Borel–Moore homology

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 ${\displaystyle H_{G}^{*}(X)=H^{*}((EG\times X)/G).}$ 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 manifolds 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.[2] As a result, there is a short exact sequence analogous to the universal coefficient theorem:

$${\displaystyle 0\to {\text{Ext}}_{\mathbb {Z} }^{1}(H_{c}^{i+1}(X,\mathbb {Z} ),\mathbb {Z} )\to H_{i}^{BM}(X,\mathbb {Z} )\to {\text{Hom}}(H_{c}^{i}(X,\mathbb {Z} ),\mathbb {Z} )\to 0.}$$

In what follows, the coefficients ${\displaystyle \mathbb {Z} }$ 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.[3]

In more detail, let ${\displaystyle C_{i}^{BM}(X)}$ be the abelian group of formal (infinite) sums

$${\displaystyle u=\sum _{\sigma }a_{\sigma }\sigma ,}$$

where σ runs over the set of all continuous maps from the standard i-simplex Δ^i to X and each a_σ is an integer, such that for each compact subset K of X, we have ${\displaystyle a_{\sigma }\neq 0}$ 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:

$${\displaystyle \cdots \to C_{2}^{BM}(X)\to C_{1}^{BM}(X)\to C_{0}^{BM}(X)\to 0.}$$

The Borel−Moore homology groups ${\displaystyle H_{i}^{BM}(X)}$ are the homology groups of this chain complex. That is,

$${\displaystyle H_{i}^{BM}(X)=\ker \left(\partial :C_{i}^{BM}(X)\to C_{i-1}^{BM}(X)\right)/{\text{im}}\left(\partial :C_{i+1}^{BM}(X)\to C_{i}^{BM}(X)\right).}$$

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 ${\displaystyle H_{i}^{BM}(X)}$ coincides with the usual singular homology ${\displaystyle H_{i}(X)}$ 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 ${\displaystyle H_{i}^{BM}(X)}$ is isomorphic to the relative homology H_i(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

$${\displaystyle H_{i}^{BM}(X)=H^{m-i}(M,M\setminus X),}$$

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

Definition via the dualizing complex

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

$${\displaystyle H_{i}^{BM}(X)=\mathbb {H} ^{-i}(X,D_{X}),}$$

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

Properties

Borel−Moore homology is a covariant functor with respect to proper maps. That is, a proper map f: X → Y induces a pushforward homomorphism ${\displaystyle H_{i}^{BM}(X)\to H_{i}^{BM}(Y)}$ 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 ${\displaystyle \mathbb {R} ^{2}\setminus \{0\}\to \mathbb {R} ^{2}.}$

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 ${\displaystyle H_{i}^{BM}(X)\to H_{i}^{BM}(U).}$

For any locally compact space X and any closed subset F, with ${\displaystyle U=X\setminus F}$ the complement, there is a long exact localization sequence:[6] $${\displaystyle \cdots \to H_{i}^{BM}(F)\to H_{i}^{BM}(X)\to H_{i}^{BM}(U)\to H_{i-1}^{BM}(F)\to \cdots }$$

Borel−Moore homology is homotopy invariant in the sense that for any space X, there is an isomorphism ${\displaystyle H_{i}^{BM}(X)\to H_{i+1}^{BM}(X\times \mathbb {R} ).}$ 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 ${\displaystyle \mathbb {R} ^{n}}$ is isomorphic to ${\displaystyle \mathbb {Z} }$ 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, $${\displaystyle H^{i}(X){\stackrel {\cong }{\to }}H_{n-i}^{BM}(X)}$$ 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: $${\displaystyle H_{c}^{i}(X){\stackrel {\cong }{\to }}H_{n-i}(X).}$$

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 ${\displaystyle [M]\in H_{n}^{BM}(M).}$ 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 ${\displaystyle M^{\text{reg}}\subset M}$ has (real) codimension at least 2, and by the long exact sequence above the top dimensional homologies of M and ${\displaystyle M^{\text{reg}}}$ are canonically isomorphic. The fundamental class of M is then defined to be the fundamental class of ${\displaystyle M^{\text{reg}}}$.[7]

Examples

Compact Spaces

Given a compact topological space ${\displaystyle X}$ its Borel-Moore homology agrees with its standard homology; that is,

$${\displaystyle H_{*}^{BM}(X)\cong H_{*}(X)}$$

Real line

The first non-trivial calculation of Borel-Moore homology is of the real line. First observe that any ${\displaystyle 0}$-chain is cohomologous to ${\displaystyle 0}$. Since this reduces to the case of a point ${\displaystyle p}$, notice that we can take the Borel-Moore chain

$${\displaystyle \sigma =\sum _{i=0}^{\infty }1\cdot [p+i,p+i+1]}$$

since the boundary of this chain is ${\displaystyle \partial \sigma =p}$ and the non-existent point at infinity, the point is cohomologous to zero. Now, we can take the Borel-Moore chain

$${\displaystyle \sigma =\sum _{-\infty <k<\infty }[k,k+1]}$$

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

$${\displaystyle H_{k}^{BM}(\mathbb {R} )={\begin{cases}\mathbb {Z} &k=1\\0&{\text{otherwise}}\end{cases}}}$$

Real n-space

The previous computation can be generalized to the case ${\displaystyle \mathbb {R} ^{n}.}$ We get

$${\displaystyle H_{k}^{BM}(\mathbb {R} ^{n})={\begin{cases}\mathbb {Z} &k=n\\0&{\text{otherwise}}\end{cases}}}$$

