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. One possible route to a proof is the acyclic model 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,

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