# Eilenberg–Zilber theorem

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 ${\displaystyle X\times Y}$ and those of the spaces ${\displaystyle X}$ and ${\displaystyle Y}$. 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.

## Statement of the theorem

The theorem can be formulated as follows. Suppose ${\displaystyle X}$ and ${\displaystyle Y}$ are topological spaces, Then we have the three chain complexes ${\displaystyle C_{*}(X)}$, ${\displaystyle C_{*}(Y)}$, and ${\displaystyle C_{*}(X\times Y)}$. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex ${\displaystyle C_{*}(X)\otimes C_{*}(Y)}$, whose differential is, by definition,

${\displaystyle \delta (\sigma \otimes \tau )=\delta _{X}\sigma \otimes \tau +(-1)^{p}\sigma \otimes \delta _{Y}\tau }$

for ${\displaystyle \sigma \in C_{p}(X)}$ and ${\displaystyle \delta _{X}}$, ${\displaystyle \delta _{Y}}$ the differentials on ${\displaystyle C_{*}(X)}$,${\displaystyle C_{*}(Y)}$.

Then the theorem says that we have chain maps

${\displaystyle F:C_{*}(X\times Y)\rightarrow C_{*}(X)\otimes C_{*}(Y),\quad G:C_{*}(X)\otimes C_{*}(Y)\rightarrow C_{*}(X\times Y)}$

such that ${\displaystyle FG}$ is the identity and ${\displaystyle GF}$ is chain-homotopic to the identity. Moreover, the maps are natural in ${\displaystyle X}$ and ${\displaystyle Y}$. Consequently the two complexes must have the same homology:

${\displaystyle H_{*}(C_{*}(X\times Y))\cong H_{*}(C_{*}(X)\otimes C_{*}(Y)).}$

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.

## Consequences

The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups ${\displaystyle H_{*}(X\times Y)}$ in terms of ${\displaystyle H_{*}(X)}$ and ${\displaystyle H_{*}(Y)}$. 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.