Borel–Moore homology

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.

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 Δⁱ to X and each a_σ is an integer, such that for each compact subset S of X, only finitely many maps σ whose image meets S have nonzero coefficient in u. 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.

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.

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]

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]

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

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

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

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

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

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

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

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

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 ${\displaystyle \mathbb {R} _{>0}.}$

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 ${\displaystyle H_{2}^{BM}}$.

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

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

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.

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