Definition of $\tilde{H}_n(X,A)$

Question

My question arises from the following sentence of Hatcher's book p.118, in particular

I do understand that $\tilde{H}_n(X,A)$ is defined to be $H_n(X,A)$ if $n \ne 0$. There is a canonical way to define the reduced homology in order to have, given a long exact sequence of complex

$$0 \longrightarrow C_n(A) \longrightarrow C_n(X) \longrightarrow C_n(X,A) \longrightarrow 0$$

A natural long exact sequence of the pair $(X,A)$ for the reduced relative homology ? i.e

$$\cdots \longrightarrow \tilde{H}_n(A) \longrightarrow \tilde{H}_n(X) \longrightarrow H_n(X,A) \longrightarrow \tilde{H}_{n-1}(A) \longrightarrow \cdots$$

Related that doesn't solve the problem since the definition seems implicit.

Answer 1

The reduced homology groups of spaces $X$ and pairs $(X,A)$ are defined as the homology groups of the augmented chain complexes $\tilde C_*(X)$ and $\tilde C_*(X,A)$. For details see my answer to Suppose that $X$ is a topological space and $x_0\in X$. Prove that $\widetilde{H_n}(X)=H_n(X,x_0)$ for all $n\geq 0$. Note that $X = \emptyset$ plays a somewhat strange role in this context since formally $\tilde H_{-1}(\emptyset) = \mathbb Z$. Therefore one usually excludes the case $X = \emptyset$ when working with reduced homology. But note that formally everything works well even if we allow $X = \emptyset$.

In fact for any pair $(X,A)$ we obtain a short exact sequence of chain complexes $$0 \to \tilde C_*(A) \to \tilde C_*(X) \to \tilde C_*(X,A) \to 0 .$$ But each short exact sequence of chain complexes $$0 \to R \to S \to T \to 0$$ gives a long exact sequence of homology groups $$ \dots \to H_n(R) \to H_n(S) \to H_n(T) \stackrel{\partial}{\to} H_{n-1}(R) \to \dots$$ This results in the natural long exact reduced homology sequence of the pair $(X,A)$.

From the definitions of $\tilde C_*(X)$ and $\tilde C_*(X,A)$ we see that

1. $\tilde H_n(X) = H_n(X)$ for $n > 0$

2. $\tilde H_n(X,A) = H_n(X,A)$ for all $n$.

3. $H_0(X) \approx \tilde H_0(X) \oplus \mathbb Z $

The long exact reduced homology sequence of a pair $(X,A)$ with $A \ne \emptyset$ is therefore identical with the long exact unreduced homology sequence except at the final terms: $$\dots \to H_1(A) \to H_1(X) \to H_1(X,A) \to \tilde H_0(A) \to \tilde H_0(X) \to H_0(X,A) \to 0$$

Also have a look at my answer to Clarification about reduced Homology.