Relative singular chains and relative homology

Overview

Extension of notes on Homology of general chain complexes. Informally, relative homology of a subspace $A\subseteq X$ counts the number of “holes” in $X$ that are not in $A$. In particular, if the inclusion $A\hookrightarrow X$ is a homotopy equivalence, then the spaces are the same from the perspective of homology, and thus the relative homology groups are all zero.

Relevant theorems:

• (Theorem) (Relative) homology is homotopy invariant
• (Theorem) The relative homology of a subset is isomorphic to the subset with excision when the closure of the excision is in the interior of the subset

---

Basic definition

Relative homology

Let $(X,A)$ be a pair of spaces where $A\subseteq X$ as a (topological) subspace. If $(Y,B)$ is another pair of spaces, a map of pairs is a continuous $f:X\to Y$ such that $f(A)\subseteq B$.

To obtain a chain complex from a pair of spaces, let $C_p(X,A)$ be the quotient group $C_p(X)/C_p(A)$ consisting of formal linear combinations of $p$-simplices, and let $\partial :C_p(X,A)\to C_{p-1}(X,A)$ be defined by the commutative diagram where $i_{\ast }$ is the map induced by inclusion $i:A\to X$ and $j_{\ast }$ is the quotient map. The relative homology of $(X,A)$ is defined as the homology of the chain complex $C_{\ast }(X,A)$:

$$H_p(X,A)=H_p(C_{\ast }(X,A),\partial ).$$

---

The long exact sequence in relative homology

Notice that

$$0\to C_p(A){\to }_{i_{\ast }}{C}_p(X){\to }_{j_{\ast }}{C}_p(X,A)\to 0$$

is a short exact sequence for all $p$, which incudes a long exact sequence of homology groups

$$\cdots \to H_p(A){\to }_{i_{\ast }}{H}_p(X){\to }_{j_{\ast }}{H}_p(X,A){\to }_{\delta }{H}_{p-1}(A)\to \cdots ,$$

where $\partial$ is the connecting homomorphism defined explicitly as follows: let $[c]\in H_p(X,A)$ be a relative homology class represented by $c\in C_p(X)$ with $\partial c\in C_{p-1}(A)$. Then $\delta ([c])=[\partial c]$ is the homology class in $H_{p-1}(A)$ defined by the boundary.

Long exact sequence of a sequence of inclusions

For $B\subseteq A\subseteq X$, the inclusions $i:(A,B)\to (X,B)$ and $j:(X,B)\to (X,A)$ induce a short exact sequence of chain complexes

$$0\to C_{\ast }(A,B){\to }_{i_{\ast }}{C}_{\ast }(X,B){\to }_{j_{\ast }}{C}_{\ast }(X,A)\to 0,$$

and hence a long exact sequence in homology

$$\cdots \to H_p(A,B){\to }_{i_{\ast }}{H}_p(X,B){\to }_{j_{\ast }}{H}_p(X,A){\to }_{\delta }{H}_{p-1}(A,B)\to \cdots .$$

Remark

If $B\subseteq X$ and the inclusion $i:B\to X$ is a homotopy equivalence, then it follows from homotopy invariance that $i:H_p(B)\to H_p(X)$ is an isomorphism for all $p$. Thus, the LES from the beginning of this section implies $H_p(X,B)=0$ for all $p$.

For a triple $B\subseteq A\subseteq X$ as in the previous theorem, this means that the connecting homomorphism $H_p(X,A)\to H_{p-1}(A,B)$ is an isomorphism for all $p$.