Eilenberg–Zilber theorem
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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 
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 
. (The argument applies equally to the simplicial or singular chain complexes.) We also have the tensor product complex 
, whose differential is, by definition,
for 
and δX, δY the differentials on C * (X),C * (Y).
Then the theorem says that we have a chain maps
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:
[edit] Consequences
The Eilenberg–Zilber theorem is a key ingredient in establishing the Künneth theorem, which expresses the homology groups 
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, doi:, MR52767.
- Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0.
