De Gua's theorem

In mathematics, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Template:Harvs.

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 $\ \xi$ 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,

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

Definition via compactifications

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

$\ H_{i}^{BM}(X)=H_{i}({\bar {X}},{\bar {X}}\setminus X),$

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

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

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

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

where in the right hand side, hypercohomology is meant.

Properties

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

$\ H_{i}^{BM}(\mathbb {R} ^{n})$

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

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

• 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 $\ [M]\in H_{top}^{BM}(M)$. 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 Template:MR
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