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Hurewicz homomorphisms Lê, Anh-Chi’ 1974

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HUREWICZ HOMOMORPHISMS BY ANH-CHI LE B.Sc.,University of B r i t i s h Columbia,1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of MATHEMATICS We accept t h i s as conforming to the required standard, THE UNIVERSITY OF BRITISH COLUMBIA May,197^ In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requ i remenr s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I ag ree 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 r e f e r e n c e and s tudy . I f u r t h e r agree 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 purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that 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 not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Depa rtment The U n i v e r s i t y o f B r i t i s h Co lumbia Vancouver 8, Canada i i Supervisor s Dr Roy Douglas ABSTRACT ; Theorem : Let X be simply connected . H q(X) be f i n i t e l y generated f o r each q JC q(X) be f i n i t e f o r each q .^ n . n > 1 Then , h q ,7Tq(X) > H q(X) has f i n i t e kernel f o r q 4 2n has f i n i t e cokernel f o r q < 2n+l ker h 2 n + 1 8 Q = ker KJ where , u i s the cup product or n+1 the square free cup product on R ( X ) depending on whether n+1 i s even or odd , respectively . ( R N + 1 ( X ) i s a quotient group of H 2 + 1 ( X ) ® H Q + 1 ( X ) to be defined i n t h i s thesis ) TABLE OF CONTENTS PAGE I Introduction 1 II The Meta-theorem 8 III The kernels and cokernels of Hurewicz homomorphisms 10 Bibliography Zk i v ACKNOWLEDGEMENTS The author thanks his supervisor , Dr Roy Rene Douglas , without whom the r e a l i z a t i o n of t h i s thesis would have been impossible , The author i s also very happy to thank the University of B r i t i s h Columbia f o r t h e i r f i n a n c i a l support ( 1 9 7 2 - 1 9 7 ^ ) and Dr Rene' Held for his many he l p f u l suggestions . 1 Chapter I INTRODUCTION The Mod ^Hurewicz Theorem provides a v i t a l l i n k between homotopy groups and homology groups . In p a r t i c u l a r , i t s importance l i e s i n the fa c t that homology groups are more r e a d i l y computable . The well-known theorem asserts that i f X i s simply connected and 7Tq(X) e ^ . a perfect and weakly complete class (Serre) of abelian groups , for q i- n ;then , fi^+1(X) i s ^-isomorphic to H n + 1 ( X ) by the Hurewicz homomorphism h n + ^ , and at one l e v e l higher , n n+2 a ^-epimorphism . This thesis extends the theorem to higher dimensions for the case \f = ft , the Serre class of f i n i t e abelian groups . We e s t a b l i s h that h q , q 4 2n , are £ isomorphisms and h 2n+l i s . epimorphism . We also compute kernel h 2n+l (Mod ft ) . Counter-examples are provided to conclude' that further extensions' are not possible , i n general . 2 l.Qn f i brat ions and the process of k i l l i n g homotopy groups. ( 1 . 1)The d e f i n i t i o n of a f i b r a t i o n . -Let p :E ^ B be a map between topological spaces. Then , the map p i s c a l l e d a f i b r a t i o n ( i n the sense of Serre) i f given any f i n i t e CW complex K,a map g :K > E and a map F :KxI B such that F(x,0)=p(g(x)) f o r a l l x in K ; then there exists a map G : K x I — E such that G(x ,0)=g(x) and pG(x,t)=F(x,t) Let * be the base point of B. Then,F=p (*) i s c a l l e d the f i b r e of the f i b r a t i o n p We have a Hurewicz f i b r a t i o n i f the above description holds when K i s an arb i t r a r y t o p o l o g i c a l space. In t h i s thesis,we s h a l l use the term " f i b r a t i o n " without specifying "Serre" or "Hurewicz" when the two are both admissable. (1.2)Remark.