# Elementary reference for Borel-Moore/locally finite homology

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).

• 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". Dec 4, 2020 at 17:24
• 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. Dec 4, 2020 at 19:35
• 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. Dec 6, 2020 at 12:52