1
$\begingroup$

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?

$\endgroup$

2 Answers 2

3
$\begingroup$

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.

$\endgroup$
0
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.