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 × Y and those of the spaces X and Y . The theorem first appeared in a 1953 paper in the American Journal of Mathematics. One possible route to a proof is the acyclic model 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 C ∗ ( 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,
δ ( σ ⊗ τ ) = δ X σ ⊗ τ + ( − 1 ) p σ ⊗ δ Y τ for σ ∈ C p ( X ) and δ X , δ Y the differentials on C ∗ ( X ) , C ∗ ( Y ) .
Then the theorem says that we have chain maps
F : C ∗ ( X × Y ) → C ∗ ( X ) ⊗ C ∗ ( Y ) , G : C ∗ ( X ) ⊗ C ∗ ( Y ) → C ∗ ( X × Y ) such that F G is the identity and G F is chain-homotopic to the identity. Moreover, the maps are natural in X and Y . Consequently the two complexes must have the same homology:
H ∗ ( C ∗ ( X × Y ) ) ≅ H ∗ ( C ∗ ( X ) ⊗ C ∗ ( Y ) ) . An important generalisation to the non-abelian case using crossed complexes is given in the paper by 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 Brown and Higgins on classifying spaces.
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups H ∗ ( 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.
