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.

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,

$\delta( \sigma \otimes \tau) = \delta_X \sigma \otimes \tau %2B (-1)^p \sigma \otimes \delta_Y \tau$

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 a 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 nonabelian 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; the answer is somewhat subtle.

References

Eilenberg, Samuel; Zilber, J. A. (1953), "On Products of Complexes", Amer. Jour. Math. (American Journal of Mathematics, Vol. 75, No. 1) 75 (1): 200–204, doi:10.2307/2372629, JSTOR 2372629, MR 52767 .
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: 199–230 .

Brown, Ronald; Higgins, Philip J. (1991), "The classifying space of a crossed complex", Proc. Camb. Phil. Soc. 110: 95–120, doi:10.1017/S0305004100070158 .