Infinite Cylinder

Using the Kunneth decomposition, we can see that the infinite cylinder ${\displaystyle S^{1}\times \mathbb {R} }$ has homology

$${\displaystyle H_{k}^{BM}(S^{1}\times \mathbb {R} )={\begin{cases}\mathbb {Z} &k=1\\\mathbb {Z} &k=2\\0&{\text{otherwise}}\end{cases}}}$$

Real n-space minus a point

Using the long exact sequence in Borel-Moore homology, we get (for ${\displaystyle n>1}$) the non-zero exact sequences

$${\displaystyle 0\to H_{n}^{BM}(\{0\})\to H_{n}^{BM}(\mathbb {R} ^{n})\to H_{n}^{BM}(\mathbb {R} ^{n}-\{0\})\to 0}$$

and

$${\displaystyle 0\to H_{1}^{BM}(\mathbb {R} ^{n}-\{0\})\to H_{0}^{BM}(\{0\})\to H_{0}^{BM}(\mathbb {R} ^{n})\to H_{0}^{BM}(\mathbb {R} ^{n}-\{0\})\to 0}$$

From the first sequence we get that

$${\displaystyle H_{n}^{BM}(\mathbb {R} ^{n})\cong H_{n}^{BM}(\mathbb {R} ^{n}-\{0\})}$$

and from the second we get that

$${\displaystyle H_{1}^{BM}(\mathbb {R} ^{n}-\{0\})\cong H_{0}^{BM}(\{0\})}$$ and

$${\displaystyle 0\cong H_{0}^{BM}(\mathbb {R} ^{n})\cong H_{0}^{BM}(\mathbb {R} ^{n}-\{0\})}$$

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

1. There is the homotopy equivalence ${\displaystyle \mathbb {R} ^{n}-\{0\}\simeq S^{n-1}.}$
2. A topological isomorphism ${\displaystyle \mathbb {R} ^{n}-\{0\}\cong S^{n-1}\times \mathbb {R} _{>0}.}$

hence we can use the computation for the infinite cylinder to interpret ${\displaystyle H_{n}^{BM}}$ as the homology class represented by ${\displaystyle S^{n-1}\times \mathbb {R} _{>0}}$ and ${\displaystyle H_{1}^{BM}}$ as ${\displaystyle \mathbb {R} _{>0}.}$

Plane with Points Removed

Let ${\displaystyle X=\mathbb {R} ^{2}-\{p_{1},\ldots ,p_{k}\}}$ have ${\displaystyle k}$-distinct points removed. Notice the previous computation with the fact that Borel-Moore homology is an isomorphism invariant gives this computation for the case ${\displaystyle k=1}$. In general, we will find a ${\displaystyle 1}$-class corresponding to a loop around a point, and the fundamental class ${\displaystyle [X]}$ in ${\displaystyle H_{2}^{BM}}$.

Double Cone

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

$${\displaystyle {\begin{aligned}H_{2}^{BM}(X)&=\mathbb {Z} ^{\oplus 2}\\H_{1}^{BM}(X)&=\mathbb {Z} \\H_{k}^{BM}(X)&=0&&{\text{ for }}k\not \in \{1,2\}\end{aligned}}}$$

Genus Two Curve with Three Points Removed

Given a genus two curve (Riemann surface) ${\displaystyle X}$ and three points ${\displaystyle F}$, we can use the long exact sequence to compute the Borel-Moore homology of ${\displaystyle U=X\setminus F.}$ This gives

$${\displaystyle {\begin{aligned}H_{2}^{BM}(F)\to &H_{2}^{BM}(X)\to H_{2}^{BM}(U)\\\to H_{1}^{BM}(F)\to &H_{1}^{BM}(X)\to H_{1}^{BM}(U)\\\to H_{0}^{BM}(F)\to &H_{0}^{BM}(X)\to H_{0}^{BM}(U)\to 0\end{aligned}}}$$

Since ${\displaystyle F}$ is only three points we have

$${\displaystyle H_{1}^{BM}(F)=H_{2}^{BM}(F)=0.}$$

This gives us that ${\displaystyle H_{2}^{BM}(U)=\mathbb {Z} .}$ Using Poincare-duality we can compute

$${\displaystyle H_{0}^{BM}(U)=H^{2}(U)=0,}$$

since ${\displaystyle U}$ 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

$${\displaystyle 0\to \mathbb {Z} ^{\oplus 4}\to H_{1}^{BM}(U)\to \mathbb {Z} ^{\oplus 3}\to \mathbb {Z} \to 0}$$

showing

$${\displaystyle H_{1}^{BM}(U)\cong \mathbb {Z} ^{\oplus 6}}$$

since we have the short exact sequence of free abelian groups

$${\displaystyle 0\to \mathbb {Z} ^{\oplus 4}\to H_{1}^{BM}(U)\to \mathbb {Z} ^{\oplus 2}\to 0}$$

from the previous sequence.

Notes

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

Survey articles

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

Books

• Borel, Armand; Moore, John C. (1960). "Homology theory for locally compact spaces". Michigan Mathematical Journal. 7 (2): 137–159. doi:10.1307/mmj/1028998385. ISSN 0026-2285. MR 0131271.
• Bredon, Glen E. (1997). Sheaf Theory (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-0-387-94905-5. MR 1481706.
• Fulton, William (1998). Intersection Theory (2nd ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-62046-4. MR 1644323.
• Iversen, Birger (1986). Cohomology of Sheaves. Berlin: Springer-Verlag. ISBN 3-540-16389-1. MR 0842190.