Borel–Moore homology

In mathematics, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Borel and Moore (1960).

For compact spaces, the Borel−Moore homology coincide with the usual singular homology, but for non-compact spaces, it usually gives homology groups with better properties.

Note: There is an equivariant cohomology theory for spaces upon which a group $G$ acts which is also called Borel cohomology and is defined as $H^*_G(X) = H^*(EG\times_G X)$ . This is not related to the subject of this article.

Definition

There are several ways to define Borel−Moore homology. They all coincide for spaces $X$ that are homotopy equivalent to a finite CW complex and admit a closed embedding into a smooth manifold $M$ such that $X$ is a retract of an open neighborhood of itself in $M$ .

Definition via locally finite chains

Let $T$ be a triangulation of $X$ . Denote by $C_i^T ((X))$ the vector space of formal (infinite) sums

\xi = \sum_{\sigma \in T^{(i)} } \xi_{\sigma } \sigma.

Note that for each element

\xi \in C_i ^T ((X)),

its support,

|\xi | = \bigcup_{\xi_{\sigma}\neq 0} \sigma,

is closed. The support is compact if and only if $ξ$ is a finite linear combination of simplices.

The space

C_i ((X))

of $i$ -chains with closed support is defined to be the direct limit of

C_i^T ((X))

under refinements of $T$ . The boundary map of simplicial homology extends to a boundary map

\partial : C_i ((X)) \to C_{i-1} ((X))

and it is easy to see that the sequence

\cdots \to C_{i+1} ((X)) \to C_i ((X)) \to C_{i-1} ((X)) \to \cdots

is a chain complex. The Borel−Moore homology of $X$ is defined to be the homology of this chain complex. Concretely,

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

Definition via compactifications

Let $X$ be a compactification of $X$ such that the pair $(X, X \ X )$ is a CW-pair. For example, one may take the one point compactification of $X$ . Then

H^{BM}_i(X)=H_i(\bar{X} , \bar{X} \setminus X),

where in the right hand side, usual relative homology is meant.

Definition via Poincaré duality

Let $X \subset M$ be a closed embedding of $X$ in a smooth manifold of dimension $m$ , such that $X$ is a retract of an open neighborhood of itself. Then

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

where in the right hand side, usual relative cohomology is meant.

Definition via the dualizing complex

Let $D X$ 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.

Properties

Borel−Moore homology is a covariant functor with respect to proper maps. Suppose $f : X \to Y$ is a proper map. Then $f$ induces a continuous map $\bar{f} : (\bar{X}, \bar{X} \setminus X ) \to (\bar {Y}, \bar{Y} \setminus Y)$ where $\bar{X}=X\cup \{ \infty \}, \bar{Y}=Y\cup \{ \infty \}$ are the one point compactifications. Using the definition of Borel−Moore homology via compactification, there is a map $\ f_*:H^{BM}_* (X)\to H^{BM}_* (Y)$ . Properness is essential, as it guarantees that the induced map on compactifications will be continuous. There is no pushforward for a general continuous map of spaces. As a counterexample, one can consider the non-proper inclusion $C * \to C$ .

If $F \subset X$ is a closed set and $U = X \ F$ is its complement, then there is a long exact sequence

\cdots \to H^{BM}_i (F) \to H^{BM}_i (X) \to H^{BM}_i (U) \to H^{BM}_{i-1} (F) \to \cdots

Borel−Moore homology is "not" homotopy invariant. Note a contractible space with non-zero Borel-Moore 1-homology. For example, as it is easy to see from the above exact sequence by removing a point p of the circle $S 1$ we have

H^{BM}_i \left (\mathbf{S}^1 \setminus p \right ) = \begin{cases} H_1\left (\mathbf{S}^1\right ) & i = 1 \\ 0 & i \neq 1 \end{cases}

However, if $f, g : X \to Y$ are two parallel homotopic proper maps then they induce the same map $\ f_* = g_*:H^{BM}_* (X)\to H^{BM}_* (Y)$ in homology.

One of the main reasons to use Borel−Moore homology is that for every orientable manifold (in particular, for every smooth complex variety) $M$ , there is a fundamental class $[M]\in H^{BM}_{top}(M)$ . This is just the sum over all top dimensional simplices in a specific triangulation. In fact, in Borel−Moore homology, one can define a fundamental class for arbitrary (i.e. possibly singular) complex varieties. In this case the set of smooth points $M reg \subset M$ has complement of (real) codimension $2$ and by the long exact sequence above the top dimensional homologies of $M$ and $M reg$ are canonically isomorphic. One then defines the fundamental class of $M$ to be the fundamental class of $M reg$ .

References

Iversen, Birger Cohomology of sheaves. Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp. ISBN 3-540-16389-1 MR 0842190
Borel, Armand; Moore, John C. (1960), "Homology theory for locally compact spaces", The Michigan Mathematical Journal 7: 137–159, doi:10.1307/mmj/1028998385, ISSN 0026-2285, MR 0131271