Borel-Moore homology

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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).

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[edit] 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 .

[edit] 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 | = \bigcup _{\xi _{\sigma}\neq 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))\to C_{i-1}((X))

and it is easy to see that the sequence

\dots \to C_{i+1} ((X)) \to C_i ((X)) \to C_{i-1} ((X)) \to \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)) \to C_{i-1} ((X)) )/ Im (\partial :C_{i+1} ((X)) \to C_i ((X)) )

[edit] Definition via compactifications

Let \ \bar{X} be a compactification of \ X such that the pair

\ (\bar{X} ,X)

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

\ H^{BM}_i(X)=H_i(\bar{X} , \bar{X} \setminus X) ,

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

[edit] 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,M\setminus X),

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

[edit] 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.

[edit] Properties

\ H^{BM}_i(\mathbb{R} ^n )

vanishes for \ i\neq n and equals \ \mathbb{R} for \ i=n .

  • Borel-Moore homology is a covariant functor with respect to proper maps. Suppose \ f:X\to Y is a proper map. Then \ f induces a continuous map \ \bar{f} :(\bar{X} , \bar{X} \setminus X )\to (\bar {Y} , \bar{Y} \setminus Y) where \bar{X}=X\cup \{ \infty \} , \bar{Y}=Y\cup \{ \infty \} are the one point compactifications. Using the definition of Borel-Moore homology via compactification, there is a map \ f_*:H^{BM}_* (X)\to H^{BM}_* (Y) .
  • If \ F \subset X is a closed set and \ U=X\setminus F is its complement, then there is a long exact sequence

\dots \to H^{BM}_i (F) \to H^{BM}_i (X) \to H^{BM}_i (U) \to H^{BM}_{i-1} (F) \to \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 \ [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} .

[edit] References

  • Iversen, Birger Cohomology of sheaves. Universitext. Springer-Verlag, Berlin, 1986. xii+464 pp. ISBN 3-540-16389-1 MR0842190
  • A, Borel, John C. Moore, Homology theory for locally compact spaces, Michigan Math. J. 7 (1960) 137-159