View source for Borel–Moore homology

{{Short description|Homology theory for locally compact spaces}}
In [[topology]], '''Borel−Moore homology''' or '''homology with closed support''' is a [[homology theory]] for [[locally compact space]]s, introduced by [[Armand Borel]] and [[John Coleman Moore |John Moore]] in 1960.{{sfn|Borel|Moore|1960}}

For reasonable [[compact space]]s, 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 <math>H^*_G(X) = H^*((EG \times X)/G).</math> That is not related to the subject of this article.

==Definition==
There are several ways to define Borel−Moore homology. They all coincide for reasonable spaces such as [[manifold]]s and locally finite [[CW complexes]].

===Definition via sheaf cohomology===
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.<ref>Birger Iversen. Cohomology of sheaves. Section IX.1.</ref> As a result, there is a [[short exact sequence]] analogous to the [[universal coefficient theorem]]:

<math display="block">0 \to \text{Ext}^1_{\Z}(H^{i+1}_c(X,\Z),\Z) \to H_i^{BM}(X,\Z) \to \text{Hom}(H^i_c(X,\Z),\Z) \to 0.</math>

In what follows, the coefficients <math>\Z</math> are not written.

===Definition via locally finite chains===
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.<ref>Glen  Bredon. Sheaf theory. Corollary V.12.21.</ref>

In more detail, let <math>C_i^{BM}(X)</math> be the abelian group of formal (infinite) sums

<math display="block"> u = \sum_{\sigma} a_{\sigma } \sigma, </math>

where ''σ'' runs over the set of all continuous maps from the standard ''i''-simplex Δ<sup>''i''</sup> to ''X'' and each ''a<sub>σ</sub>'' is an integer, such that for each compact subset ''K'' of ''X'', we have  <math>a_\sigma\neq 0</math> 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:

<math display="block"> \cdots \to C_2^{BM}(X) \to C_1^{BM}(X) \to C_0^{BM}(X) \to 0.</math>

The Borel−Moore homology groups <math>H_i^{BM}(X)</math> are the homology groups of this chain complex. That is,

<math display="block">H^{BM}_i (X) = \ker \left (\partial : C_i^{BM}(X) \to C_{i-1}^{BM}(X) \right )/ \text{im} \left (\partial :C_{i+1}^{BM}(X) \to C_i^{BM}(X) \right ). </math>

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 <math>H_i^{BM}(X)</math> coincides with the usual singular homology <math>H_i(X)</math> for ''X'' compact.

===Definition via compactifications===
Suppose that ''X'' is homeomorphic to the complement of a closed subcomplex ''S'' in a finite CW complex ''Y''. Then Borel–Moore homology <math>H_i^{BM}(X)</math> is isomorphic to the [[relative homology]] ''H''<sub>''i''</sub>(''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===
Let ''X'' be any locally compact space with a closed embedding into an oriented [[manifold]] ''M'' of dimension ''m''. Then

<math display="block">H^{BM}_i(X)= H^{m-i}(M,M\setminus X),</math>

where in the right hand side, relative [[cohomology]] is meant.<ref>Birger Iversen. Cohomology of sheaves. Theorem IX.4.7.</ref>

===Definition via the dualizing complex===
For any locally compact space ''X'' of finite dimension, let {{math|''D<sub>X</sub>''}} be the [[Verdier duality|dualizing complex]] of {{mvar|X}}. Then

<math display="block">H^{BM}_i (X)=\mathbb{H}^{-i} (X, D_X), </math>

where in the right hand side, [[hypercohomology]] is meant.<ref>Birger Iversen. Cohomology of sheaves. Equation IX.4.1.</ref>

==Properties==
Borel−Moore homology is a [[covariant functor]] with respect to [[proper map]]s. That is, a proper map ''f'': ''X'' → ''Y'' induces a '''pushforward''' homomorphism <math>H_i^{BM}(X) \to H_i^{BM}(Y)</math> 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 <math>\R^2 \setminus \{0\} \to \R^2.</math>

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 <math>H_i^{BM}(X) \to H_i^{BM}(U).</math>

For any locally compact space ''X'' and any closed subset ''F'', with <math>U = X\setminus F</math> the complement, there is a long exact '''localization''' sequence:<ref>Birger Iversen. Cohomology of sheaves. Equation IX.2.1.</ref>
<math display="block"> \cdots \to H^{BM}_i (F) \to H^{BM}_i (X) \to H^{BM}_i (U) \to H^{BM}_{i-1} (F) \to \cdots </math>

