Eilenberg–Zilber 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

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

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

• Acyclic model

• Eilenberg, Samuel; Zilber, Joseph A. (1953), "On Products of Complexes", American Journal of Mathematics, 75 (1), pp. 200–204, doi:10.2307/2372629, JSTOR 2372629, MR 0052767.
• Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 978-0-521-79540-1.
• Tonks, Andrew (2003), "On the Eilenberg–Zilber theorem for crossed complexes", Journal of Pure and Applied Algebra, 179 (1–2), pp. 199–230, doi:10.1016/S0022-4049(02)00160-3, MR 1958384.
• Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Mathematical Proceedings of the Cambridge Philosophical Society, 110, pp. 95–120, CiteSeerX 10.1.1.145.9813, doi:10.1017/S0305004100070158.