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If $X$ is a CW-complex and $f:X\rightarrow X$ a cellular map. Then why it induces a map of chain complexes $f_*:C_*(X)\rightarrow C_*(X)$. (Why it commutes with the differential?).

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    Have you looked in any algebraic topology textbook that offers a proof of this? Bredon, Hatcher, May, Whitehead, etc.. Once you interpret what the chain complex *is*, your question boils down to the fact that the degree of a map $S^n \to S^n$ is multiplicative under composition. The proof of this is fairly simple. There are transversality arguments, and homological arguments.2011-10-03

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