-Any map A =»B can be i d e n t i f i e d up to homotopy with a f i b r a t i o n (See [ s]p.99 ) E p > B such that the following diagram i s commutative : (homotopy equivalences) By t h e ( a r t i f i c i a l ) f i b r e of f , we mean the f i b r e of p . 3 (1 . 3)Annihilating homotopy groups .-Let X be a CW-complex and denote i t s n-skeleton by X n For the following construction , l e t us assume that X i s a connected CW-complex with X°= pt . Let L * ^ i f e i be a set of generators of the homotopy group 7 ^ + 1 ( X ) . By c e l l u l a r approximation , we may assume that : * . : s n + 1 » x n + 1 l We now attach an n+2 - c e l l corresponding to each<^ =inclusion X C > Cc< C l e a r l y , 71^(X)S K±(C* ) f o r (KUn and •^ n + 1(C c <)=0 In order to " k i l l o f f " a l l the homotopy groups -^(X) for q>n , we it e r a t e t h i s process u n t i l we arrive at a space X(l,...,n) . where : the inclusion X ^ > X(l,...,n) induces; K_(X)=/T(X(1,...,n))for q*n and , moreover , 7T (X(l,...,n))=0 f o r q>n Now , the Cartan-Serre -Whitehead construction i s to take the fi b r e (See (1.2)) of X C »X(l,...,n) and , we obtain : X(n+1,.. . ,-) > X > X ( l , . . , , n ) where TC (X(n+1,... ,°»)£0 i f qfen = 7 T ( X ) i f q^n+1 Taking the f i b r e X(n+1,...,«) and k i l l i n g off i t s homotopy groups i n dimensions greater than m gives a space denote by X(n+l,...,m) Cl e a r l y , TT (X(n,...,m))=0 i f q^n-1 S t t _ ( X ) i f n*q*=m =0 i f q>m Observation.-Given any map Y: X => X(i,...,m) «* B we can p u l l back through the path space PB of B , and have the f i b r a t i o n : SIB ^ W=X(m+l,.. . ,co) P X 2.The Serre - Leray Spectral Sequence of a f i b r a t i o n . (2.1) A-spectral sequence i s a sequence of abelian groups E n together with a sequence of d i f f e r e n t i a l endomorphisms : d n 5 E n > E n d n d n = ° » ^ such that E n + l = H ( E n , d n ^ * k e r d n / i m d n (2.2) We s h a l l describe the Serre-Leray spectral sequence which arises from a Serre f i b r a t i o n 0 = ( F * X P >B ) We assume B to be a simply - connected CW -complex with f i n i t e skeleta and X of the homotopy type of a f i n i t e CW-complex Remark.-Simply - connectednesss of the base space B i s imposed i n order-to avoid the discussion of " l o c a l c o e f f i c i -ents" i n t h i s general s e t t i n g . The t o t a l space i s f i l t e r e d by the inverse p-images of the n-skeleton B n of the base space B . Set X n=p" 1(B n) Let H* stand for the ordinary " c e l l u l a r " cohomology theory . We are now going to define the Serre - Leray Spectral Sequence (E r($£) td r) of j£ i n terms of the following exact couple : D P , q = H p + q ( x p ) The exactnesss axiom of H : ^ P ^ X ^ X ^ ) h l , HP + (^(X p) f 1 > H P ^ X p ^ ) > H P + ( i + 1 ( x p , x p _ 1 ) , . . . enables us to derive (inductively) the following exact couple : r ' / r "r where , D r + 1=im f r , f r = d r=S r hr » E r + l = H ( E r ! d r ) deg(d r)=(r,l-r) r Proposition.-( X ) a) E p ' q = H p ( B j H q ( F ) ) E P ' q = ker ( H p + q ( X ) > H p + q ( X ^ ) _ G r Hp+q ker ( H p + q ( X ) > H p + q ( X „ ) ) b) H = H ( ;G) i s a m u l t i p l i c a t i v e cohomology theory i f G i s a commutative r i n g with unit . Therefore , the spectral sequence in h e r i t s a canonical m u l t i p l i c a t i v e structure .With respect to t h i s m u l t i p l i c a t i o n , the d i f f e r e n t i a l s d r turn out to be derivations , i . e . d r(ab) = d r(a)b + ( - l ) p + q a d r ( b ) where a e E p ' q r c)The edge homomorphisms f , p can be computed as follows H n ( B ) -> H n ( X ) ^ l £ ' 0 < $ ) H n ( X ) J L _ _ ^ H n ( F) ( 2 . 