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.

1 Definition
2 Properties
3 References

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 $\ \xi$ 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

$\dots \to C_{i%2B1} ((X)) \to C_i ((X)) \to C_{i-1} ((X)) \to \dots$

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

$H^{BM} _i (X) =Ker (\partial�:C_i ((X)) \to C_{i-1} ((X)) )/ Im (\partial�:C_{i%2B1} ((X)) \to C_i ((X)) ).$

Definition via compactifications

Let $\ \bar{X}$ be a compactification of $\ X$ such that the pair

$\ (\bar{X} ,X)$

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

$\ \mathbb{D} _X$

be the dualizing complex of $\ \ X$ . Then

$\ H^{BM}_i (X)=H^{-i} (X,\mathbb{D} _X),$

where in the right hand side, hypercohomology is meant.

Properties

Borel−Moore homology is not homotopy invariant. For example,

$\ H^{BM}_i(\mathbb{R} ^n )$

vanishes for $\ i\neq n$ and equals $\ \mathbb{R}$ for $\ i=n$ .

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 $\mathbb{C}^* \to \mathbb{C}$ .

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

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

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, ISSN 0026-2285, MR 0131271, http://projecteuclid.org/euclid.mmj/1028998385