Eilenberg–Zilber theorem explained

X x Y

and those of the spaces

and

. The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem.

Statement of the theorem

The theorem can be formulated as follows. Suppose

and

are topological spaces, Then we have the three chain complexes

C_*(X)

C_*(Y)

, and

C_*(X x Y)

. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex

C_*(X) ⊗ C_*(Y)

, whose differential is, by definition,

\partial
	C_(X) ⊗ C_(Y)

(\sigma ⊗ \tau)=\partial_X\sigma ⊗ \tau+(-1)^p\sigma ⊗ \partial_Y\tau

for

\sigma\inC_p(X)

and

\partial_X

\partial_Y

the differentials on

C_*(X)

C_*(Y)

Then the theorem says that we have chain maps

F\colonC_*(X x Y) → C_*(X) ⊗ C_*(Y), G\colonC_*(X) ⊗ C_*(Y) → C_*(X x Y)

such that

is the identity and

is chain-homotopic to the identity. Moreover, the maps are natural in

and

. Consequently the two complexes must have the same homology:

H_*(C_*(X x Y))\congH_*(C_*(X) ⊗ C_*(Y)).

Statement in terms of composite maps

The original theorem was proven in terms of acyclic models but more mileage was gotten in a phrasing by Eilenberg and Mac Lane using explicit maps. The standard map

they produce is traditionally referred to as the Alexander - Whitney map and

the Eilenberg - Zilber map. The maps are natural in both

and

and inverse up to homotopy: one has

FG=

id
	C_(X) ⊗ C_(Y)

, GF-

id
	C_*(X x Y)

\partial
	C_(X) ⊗ C_(Y)

H+H\partial
	C_(X) ⊗ C_(Y)

for a homotopy

natural in both

and

such that further, each of

, and

is zero. This is what would come to be known as a contraction or a homotopy retract datum.

The coproduct

The diagonal map

\Delta\colonX\toX x X

induces a map of cochain complexes

C_*(X)\toC_*(X x X)

which, followed by the Alexander - Whitney

yields a coproduct

C_*(X)\toC_*(X) ⊗ C_*(X)

inducing the standard coproduct on

H_*(X)

. With respect to these coproducts on

and

, the map

H_*(X) ⊗ H_*(Y)\toH_*(C_*(X) ⊗ C_{*(Y)) \overset\sim\to H}_*(X x Y)

also called the Eilenberg - Zilber map, becomes a map of differential graded coalgebras. The composite

C_*(X)\toC_*(X) ⊗ C_*(X)

itself is not a map of coalgebras.

Statement in cohomology

The Alexander - Whitney and Eilenberg - Zilber maps dualize (over any choice of commutative coefficient ring

with unity) to a pair of maps

G^*\colonC^*(X x Y) → (C_*(X) ⊗

	*,
C
	*(Y))

F^*\colon(C_*(X) ⊗

	* →
C
	*(Y))

C^*(X x Y)

which are also homotopy equivalences, as witnessed by the duals of the preceding equations, using the dual homotopy

H^*

. The coproduct does not dualize straightforwardly, because dualization does not distribute over tensor products of infinitely-generated modules, but there is a natural injection of differential graded algebras

i\colonC^*(X) ⊗ C^*(Y)\to(C_*(X) ⊗

	*
C
	*(Y))

given by

\alpha ⊗ \beta\mapsto(\sigma ⊗ \tau\mapsto\alpha(\sigma)\beta(\tau))

, the product being taken in the coefficient ring

. This

induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps

C^*(X) ⊗

	(X) \overset{i}{\to}* (C
C
	*(X)

⊗

	* \overset{G
C
	*(X))

^{*}{\leftarrow} C}^*(X x X)\overset{C^{*(\Delta)}{\to}}C^*(X)

inducing a product

\smile\colonH^*(X) ⊗ H^*(X)\toH^*(X)

in cohomology, known as the cup product, because

H^*(i)

and

H^*(G)

are isomorphisms. Replacing

G^*

with

F^*

so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by

\alpha ⊗ \beta\mapsto(\sigma\mapsto(\alpha ⊗ \beta)(F^*\Delta^*\sigma)

	\dim\sigma
= \sum
	p=0

\alpha(\sigma\|
	\Delta^[0,p]

) ⋅

\beta(\sigma\|
	\Delta^[p,\dim

))

which, since cochain evaluation

C^p(X) ⊗ C_q(X)\tok

vanishes unless

p=q

, reduces to the more familiar expression.

Note that if this direct map

C^*(X) ⊗ C^*(X)\toC^*(X)

of cochain complexes were in fact a map of differential graded algebras, then the cup product would make

C^*(X)

a commutative graded algebra, which it is not. This failure of the Alexander - Whitney map to be a coalgebra map is an example the unavailability of commutative cochain-level models for cohomology over fields of nonzero characteristic, and thus is in a way responsible for much of the subtlety and complication in stable homotopy theory.

Generalizations

An important generalisation to the non-abelian case using crossed complexes is given in the paper by Andrew Tonks below. This give full details of a result on the (simplicial) classifying space of a crossed complex stated but not proved in the paper by Ronald Brown and Philip J. Higgins on classifying spaces.

Consequences

The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups

H_*(X x Y)

in terms of

H_*(X)

and

H_*(Y)

. In light of the Eilenberg–Zilber theorem, the content of the Künneth theorem consists in analysing how the homology of the tensor product complex relates to the homologies of the factors.