HUREWICZ HOMOMORPHISMS by ANH-CHI LE B.Sc.,University Of

HUREWICZ HOMOMORPHISMS

BY

ANH-CHI LE

B.Sc.,University of British 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 this as conforming to

the required standard,

THE UNIVERSITY OF BRITISH COLUMBIA

May,197^ In presenting this thesis in partial fulfilment of the requiremenrs 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.

Depa rtment

The University of British Columbia Vancouver 8, Canada ii

Supervisor s Dr Roy Douglas

ABSTRACT ;

Theorem :

Let X be simply connected .

Hq(X) be finitely generated for each q

JCq(X) be finite for each q ^. n .

n > 1

Then ,

hq ,7Tq(X) > Hq(X)

has finite kernel for q 4 2n

has finite cokernel for q < 2n+l

ker h2n+1 8 Q = ker KJ

where , u is the cup product or n+1

the square free cup product on R (X)

depending on whether n+1 is even or odd ,

respectively .

( RN+1(X) is a quotient group of

H2+1(X)® HQ+1(X) to be defined in this

thesis ) TABLE OF CONTENTS

PAGE

I Introduction 1

II The Meta-theorem 8

III The kernels and cokernels

of Hurewicz homomorphisms 10

Bibliography Zk iv

ACKNOWLEDGEMENTS

The author thanks his supervisor , Dr Roy Rene

Douglas , without whom the realization of this thesis would have been impossible ,

The author is also very happy to thank the

University of British Columbia for their financial

support (1972-197^) and Dr Rene' Held for his many helpful suggestions . 1

Chapter I

INTRODUCTION

The Mod ^Hurewicz Theorem provides a vital link between homotopy groups and homology groups . In particular , its importance lies in the fact that homology groups are more readily computable .

The well-known theorem asserts that if X is simply

connected and 7Tq(X) e^.a perfect and weakly complete class (Serre) of abelian groups , for q i- n ;then ,

fi^+1(X) is ^-isomorphic to Hn+1(X) by the Hurewicz

n a homomorphism hn+^ , and at one level higher , n+2

^-epimorphism .

This thesis extends the theorem to higher dimensions

for the case \f = ft , the Serre class of finite abelian

groups . We establish that hq , q 4 2n , are £ isomorphisms

and h2n+l is. epimorphism . We also compute kernel h2n+l

(Mod ft ) . Counter-examples are provided to conclude' that

further extensions' are not possible , in general . 2

l.Qn fi brat ions and the process of killing homotopy groups.

(1.1)The definition of a fibration.-

Let p :E ^ B be a map between topological spaces.

Then , the map p is called a fibration (in the sense of

Serre) if given any finite CW complex K,a map g :K > E and a map F :KxI B such that F(x,0)=p(g(x)) for all x

in K ; then there exists a map G:KxI—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 (*) is called the fibre of the fibration p

We have a Hurewicz fibration if the above description

holds when K is an arbitrary topological space.

In this thesis,we shall use the term "fibration" without

specifying "Serre" or "Hurewicz" when the two are both

admissable.

(1.2)Remark.- Any map A =»B can be identified up to homotopy with

a fibration (See [s]p.99 ) E p > B such that the following

diagram is commutative :

(homotopy equivalences)

By the(artificial)fibre of f , we mean the fibre of p . 3

(1.3)Annihilating homotopy groups .-

Let X be a CW-complex and denote its n-skeleton by Xn

For the following construction , let us assume that X is a connected CW-complex with X°= pt .

Let L*^ifei be a set of generators of the homotopy

group 7^+1(X) .

By cellular approximation , we may assume that :

n+1 n+1 *. :s » x l We now attach an n+2 -cell corresponding to each<^

=inclusion

X C > Cc<

Clearly , 71^(X)S K±(C* ) for (KUn

and •^n+1(Cc<)=0

In order to "kill off" all the homotopy groups -^(X) for q>n , we iterate this process until 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 for q>n

Now , the Cartan-Serre -Whitehead construction is to take the fibre (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 if qfen

= 7T(X) if q^n+1

Taking the fibre X(n+1,...,«) and killing off its homotopy groups in dimensions greater than m gives a space denote by X(n+l,...,m)

Clearly ,

TT (X(n,...,m))=0 if q^n-1

Stt_(X) if n*q*=m

=0 if q>m

Observation.-

Given any map Y: X => X(i,...,m) «* B

we can pull back through the path space PB of B , and have

the fibration : SIB ^ W=X(m+l,.. . ,co)

P

X

2.The Serre - Leray Spectral Sequence of a fibration .

