Borel–Moore homology

In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Borel and Moore (1960). In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Borel and Moore (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 H G ∗ ( X ) = H ∗ ( ( E G × X ) / G ) . {displaystyle H_{G}^{*}(X)=H^{*}((EG imes X)/G).} That is not related to the subject of this article. There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as manifolds and locally finite CW complexes. 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 Z {displaystyle mathbb {Z} } are not written. 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 C i B M ( X ) {displaystyle C_{i}^{BM}(X)} 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: