§3 Covering Spaces

23

§3 COVERING SPACES

In this chapter we study the covering spaces of finite H-spaces. Our main tool is the Serre spectral sequence. Notably, we will demonstrate the usefulness of introducing a Hopf algebra structure into this spectral sequence.

fhe reader should observe how the DHA Lemma of §1-6 is used to

control the action of differentials. Our arguments are based on the work of Browder [lJ. §3-1:Main Results It is a fact of elementary topology that the covering spaces of X correspond to the subgroups of "l(X), Given H covering space p:X ~ X satisfies "l(X)

"l(X) then the corresponding

~

= H.

See, for example, Massey [lJ.

Regarding "1 of a H-space we have: THEOREM A: For any H-space

(X,~),

"l(X) is abelian.

Proof: Given a,~: Sl~ X we need a homotopy of the form

We do this in two stages

The inner diamond is obtained from the map 8 outer triangles are based on the fact that ~(xO.~(t».

1

x 8

aCt)

1

~ X x X ~ X. The

~ ~(a(t).xO)

and

~(t) ~

Q.E.D.

In the "1 abelian case we can obtain the covering space (up to homotopy

24

The Homology of Hopf Spaces

type) as the fibres of Eilenberg-Maclane spaces. Namely, given He ITI(X) the

corresponding

covering

space

is

the

fibre

of

the

map

X-->

K(IT1(X)/H,l). In particular, the universal covering space is simply connected and is the fibre of the map X --> K(IT1(X),l). We want to show: THEOREM B: The covering spaces of finite H-spaces are also finite H-spaces The proof is a Serre spectral sequence argument and will be done in the next section. §3-2:Proof of Theorem B

To prove Theorem B we use H-fibrations i. e.

fibrations F.!..... E

i..... B

where F.E,B are H-spaces and i.f are H-maps (see §6-4 for the definition of a H-map). We have the following basic facts regarding such fibrations LEMMA A: Given a fibration F .!..... E

i..... B where

E,B are H-spaces and f is a

H-map then F is a H-space and i is a H-map. LEMMA B: If

(X,~)

is a H-space then f:X --> K(G,n) is a H-map if and only

if [f] E Hn(X;G) is primitive So. in our case, the map X --> K(IT1(X)/H,l) will be H-maps and the corresponding covering space will be a H-space. We are left wi th demonstrating the fini teness of the covering space. First of all, we can reduce to the case IT

1(X)

finite.

PROPOSITION: For any H-space (X,~) X '" Sl x .. x

s'

fini te Proof: If rank IT

r we define Xby a fibration of the form

1(X)

-

X -+ X --> S

(recall S

1

K(1,1»

x X where IT 1 (X) is

1

X ..•

x S

1

We then use the multiplication on X to define a map

s'

X ••• x

Sl x X --> Xx ... x X

---->

X

which is obviously an equivalence in IT*( ). Q.E.D. To prove the finiteness of our covering space in the IT l finite case we use the Serre spectral sequence with mod p coefficients. Given a H-fibration F --> E --> B the resulting spectral sequence

25

I: Hopf Algebras

H*(F;W ) 0 H*(B;W ) p

p

==>

H*(E;W ) P

is a spectral sequence of commutative associative Hopf algebras. This Hopf algebra structure imposes strong restrictions on the action of differentials in the spectral sequence. Notably, we have the DHA Lemma of §1-6 Since we have reduced to the case of ITl(X) finite (and abelian) we need only demonstrate for H-fibrations of the form

that X is finite whenever X is finite (up to homotopy type). Let us deal with the case p odd until further notice. The Serre spectral sequence is of the form E 2 where lal

=1

= H*(X;Wp )

=2

and Ibl

==> H*(X;Wp )

0 E(a) 0 W [b)

p

(see Appendix B for the cohomology of Eilenberg-

Maclane spaces). Since H*(X;W ) is finite there must be a non trivial difp

ferential in the spectral sequence. Suppose d.

=0

d

"# 0

1

We know dra

= drb = O.

r


when i

So the action of d

effect on H*(X;W We will use the p).

r

is completely determined by its

DHA Lemma of

§1-6

to show that d

r

first acts non trivially in a transgressive manner i.e. in the following

manner H*(X;W )

d~ r

E(a) 0 W[b) p

Moreover, this transgressive action completely determines the action of d on E

r.

(i) r

Regarding the transgressive action we have:

= 2ps

for some s ~ 1

(ii) there exists indecomposable x €

~s-l(X;W

To see this pick the smallest i such that d Lemma d (Hi(X;W r

r

p

»

p

) such that d i -

r

"# 0 on H (X;W

2ps

p).