(2.1) A-spectral sequence is a sequence of abelian

groups En together with a sequence of differential

endomorphisms :

dn 5 En >E n dndn=° » ^

E =H(E ,d ker d im d such that n+l n n^ * n / n

(2.2) We shall describe the Serre-Leray spectral

sequence which arises from a Serre fibration

0 = ( F * X P >B )

We assume B to be a simply - connected CW -complex with finite skeleta and X of the homotopy type of a finite

CW-complex

Remark.-

Simply - connectednesss of the base space B is

imposed in order-to avoid the discussion of "local coeffici

-ents" in this general setting .

The total space is filtered by the inverse p-images

of the n-skeleton Bn of the base space B .

1 n Set Xn=p" (B )

Let H* stand for the ordinary "cellular" cohomology

theory .

We are now going to define the Serre - Leray Spectral

Sequence (Er($£)tdr) of j£ in terms of the following exact

couple :

p+q DP,q= H (xp)

The exactnesss axiom of H :

h +( f ^P^X^X^) l , HP ^(Xp) 1 > HP^Xp^)

+( +1 >HP i (xp,xp_1) , . . .

enables us to derive (inductively) the following

exact couple :

r ' / r

"r

where , Dr+1=im fr , fr = r

h E =H(E d ) dr=Sr r » r+l r! r

deg(dr)=(r,l-r) Proposition.-

a) Ep'q =Hp(BjHq(F))

P q p q p+q E ' = ker (H + (X) >H (X ^ ) _ Gr Hp+q (X) ker (Hp+q(X) > Hp+q(X„) )

b) H = H ( ;G) is a multiplicative cohomology theory if G is a commutative ring with unit . Therefore , the spectral sequence inherits a canonical multiplicative structure .With

respect to this multiplication , the differentials dr turn out to be derivations , i.e.

p+q dr(ab) = dr(a)b +(-l) adr(b)

where a eEp'q r c)The edge homomorphisms f , p can be computed as follows

Hn(B) -> Hn(X)

^ l£'0<$)

Hn (X) JL__^ Hn ( F)

(2.3)Pictorially , the Eg term of the spectral sequence can be represented as follows :

H (F) • r-l 2

H (B) 7

The following proposition can be shown using the Serre

-Leray spectral sequence

Proposition .-

Let F •—£-» X —>B be a fibration

B be m-connected , F n-connected

Then , we have Serre homology and cohomology exact sequences of finite lengths :

H P H B F m+n+l< '-^— Vn+l^ m+n+1< '—^ W > > ...... „ Hm+n(F) , Hm+n+1(B)_P!^Hm+n+1(X)_£l Hm+n+1(F) Chapter II

THE META-THEOREM

Let us define a homomorphism

X) (n X) hn(X) : 7Tn(X) >V - relating homotopy and homology groups of a space X .

This homomorphism is given by :

hn(X)([fl)=hn([fl>)=f^(gn)

11 for any homotopy class [f] € JT^CS ) where gnis a generator

n of Hn(S ;Z)

hn is natural and is called "Hurewicz*homomorphism" .

The following is a crucial observation as far as studying the Hurewicz homomorphisms is concerned .

We shall call it "THE META-THEOREM" 9

Let P(k,n) stand for : "h^^has finite kernel and cokernel" . This notation is used to formulate the theorem

(3.1) The Meta-Theorem.-

Let k £ 1 , n >, max(k-l,l) .

If P(k,n) is true for any n-connected space , then , P(k,n+1) is true for any n+1 -connected space .

PROOF :

Suppose Y is n+1 -connected .

We get , from the Serre exact sequence of the path fibration ftY > PY > Y , the following commutative diagram , the rows of which being exact s

{PY) » \+kUY) rrn+k(PY) ren+k + n+k+ h n+k+1

H {PY) H. -> H. (ay) PY ^ n+k+ n+k+ n+k * W >: 0' ^ o

Since n+k+1 4 n+l+n+1 = 2n+2 , the above commutative diagram is always true given the hypothesis of the theorem.

In particular , by the square [L].hn+k(£Y) is equivalent to

hn+jc+^(Y) . Since D.Y is n-connected , P(k,n) is true for

hn+k(S.Y) . Hence , P(k,n+1) is true for hn+k+1( Y) .

Q.E.D.

Remark .-

The proof shows that P(k,n) can stand for a more

general algebraic statement about h +^ • 10

Chapter III

THE KERNELS AND COKERNELS

OF HUREWICZ HOMOMORPHISMS

Henceforth , we assume that H^(X) is finitely generated for all q and for any space X . In particular , this means that H (X) is finite if and only if H (X) 0 Q =0

We need two lemmas and two definitions .

(4.1) Lemma ( Hopf theorem )

Let X "be an H space .

Then , H (XjQ) is a free graded commutative Q- algebra i.e

H (X;Q) is isomorphicE.to the tensor product of an exterior algebra ( with odd dimensional generators ) and a polynomial algebra ( with even dimensional generators ) over Q .

