Borel–Moore homology

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:

0 → Ext Z 1 ( H c i + 1 ( X , Z ) , Z ) → H i B M ( X , Z ) → Hom ( H c i ( X , Z ) , Z ) → 0.

In what follows, the coefficients Z 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. "Reasonable" means here that X is locally contractible, σ-compact, and of finite dimension.

In more detail, let C_i^BM(X) be the abelian group of formal (infinite) sums

u = ∑ σ a σ σ ,

where σ runs over the set of all continuous maps from the standard i-simplex Δⁱ to X and each a_σ 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:

⋯ → C 2 B M ( X ) → C 1 B M ( X ) → C 0 B M ( X ) → 0.

The Borel−Moore homology groups H_i^BM(X) are the homology groups of this chain complex. That is,

H i B M ( X ) = ker ⁡ ( ∂ : C i B M ( X ) → C i − 1 B M ( X ) ) / im ( ∂ : C i + 1 B M ( X ) → C i B M ( X ) ) .

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 H_i^BM(X) coincides with the usual singular homology H_i(X) 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 H_i^BM(X) is isomorphic to the relative homology H_i(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

H i B M ( X ) = H m − i ( M , M ∖ X ) ,

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 D_X be the dualizing complex of X. Then

H i B M ( X ) = H − i ( X , D X ) ,

where in the right hand side, hypercohomology is meant.

Example

Let X = R² − 0, the plane minus the origin. The usual homology of X is Z in degrees 0 and 1 (and zero otherwise), while the Borel−Moore homology of X is Z in degrees 1 and 2 (and zero otherwise). In this example, the natural homomorphism H_i(X) → H_i^BM(X) is zero for all integers i. For example, the usual homology H₁(X) is generated by a circle around the origin, but this maps to zero in Borel–Moore homology, because a finite 1-chain representing the circle is the boundary of a locally finite 2-chain whose image is the region inside the circle. The Borel–Moore homology H₁^BM(X) is generated by a locally finite 1-chain whose image is (for example) the ray from 0 to ∞ in the real line R inside R².

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 H_i^BM(X) → H_i^BM(Y) 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 R² − 0 → R².

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 H_i^BM(X) → H_i^BM(U).

For any locally compact space X and any closed subset F, with U = X − F 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 H_i^BM(X) → H_i+1^BM(X × R). 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 Rⁿ is isomorphic to Z 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: H c i ( X ) → ≅ H n − i ( X ) .

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 [M] ∈ H_n^BM(M). 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 set of smooth points M^reg ⊂ M has complement of (real) codimension at least 2, and by the long exact sequence above the top dimensional homologies of M and M^reg are canonically isomorphic. The fundamental class of M is then defined to be the fundamental class of M^reg.