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)


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?


3 Answers 3


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'').

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    $\begingroup$ Welcome to math.SE, Professor May! $\endgroup$ Jul 21, 2011 at 1:32

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.

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    $\begingroup$ Reduced homology is really the relative homology of $X$ relative to a point, right? So I guess it's natural for it to be related to wedge sum. $\endgroup$ Jul 13, 2011 at 2:03
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    $\begingroup$ The difference lies in that reduced homology is independent of the choice of a point, though :) $\endgroup$ Jul 13, 2011 at 2:08
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    $\begingroup$ @Olivier: The isomorphism $\tilde H_0(X)\oplus\mathbb Z\cong H_0(X)$ you mention comes from an exact sequence $$0\to \tilde H_0(X)\to H_0(X) \to\mathbb Z\to 0$$ This sequence splits (because $\mathbb Z$ is a projective abelian group, say) (and this splitting gives your isomorphism) but the splitting is not canonical. A splitting corresponds, more or less, to picking an element in $H_0(X)$ which is part of a basis, and this cannot be done, in general, in a natural way. $\endgroup$ Jul 13, 2011 at 20:32
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    $\begingroup$ Likewise, The isomorphism $\tilde{H}_*(X)\approx H_*(X,\lbrace \mathrm{pt}\rbrace)$ obviousl depends on the choice of the point $\mathrm{pt}$. $\endgroup$ Jul 13, 2011 at 20:34
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    $\begingroup$ The isomorphism $\bar{H}_0(X)\oplus \mathbb{Z} \cong H_0(X)$ is indeed unnatural, but somewhat counterintuitively (at least for me) is that the isomorphism $\bar{H}_0(X) \cong H_0(X,*)$ is natural as functors $Top_* \rightarrow Ab$. And similarly as functors $Top_* \rightarrow Ab$, $\bar{H}_0(X)\oplus \mathbb{Z} \cong H_0(X)$, naturally. So we have an example of two objectwise isomorphic functors that are not naturally isomorphic, that become naturally isomorphic when moving to a slice category. $\endgroup$ Aug 7, 2020 at 1:59

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!


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