Notation .-

FQ(X^,...,xn) denotes the free graded commutative

Q- algebra over Q with generators x^,...,x .

(4.2) Lemma

Let B = X(n,...,m)

Then ,

HQ>B) = H*(T)

where 'm-n ° : T = K(7L .X,n+i-l) {Jo 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 rational homotopy type ( by an induction argument ; induction on m-n )

0.3) Definition .-

Let AM(X) be the subspace in HJJ(X) 8 HJJ(X) generated by x. 0 x. , where the x. are the generators of HM(X) . Ill w Then , RM(X) is defined to be :

a) H^(X) 0 Hjl'(X) if m is even .

AM(X) 0.4) Definition .-

vT : RM(X) > H*m(X)

is defined to be the homomorphism induced by u :H™(X) 0 HjjU) > H*m(X)

In the case m is odd we call o the square free cup product on R (X) . 11

We call the following theorem P(n,n) , thus referring to the "Meta - theorem "

(5.1) Theorem P(n,n)

If X is n-connected , n >/ 2

then h X ' n+n i ^n^ > *W > has finite kernel and cokernel .

PROOF ;

Consider the fibration :

_P_B > W=X(2n, . . . ,oo) 2—> x

with B = X(n+1,...,2n-l)

and its Serre spectral sequence with Q

coefficients : The bottom row of the following commutative diagram

H2N(X) -> H2n(W)

2n 0 >E »° ±- 2n 2R 0 > HQ (W)-~ -^E°' L_>

is exact .

( Because : XLB is n-1 connected

q —\ HQ (ilB) * 0 , 1 ^q^n-l ^> EP»q = 0 if p,q£0 , p+q=2n

H2n(W) £ E£n'°

Note that JlB .and ^0 K(7^+i+1X,n+i) have the same rational

homotopy type . Therefore ,

n 2n R (-flJ3) = HQ (i*B) Consider

v pn+l.n^n+l,n _£, Vl1 HQN(^B) ^ n+l 2

=H£+1(X) 6 HQ(£B) Let x «y (-RB)

i.e. xeHg(fiB) ; yeH^C-RB)

Then , dn+1(x) / 0 ^ dn+1(y)

n n 0 by the isomorphism dn+1 : E°' =E°; =E^];' ;n+l, 0

Hence ,

dn+1(x.y) = dn+1(x)®y + (-l)V(y) ji 0

unless n is odd and x=y

Therefore ,

2n ker dn+1 = 0 and E°' = 0

Which implies that in diagram pP~[ , i is isomorphism . We now again use the fact :

H*UB) = FQ(xn .. ,xn )

where , n ^ n^ 2n-2 ; i = 1». .. »k to obtain : HQ^UB) =0

W which implies = ^Zn+1 Q

The diagram becomes :

H*n(X)

,2n,0 c 4> H^n(W)

i.e. p^ is an isomorphism

i.e. p* : H^n(W) > H^(X)

is also an isomorphism .

i.e. ker p^ = coker p# - 0

Hence ,

ker [p*:H2n(W) > H2n(X^

and

H (X coker jp*:H2n(W) > 2n ?l

are finite .

The commutative diagram :

ru2n(w)

2n h2n(w) h (x)

H2n(w) -> H2n(X)

completes the proof . 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 all n >/ k

We state P(k,n) in the following

(5.2) Theorem P(k,n)

Let X be n - connected , n V 2

Then ,

hn+k s7WX> >WX)

k ^ n

has finite kernel and cokernel .

(6.1) Theorem R(n+l,n)

Let X be n - connected , n >/ 2 .

Then ,

h sJr X 2n+l 2n+lW > *W > has finite cokernel .

PROOF.-

Consider:;"the fibration :

ilE > W

P

X

where,

E * X(n+1,...,2n)

W =X(2n+l,. . . ,oo) 15

2n

o NN\.

H*(£lE )

0 0

Q 0 0 n+i +

HQ(X)

As in (5.1) » we have

H*n(QE) "S Rn(-RE)

Thus , in

^0.2n d ... Fn+l,n

E Jnil ^ » n+1

»0,2n gn+l.n £ IIinI Rn(ilE) ' . • Hn(-£E) ® Hn+1(X)

dn+^ is monic (by an argument similar to the one in the proof of P(n,n) ) .

Therefore ,

n+2 "u _£,2n+l

+1 2N+1 i.e. E2n .°firH (X,Q) 16

H*n+1(X) -> Hf+1(W)

2n+l,0 E implies that p is a monomorphism .