Borel−Moore homology is [[homotopy invariant]] in the sense that for any space ''X'', there is an isomorphism <math>H_i^{BM}(X) \to H_{i+1}^{BM}(X\times \R).</math> 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 <math>\R^n</math> is isomorphic to <math>\Z</math> 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,
<math display="block">H^i(X) \stackrel{\cong}{\to} H_{n-i}^{BM}(X)</math>
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:
<math display="block">H^i_c(X) \stackrel{\cong}{\to} H_{n-i}(X).</math>

A key advantage of Borel−Moore homology is that every [[orientable manifold|oriented manifold]] ''M'' of dimension ''n'' (in particular, every [[smooth scheme|smooth]] complex [[algebraic variety]]), not necessarily compact, has a [[fundamental class]] <math>[M] \in H_n^{BM}(M).</math> If the manifold ''M'' has a [[triangulation (topology)|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 <math>M^{\text{reg}} \subset M</math> has (real) [[codimension]] at least 2, and by the long exact sequence above the top dimensional homologies of {{mvar|M}} and <math>M^{\text{reg}}</math> are canonically isomorphic. The fundamental class of {{mvar|M}} is then defined to be the fundamental class of <math>M^{\text{reg}}</math>.<ref>William Fulton. Intersection theory. Lemma 19.1.1.</ref>

==Examples==

===Compact Spaces===
Given a compact topological space <math>X</math> its Borel-Moore homology agrees with its standard homology; that is,

<math display="block">H^{BM}_*(X) \cong H_*(X)</math>

===Real line===
The first non-trivial calculation of Borel-Moore homology is of the real line. First observe that any <math>0</math>-chain is cohomologous to <math>0</math>. Since this reduces to the case of a point <math>p</math>, notice that we can take the Borel-Moore chain

<math display="block">\sigma = \sum_{i=0}^\infty 1\cdot [p+i,p+i+1]</math>

since the boundary of this chain is <math>\partial\sigma = p</math> and the non-existent point at infinity, the point is cohomologous to zero. Now, we can take the Borel-Moore chain

<math display="block">\sigma = \sum_{-\infty<k<\infty} [k, k+1]</math>

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

<math display="block"> H_k^{BM}(\R) = \begin{cases} \Z & k = 1 \\ 0 & \text{otherwise} \end{cases} </math>

===Real n-space===
The previous computation can be generalized to the case <math>\R^n.</math> We get

<math display="block"> H_k^{BM}(\R^n) = \begin{cases} \Z & k = n \\ 0 & \text{otherwise} \end{cases} </math>

===Infinite Cylinder===
Using the Kunneth decomposition, we can see that the infinite cylinder <math>S^1\times\R</math> has homology

<math display="block"> H_k^{BM}(S^1\times \R ) = \begin{cases}
\Z & k = 1 \\
\Z & k = 2 \\
0 & \text{otherwise}
\end{cases}
</math>

===Real n-space minus a point===
Using the long exact sequence in Borel-Moore homology, we get (for <math>n>1</math>) the non-zero exact sequences

<math display="block">0 \to H_n^{BM}(\{0\}) \to H_n^{BM}(\R ^n) \to H_n^{BM}(\R ^n-\{0\}) \to 0</math>

and

<math display="block">0 \to H_1^{BM}(\R ^n-\{0\}) \to H_0^{BM}(\{0\}) \to H_0^{BM}(\R ^n) \to H_0^{BM}(\R ^n-\{0\}) \to 0</math>

From the first sequence we get that

<math display="block">H_n^{BM}(\R ^n) \cong H_n^{BM}(\R ^n-\{0\})</math>

and from the second we get that

<math display="block">H_1^{BM}(\R ^n-\{0\}) \cong H_0^{BM}(\{0\})</math>
and

<math display="block">0 \cong H_0^{BM}(\R ^n) \cong H_0^{BM}(\R ^n-\{0\})</math>

We can interpret these non-zero homology classes using the following observations:
# There is the homotopy equivalence <math>\R ^n-\{0\} \simeq S^{n-1}.</math> 
# A topological isomorphism <math>\R ^n-\{0\} \cong S^{n-1} \times \R _{>0}.</math>

hence we can use the computation for the infinite cylinder to interpret <math>H_n^{BM}</math> as the homology class represented by <math>S^{n-1}\times\R _{>0}</math> and <math>H_1^{BM}</math> as <math>\R _{>0}.</math>

