A singular cochain has compact support when it vanishes on every simplex whose image lies outside some compact set. Such cochains form a subcomplex because the faces of a simplex outside that compact set also lie outside it. Its cohomology is <compactly supported cohomology> $H_c^*(X)$.

If $f:X\to Y$ is <proper map>[proper] and a cochain on $Y$ is supported in the compact set $K$, its pullback is supported in the compact set $f^{-1}(K)$. Pullback commutes with the coboundary, so it induces a well-defined map
$$
f^*:H_c^*(Y)\longrightarrow H_c^*(X).
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 114 / 3 / a / Solution

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A continuous map is proper when inverse images of compact sets are compact. Proper maps induce contravariant maps on compactly supported cohomology.


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