Borel-Moore homology

Borel-Moore homology

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 John C. Moore (1960).

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 $C_i ^T \left(\left(X\right)\right)$ the vector space of formal (infinite) sums

:$xi = sum _\left\{sigma in T^\left\{\left(i\right)\right\} \right\} xi _\left\{sigma \right\} sigma$.

Note that for each element

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

its support,

:,

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

The space

:$C_i \left(\left(X\right)\right)$

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

:$C_i ^T \left(\left(X\right)\right)$

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

:$partial :C_i\left(\left(X\right)\right) o C_\left\{i-1\right\}\left(\left(X\right)\right)$

and it is easy to see that the sequence

:$dots o C_\left\{i+1\right\} \left(\left(X\right)\right) o C_i \left(\left(X\right)\right) o C_\left\{i-1\right\} \left(\left(X\right)\right) o dots$

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

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

Definition via compactifications

Let be a compactification of $X$ such that the pair

:

is a CW-pair. For example, one may take the one point compactification of $X$. Then

:,

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

:$H^\left\{BM\right\}_i\left(X\right)= H^\left\{m-i\right\}\left(M,Msetminus X\right)$,

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

Definition via the dualizing complex

Let

:$mathbb\left\{D\right\} _X$

be the dualizing complex of $X$. Then

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

where in the right hand side, hypercohomology is meant.

Properties

* Borel-Moore homology is not homotopy invariant. For example,

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

vanishes for $i eq n$ and equals $mathbb\left\{R\right\}$ for $i=n$.

* Borel-Moore homology is a covariant functor with respect to proper maps. Suppose $f:X o Y$ is a proper map. Then $f$ induces a continuous map where are the one point compactifications. Using the definition of Borel-Moore homology via compactification, there is a map $f_*:H^\left\{BM\right\}_* \left(X\right) o H^\left\{BM\right\}_* \left(Y\right)$. 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 $mathbb\left\{C\right\}^* o mathbb\left\{C\right\}$ .

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

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

* 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 $\left[M\right] in H^\left\{BM\right\}_\left\{top\right\}\left(M\right)$. 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 $M^\left\{reg\right\} subset M$ has complement of (real) codimension 2 and by the long exact sequence above the top dimensional homologies of $M$ and $M^\left\{reg\right\}$ are canonically isomorphic. One then defines the fundamental class of $M$ to be the fundamental class of $M^\left\{reg\right\}$.

References

*Iversen, Birger "Cohomology of sheaves." Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp. ISBN 3-540-16389-1 MathSciNet|id=0842190
*A, Borel, John C. Moore, "Homology theory for locally compact spaces", Michigan Math. J. 7 (1960) 137-159

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