relative singular homology

Overview

Singular homology can be very hard to compute directly. Therefore, it is sometimes nice to “ignore” some parts of the topological space and make them trivial in some homology that is related. Relative homology is just that. It is the homology that comes from quotienting out the part of the homology that comes from a subspace

Relative homology definition

Given a pair $(X,A)$, we have

${\text{Sing}}_n(A)\subset {\text{Sing}}_n(X)⟹C_n(A)⊴C_n(X).$

Therefore, we define the singular chains as

$C_n(X,A)=C_n(X)/C_n(A)$

For the boundary map, we can use the universal property of quotient groups

Thus, $(C_{\bullet }(X,A),\partial )$ is the relative singular chain complex.

$H_{\bullet }(X,A)$ is the relative homology of $(X,A)$.

Chain maps

Note that since $i:A\hookrightarrow X$, we get chain map

$i_{\ast }:H_{\bullet }(A)⟶H_{\bullet }(X).$

Similarly, since we have the quotient map $q$ between complexes, we get the chain map

$q_{\ast }:H_{\bullet }(X)⟶H_{\bullet }(X,A)$

Next, we want to construct a new map

$${\partial }_n:H_n(X,A)⟶H_{n-1}(A)$$

Let $x\in H_n(X,A)$. We can then pick a representative $[\sigma ]\in Z_n(X,A)$. This is itself an equivalence class, so pick $\sigma \in C_n(X)$ which represents $[\sigma ]$.

We know that $\partial \sigma \in C_{n-1}(A)$, then $\partial \partial \sigma =0$

Boundary maps

Note that technically the two boundary maps in $\partial \partial \sigma$ above are different boundary maps, since one is in $C_{\bullet }(A)$ and the other is in $C_{\bullet }(X)$ but we can bring them both into the higher space and get the same result

Thus, we define $\partial x=[\partial \sigma ]$

\sigma is well defined

Proof:todo

Relation to singular homology

There is a exact sequence

Proof

todo

Resources

• @hatcher2002 - Chapter 2