- 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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