Relative homology

In

comes from which subspace.

Definition

Given a subspace ${\displaystyle A\subseteq X}$, one may form the

${\displaystyle 0\to C_{\bullet }(A)\to C_{\bullet }(X)\to C_{\bullet }(X)/C_{\bullet }(A)\to 0,}$

where ${\displaystyle C_{\bullet }(X)}$ denotes the

on the space X. The boundary map on ${\displaystyle C_{\bullet }(X)}$ descends^a to ${\displaystyle C_{\bullet }(A)}$ and therefore induces a boundary map ${\displaystyle \partial '_{\bullet }}$ on the quotient. If we denote this quotient by ${\displaystyle C_{n}(X,A):=C_{n}(X)/C_{n}(A)}$, we then have a complex

${\displaystyle \cdots \longrightarrow C_{n}(X,A)\xrightarrow {\partial '_{n}} C_{n-1}(X,A)\longrightarrow \cdots .}$

By definition, the n^th relative homology group of the pair of spaces ${\displaystyle (X,A)}$ is

${\displaystyle H_{n}(X,A):=\ker \partial '_{n}/\operatorname {im} \partial '_{n+1}.}$

One says that relative homology is given by the relative cycles, chains whose boundaries are chains on A, modulo the relative boundaries (chains that are homologous to a chain on A, i.e., chains that would be boundaries, modulo A again).^[1]

Properties

The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the

${\displaystyle \cdots \to H_{n}(A){\stackrel {i_{*}}{\to }}H_{n}(X){\stackrel {j_{*}}{\to }}H_{n}(X,A){\stackrel {\partial }{\to }}H_{n-1}(A)\to \cdots .}$

The connecting map ${\displaystyle \partial }$ takes a relative cycle, representing a homology class in ${\displaystyle H_{n}(X,A)}$, to its boundary (which is a cycle in A).^[2]

It follows that ${\displaystyle H_{n}(X,x_{0})}$, where ${\displaystyle x_{0}}$ is a point in X, is the n-th reduced homology group of X. In other words, ${\displaystyle H_{i}(X,x_{0})=H_{i}(X)}$ for all ${\displaystyle i>0}$. When ${\displaystyle i=0}$, ${\displaystyle H_{0}(X,x_{0})}$ is the free module of one rank less than ${\displaystyle H_{0}(X)}$. The connected component containing ${\displaystyle x_{0}}$ becomes trivial in relative homology.

The excision theorem says that removing a sufficiently nice subset ${\displaystyle Z\subset A}$ leaves the relative homology groups ${\displaystyle H_{n}(X,A)}$ unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that ${\displaystyle H_{n}(X,A)}$ is the same as the n-th reduced homology groups of the quotient space ${\displaystyle X/A}$.

Relative homology readily extends to the triple ${\displaystyle (X,Y,Z)}$ for ${\displaystyle Z\subset Y\subset X}$.

One can define the Euler characteristic for a pair ${\displaystyle Y\subset X}$ by

${\displaystyle \chi (X,Y)=\sum _{j=0}^{n}(-1)^{j}\operatorname {rank} H_{j}(X,Y).}$

The exactness of the sequence implies that the Euler characteristic is additive, i.e., if ${\displaystyle Z\subset Y\subset X}$, one has

${\displaystyle \chi (X,Z)=\chi (X,Y)+\chi (Y,Z).}$

Local homology

The ${\displaystyle n}$-th local homology group of a space ${\displaystyle X}$ at a point ${\displaystyle x_{0}}$, denoted

${\displaystyle H_{n,\{x_{0}\}}(X)}$

is defined to be the relative homology group ${\displaystyle H_{n}(X,X\setminus \{x_{0}\})}$. Informally, this is the "local" homology of ${\displaystyle X}$ close to ${\displaystyle x_{0}}$.

