TITLE: smoothing locally-finite (Borel-Moore chains)
QUESTION [5 upvotes]: Let $M$ be a smooth manifold. As is recorded in (for example) Lee's book, de Rham proved that one can calculated singular homology, $H_*(M)$ using smooth simplices. Does the result extend to Borel-Moore homology $H^{BM}_{*}(M)$? One might guess that the argument in Lee's book even proves this, but there are a lot of details there and even if it goes through, it would be good to have a reference that has already checked every line.

REPLY [3 votes]: I do not know a reference, let me give you some ideas on how to prove it:
1) First let us remark that we have an isomorphism:
$$\rho:C_*^{BM}(M)\rightarrow lim_KC_*(M,M-K)$$
between Borel-Moore chains and the limit of relative chains taken over all $K$ where $K\subset M$ is a compact subset.
2) Similarly you can build an isomorphism  
$$\rho^{sm}:C_*^{BM,sm}(M)\rightarrow lim_KC^{sm}_*(M,M-K)$$
between Borel-Moore smooth chains and the limit of relative smooth chains.
3) For any subset $U\subset M$ you have a canonical inclusion:
$$\phi:C^{sm}_*(U)\rightarrow C_*(U)$$
that induces a canonical morphism
$$\phi:lim_KC^{sm}_*(M,M-K)\rightarrow lim_KC_*(M,M-K).$$
4) For each compact set you have a quasi-isomorphism (by the 5-lemma):
$$C^{sm}_*(M,M-K)\rightarrow C_*(M,M-K)$$
the inverse limit of these morphisms is the map $\phi$. To conclude that the inverse limit is also a quasi-isomorphism you have to use homological properties of the inverse limit functor.