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).

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*((EG × X)/G). 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.[1] As a result, there is a short exact sequence analogous to the universal coefficient theorem:

0 \to \text{Ext}^1_{\mathbf Z}(H^{i+1}_c(X,\mathbf Z),\mathbf Z) \to H_i^{BM}(X,\mathbf Z) \to \text{Hom}(H^i_c(X,\mathbf Z),\mathbf Z) \to 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.[2]

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

 u = \sum_{\sigma} a_{\sigma } \sigma,

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:

 \cdots \to C_2^{BM}(X) \to C_1^{BM}(X) \to C_0^{BM}(X) \to 0.

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

 H^{BM}_i (X) = \ker \left (\partial : C_i^{BM}(X) \to C_{i-1}^{BM}(X) \right )/ \text{im} \left (\partial :C_{i+1}^{BM}(X) \to C_i^{BM}(X) \right ).

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 HiBM(X) coincides with the usual singular homology Hi(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 HiBM(X) 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

H^{BM}_i(X)= H^{m-i}(M,M\setminus X),

where in the right hand side, relative cohomology is meant.[3]

Definition via the dualizing complex

For any locally compact space X of finite dimension, let DX be the dualizing complex of X. Then

H^{BM}_i (X)=H^{-i} (X, D_X),

where in the right hand side, hypercohomology is meant.[4]

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 Hi(X) → HiBM(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 H1BM(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

 \cdots \to H^{BM}_i (F) \to H^{BM}_i (X) \to H^{BM}_i (U) \to H^{BM}_{i-1} (F) \to \cdots
H^i(X) \stackrel{\cong}{\to} H_{n-i}^{BM}(X)
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^i_c(X) \stackrel{\cong}{\to} H_{n-i}(X).

Notes

  1. B. Iversen. Cohomology of sheaves. Section IX.1.
  2. G. Bredon. Sheaf theory. Corollary V.12.21.
  3. B. Iversen. Cohomology of sheaves. Theorem IX.4.7.
  4. B. Iversen. Cohomology of sheaves. Equation IX.4.1.
  5. B. Iversen. Cohomology of sheaves. Equation IX.2.1.
  6. W. Fulton. Intersection theory. Lemma 19.1.1.

References

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