---
tag: T1N6E5Q8Z2
kind: concept
name: Borel–Moore homology
msc: [55N35, 55N30]
sources:
  - url: https://en.wikipedia.org/wiki/Borel–Moore_homology
same_as:
  wikidata: Q4944924
  wikipedia: 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.

## Explanations

### L1 human

In ordinary homology a cycle is a finite sum of small pieces.
In Borel–Moore homology a cycle may have infinitely many pieces, provided only finitely many of them crowd near any one point.
The whole real line, cut into the intervals $[k, k+1]$ for all integers $k$, is such a cycle.
It has no boundary, because each endpoint is shared by two intervals and cancels.
So the line has a fundamental class in Borel–Moore homology, though it has none in ordinary homology.