Use of Reduced Homology

Question

I've been reading Hatcher's Algebraic Topology, specifically the paragraph about reduced homology $\tilde{H}_*$ (for singular homology of topological spaces). Can someone please provide reasons why reduced homology is defined and studied?

I understand the following facts, which are all found in Hatcher's book :

$-$ The reduced homology of a point is $0$.

$-$ The reduced homology is the same in all degrees $*$ as the usual singular homology for pairs of spaces $(X,A)$ with $A\neq \emptyset$ : $\tilde{H}_*(X,A)= H_*(X,A)$, and in positive degrees $(*=n>0)$ for single spaces $X$ (that is when $A=\emptyset$). There is the same long exact sequence in reduced homology for a pair of spaces as in standard homology.

$-$ In degree $0$, one has $\tilde{H}_0(X)\oplus\mathbb{Z}\cong H_0(X)$, with the coefficients for homology in $\mathbb{Z}$.

$-$ For any space $X$, and any point $\mathrm{pt}\in X$, there is an isomorphism $\tilde{H}_*(X)\cong H_*(X,\lbrace \mathrm{pt}\rbrace)$

$-$ This in turn implies that when $A\subset U\subset X$ is such that $A$ is closed, $U$ is open, and $A$ is a strong deformation retract of $U$, then there is an exact sequence in reduced homology (that stems from an exact sequence for standard singular homology)

$$\cdots\rightarrow\tilde{H}_*(A)\rightarrow\tilde{H}_*(X)\rightarrow\tilde{H}_*(X/A)\rightarrow\cdots$$

All of this is straightforward to prove, but it doesn't tell me why reduced homology is defined and when it is used. Can someone please shed some light on this matter?

Answer 1

The essential reason for preferring reduced homology (as experts do) is that the suspension axiom holds in all degrees, as it must when one generalizes from spaces to spectra and studies generalized homology theories. Also, when using reduced homology, one need not explicitly use pairs of spaces since $H_*(X,A)$ is the reduced homology of the cofiber $Ci$ of the inclusion $i\colon A\to X$.
The Eilenberg-Steenrod axioms for homology theories have a variant version for reduced theories, and the reduced and unreduced theories determine each other. (See for example my book ``A concise course in algebraic topology'').

Answer 2

Reduced homology is used, mostly, to simplify statements.

For example, it is not true that the homology of a wedge of two spaces $X\vee Y$ is the direct sum of the homologies of $X$ and of $Y$, but the only problem is actually in degree $0$. It is true, on the other hand, that the reduced homology of $X\vee Y$ is the direct sum of the reduced homologies of $X$ and of $Y$. This happens in various other contexts.

N.B.: You should be careful with those isomorphisms you mention, for they are generally not natural.

Answer 3

Firstly $H_0$ isn't particularly interesting since we frequently deal with connected spaces anyway. Apart from that, in various exact sequences, such as Mayer-Vietoris, using standard homology leaves us with a bunch of $\mathbb{Z}$'s at the end. But exact sequences are so much nicer with $0$'s instead! Reduced homology gives us 'exact'ly this!