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 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 ${\displaystyle X}$ 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.

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.

See also

• Acyclic model

References

• 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.