Eilenberg–Zilber theorem

From Wikipedia, the free encyclopedia

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.

[edit] 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 $δ X$ , $δ Y$ 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 $F G$ is the identity and $G F$ is chain-homotopic to the identity. Moreover, the maps are natural in $X$ and $Y$ . Consequently the two complexes must have the same homology:

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

[edit] 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.

[edit] References

Eilenberg, Samuel & Zilber, J. A. (1953), “On Products of Complexes”, Amer. Jour. Math. 75 (1): 200–204, MR 52767 .
Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0 .