worked out examples in borel-moore homology

I'm trying to learn about BM homology. I've found a few references and they are all quite abstract and high-powered. I am actually familiar with derived categories and sheaves so I don't mind thinking about those, but I would appreciate some (easy and less easy) worked out examples just to understand what I'm looking at first.

Does such a thing exist anywhere?

• The question was asked a while ago, but there is a nice section about Borel-Moore homology in the book "Representation theory and complex geometry" by Chriss and Ginzburg. Also, there is a nice picture in Alberto Arabia's lecture notes on perverse sheaves (available on his webpage), which explains why one can sheafify Borel-Moore chains (taking successive barycentric subdivisions of the simplices when restricting to an open subset). – Daniel Juteau Apr 4 '15 at 7:38
• A nice source of examples are algebraic curves with points removed from them. Even playing around with $\mathbb{P}^1_\mathbb{C}$ gives a lot of nice computations. Also, try computing borel moore homology of double cones. – 54321user Dec 14 '16 at 17:09

While not so easy to work out directly, on "nice enough" spaces, Borel-Moore homology can be interpreted as what you get by allowing singular chains that are formal linear combinations of a possibly infinite number of singular simplices, subject only to the constraint that every point of the space should have a neighborhood that intersects only finitely many of these simplices, i.e. it is the homology of $S^{\infty}_\ast(X)$, the complex of "locally-finite chains".
So here's a good starter example: let $X=\mathbb{R}$. I claim then that $H_0^{BM}(X)=0$ but $H_1^{BM}(X)=\mathbb{Z}$. To see this, suppose that $v$ is a $0$-simplex, which is a cycle. In ordinary homology, $v$ does not bound. However, in this setting, $v$ does bound: it bounds the chain made up of an infinite number of unit length intervals (interpreted as singular 1-simplices) marching off to infinity (in one direction and with everything oriented the right way). On the other hand, $H_1^{BM}(X)=\mathbb{Z}$, generated by a chain that you can imagine as the sum of all the unit intervals $[n,n+1]$, interpreted as the images of singular $1$-simplices. It takes a bit of thought to see that this isn't a boundary, but it's not too hard.
After you play around with this for awhile, you'll start to notice that, again for nice enough spaces, you're basically getting something like $H_*(X,X_\infty)$, where $X_\infty$ is conceptually the "ends" of the space $X$ (equivalently, for a nice enough space, this is the homology of the one point compactification releative to the point at infinity). In fact, this interpretation, further reinterpreted perhaps in terms of some sort of limiting process (probably inverse limits), starts looking comparable in some ways to cohomology with compact supports, and that's probably the route to take toward understanding the definition of Borel-Moore homology as defined in the original paper of Borel and Moore.