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 $\tilde{H}_*(X)\oplus\mathbb{Z}\approx H_*(X)$ (here coefficients for homology are in $\mathbb{Z}$) (EDIT : it should read $\tilde{H}_0(X)\oplus\mathbb{Z}\approx H_0(X)$)
$-$ For any space $X$, and any point $\mathrm{pt}\in X$, there is an isomorphism $\tilde{H}_*(X)\approx H_*(X,\lbrace \mathrm{pt}\rbrace)$
$-$ 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?
Best Answer
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'').