(x)

By the DHA

C peE ). By Lemma 1-5 we have an exact sequence r

= bPS

26

The Homology of Hopf Spaces

PH*{X;F } ~ peE } ~ P{E(a) @ F [b]) p

p

r

Obviously d Hi(X;F } n Hi(X;IF} = O. Also P(E(a) I» IF [b]) has basis r p p p 2 {a,b,bP,bP , ... } (see the discussion following Theorem 1-5B). Next, the relation d d

2ps

on E 2ps

(iii) E2ps

s

= bP

(x)

2p More precisly,

*= H (X;lFp)//E(x)

s

completely determines the action of

where d 2p s (x )

@ E{a) I» E{x) @ IFp[b]

= bP

s

This identity holds as differential Hopf algebras. The verification of (iii) uses strongly the fact that p is odd.

We can certainly choose such

a decomposition as Hopf algebras. (since p is odd x

2

= 0).

Regarding the

action of d2ps we have d In other words, d E2p s

/

2ps

_ = H* (X)

s

(bP )

s

C bP oE 2ps(E2ps) 2ps

acts trivially on the quotient Hopf algebra IF [b]

I» E(a) @ p

/

s. We establish the fact by an argu-

(bP )

ment similar to the previous one. Next. choose a Borel decomposition of E2ps

including {x,a,b} as generators and with the remaining generators

from ker d2ps. We place y by height of

Y= y

can

always do so. For, if d2ps{y)

- xz. (Observe that since x

2

=0

= bP

s

z T- 0 we

can

re-

=

the

the height of y

y)

It follows from (iii) that (iv) E2ps+ l

*-

=H

(X;lFp)//E(X) I» E(a) @ IFp[b]/(bps)

Finally

To prove this we argue as before that if d. T- 0 for i act non

trivially in

a

1

trangressive manner

IF [b] s E{a) I» p /(bP ) ' However,

> 2ps

and hit a

it would first primi t i ve

of

there are no primitives in the appropriate

degree. The identity

*-

H (X;IF )// p

E(x)

@ E( ) I» IF [b]/

a

p

s (b P)

27

I: Hopf Algebras

now establishes that H*(X;W ) is a W vector space of finite rank. p

p

This concludes the proof of Theorem B for p odd. When p =2 we have the extra

complication

of

odd

degree

elements

having

possible

non

zero

squares. But we can repeat the above argument if we alter the algebraic structure. Namely, assume that odd degree elements have zero squares by introducing new indecomposables to replace the non zero square of any odd degree indecomposable. §3-3:Examples The above proof enables one to explicitly calculate the cohomology of covering spaces. First, consider rational cohomology. Suppose H*(X;Ill) = E(x

1,

.. ,x

r).

Splitting off copies of S1 from X amounts to removing the ex-

terior algebra generators of degree 1. Fibering X using K(l/pk,1) does not affect H*(X;Ill). Next,

o

consider W cohomology. p

Pick a Borel decomposi tion H*(X;W ) = p

A,. As we split off copies of S1 we remove exterior algebras E(a) where 1

lal = 1 from H*(X;W ). When we fibre X using K(l/pk,1) we replace factors p

of the form E(a) 0 Wp[b]/(bp s) where lal = 1 and Ibl = 2 by an exterior algebra E(x) where Ixl = 2ps-1. Examples (1)(Baum-Browder[1]) Let X 1I1 (X) (p ) =

PU(n). Write n

l/l

*

H (X;Wp) = E(x 1,x3,·· .,x2pk+1,·· .,x2n+1) 0

W[b] p

If X is given by the fibration X ~ X ~ K(l/pk,1) then

(2)(Ishitoya-Kono-Mimura [1]) Let X = 1I 1(X)

1, We have

pkm where (p.m)

= r/z

Ad(~).

Then

k

/(bP)

28

The Homology of Hopf Spaces

Then X

= E7

is given by the fibration X --+ X --+ K(Z/2, 1) and

This example illustrates the fact that the Serre spectral sequence does not give complete information about the algebra structure of H*(X:f ) when p

p

= 2.

If we follow our recipe as outlined above for calculating H*(X;f

from H*(X;JF we would replace JF2 [ X1]/(xi) by E(x However, 2) 3). also a non trivial algebra extension obtained by f 2[x3]/

4. This extension is not picked up by

(x 3 )

replacing E(x

2)

there is 3,x6)

by

the Serre spectral se-

quence. Observe. also. that this type of extension can not happen in the p odd case.