Relative homology

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

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

where ${\displaystyle C_{\bullet }(X)}$ denotes the singular chains on the space X. The boundary map on ${\displaystyle C_{\bullet }(X)}$ leaves ${\displaystyle C_{\bullet }(A)}$ invariant^a and therefore descends to 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]

The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence

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

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

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

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 ${\displaystyle x_{0}}$.

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]

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

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

• Excision Theorem
• Mayer–Vietoris sequence

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

• "Relative homology groups". PlanetMath.
• Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1

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