3 ) P i c t o r i a l l y , the Eg term of the spectral sequence can be represented as follows : H ( F ) • 2 r - l H ( B ) 7 The f o l l o w i n g p r o p o s i t i o n can be shown u s i n g the Serre -Leray s p e c t r a l sequence P r o p o s i t i o n .-L e t F •—£-» X — > B be a f i b r a t i o n B be m-connected , F n-connected Then , we have Serre homology and cohomology exact sequences of f i n i t e l e ngths : Hm + n + l < P ' - ^ — V n + l ^ H m + n + 1 < B ' — ^ W F > > . . . ... „ H m + n ( F ) , H m + n + 1 ( B ) _ P ! ^ H m + n + 1 ( X ) _ £ l H m + n + 1 ( F ) Chapter II THE META-THEOREM Let us define a homomorphism h n(X) : 7T n(X) > V X ) ( n - X ) r e l a t i n g homotopy and homology groups of a space X . This homomorphism i s given by : h n(X)([fl)=h n([fl>)=f^(g n) for any homotopy class [f] € JT^CS 1 1) where g n i s a generator of H n(S n;Z) h n i s natural and i s c a l l e d "Hurewicz*homomorphism" . The following i s a c r u c i a l observation as f a r as studying the Hurewicz homomorphisms i s concerned . We s h a l l c a l l i t "THE META-THEOREM" 9 Let P(k,n) stand for : "h^^has f i n i t e kernel and cokernel" . This notation i s used to formulate the theorem (3.1) The Meta-Theorem.-Let k £ 1 , n >, max(k-l,l) . If P(k,n) i s true for any n-connected space , then , P(k,n+1) i s true f o r any n+1 -connected space . PROOF : Suppose Y i s n+1 -connected . We get , from the Serre exact sequence of the path f i b r a t i o n ftY > PY > Y , the following commutative diagram , the rows of which being exact s re n+k+ {PY) ^ Hn+k+ {PY) n+k+ h n+k+1 » \ + k U Y ) r r n + k ( P Y ) H. n+k+ -> H. n+k (ay) * W P Y > : 0 ' ^ o Since n+k+1 4 n+l+n+1 = 2n+2 , the above commutative diagram i s always true given the hypothesis of the theorem. In p a r t i c u l a r , by the square [L].h n + k(£Y) i s equivalent to h n + j c + ^ ( Y ) . Since D.Y i s n-connected , P(k,n) i s true for h n + k(S.Y) . Hence , P(k,n+1) i s true f o r h n + k + 1 ( Y) . Q.E.D. Remark .-The proof shows that P(k,n) can stand f o r a more general algebraic statement about h + ^ • 10 Chapter III THE KERNELS AND COKERNELS OF HUREWICZ HOMOMORPHISMS Henceforth , we assume that H^(X) i s f i n i t e l y generated for a l l q and f o r any space X . In p a r t i c u l a r , t h i s means that H (X) i s f i n i t e i f and only i f H (X) 0 Q =0 We need two lemmas and two d e f i n i t i o n s . (4.1) Lemma ( Hopf theorem ) Let X "be an H space . Then , H (XjQ) i s a free graded commutative Q- algebra i . e H (X;Q) i s isomorphicE.to the tensor product of an exterior algebra ( with odd dimensional generators ) and a polynomial algebra ( with even dimensional generators ) over Q . Notation .-F Q ( X ^ , . . . , x n ) denotes the free graded commutative Q- algebra over Q with generators x^,...,x . (4.2) Lemma Let B = X(n,...,m) Then , H Q > B ) = H*(T) w h e r e ' m - n ° : T = K(7L .X,n+i-l) { J o n 1 10a PROOF Thorn »|_TJ , has observed that the Postnikov invariants of an H-space are torsion cohomology classes . Therefore , one can show that SLB and T have the same r a t i o n a l homotopy type ( by an induction argument ; induction on m-n ) 0.3) D e f i n i t i o n .-Let A M ( X ) be the subspace i n H J J ( X ) 8 H J J ( X ) generated by x. 0 x. , where the x. are the generators of H M ( X ) . I l l w Then , R M ( X ) i s defined to be : a) H ^ ( X ) 0 Hjl'(X) i f m i s even . A M ( X ) 0.4) D e f i n i t i o n .