that statement (4.8) contradicts statement (4.7), and our theorem i s proved. (4.9) Remark. Theorem (4.3) i s of course true under the weaker condition : X i s of the homotopy type of a connected CW-complex, H^(X; 2) i s f i n i t e l y generated, and Y i s of the homotopy type of an arbitrary connected CW-complex. - 29 -§5. Examples of i n f i n i t e CW-complexes X and Y ; such that X V Y i s  an H-space . In this l a s t section, we i l l u s t r a t e by an example that our main theorem i s no longer true, i f one drops the assumption " X i s a CW-complex, with f i n i t e l y generated homology H^(X; 2?) "• Since the example, which is.given here, consists of Eilenberg-MacLane spaces, we f i r s t give some facts about Eilenberg-MacLane spaces below : (5.1) An Eilenberg-MacLane space K(2" , m) with base point x , i s a w 1 p o connected CW-complex having only one non-vanishing homotopy group; namely TT m(K(Zp, m)) - . According to [5, Ch.8], K(2 p, m) has the following properties : ( i ) K(2 , m) i s a CW-complex with f i n i t e skeletons . P ( i i ) (K(Z , m), x ) admits an H-space structure, p o ( i i i ) If p i s a prime, then the reduced homology groups H n(X, X Q; 2) are f i n i t e l y generated, and have only p-torsion. (This f o l -lows from [5, 9.6.15]. T r i v i a l l y the homotopy groups of K(2^, m) are of Serre-class C^  , and hence for m >_ 2, so are the homology groups . For m = 1, see [5, 9.5.7]. Here C denotes the class of f i n i t e abelian P groups having only p-torsion.). - 30 -(5.2) Lemma. Let p and q be different primes, and l e t m, n be integers >_ 2. Then the canonical inclusion j : K(Z p, m) v K(2 q, n) —> K ( 2 p J m) x K(2 q, n) i s a homotopy equivalence . Proof : Let X = K(2 , m), and Y = K(2 , n) with base points x and P q o y Q respectively. The Kiinneth formula for the pairs (X, X q) and (Y, y Q) over integer coefficients (1.2) gives H (X x . Y, X V Y) = 9 H.(X, x ) 0 H .(Y, y ) 9 n . _ l o n - i o 1=0 n 9 Tor(H (X, x ); H (Y, y )) . „ l o n - i - i o 1=0 Property ( 5 . 1 ( i i i ) ) then implies that : Tor(H.(X, x ); H . .(Y, y )) = 0 , for a l l i and n , l ' o n-i-1 '  Jo ' and H.(X, x ) 0 H .(Y, y ) = 0 , for a l l i and n . Hence l o n - i • o (5.3) H n(X x Y, X V Y) = 0 for a l l n . We now consider the exact homology sequence of the pair (X x Y, X v Y) : . . . —> H ,, (X x Y, X V Y) -^ -> H (X V Y) — — > H (X x Y) n+i n n H (X x Y, X V Y) —> ... n - 31 -By (5.3), the map j induces an isomorphism i n homology. Since X, Y are simply-connected CW-complexes (m, n >_ 2), we conclude with Whitehead's Theorem (1.4), that j i s a homotopy equivalence, and our lemma i s proved. Since by ( 5 . 1 ( i i ) ) , X and Y are both H-spaces, X x y i s an H-space too (2.8). Then lemma (5.2) together (2.6) implies that X V Y = K(Z , m)V K(2 , n) P q i s an H-space, i f p and q are different primes and both m, n ^  2 . - 32 -Bibliography [1]. A. Borel ; "Topics i n the Homology Theory of Fibre Bundles", Springer-Verlag, B e r l i n , Heidelberg, New York 36 (1967). [2]. W. Browder ; "Torsion i n H-spaces", Ann. of Math. 74 (1961), 24-51. [3]. C. H. Dowker ; "Topology of Metric Complexes", Ann. of Math. 74 (1961). [4]. P. J. Hil t o n ; "An Introduction to Homotopy Theory", Cambridge Tracts i n Mathematics and Mathematical Physica, No. 43. [5]. E. H. Spanier ; "Algebraic Topology", McGraw-Hill Series i n Higher Mathematics. [6]. N. E. Steenrod ; "The Topology of Fibre Bundles", Princeton University Press (1951). [7]. N. E. Steenrod ; "Cohomology Operations", Princeton University Press (1962). 

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