Borel–Moore homology

Summary

Borel–Moore homology is a homology theory for locally compact spaces in which cycles may be infinite, as long as they are locally finite. It is also called homology with closed supports. For an oriented $n$-manifold $M$ it is dual to ordinary cohomology, $H^{\mathrm{BM}}_i(M) \cong H^{n-i}(M)$, even when $M$ is not compact. It is the natural home of fundamental classes of non-compact manifolds and of closed subvarieties of complex varieties.