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
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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 the paper by Hans Munkholm: A chain level transfer homomorphism for PL fibrations, Math. Z. 166, 183-186 (1979). ZBL0404.55006.
answered Jun 24, 2010 at 3:25
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No, there is a pullback on singular cochains, given by composition.
answered Jun 22, 2010 at 22:20