===Plane with Points Removed===
Let <math>X = \R^2 - \{p_1,\ldots, p_k \}</math> have <math>k</math>-distinct points removed. Notice the previous computation with the fact that Borel-Moore homology is an isomorphism invariant gives this computation for the case <math>k = 1</math>. In general, we will find a <math>1</math>-class corresponding to a loop around a point, and the fundamental class <math>[X]</math> in <math>H_2^{BM}</math>.
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Can this can be shown using Meyer-Vietoris?
-->

===Double Cone===
Consider the double cone <math>X = \mathbb{V}(x^2 + y^2 - z^2) \subset \R ^3</math>. If we take <math>U = X \setminus \{0\}</math> then the long exact sequence shows

<math display="block">\begin{align}
H_2^{BM}(X) &= \Z^{\oplus 2} \\
H_1^{BM}(X) &= \Z \\
H_k^{BM}(X) &= 0 && \text{ for } k \not\in \{1,2\}
\end{align}</math>

=== Genus Two Curve with Three Points Removed ===
Given a genus two curve ([[Riemann surface]]) <math>X</math> and three points <math>F</math>, we can use the long exact sequence to compute the Borel-Moore homology of <math>U = X \setminus F.</math> This gives

<math display="block">\begin{align}
    H_2^{BM}(F) \to & H_2^{BM}(X) \to H_2^{BM}(U)     \\
\to H_1^{BM}(F) \to & H_1^{BM}(X) \to H_1^{BM}(U)     \\
\to H_0^{BM}(F) \to & H_0^{BM}(X) \to H_0^{BM}(U) \to  0
\end{align}</math>

Since <math>F</math> is only three points we have 

<math display="block">H_1^{BM}(F) = H_2^{BM}(F) =0.</math> 

This gives us that <math>H_2^{BM}(U) = \Z.</math> Using Poincare-duality we can compute 

<math display="block">H_0^{BM}(U) = H^2(U) = 0,</math> 

since <math>U</math> 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

<math display="block">0 \to \Z ^{\oplus 4} \to H_1^{BM}(U) \to \Z ^{\oplus 3} \to \Z \to 0</math>

showing

<math display="block">H_1^{BM}(U) \cong \Z ^{\oplus 6}</math>

since we have the short exact sequence of free abelian groups

<math display="block">0 \to \Z ^{\oplus 4} \to H_1^{BM}(U) \to \Z ^{\oplus 2} \to 0</math>

from the previous sequence.

==Notes==
{{reflist}}

==References==
=== Survey articles ===
*{{Citation | last=Goresky | first=Mark | authorlink=Mark Goresky| title=Primer on Sheaves | url=http://www.math.ias.edu/~goresky/pdf/Sheaves.pdf | archive-url=https://web.archive.org/web/20170927040809/http://www.math.ias.edu/~goresky/pdf/Sheaves.pdf | url-status=dead | archive-date=2017-09-27 }}

=== Books ===
*{{cite journal| last1=Borel | first1=Armand | author1-link=Armand Borel | last2=Moore | first2=John C. | author2-link=John Coleman Moore | title=Homology theory for locally compact spaces | url=http://projecteuclid.org/euclid.mmj/1028998385 | mr=0131271 | year=1960 | journal=[[Michigan Mathematical Journal]] | issn=0026-2285 | volume=7 | issue=2 | pages=137–159 | doi=10.1307/mmj/1028998385| doi-access=free }}
*{{cite book| last1=Bredon | first1=Glen E. | author1-link = Glen Bredon | title=Sheaf Theory | publisher=[[Springer-Verlag]] | location =Berlin, New York | isbn=978-0-387-94905-5 | mr=1481706 | edition=2nd | year=1997}}
*{{cite book| last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection Theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-62046-4 | mr=1644323 | year=1998 | edition=2nd}}
*{{cite book| last1=Iversen | first1= Birger | title=Cohomology of Sheaves | publisher=[[Springer-Verlag]] | location=Berlin | year =1986 |isbn=3-540-16389-1 | mr=0842190}}

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[[Category:Homology theory]]
[[Category:Sheaf theory]]