-vT : R M ( X ) > H* m(X) i s defined to be the homomorphism induced by u :H™(X) 0 HjjU) > H* m(X) <j i s the usual cup product . In the case m i s odd we c a l l o the square free cup product on R (X) . 11 We c a l l the following theorem P(n,n) , thus r e f e r r i n g to the "Meta - theorem " (5.1) Theorem P(n,n) If X i s n-connected , n >/ 2 t h e n ' h n + n i ^ n ^ > *WX> has f i n i t e kernel and cokernel . PROOF ; Consider the f i b r a t i o n : _P_B > W=X(2n, . . . ,oo) 2—> x with B = X(n+1, . . . ,2n-l) and i t s Serre spectral sequence with Q c o e f f i c i e n t s : The bottom row of the following commutative diagram H 2 N ( X ) -> H 2 n(W) 0 >E 2 n»° ±- > H Q 2 n(W)-~ i s exact . ( Because : XLB i s n - 1 connected — \ H Q q(ilB) * 0 , 1 ^ q ^ n - l -^E°' 2 RL_> 0 n-2 ^> EP» q = 0 i f p,q£0 , p+q=2n H 2 n(W) £ E£ n'° <B E ° ' 2 n ) Note that JlB .and ^ 0 K ( 7^ + i + 1X,n+i) have the same r a t i o n a l homotopy type . Therefore , Consider V l 1 H Q N ( ^ B ) Let x «y (-RB) i.e . xeHg(fiB) ; yeH^C-RB) Then , d n + 1 ( x ) / 0 ^  d n + 1 ( y ) by the isomorphism d n + 1 : E°' n=E°; n=E^];' 0 R n(-flJ3) = H Q 2 n(i*B) v pn+l.n^n+l,n ^ n+l _ £ , 2 = H £ + 1 ( X ) 6 H Q ( £ B ) ;n+l, 0 Hence , d n + 1 ( x . y ) = d n + 1(x)®y + ( - l ) V ( y ) ji 0 unless n i s odd and x=y Therefore , ker d n + 1 = 0 and E ° ' 2 n = 0 Which implies that i n diagram pP~[ , i i s isomorphism . We now again use the fac t : H*UB) = F Q ( x n .. ,x n ) where , n ^ n^ 2n-2 ; i = 1». .. »k to obtain : H Q ^ U B ) =0 which implies = ^ Zn+1 Q W The diagram becomes : H* n(X) ,2n,0 c 4> H^n(W) i.e . p^ i s an isomorphism i . e . p* : H^n(W) i s also an isomorphism . i.e . ker p^ = coker p # - 0 > H^(X) Hence , ker [p*:H2n(W) > H 2 n ( X ^ and coker jp*:H 2 n(W) > H2n ( X?l are f i n i t e . The commutative diagram : ru 2 n(w) h 2 n(w) H 2 n(w) completes the proof . h 2 n ( x ) -> H 2 n ( X ) 14 Thus , we have proved. P(k,k) for k >/ 2 . Apply the Meta - theorem : P(k,k) P(k,k+1)=^ ... =>P(k,n) ,for a l l n >/ k We state P(k,n) i n the following (5.2) Theorem P(k,n) Let X be n - connected , n V 2 Then , hn+k s 7 W X > > W X ) k ^ n has f i n i t e kernel and cokernel . (6.1) Theorem R(n+l,n) Let X be n - connected , n >/ 2 . Then , h 2 n + l s J r2n +lW > * W X > has f i n i t e cokernel . PROOF.-Consider:;"the f i b r a t i o n : i l E > W P X where, E * X(n+1,...,2n) W =X(2n+l,. . . ,oo) 15 2n H*(£lE ) o N N \ . 0 0 Q 0 0 n + i + H Q ( X ) As i n (5.1) » we have Thus , i n H * n ( Q E ) "S R n ( - R E ) ^0.2n d ... Fn+l,n J n i l ^ » En + 1 »0,2n gn + l .n £ in III R n ( i l E ) ' . • H n ( - £ E ) ® H n + 1 ( X ) d n + ^ i s monic (by an argument sim i l a r to the one i n the proof of P(n,n) ) . Therefore , n+2 " u _ £ , 2n+l i.e. E2n + 1 . ° f i r H 2 N + 1(X,Q) 16 H* n + 1(X) -> H f + 1 ( W ) E 2n+l,0 implies that p i s a monomorphism . Hence , H « n + 1 ( W ) H « n + 1 ( X ) or i s an epimorphism , coker ( p # s H 2 n + l ^ W ^ i s f i n i t e H2n+l ( X> > The commutative diagram : ^ i ( » > S * ^ » H » ) H 2 n + l ( W ) * H 2 n + 1 ( X ) completes the proof 17 7. The kernel of { 1 ( 7 . 