The cap product

H Cx x HUH H Ux

On homology we have a map

Hp X x HqCX Hpq

xxx

bilinear come up in proofof homotopy naiace

Ideally we want a map ftp.q xx x Haq

coming from a map x x X

But cohomology is contravariant so to get a apft

d'tH t xxx 2 H Ux 2

This comes from a map X Xxx

so we can use the diagonalmap d X xxxx te x c

So now we need a mapRecall thx HIx D

H G xHKx 2 H t xxx

we already have one goingthe other way coming

from Cp X xCq Cp gxxx

gives us Home lk z xHomCCqCD.z z HomCG.qGxDR

We want a map goingthe other way to do

this we will define a chain map0 C Xxx CpG CgePtg n

Ideas are the same as defining the bilinear map

Cp G x Cga CpqCxxY

Lemma If X Y or e contractible then

the chain ox qnCp X Cqc'D 8 is acyclic

Namely Itn Hq Glx GM LERecall V d B Ex p t J x 2B

Pf Follows from the proof that CpG CqGD one

acyclic To prove this we constructed maps

f Cp x CpG and D Cp x C Cx

f

f o tp 0 and that D2 2 D id fflail x x p o

ThmF a chain map 0 CnCxxD stOCpCx7xoCgCDPtg

nS_t.lJForno Cx.y x y xEX yet

2 If f X X get Y then

fxg Xxx x'xx OoCfxgb fa ga O

or the following diagram commutes

fxgbCn xxx Cn xD

t tocpcxixocq.DK tOCpCx9x0CgCtJ

Idea For 0 xxx this is a path Gayto

KcyDProj to X t weget of x octroi

and Oy I Y yay we can putthesetogether o y x of epg CpG Cqc'D

I

Pf using acyclicmodels

Let dn D A xD be the diagonalmapK 1 7 Chi c

so Ine Cn CD xD

By induction 0 Ordn is defined p.IQGesnsxocqCE

O is a chainmap

Now Vloadn 5062 du o so Ocadn

defines andtofttn.fig cpCa7xoCqC5 0

So I TE Cp bn CgcDn S t 8C OC2dnPtg n

Define 0 Cdn to be anysuch 2

To define 0 elsewhere we proceed as follows

Given O D Xx't OE Cn xxx

17 00 C Cn X and t.yooEC.CI Tix Xxx X

projection

and we can write 0 as qq.otxexoot.DE's

E Hoo x CTyoo In xxx

G Hoo x Hoo Ldn

So we are forced to define OCD as follows

o O ooxthoob da

Hoo Hoo Cdn

checkthis is a chain map I

Th m Eilenberg Zilber Thus

The chain maps 0 Cucxxx Go Cqc'tPtq n

and pq.ncpcxtxocqD cnlx.at

ore chain homotopy equivalences and homotopyinverses to eachother

Thus Hn xxx Hn Fer Glx GCD

Hn X xD E Hn Huffy GG GED KD

Let m 7407L 7L This is an isomorphisma b 7 ab

Given f CpG Z and g Cq Z

we get Cp Cgc1 2 072 72

O t to flaxogcesteriorfcogtc

This is a homomorphism Cp GCD 77L

This gives a map

c c x 2 Ict 2 IK xxx 2

f y 1 m ofxogotf.gg 0

This map is defined on chains check

5 cCf g CESf g teh's 5g

to get c H'Tx 2 Hkt 2 H acxxx 2

In the case X Y we can further compose

with d to getu H x 2 x HMX's 2 H Ux's 2

Cf g to d ECf g

This is usually written as fugSomefactsabout u

D fog 459gofD f og fog f ogfucgtg fog t f og

This turns to HP x 24 this becomes a ringPBO with as multiplication

This ring structureis a top invariant of X