In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.
For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages. In particular, a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact.
Note: Borel equivariant cohomology is an invariant of spaces with an action of a group G; it is defined as That is not related to the subject of this article.
Definition
There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as manifolds and locally finite CW complexes.
Definition via sheaf cohomology
For any locally compact space X, Borel–Moore homology with integral coefficients is defined as the cohomology of the dual of the chain complex which computes sheaf cohomology with compact support. As a result, there is a short exact sequence analogous to the universal coefficient theorem:
In what follows, the coefficients are not written.
Definition via locally finite chains
The singular homology of a topological space X is defined as the homology of the chain complex of singular chains, that is, finite linear combinations of continuous maps from the simplex to X. The Borel−Moore homology of a reasonable locally compact space X, on the other hand, is isomorphic to the homology of the chain complex of locally finite singular chains. Here "reasonable" means X is locally contractible, σ-compact, and of finite dimension.
In more detail, let be the abelian group of formal (infinite) sums
where σ runs over the set of all continuous maps from the standard i-simplex Δi to X and each aσ is an integer, such that for each compact subset S of X, only finitely many maps σ whose image meets S have nonzero coefficient in u. Then the usual definition of the boundary ∂ of a singular chain makes these abelian groups into a chain complex:
The Borel−Moore homology groups are the homology groups of this chain complex. That is,
If X is compact, then every locally finite chain is in fact finite. So, given that X is "reasonable" in the sense above, Borel−Moore homology coincides with the usual singular homology for X compact.
Definition via compactifications
Suppose that X is homeomorphic to the complement of a closed subcomplex S in a finite CW complex Y. Then Borel–Moore homology is isomorphic to the relative homology Hi(Y, S). Under the same assumption on X, the one-point compactification of X is homeomorphic to a finite CW complex. As a result, Borel–Moore homology can be viewed as the relative homology of the one-point compactification with respect to the added point.
Definition via Poincaré duality
Let X be any locally compact space with a closed embedding into an oriented manifold M of dimension m. Then
where in the right hand side, relative cohomology is meant.
Definition via the dualizing complex
For any locally compact space X of finite dimension, let be the dualizing complex of . Then
where in the right hand side, hypercohomology is meant.
Properties
Borel−Moore homology is a covariant functor with respect to proper maps. That is, a proper map f: X → Y induces a pushforward homomorphism for all integers i. In contrast to ordinary homology, there is no pushforward on Borel−Moore homology for an arbitrary continuous map f. As a counterexample, one can consider the non-proper inclusion
Borel−Moore homology is a contravariant functor with respect to inclusions of open subsets. That is, for U open in X, there is a natural pullback or restriction homomorphism
For any locally compact space X and any closed subset F, with the complement, there is a long exact localization sequence:
Borel−Moore homology is homotopy invariant in the sense that for any space X, there is an isomorphism The shift in dimension means that Borel−Moore homology is not homotopy invariant in the naive sense. For example, the Borel−Moore homology of Euclidean space is isomorphic to in degree n and is otherwise zero.
Poincaré duality extends to non-compact manifolds using Borel–Moore homology. Namely, for an oriented n-manifold X, Poincaré duality is an isomorphism from singular cohomology to Borel−Moore homology,
for all integers i. A different version of Poincaré duality for non-compact manifolds is the isomorphism from cohomology with compact support to usual homology:
