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\begin{document}

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\section*{reduced homology}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{reduced_singular_homology}{Reduced singular homology}\dotfill \pageref*{reduced_singular_homology} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToOrdinaryHomology}{Relation to ordinary homology}\dotfill \pageref*{RelationToOrdinaryHomology} \linebreak
\noindent\hyperlink{RelationToRelativeHomology}{Relation to relative homology}\dotfill \pageref*{RelationToRelativeHomology} \linebreak
\noindent\hyperlink{RelationToWedgeSums}{Relation to wedge sums}\dotfill \pageref*{RelationToWedgeSums} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{for_singular_homology}{For singular homology}\dotfill \pageref*{for_singular_homology} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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 \emph{reduced homology}.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{reduced_singular_homology}{}\subsubsection*{{Reduced singular homology}}\label{reduced_singular_homology}

We discuss the reduced version of [[singular homology]].

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

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{augmentation map} is the homomorphism of abelian groups

\begin{displaymath}
\epsilon \colon C_0(X) \to \mathbb{Z}
\end{displaymath}
which adds up all the coefficients of all 0-chains:

\begin{displaymath}
\epsilon \colon \colon \sum_{i} n_i \sigma_i \mapsto \sum_i n_i
  \,.
\end{displaymath}
Since the boundary of a 1-chain is in the [[kernel]] of this map, it constitutes a [[chain map]]

\begin{displaymath}
\epsilon \colon C_\bullet(X) \to \mathbb{Z}
  \,,
\end{displaymath}
where now $\mathbb{Z}$ is regarded as a chain complex concentrated in degree 0.

\end{defn}
\begin{defn}
\label{ReducedSingularChainComplex}\hypertarget{ReducedSingularChainComplex}{}
The \textbf{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]]

\begin{displaymath}
0 \to \tilde C_\bullet(X) \to C_\bullet(X) \stackrel{\epsilon}{\to} \mathbb{Z} \to 0
  \,.
\end{displaymath}
The \textbf{reduced singular homology} $\tilde H_\bullet(X)$ of $X$ is the [[chain homology]] of the reduced singular chain complex

\begin{displaymath}
\tilde H_\bullet(X) \coloneqq H_\bullet(\tilde C_\bullet(X))
  \,.
\end{displaymath}
\end{defn}
Equivalently:

\begin{defn}
\label{}\hypertarget{}{}
The \textbf{reduced singular homology} of $X$, denoted $\tilde H_\bullet(X)$, is the [[chain homology]] of the [[augmentation|augmented]] chain complex

\begin{displaymath}
\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
  \,.
\end{displaymath}
\end{defn}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToOrdinaryHomology}{}\subsubsection*{{Relation to ordinary homology}}\label{RelationToOrdinaryHomology}

Let $X$ be a [[topological space]], $H_\bullet(X)$ its [[singular homology]] and $\tilde H_\bullet(X)$ its reduced singular homology, def. \ref{ReducedSingularChainComplex}.

\begin{prop}
\label{RelationBetweenReducedSingularAndSingular}\hypertarget{RelationBetweenReducedSingularAndSingular}{}
For $n \in \mathbb{N}$ there is an [[isomorphism]]

\begin{displaymath}
H_n(X)
  \simeq
  \left\{
    \itexarray{
      \tilde H_n(X) & for \; n \geq 1
      \\
      \tilde H_0(X) \oplus \mathbb{Z} & for\; n = 0
    }
  \right.
\end{displaymath}
\end{prop}
\begin{proof}
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

\begin{displaymath}
\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 
  \,.
\end{displaymath}
Here [[exact sequence|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]]

\begin{displaymath}
0 \to \tilde H_0(X) \to H_0(X) \to \mathbb{Z} \to 0
\end{displaymath}
is [[split exact sequence|split]] (as discussed there) and hence $H_0(X) \simeq \tilde H_0(X) \oplus \mathbb{Z}$.

\end{proof}
\begin{prop}
\label{}\hypertarget{}{}
For $X = *$ the [[point]], the morphism

\begin{displaymath}
H_0(\epsilon) \colon H_0(X) \to \mathbb{Z}
\end{displaymath}
is an [[isomorphism]]. Accordingly the reduced homology of the point vanishes in every degree:

\begin{displaymath}
\tilde H_\bullet(*) \simeq 0
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
By the discussion at \emph{\href{singular%20homology#RelationToHomotopyGroups}{Singular homology -- Relation to homotopy groups}} we have that

\begin{displaymath}
H_n(*) \simeq
  \left\{
    \itexarray{
       \mathbb{Z} & for \; n = 0
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
Moreover, it is clear that $\epsilon \colon C_0(*) \to \mathbb{Z}$ is the [[identity]] map.

