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What does it mean for a map to "descend" to homology?

If we have a map $f:X \rightarrow Y$ then we may define a map $f_{\sharp}:C_n(X) \rightarrow C_n(Y)$ by $f_{\sharp}(\gamma) = f \circ \gamma$. We can then define $f_{\ast} : H_n(X) \rightarrow H_n(Y)$ by $f_{\ast}([\gamma]) = [f_{\sharp}(\gamma)] = [f\circ\gamma]$. It can be proved that this is a group homomorphism.

Christine Devena 2003-07-25