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Do we have a pullback operation on singular simplicial chains,that is if f:X-->Y is a continuous map between topological space X and Y,and C is a singular simplicial chain on Y,then do we have a singular simplicial chain on X which is the pullback of C along f?

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2 Answers

No, there is a pullback on singular cochains, given by composition.

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For a general map, there is no such pullback operation, but there are things you can do in special cases. For example, if $f\colon X\to Y$ is a finite cover, there is a chain homomorphism $C(Y)\to C(X)$ that sends a singular simplex in $Y$ to the sum of its lifts in $X$. This induces the transfer homomorphism in homology.

There are more general versions of the transfer that can be realized on chain level. See for example this paper of H. Munkholm

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