Borel-Moore homology
Encyclopedia
In mathematics
, Borel−Moore homology or homology with closed support is a homology theory
for locally compact space
s, introduced by .
For compact space
s, 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.
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 
.
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,
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.
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.
be the dualizing complex
of
. Then
where in the right hand side, hypercohomology is meant.
vanishes for
and equals 
for 
.
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Borel−Moore homology or homology with closed support is a homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
for locally compact space
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
s, introduced by .
For compact space
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
s, the Borel−Moore homology coincide with the usual singular homology
Singular homology
In algebraic topology, a branch of mathematics, singular homology refers to the study of a certain set of algebraic invariants of a topological space X, the so-called homology groups H_n....
, but for non-compact spaces, it usually gives homology groups with better properties.
Note: There is an equivariant cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
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 complexCW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
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 triangulationSimplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
of
. Denote by the vector space of formal (infinite) sumsNote that for each element
its support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
,
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
Direct limit
In mathematics, a direct limit is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.- Algebraic objects :In this section objects are understood to be...
of
under refinements of
. The boundary map of simplicial homology extends to a boundary mapand it is easy to see that the sequence
is a chain complex
Chain complex
In mathematics, chain complex and cochain complex are constructs originally used in the field of algebraic topology. They are algebraic means of representing the relationships between the cycles and boundaries in various dimensions of some "space". Here the "space" could be a topological space or...
. The Borel−Moore homology of X is defined to be the homology of this chain complex. Concretely,
Definition via compactifications
Let be a compactificationCompactification (mathematics)
In mathematics, compactification is the process or result of making a topological space compact. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".-An...
of
such that the pairis a CW-pair. For example, one may take the one point compactification of
. Thenwhere in the right hand side, usual relative homology
Relative homology
In algebraic topology, a branch of mathematics, the homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways...
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. Thenwhere in the right hand side, usual relative cohomology is meant.
Definition via the dualizing complex
Letbe the dualizing complex
Verdier duality
In mathematics, Verdier duality is a generalization of the Poincaré duality of manifolds to locally compact spaces with singularities. Verdier duality was introduced by , as an analog for locally compact spaces of the coherent duality for schemes due to Grothendieck...
of
. Thenwhere in the right hand side, hypercohomology is meant.
Properties
- Borel−Moore homology is not homotopy invariant. For example,
vanishes for
and equals for .- Borel−Moore homology is a covariant functor with respect to proper mapProper mapIn mathematics, a continuous function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometry, the analogous concept is called a proper morphism.- Definition :...
s. 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 classFundamental classIn mathematics, the fundamental class is a homology class [M] associated to an oriented manifold M, which corresponds to "the whole manifold", and pairing with which corresponds to "integrating over the manifold"...
. 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) codimensionCodimensionIn mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, and also to submanifolds in manifolds, and suitable subsets of algebraic varieties.The dual concept is relative dimension.-Definition:...
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 
.