- Borel-Moore homology
In

mathematics ,**Borel-Moore homology**or**homology with closed support**is ahomology theory forlocally compact space s.For

compact space s, the Borel-Moore homology coincide with the usualsingular 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 andJohn 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 aretract 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

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

its support,

:$|xi\; |\; =\; igcup\; \_\{xi\; \_\{sigma\}\; eq\; 0\}sigma$,

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

:$partial\; :C\_i((X))\; o\; C\_\{i-1\}((X))$

and it is easy to see that the sequence

:$dots\; o\; C\_\{i+1\}\; ((X))\; o\; C\_i\; ((X))\; o\; C\_\{i-1\}\; ((X))\; o\; dots$

is a

chain complex . The**Borel-Moore homology**of X is defined to be the homology of this chain complex. Concretely,:$H^\{BM\}\; \_i\; (X)\; =Ker\; (partial\; :C\_i\; ((X))\; o\; C\_\{i-1\}\; ((X))\; )/\; Im\; (partial\; :C\_\{i+1\}\; ((X))\; o\; C\_i\; ((X))\; )$

**Definition via compactifications**Let $ar\{X\}$ be a compactification of $X$ such that the pair

:$(ar\{X\}\; ,X)$

is a

CW-pair . For example, one may take theone point compactification of $X$. Then:$H^\{BM\}\_i(X)=H\_i(ar\{X\}\; ,\; ar\{X\}\; setminus\; X)$,

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

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 $i\; eq\; n$ and equals $mathbb\{R\}$ for $i=n$.

* Borel-Moore homology is a

covariant functor with respect toproper map s. Suppose $f:X\; o\; Y$ is a proper map. Then $f$ induces a continuous map $ar\{f\}\; :(ar\{X\}\; ,\; ar\{X\}\; setminus\; X\; )\; o\; (ar\; \{Y\}\; ,\; ar\{Y\}\; setminus\; Y)$ where $ar\{X\}=Xcup\; \{\; infty\; \}\; ,\; ar\{Y\}=Ycup\; \{\; infty\; \}$ are the one point compactifications. Using the definition of Borel-Moore homology via compactification, there is a map $f\_*:H^\{BM\}\_*\; (X)\; o\; H^\{BM\}\_*\; (Y)$. 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 $mathbb\{C\}^*\; o\; mathbb\{C\}$ .* If $F\; subset\; X$ is a closed set and $U=Xsetminus\; F$ is its complement, then there is a long exact sequence

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

* 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 afundamental 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 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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