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 $X\times Y$ 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.

Statement of the 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 $C_{*}(X\times Y)$ . (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex $C_{*}(X)\otimes C_{*}(Y)$ , whose differential is, by definition,

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

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

Then the theorem says that we have chain maps

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 $FG$ is the identity and $GF$ 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\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 $H_{*}(X\times Y)$ 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.

References

Eilenberg, Samuel; Zilber, J. A. (1953), "On Products of Complexes", Amer. Jour. Math., American Journal of Mathematics, Vol. 75, No. 1, 75 (1), pp. 200–204, JSTOR 2372629, MR 52767, doi:10.2307/2372629 .
Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0 .
Tonks, Andrew (2003), "On the Eilenberg-Zilber theorem for crossed complexes", Jour. Pure Applied Algebra, 179, pp. 199–230 .
Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Proc. Camb. Phil. Soc., 110, pp. 95–120, doi:10.1017/S0305004100070158 .

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Eilenberg–Zilber theorem

Statement of the theorem

Consequences

See also

References