- Borel-Moore homology
mathematics, Borel-Moore homology or homology with closed support is a homology theoryfor locally compact spaces.
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 Boreland John C. Moore(1960).
There are several ways to define Borel-Moore homology. They all coincide for spaces that are homotopy equivalent to a finite
CW complexand admit a closed embedding into a smooth manifold such that is a retractof an open neighborhood of itself in .
Definition via locally finite chains
Let be a triangulation of . Denote by the vector space of formal (infinite) sums
Note that for each element
is closed. The support is compact if and only if is a finite linear combination of simplices.
of i-chains with closed support is defined to be the
under refinements of . The boundary map of simplicial homology extends to a boundary map
and it is easy to see that the sequence
chain complex. The Borel-Moore homology of X is defined to be the homology of this chain complex. Concretely,
Definition via compactifications
Let be a compactification of such that the pair
CW-pair. For example, one may take the one point compactificationof . Then
where in the right hand side, usual
relative homologyis meant.
Let be a closed embedding of in a
smooth manifoldof dimension "m", such that is a retract of an open neighborhood of itself. Then
where in the right hand side, usual relative cohomology is meant.
Definition via the dualizing complex
be the dualizing complex of . Then
where in the right hand side,
* Borel-Moore homology is not
homotopy invariant. For example,
vanishes for and equals for .
* Borel-Moore homology is a
covariant functorwith respect to proper maps. Suppose is a proper map. Then induces a continuous map where are the one point compactifications. Using the definition of Borel-Moore homology via compactification, there is a map . 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 .
* If is a closed set and is its complement, then there is a long exact sequence
* One of the main reasons to use Borel-Moore homology is that for every
orientable manifold(in particular, for every smooth complex variety) , there is a fundamental class. 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 has complement of (real) codimension2 and by the long exact sequence above the top dimensional homologies of and are canonically isomorphic. One then defines the fundamental class of to be the fundamental class of .
*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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