Borel–Moore homology

Remarks

The boundary of a locally finite chain is locally finite, because each face of $\sigma$ has image inside the image of $\sigma$. If $X$ is compact, a locally finite chain is a finite chain, so $H^{\mathrm{BM}}_i(X; A) = H_i(X; A)$. Some authors define the theory with sheaves. If $X = Y \setminus Z$ for a finite CW complex $Y$ and a subcomplex $Z$, then $H^{\mathrm{BM}}_i(X; A) \cong H_i(Y, Z; A)$.

Example: on $X = \R$, the chain $c = \sum_{k \in \Z} \sigma_k$, where $\sigma_k$ is the affine path from $k$ to $k+1$, is locally finite, and $\partial c = \sum_k (\set{k+1} - \set{k}) = 0$. Its class generates $H^{\mathrm{BM}}_1(\R; \Z) \cong \Z$, while $H^{\mathrm{BM}}_0(\R; \Z) = 0$.

Non-example: the infinite sum $\sum_{k \ge 1} \tau_k$, where $\tau_k$ is the affine path from $0$ to $1/k$ in $\R$, is not locally finite: every neighbourhood of $0$ meets all the $\tau_k$.

For an oriented $n$-manifold $M$, Poincaré duality takes the form $H^{\mathrm{BM}}_i(M; \Z) \cong H^{n-i}(M; \Z)$.