There is a homology theory called "Borel-Moore" or "locally finite" homology, which can either be constructed by using locally-finite chains or by more advanced sheaf-theoretic methods. Here are two useful facts about it:

  1. If $X$ is locally compact and $\tilde X$ is a reasonable compactification, then $H^{lf}(X) \cong H(X, \tilde X \setminus X)$ agrees with the ordinary homology relative to infinity.
  2. If $X$ is an orientable manifold with boundary (maybe weaker hypotheses suffice), not necessarily closed, then $H^{lf}_k(X)$ is dual to $H_{n-k}(X, \partial X)$.

Either of these results might be stated slightly wrong, which is partially what my question is about.

I am looking for a reference that constructs $H^{lf}$ and proves one or both of these results, but using less technical machinery than last chapter of Bredon's Sheaf Theory (which I think has what I want, but I don't know enough sheaf theory to understand it.) Hatcher only mentions $H^{lf}$ in an exercise, and Bredon's Topology and Geometry doesn't discuss it at all.

I see that there are some related questions (here and here) that suggest there are no good references, so maybe this is a lost cause, but I notice neither mentions the duality property (2).

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    $\begingroup$ For constant coefficients the standard reference is Massey "Homology and cohomology theory". As you say, the reference for local coefficients is Bredon's "Sheaf theory". $\endgroup$ – Igor Belegradek Dec 4 '20 at 17:24
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    $\begingroup$ Not sure if this is what you want, but the appendix to Fulton's "Young Tableaux" takes an elementary approach by defining $H_*^{BM}(X)$ to be $H^*(\mathbb R^n , \mathbb R^n - X)[n]$ where $X \to \mathbb R^n$ is a closed embedding. You can prove that this is independent of the choice of embedding. It seems likely that you can then prove $(1)$ and $(2)$ directly using Alexander/Lefschetz duality. $\endgroup$ – Phil Tosteson Dec 4 '20 at 19:35
  • $\begingroup$ Massey wrote down a bunch of stuff in his book Homology and Cohomology Theory. He doesn't call it Borel-Moore, but he cites their paper and constructs some kind of locally-finite cohomology. I don't know how much he does. $\endgroup$ – Tyrone Dec 6 '20 at 12:52

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