Local homology of the cone CX at the origin

One easy example of local homology is calculating the local homology of the cone (topology) of a space at the origin of the cone. Recall that the cone is defined as the quotient space

${\displaystyle CX=(X\times I)/(X\times \{0\}),}$

where ${\displaystyle X\times \{0\}}$ has the subspace topology. Then, the origin ${\displaystyle x_{0}=0}$ is the equivalence class of points ${\displaystyle [X\times 0]}$. Using the intuition that the local homology group ${\displaystyle H_{*,\{x_{0}\}}(CX)}$ of ${\displaystyle CX}$ at ${\displaystyle x_{0}}$ captures the homology of ${\displaystyle CX}$ "near" the origin, we should expect this is the homology of ${\displaystyle H_{*}(X)}$ since ${\displaystyle CX\setminus \{x_{0}\}}$ has a

to ${\displaystyle X}$. Computing the local homology can then be done using the long exact sequence in homology

${\displaystyle {\begin{aligned}\to &H_{n}(CX\setminus \{x_{0}\})\to H_{n}(CX)\to H_{n,\{x_{0}\}}(CX)\\\to &H_{n-1}(CX\setminus \{x_{0}\})\to H_{n-1}(CX)\to H_{n-1,\{x_{0}\}}(CX).\end{aligned}}}$

Because the cone of a space is contractible, the middle homology groups are all zero, giving the isomorphism

${\displaystyle {\begin{aligned}H_{n,\{x_{0}\}}(CX)&\cong H_{n-1}(CX\setminus \{x_{0}\})\\&\cong H_{n-1}(X),\end{aligned}}}$

since ${\displaystyle CX\setminus \{x_{0}\}}$ is contractible to ${\displaystyle X}$.

In algebraic geometry

Note the previous construction can be proven in algebraic geometry using the affine cone of a projective variety ${\displaystyle X}$ using Local cohomology.

Local homology of a point on a smooth manifold

Another computation for local homology can be computed on a point ${\displaystyle p}$ of a manifold ${\displaystyle M}$. Then, let ${\displaystyle K}$ be a compact neighborhood of ${\displaystyle p}$ isomorphic to a closed disk ${\displaystyle \mathbb {D} ^{n}=\{x\in \mathbb {R} ^{n}:|x|\leq 1\}}$ and let ${\displaystyle U=M\setminus K}$. Using the excision theorem there is an isomorphism of relative homology groups

${\displaystyle {\begin{aligned}H_{n}(M,M\setminus \{p\})&\cong H_{n}(M\setminus U,M\setminus (U\cup \{p\}))\\&=H_{n}(K,K\setminus \{p\}),\end{aligned}}}$

hence the local homology of a point reduces to the local homology of a point in a closed ball ${\displaystyle \mathbb {D} ^{n}}$. Because of the homotopy equivalence

${\displaystyle \mathbb {D} ^{n}\setminus \{0\}\simeq S^{n-1}}$

and the fact

${\displaystyle H_{k}(\mathbb {D} ^{n})\cong {\begin{cases}\mathbb {Z} &k=0\\0&k\neq 0,\end{cases}}}$

the only non-trivial part of the long exact sequence of the pair ${\displaystyle (\mathbb {D} ,\mathbb {D} \setminus \{0\})}$ is

${\displaystyle 0\to H_{n,\{0\}}(\mathbb {D} ^{n})\to H_{n-1}(S^{n-1})\to 0,}$

hence the only non-zero local homology group is ${\displaystyle H_{n,\{0\}}(\mathbb {D} ^{n})}$.

Functoriality

Just as in absolute homology, continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but it descends to the quotient.

Let ${\displaystyle (X,A)}$ and ${\displaystyle (Y,B)}$ be pairs of spaces such that ${\displaystyle A\subseteq X}$ and ${\displaystyle B\subseteq Y}$, and let ${\displaystyle f\colon X\to Y}$ be a continuous map. Then there is an induced map ${\displaystyle f_{\#}\colon C_{n}(X)\to C_{n}(Y)}$ on the (absolute) chain groups. If ${\displaystyle f(A)\subseteq B}$, then ${\displaystyle f_{\#}(C_{n}(A))\subseteq C_{n}(B)}$. Let

${\displaystyle {\begin{aligned}\pi _{X}&:C_{n}(X)\longrightarrow C_{n}(X)/C_{n}(A)\\\pi _{Y}&:C_{n}(Y)\longrightarrow C_{n}(Y)/C_{n}(B)\\\end{aligned}}}$

be the natural projections which take elements to their equivalence classes in the quotient groups. Then the map ${\displaystyle \pi _{Y}\circ f_{\#}\colon C_{n}(X)\to C_{n}(Y)/C_{n}(B)}$ is a group homomorphism. Since ${\displaystyle f_{\#}(C_{n}(A))\subseteq C_{n}(B)=\ker \pi _{Y}}$, this map descends to the quotient, inducing a well-defined map ${\displaystyle \varphi \colon C_{n}(X)/C_{n}(A)\to C_{n}(Y)/C_{n}(B)}$ such that the following diagram commutes:^[3]

Chain maps induce homomorphisms between homology groups, so ${\displaystyle f}$ induces a map ${\displaystyle f_{*}\colon H_{n}(X,A)\to H_{n}(Y,B)}$ on the relative homology groups.^[2]

Examples

One important use of relative homology is the computation of the homology groups of quotient spaces ${\displaystyle X/A}$. In the case that ${\displaystyle A}$ is a subspace of ${\displaystyle X}$ fulfilling the mild regularity condition that there exists a neighborhood of ${\displaystyle A}$ that has ${\displaystyle A}$ as a deformation retract, then the group ${\displaystyle {\tilde {H}}_{n}(X/A)}$ is isomorphic to ${\displaystyle H_{n}(X,A)}$. We can immediately use this fact to compute the homology of a sphere. We can realize ${\displaystyle S^{n}}$ as the quotient of an n-disk by its boundary, i.e. ${\displaystyle S^{n}=D^{n}/S^{n-1}}$. Applying the exact sequence of relative homology gives the following:
${\displaystyle \cdots \to {\tilde {H}}_{n}(D^{n})\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow {\tilde {H}}_{n-1}(S^{n-1})\rightarrow {\tilde {H}}_{n-1}(D^{n})\to \cdots .}$

Because the disk is contractible, we know its reduced homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:

${\displaystyle 0\rightarrow H_{n}(D^{n},S^{n-1})\rightarrow {\tilde {H}}_{n-1}(S^{n-1})\rightarrow 0.}$

Therefore, we get isomorphisms ${\displaystyle H_{n}(D^{n},S^{n-1})\cong {\tilde {H}}_{n-1}(S^{n-1})}$. We can now proceed by induction to show that ${\displaystyle H_{n}(D^{n},S^{n-1})\cong \mathbb {Z} }$. Now because ${\displaystyle S^{n-1}}$ is the deformation retract of a suitable neighborhood of itself in ${\displaystyle D^{n}}$, we get that ${\displaystyle H_{n}(D^{n},S^{n-1})\cong {\tilde {H}}_{n}(S^{n})\cong \mathbb {Z} }$.

Another insightful geometric example is given by the relative homology of ${\displaystyle (X=\mathbb {C} ^{*},D=\{1,\alpha \})}$ where ${\displaystyle \alpha \neq 0,1}$. Then we can use the long exact sequence

${\displaystyle {\begin{aligned}0&\to H_{1}(D)\to H_{1}(X)\to H_{1}(X,D)\\&\to H_{0}(D)\to H_{0}(X)\to H_{0}(X,D)\end{aligned}}={\begin{aligned}0&\to 0\to \mathbb {Z} \to H_{1}(X,D)\\&\to \mathbb {Z} ^{\oplus 2}\to \mathbb {Z} \to 0\end{aligned}}}$

Using exactness of the sequence we can see that ${\displaystyle H_{1}(X,D)}$ contains a loop ${\displaystyle \sigma }$ counterclockwise around the origin. Since the cokernel of ${\displaystyle \phi \colon \mathbb {Z} \to H_{1}(X,D)}$ fits into the exact sequence

${\displaystyle 0\to \operatorname {coker} (\phi )\to \mathbb {Z} ^{\oplus 2}\to \mathbb {Z} \to 0}$

it must be isomorphic to ${\displaystyle \mathbb {Z} }$. One generator for the cokernel is the ${\displaystyle 1}$-chain ${\displaystyle [1,\alpha ]}$ since its boundary map is

${\displaystyle \partial ([1,\alpha ])=[\alpha ]-[1]}$

See also

• Excision Theorem
• Mayer–Vietoris sequence

Notes

^ i.e., the boundary ${\displaystyle \partial \colon C_{n}(X)\to C_{n-1}(X)}$ maps ${\displaystyle C_{n}(A)}$ to ${\displaystyle C_{n-1}(A)}$

References

• "Relative homology groups". PlanetMath.
• ISBN 0-387-96678-1

Specific