reduced homology

(also nonabelian homological algebra)

In general, the homology of a point is not trivial but is concentrated in degree 0 on the given coefficient object. For some applications, though, it is convenient to divide out that contribution such as to have the homology of the point be entirely trivial. This is called *reduced homology*.

We discuss the reduced version of singular homology.

Let $X$ be a topological space. Write $C_\bullet(X)$ for its singular chain complex.

The **augmentation map** is the homomorphism of abelian groups

$\epsilon \colon C_0(X) \to \mathbb{Z}$

which adds up all the coefficients of all 0-chains:

$\epsilon \colon \colon \sum_{i} n_i \sigma_i \mapsto \sum_i n_i
\,.$

Since the boundary of a 1-chain is in the kernel of this map, it constitutes a chain map

$\epsilon \colon C_\bullet(X) \to \mathbb{Z}
\,,$

where now $\mathbb{Z}$ is regarded as a chain complex concentrated in degree 0.

The **reduced singular chain complex** $\tilde C_\bullet(X)$ of $X$ is the kernel of the augmentation map, the chain complex sitting in the short exact sequence

$0 \to \tilde C_\bullet(X) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0
\,.$

The **reduced singular homology** $\tilde H_\bullet(X)$ of $X$ is the chain homology of the reduced singular chain complex

$\tilde H_\bullet(X) \coloneqq H_\bullet(\tilde C_\bullet(X))
\,.$

Equivalently:

The **reduced singular homology** of $X$, denoted $\tilde H_\bullet(X)$, is the chain homology of the augmented chain complex

$\cdots \to C_2(X) \stackrel{\partial_1}{\to} C_1(X) \stackrel{\partial_0}{\to} C_0(X) \stackrel{\epsilon}{\to}
\mathbb{Z} \to 0
\,.$

Let $X$ be a topological space, $H_\bullet(X)$ its singular homology and $\tilde H_\bullet(X)$ its reduced singular homology, def. .

For $n \in \mathbb{N}$ there is an isomorphism

$H_n(X)
\simeq
\left\{
\array{
\tilde H_n(X) & for \; n \geq 1
\\
\tilde H_0(X) \oplus \mathbb{Z} & for\; n = 0
}
\right.$

The homology long exact sequence of the defining short exact sequence $\tilde C_\bullet(C) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z}$ is, since $\mathbb{Z}$ here is concentrated in degree 0, of the form

$\cdots \to \tilde H_n(X) \to H_n(X) \to 0 \to \cdots \to
0 \to
\cdots \to \tilde H_1(X) \to H_1(X) \to 0 \to
\tilde H_0(X) \to H_0(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0
\,.$

Here exactness says that all the morphisms $\tilde H_n(X) \to H_n(X)$ for positive $n$ are isomorphisms. Moreover, since $\mathbb{Z}$ is a free abelian group, hence a projective object, the remaining short exact sequence

$0 \to \tilde H_0(X) \to H_0(X) \to \mathbb{Z} \to 0$

is split (as discussed there) and hence $H_0(X) \simeq \tilde H_0(X) \oplus \mathbb{Z}$.

For $X = *$ the point, the morphism

$H_0(\epsilon) \colon H_0(X) \to \mathbb{Z}$

is an isomorphism. Accordingly the reduced homology of the point vanishes in every degree:

$\tilde H_\bullet(*) \simeq 0
\,.$

By the discussion at *Singular homology – Relation to homotopy groups* we have that

$H_n(*) \simeq
\left\{
\array{
\mathbb{Z} & for \; n = 0
\\
0 & otherwise
}
\right.
\,.$

Moreover, it is clear that $\epsilon \colon C_0(*) \to \mathbb{Z}$ is the identity map.

For $X$ an inhabited topological space, its reduced singular homology, def. , coincides with its relative singular homology relative to any base point $x \colon * \to X$:

$\tilde H_\bullet(X)
\simeq
H_\bullet(X,*)
\,.$

Consider the sequence of topological subspace inclusions

$\emptyset \hookrightarrow * \stackrel{x}{\hookrightarrow} X
\,.$

By the discussion at *Relative homology - long exact sequences* this induces a long exact sequence of the form

$\cdots
\to
H_{n+1}(*) \to H_{n+1}(X) \to H_{n+1}(X,*)
\to
H_n(*) \to H_n(X) \to H_n(X,*)
\to
\cdots
\to
H_1(X) \to H_1(X,*) \to H_0(*) \stackrel{H_0(x)}{\to} H_0(X) \to H_n(X,*)
\to 0
\,.$

Here in positive degrees we have $H_n(*) \simeq 0$ and therefore exactness gives isomorphisms

$H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}$

and hence with prop. isomorphisms

$\tilde H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}
\,.$

It remains to deal with the case in degree 0. To that end, observe that $H_0(x) \colon H_0(*) \to H_0(X)$ is a monomorphism: for this notice that we have a commuting diagram

$\array{
H_0(*) &\stackrel{id}{\to}& H_0(*)
\\
{}^{\mathllap{H_0(x)}}\downarrow &{}^{\mathllap{H_0(f)}}\nearrow& \downarrow^{\mathrlap{H_0(\epsilon)}}_\simeq
\\
H_0(X) &\stackrel{H_0(\epsilon)}{\to}& \mathbb{Z}
}
\,,$

where $f \colon X \to *$ is the terminal map. That the outer square commutes means that $H_0(\epsilon) \circ H_0(x) = H_0(\epsilon)$ and hence the composite on the left is an isomorphism. This implies that $H_0(x)$ is an injection.

Therefore we have a short exact sequence as shown in the top of this diagram

$\array{
0 &\to& H_0(*) &\stackrel{H_0(x)}{\hookrightarrow}&
H_0(X) &\stackrel{}{\to}& H_0(X,*)
&\to&
0
\\
&& & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{H_0(\epsilon)}} &
\\
&& && \mathbb{Z}
}
\,.$

Using this we finally compute

$\begin{aligned}
\tilde H_0(X)
& \coloneqq
ker H_0(\epsilon)
\\
& \simeq coker( H_0(x) )
\\
& \simeq H_0(X,*)
\end{aligned}
\,.$

Moreover, for “good” inclusions $A \hookrightarrow X$ of topological space, the reduced singular homology of the quotient $X/A$ is isomorphic to the $A$-relative singular homology of $X$.

See at *Relative homology - Relation to reduced homology of quotient topological spaces*.

Let $\{* \to X_i\}_i$ be a set of pointed topological spaces. Write $\vee_i X_i \in Top$ for their wedge sum and write $\iota_i \colon X_i \to \vee_i X_i$ for the canonical inclusion functions.

For each $n \in \mathbb{N}$ the homomorphism

$(\tilde H_n(\iota_i))_i \colon \oplus_i \tilde H_n(X_i) \to \tilde H_n(\vee_i X_i)$

is an isomorphism.

For instance (Hatcher, cor. 2.25).

This follows with *this proposition* at *relative homology*.

For $X$ a topological space, write $H_n(X)$ for its singular homology with integer coefficients.

If $X$ is a contractible topological space, then for all $n \in \mathbb{N}$

$\tilde H_n(X) \simeq 0
\,.$

The reduced singular homology of the 0-sphere $S^0 \simeq {*} \coprod {*}$ is

$\tilde H_n(S^0) \simeq
\left\{
\array{
\mathbb{Z} & if \; n = 0
\\
0 & otherwise
}
\right.
\,.$

Reduced singular homology is discussed for instance around p. 119 of