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
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.
are topological spaces, Then we have the three chain complexes
(The argument applies equally to the simplicial or singular chain complexes.)
We also have the tensor product complex
Then the theorem says that we have chain maps such that
Consequently the two complexes must have the same homology: The original theorem was proven in terms of acyclic models but explicit formulas for the map
were later found by Eilenberg and Mac Lane.
they produce is traditionally referred to as the Alexander–Whitney map and
This is what would come to be known as a contraction or a homotopy retract datum.
induces a map of cochain complexes
inducing the standard coproduct on
The Alexander–Whitney and Eilenberg–Zilber maps dualize (over any choice of commutative coefficient ring
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
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
α ⊗ β ↦ ( σ ⊗ τ ↦ α ( σ ) β ( τ ) )
, the product being taken in the coefficient ring
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
so the maps all go the same way, one gets the standard cup product on cochains, given explicitly by which, since cochain evaluation
, 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
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.
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.
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups
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.