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Let $X$ be a nice enough topological space (locally compact, Hausdorff, second-countable). The $G$-module $C_i^\infty(X,G)$ of locally finite singular $i$-chains is the submodule of $\prod_\sigma G$, $\sigma$ running over the singular $i$-simplices in $X$, consisting of those elements $\varphi = \sum_\sigma a_\sigma \sigma$ such that the family $\lbrace \mathrm{Im}(\sigma) \mid a_\sigma \neq 0 \rbrace$ is locally finite. The submodule $C_i^c(X,G)$ is the one consisting of the finite chains, i.e., $\lbrace \sigma \mid a_\sigma \neq 0 \rbrace$ is finite (equivalent to $\mathrm{Supp}(\varphi)$ being compact).

Let $C^i_\infty(X,G) = \mathrm{Hom}_G(C_i^c(X,G),G)$. Let $C^i_c(X,G)$ be the submodule of $C^i_\infty(X,G)$ consisting of compactly supported cochains, i.e, those cochains $\xi : C_i^c(X,G) \rightarrow G$ such that there exists a compact subset $K$ of $X$ with $\xi(\varphi) = 0$ whenenver $\mathrm{Supp}(\varphi) \subseteq X \setminus K$.

Questions.

a) Is $C^i_c(X,G)$ isomorphic to $\mathrm{Hom}_G(C_i^\infty(X,G),G)$? If not, are they quasi-isomorphic?

b) Is $C_i^\infty(X,G)$ isomorphic to $\mathrm{Hom}_G(C^i_c(X,G),G)$? If not, are they quasi-isomorphic?

The motivation for the above questions is that I would like to know if Borel–Moore homology and singular cohomology with compact supports are related by taking duals of a (co)complex.

Improve this question
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  • $\begingroup$ I feel as if they are only quasi-isomorphic. It is certainly true that the dual of compactly-supported cochains is quasi-isomorphic to locally finite chains (see the Wiki page on Borel-Moore homology, for instance). But this is "dual" to your claim. $\endgroup$
    – Mark Grant
    May 12, 2017 at 17:12
  • $\begingroup$ May the one who down voted have a reference for whatever the answer is? :-) $\endgroup$
    – user78294
    May 12, 2017 at 22:44
  • $\begingroup$ @Mark: Do you know how to show that the complexes are quasi-isomorphic? $\endgroup$
    – user78294
    May 12, 2017 at 22:44
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    $\begingroup$ Have a look at Proposition 2.8 of maths.ed.ac.uk/~aar/papers/laitinen.pdf It doesn't do exactly what you want, but the proof there probably dualizes. There is also a remark that locally finite chains are the dual of compactly supported cochains "on the nose". $\endgroup$
    – Mark Grant
    May 13, 2017 at 6:59
  • $\begingroup$ Thanks for the reference! An important point in the proof seems to be Prop. 1.3 stating that for a locally finite (co)complex the morphism into its bidual is a quasi-isomorphism. Locally finite means here finitely generated (co)homology in each degree. Is this true for singular cohomology with compact support? $\endgroup$
    – user78294
    May 13, 2017 at 9:32

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