\end{proof}
\hypertarget{RelationToRelativeHomology}{}\subsubsection*{{Relation to relative homology}}\label{RelationToRelativeHomology}

\begin{prop}
\label{}\hypertarget{}{}
For $X$ an [[inhabited set|inhabited]] [[topological space]], its reduced singular homology, def. \ref{ReducedSingularChainComplex}, coincides with its [[relative singular homology]] relative to any base point $x \colon * \to X$:

\begin{displaymath}
\tilde H_\bullet(X)
  \simeq
   H_\bullet(X,*)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
Consider the sequence of [[topological subspace]] inclusions

\begin{displaymath}
\emptyset \hookrightarrow * \stackrel{x}{\hookrightarrow} X
  \,.
\end{displaymath}
By the discussion at \emph{\href{relative%20homology#LongExactSequences}{Relative homology - long exact sequences}} this induces a [[long exact sequence]] of the form

\begin{displaymath}
\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
  \,.
\end{displaymath}
Here in positive degrees we have $H_n(*) \simeq 0$ and therefore [[exact sequence|exactness]] gives [[isomorphisms]]

\begin{displaymath}
H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}
\end{displaymath}
and hence with prop. \ref{RelationBetweenReducedSingularAndSingular} isomorphisms

\begin{displaymath}
\tilde H_n(X) \stackrel{\simeq}{\to} H_n(X,*)\;\; \forall_{n \geq 1}  
  \,.
\end{displaymath}
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]]

\begin{displaymath}
\itexarray{
     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}
  }
  \,,
\end{displaymath}
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

\begin{displaymath}
\itexarray{
    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}
  }
  \,.
\end{displaymath}
Using this we finally compute

\begin{displaymath}
\begin{aligned}
    \tilde H_0(X)
    & \coloneqq 
    ker H_0(\epsilon)
    \\
    & \simeq coker( H_0(x) )
    \\
    & \simeq H_0(X,*)
  \end{aligned}
  \,.
\end{displaymath}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
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 \emph{\href{relative+homology#RelationToQuotientTopologicalSpaces}{Relative homology - Relation to reduced homology of quotient topological spaces}}.

\end{remark}
\hypertarget{RelationToWedgeSums}{}\subsubsection*{{Relation to wedge sums}}\label{RelationToWedgeSums}

Let $\{* \to X_i\}_i$ be a set of [[pointed object|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.

\begin{prop}
\label{}\hypertarget{}{}
For each $n \in \mathbb{N}$ the homomorphism

\begin{displaymath}
(\tilde H_n(\iota_i))_i \colon \oplus_i \tilde H_n(X_i) \to \tilde H_n(\vee_i X_i)
\end{displaymath}
is an [[isomorphism]].

\end{prop}
For instance (\hyperlink{Hatcher}{Hatcher, cor. 2.25}).

\begin{proof}
This follows with \emph{\href{relative+homology#HomologyOfQuotientSpace}{this proposition}} at \emph{[[relative homology]]}.

\end{proof}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\hypertarget{for_singular_homology}{}\subsubsection*{{For singular homology}}\label{for_singular_homology}

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

\begin{example}
\label{ReducedHomologyOfPoints}\hypertarget{ReducedHomologyOfPoints}{}
If $X$ is a [[contractible topological space]], then for all $n \in \mathbb{N}$

\begin{displaymath}
\tilde H_n(X) \simeq 0
  \,.
\end{displaymath}
\end{example}
\begin{example}
\label{ReducedHomologyOf0Sphere}\hypertarget{ReducedHomologyOf0Sphere}{}
The reduced singular homology of the 0-[[sphere]] $S^0 \simeq {*} \coprod {*}$ is

\begin{displaymath}
\tilde H_n(S^0) \simeq 
  \left\{
    \itexarray{
       \mathbb{Z} & if \; n = 0
       \\
       0 & otherwise
    }
  \right.
  \,.
\end{displaymath}
\end{example}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[reduced cohomology]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

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

\begin{itemize}%
\item [[Alan Hatcher]], \emph{\href{http://www.math.cornell.edu/~hatcher/AT/ATpage.html}{Algebraic Topology}}

\end{itemize}
[[!redirects reduced singular homology]]



\end{document}