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 acts which is also called Borel cohomology and is defined as . 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 that are homotopy equivalent to a finite CW complex and admit a closed embedding into a smooth manifold such that is a retract of 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
its support,
is closed. The support is compact if and only if is a finite linear combination of simplices.
The space
of i-chains with closed support is defined to be the direct limit of
under refinements of . The boundary map of simplicial homology extends to a boundary map
and it is easy to see that the sequence
is a 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 is a CW-pair. For example, one may take the one point compactification of . Then
where in the right hand side, usual relative homology is meant.
Definition via Poincaré duality
Let be a closed embedding of in a smooth manifold of 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
Let be the dualizing complex of . Then
where in the right hand side, hypercohomology is meant.
Properties
- Borel−Moore homology is not homotopy invariant. For example,
- Borel−Moore homology is a covariant functor with 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 .
- 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) codimension 2 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 .
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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