1 ) Theorem S(n+l,ri) Let X he n-connected , n >/ 1 . Then , ker h 2 n + l 0 Q " k e r ^ u° i s the cup product or the square free cup product on R (X) depending on whether n+l i s even or odd , respectively . PROOF ; Consider the f i b r a t i o n : W -4-* X -^ ->K = K(Jt n + 1X,n+l) W = n+l connective covering and i t s Serre Spectral Sequence i 2n+l % \ 0 H2n+2 n+2 HQ(W) 0 0 0 0 0 Q 1T+T H Q(K) 18 The f i r s t possibly non vanishing d i f f e r e n t i a l coming to E 2 n + 2 , ° i s d 2 n + 2 which i s also the f i r s t possibly vanishing d i f f e r e n t i a l from E ° ' 2 n + 1 . non Hence , E. 2n+2,0 u2n+2 r„x l f f i d2n+2 E ^ 2 ^ ~ = k e r d 2 n + 2 Thus , the edge ihomomorphisms : X* : H 2 n + 2 ( K ) • f :H 2 n + 1(X) Therefore , has k e r V* = im d 2 n + 2 > H 2 n + 1(W) „0,2n+l has c o k e r = im d 2 n + 2 * «v ker* = coker |> The commutativity of i a2n+l (W) implies that of "H2n+1 * H 2 n + l ( X ) 2n+l •>> H 2 n + 1(W) 0 Q 7 c 2 n + 1 ( x ) 0 Q h 2 n + 1 « Q -> H 2 n + 1 ( X ) 0 Q Since W i s n+l connected , "by P(k,n) we have K 2 n + 1 ( X ) « Q h 2n+l 8 Q H 2 n + 1 ( X ) 8 Q Hence , by i s an isomorphism . I * | , we obtain : PH ker ^  « Q = ker h 2 n + 1 ( X ) ft Q We also note that : ker £ # e Q = coker Combine {T] , k e r h 2 n + l ( X ) 8 Q = k e r * S i m d2n+2 Apply Hopf theorem to H Q ( K ) and dis t i n g u i s h two cases :  If n+l i s even : H 2n+2 ( K ) ~ H j J + 1 ( K ) 0 H g + 1 ( K ) = H £ + 1 ( X ) 0 H £ + 1 ( X ) And , since 5 i s a ri n g homomorphism , the following 20 diagram i s commutative : H 2 n + 2 ( x ) H 2 n + 2 ( K ) rn+l n+1 rn+l which implies 5~j ker"** = kernel of cup product on H Q + 1 ( X ) 0 H Q + 1 ( X ) If n+1 i s odd ; H 2 n + 2 ( X ) ^ - •H 2 n + 2(K) rn+l rn+l H 2 + 1 ( X ) 8 H f ^ X ) ^ ^ - H ^ ( K ) 0 H ^ ( K ) which implies : 6 ker^ = kernel of square free cup product on R n + 1 ( X ) i t » L~5J » Zl complete: the proof . Combine P(k,n), R(n+l,n), S(n+l,n) and Hurewicz theorem, we have the following (8.1) Theorem: Let X be n-connected, n J> 1 . H (X) be f i n i t e l y generated for each q . Then, & 1 K q ( X ) -> H q(X) has f i n i t e kernel for q ^ 2n has f i n i t e cokernel i f q < 2n + 1 and, h 2 n + 1 has ker h 2 n + 1 8 Q = ker vT 21 WHERE , u° i s the cup product or the square free cup product on R n +^(X) ' depending on whether n+l i s even or odd , respectively . (8.2) Corollary .-Let X be simply - connected . H q(X) be f i n i t e l y generated for each q . TT. (x) be f i n i t e for q ^ n . q Then , a l l conclusions i n Theorem (8.1) are s t i l l true PROOF : Let F be the Serre class of f i n i t e abelian groups . Consider the f i b r a t i o n : X(n+1,. .. ,oo) -> X -»X(1,.. . ,n) and i t s Serre Spectral Sequence : H Q(X(n+l oo)) Q 0 We denote fi- isomorphism f by = ( i . e . ker f , coker f e £ ) H Q(X(1,...,n)) 22 H^(X(1,...,n)) are a l l 0 because : TC q(X(l n)) = i y X ) i f q 4 n = 0 i f q > n+l By Mod Hurewicz theorem s 0 = j y x d , . . . ,n)) s H q ( X ( l n)) Therefore , i * : H?(X) = H*(X(n+l,.. . ,oo)) Hence , i * : H q(X(n+l *>) )S H (X) The commutative diagram : 7Tq(X(n+l )) 1 > TlqCX) H q(X(n+l )) 1 >H q(X) completes the proof . 23 9. A remark .-One naturally asks at t h i s point : " Can the r e s u l t s be extended any further ? " . The answer , i n the negative , i s provided by the following examples^: (9.D Let X = S n + 1 x S n + 1 Then , X i s n-connected . T f 2 n + 2 ( X ) i s f i n i t e . H 2n+2 ( X ) = 2 ' Therefore, cokernel h 2 n + 2 ^ ^ f i n i t e . (9.2) Let X = S n + 1 v S n + 1 Then , X i s n-connected J^2 n + 1(X) i s i n f i n i t e H^ n + 1(X) = 0 Therefore , i . e . ker h 2 n + 1 ( X ) & Q ^ 0 h 2 n + 1 ( X ) does not have f i n i t e kernel . 24 BIBLIOGRAPHY [sl Spanier E.H. , " Algebraic Topology " , McGraws-Hill, 1966 , pp .99 . Chapters 7 , 9 [ T j Thorn R. , L'homologie des espaces fonctionels , Colloque de topologie algebrique , pp.29-39 f Louvain , 1956 . 

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