Borel-Moore homology

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In mathematics, Borel-Moore homology or homology with closed support is a homology theory for locally compact spaces.

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. The theory was developed by (and is named after) Armand Borel and Calvin Moore.

1 Definition
2 Properties

[edit] 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 $\ M$ .

[edit] 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

$\ \xi \in C((X)) _i ^T$ ,

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

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

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

$H^{BM} _i (X) =Ker (\partial :C_i ((X)) \to C_{i-1} ((X)) )/ Im (\partial :C_{i+1} ((X)) \to C_i ((X)) )$

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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 $\ f_*:H^{BM}_* (X)\to H^{BM}_* (Y)$ .

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 smooth orientable manifold $\ M$ , 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. If $\ M$ is a complex variety, one can discard the smoothness assumption: 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. In this case, define the fundamental class of $\ M$ to be the fundamental class of $\ M^{reg}$ .