Hence ,

H«n+1(W) H«n+1(X)

is an epimorphism ,

or

sH W H (X coker ( p# 2n+l^ ^ 2n+l > >

is finite

The commutative diagram :

^i(»> S * ^»H»)

H (W) * H2n+1(X) 2n+l

completes the proof 17

7. The kernel of {1

(7.1) Theorem S(n+l,ri)

Let X he n-connected , n >/ 1 .

0 Q ker Then , ker h2n+l " ^

u° is the cup product or the square free

cup product on R (X) depending on whether n+l

is even or odd , respectively .

PROOF ;

Consider the fibration : W -4-* X -^->K = K(Jtn+1X,n+l)

W = n+l connective covering

and its Serre Spectral Sequence i

2n+l % \ 0 H2n+2

n+2

HQ(W) 0 0

0 0

0 Q 1T+T

HQ(K) 18

The first possibly non vanishing differential coming

2n+2, to E ° is d2n+2 which is also the first possibly non vanishing differential from E°'2n+1.

Hence ,

2n+2,0 u2n+2r„x E. lffi d2n+2

2 E^ ^~=kerd2n+2

Thus , the edge ihomomorphisms :

X* : H2n+2(K) •

has ker V* = im d2n+2

f :H2n+1(X) > H2n+1(W)

„0,2n+l

has coker = im d2n+2

Therefore , * «v

ker* = coker |>

The commutativity of (W) i a2n+l

"H2n+1 * H2n+l(X) implies that of •>> H2n+1(W) 0 Q 2n+l

7c2n+1(x) Q 0 -> H2n+1(X) 0 Q h2n+1« Q

Since W is n+l connected , "by P(k,n) we have

h2n+l 8 Q K2n+1(X) « Q H2n+1(X) 8 Q

is an isomorphism .

Hence , by I * | , we obtain :

PH ker ^ « Q = ker h2n+1(X) ft Q

We also note that :

ker £# e Q = coker

Combine {T] ,

ker h2n+l(X) 8 Q = ker* S im d2n+2

Apply Hopf theorem to HQ(K) and distinguish two cases :

If n+l is even :

+1 +1 H2n+2(K) ~ HjJ (K) 0 Hg (K)

= H£+1(X) 0 H£+1(X)

And , since 5 is a ring homomorphism , the following 20

diagram is commutative :

2n+2 H2n+2(x) H (K)

rn+l n+1 rn+l

which implies

+1 5~j ker"** = kernel of cup product on HQ+1(X) 0 HQ (X)

If n+1 is odd ;

H2n+2(X)^- •H2n+2(K)

rn+l rn+l H2+1(X) 8 Hf^X)^^- H^(K) 0 H^(K)

which implies :

6 ker^ = kernel of square free cup product on

Rn+1(X)

it » 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 finitely generated for each q .

Then,

-> Hq(X) & 1 Kq(X) has finite kernel for q ^ 2n

has finite cokernel if q < 2n + 1

and, h2n+1 has ker h2n+1 8 Q = ker vT 21

WHERE ,

u° is the cup product or the square free cup product on Rn+^(X) ' depending on whether n+l is

even or odd , respectively .

(8.2) Corollary .-

Let X be simply - connected .

Hq(X) be finitely generated for each q .

TT. (x) be finite for q ^ n . q

Then , all conclusions in Theorem (8.1) are still true

PROOF :

Let F be the Serre class of finite abelian groups .

Consider the fibration : X(n+1,. .. ,oo) -> X -»X(1,.. . ,n)

and its Serre Spectral Sequence :

HQ(X(n+l oo))

0

Q

HQ(X(1,...,n))

We denote fi- isomorphism f by =

( i.e. ker f , coker f e £ ) 22

H^(X(1,...,n)) are all 0 because :

TCq(X(l n)) =iyX) if q 4 n

= 0 if q > n+l

By Mod Hurewicz theorem s

0 = jyxd,. . . ,n)) s Hq(X(l n))

Therefore ,

i* : H?(X) = H*(X(n+l,.. . ,oo))

Hence ,

i* : Hq(X(n+l *>) )S H (X)

The commutative diagram :

7Tq(X(n+l )) 1 > TlqCX)

Hq(X(n+l )) 1 >Hq(X)

completes the proof . 23

9. A remark .-

One naturally asks at this point : " Can the results be extended any further ? " . The answer , in the negative , is provided by the following examples^:

(9.D

Let X = Sn+1xSn+1

Then , X is n-connected .

Tf2n+2(X) is finite .

H2n+2(X) = 2 ' Therefore,

cokernel h2n+2^ ^ finite . (9.2)

Let X = Sn+1v Sn+1

Then , X is n-connected

J^2n+1(X) is infinite

H^n+1(X) = 0

Therefore ,

ker h2n+1(X) & Q ^ 0

i.e.

h2n+1(X) does not have finite 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 .