Borel–Moore homology

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In topology, Borel−Moore homology or homology with closed support is a homology theory for locally compact spaces, introduced by Armand Borel and John Moore in 1960.[1]

For reasonable compact spaces, Borel−Moore homology coincides with the usual singular homology. For non-compact spaces, each theory has its own advantages. In particular, a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact.

Note: Borel equivariant cohomology is an invariant of spaces with an action of a group G; it is defined as That is not related to the subject of this article.

Definition

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There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as manifolds and locally finite CW complexes.

Definition via sheaf cohomology

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For any locally compact space X, Borel–Moore homology with integral coefficients is defined as the cohomology of the dual of the chain complex which computes sheaf cohomology with compact support.[2] As a result, there is a short exact sequence analogous to the universal coefficient theorem:

 

In what follows, the coefficients   are not written.

Definition via locally finite chains

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The singular homology of a topological space X is defined as the homology of the chain complex of singular chains, that is, finite linear combinations of continuous maps from the simplex to X. The Borel−Moore homology of a reasonable locally compact space X, on the other hand, is isomorphic to the homology of the chain complex of locally finite singular chains. Here "reasonable" means X is locally contractible, σ-compact, and of finite dimension.[3]

In more detail, let   be the abelian group of formal (infinite) sums

 

where σ runs over the set of all continuous maps from the standard i-simplex Δi to X and each aσ is an integer, such that for each compact subset K of X, we have   for only finitely many σ whose image meets K. Then the usual definition of the boundary ∂ of a singular chain makes these abelian groups into a chain complex:

 

The Borel−Moore homology groups   are the homology groups of this chain complex. That is,

 

If X is compact, then every locally finite chain is in fact finite. So, given that X is "reasonable" in the sense above, Borel−Moore homology   coincides with the usual singular homology   for X compact.

Definition via compactifications

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Suppose that X is homeomorphic to the complement of a closed subcomplex S in a finite CW complex Y. Then Borel–Moore homology   is isomorphic to the relative homology Hi(Y, S). Under the same assumption on X, the one-point compactification of X is homeomorphic to a finite CW complex. As a result, Borel–Moore homology can be viewed as the relative homology of the one-point compactification with respect to the added point.

Definition via Poincaré duality

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Let X be any locally compact space with a closed embedding into an oriented manifold M of dimension m. Then

 

where in the right hand side, relative cohomology is meant.[4]

Definition via the dualizing complex

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For any locally compact space X of finite dimension, let DX be the dualizing complex of X. Then

 

where in the right hand side, hypercohomology is meant.[5]

Properties

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Borel−Moore homology is a covariant functor with respect to proper maps. That is, a proper map f: XY induces a pushforward homomorphism   for all integers i. In contrast to ordinary homology, there is no pushforward on Borel−Moore homology for an arbitrary continuous map f. As a counterexample, one can consider the non-proper inclusion  

Borel−Moore homology is a contravariant functor with respect to inclusions of open subsets. That is, for U open in X, there is a natural pullback or restriction homomorphism  

For any locally compact space X and any closed subset F, with   the complement, there is a long exact localization sequence:[6]  

Borel−Moore homology is homotopy invariant in the sense that for any space X, there is an isomorphism   The shift in dimension means that Borel−Moore homology is not homotopy invariant in the naive sense. For example, the Borel−Moore homology of Euclidean space   is isomorphic to   in degree n and is otherwise zero.

Poincaré duality extends to non-compact manifolds using Borel–Moore homology. Namely, for an oriented n-manifold X, Poincaré duality is an isomorphism from singular cohomology to Borel−Moore homology,   for all integers i. A different version of Poincaré duality for non-compact manifolds is the isomorphism from cohomology with compact support to usual homology:  

A key advantage of Borel−Moore homology is that every oriented manifold M of dimension n (in particular, every smooth complex algebraic variety), not necessarily compact, has a fundamental class   If the manifold M has a triangulation, then its fundamental class is represented by the sum of all the top dimensional simplices. In fact, in Borel−Moore homology, one can define a fundamental class for arbitrary (possibly singular) complex varieties. In this case the complement of the set of smooth points   has (real) codimension at least 2, and by the long exact sequence above the top dimensional homologies of M and   are canonically isomorphic. The fundamental class of M is then defined to be the fundamental class of  .[7]

Examples

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Compact Spaces

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Given a compact topological space   its Borel-Moore homology agrees with its standard homology; that is,

 

Real line

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The first non-trivial calculation of Borel-Moore homology is of the real line. First observe that any  -chain is cohomologous to  . Since this reduces to the case of a point  , notice that we can take the Borel-Moore chain

 

since the boundary of this chain is   and the non-existent point at infinity, the point is cohomologous to zero. Now, we can take the Borel-Moore chain

 

which has no boundary, hence is a homology class. This shows that

 

Real n-space

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The previous computation can be generalized to the case   We get

 

Infinite Cylinder

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Using the Kunneth decomposition, we can see that the infinite cylinder   has homology

 

Real n-space minus a point

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Using the long exact sequence in Borel-Moore homology, we get (for  ) the non-zero exact sequences

 

and

 

From the first sequence we get that

 

and from the second we get that

  and

 

We can interpret these non-zero homology classes using the following observations:

  1. There is the homotopy equivalence  
  2. A topological isomorphism  

hence we can use the computation for the infinite cylinder to interpret   as the homology class represented by   and   as  

Plane with Points Removed

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Let   have  -distinct points removed. Notice the previous computation with the fact that Borel-Moore homology is an isomorphism invariant gives this computation for the case  . In general, we will find a  -class corresponding to a loop around a point, and the fundamental class   in  .

Double Cone

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Consider the double cone  . If we take   then the long exact sequence shows

 

Genus Two Curve with Three Points Removed

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Given a genus two curve (Riemann surface)   and three points  , we can use the long exact sequence to compute the Borel-Moore homology of   This gives

 

Since   is only three points we have

 

This gives us that   Using Poincare-duality we can compute

 

since   deformation retracts to a one-dimensional CW-complex. Finally, using the computation for the homology of a compact genus 2 curve we are left with the exact sequence

 

showing

 

since we have the short exact sequence of free abelian groups

 

from the previous sequence.

Notes

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  1. ^ Borel & Moore 1960.
  2. ^ Birger Iversen. Cohomology of sheaves. Section IX.1.
  3. ^ Glen Bredon. Sheaf theory. Corollary V.12.21.
  4. ^ Birger Iversen. Cohomology of sheaves. Theorem IX.4.7.
  5. ^ Birger Iversen. Cohomology of sheaves. Equation IX.4.1.
  6. ^ Birger Iversen. Cohomology of sheaves. Equation IX.2.1.
  7. ^ William Fulton. Intersection theory. Lemma 19.1.1.

References

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Survey articles

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  • Goresky, Mark, Primer on Sheaves (PDF), archived from the original (PDF) on 2017-09-27

Books

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