Note: Borel equivariant cohomology is an invariant of spaces with an action of a group G; it is defined as
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.
[2] As a result, there is a short exact sequence analogous to the universal coefficient theorem:
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.
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 K of X, we have
for only finitely many σ whose image meets K. Then the usual definition of the boundary ∂ of a singular chain makes these abelian groups into a chain complex:
Suppose that X is homeomorphic to the complement of a closed subcomplex S in a finite CW complex Y.
Under the same assumption on X, the one-point compactification of X is homeomorphic to a finite CW complex.
Let X be any locally compact space with a closed embedding into an oriented manifold M of dimension m. Then
[4] For any locally compact space X of finite dimension, let DX be the dualizing complex of X.
[5] Borel−Moore homology is a covariant functor with respect to proper maps.
Borel−Moore homology is a contravariant functor with respect to inclusions of open subsets.
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.
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,
A different version of Poincaré duality for non-compact manifolds is the isomorphism from cohomology with compact support to usual homology:
A key advantage of Borel−Moore homology is that every oriented manifold M of dimension n (in particular, every smooth complex algebraic variety), not necessarily compact, has a fundamental class
If the manifold M has a triangulation, then its fundamental class is represented by the sum of all the top dimensional simplices.
In fact, in Borel−Moore homology, one can define a fundamental class for arbitrary (possibly singular) complex varieties.
In this case the complement of the set of smooth points
has (real) codimension at least 2, and by the long exact sequence above the top dimensional homologies of M and
The first non-trivial calculation of Borel-Moore homology is of the real line.
, we can use the long exact sequence to compute the Borel-Moore homology of
Finally, using the computation for the homology of a compact genus 2 curve we are left with the exact sequence
since we have the short exact sequence of free abelian groups