In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X \times Y and those of the spaces X and Y. 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 X and Y are topological spaces, Then we have the three chain complexes C_(X), C_(Y), and. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex, whose differential is, by definition, for and \partial_X, \partial_Y the differentials on C_(X),C_(Y). Then the theorem says that we have chain maps such that FG is the identity and GF is chain-homotopic to the identity. Moreover, the maps are natural in X and Y. Consequently the two complexes must have the same homology:

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 F they produce is traditionally referred to as the Alexander–Whitney map and G the Eilenberg–Zilber map. The maps are natural in both X and Y and inverse up to homotopy: one has for a homotopy H natural in both X and Y such that further, each of HH, FH, and HG is zero. This is what would come to be known as a contraction or a homotopy retract datum.

The coproduct

The diagonal map induces a map of cochain complexes which, followed by the Alexander–Whitney F yields a coproduct inducing the standard coproduct on H_*(X). With respect to these coproducts on X and Y, the map also called the Eilenberg–Zilber map, becomes a map of differential graded coalgebras. The composite 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 k with unity) to a pair of maps 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 given by, the product being taken in the coefficient ring k. This i induces an isomorphism in cohomology, so one does have the zig-zag of differential graded algebra maps inducing a product 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 which, since cochain evaluation vanishes unless p=q, reduces to the more familiar expression. Note that if this direct map 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 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.