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\begin{document}

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\section*{long exact sequence in homology}

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

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

[[!include homological algebra - contents]]

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToHomotopyFiberSequences}{Relation to homotopy fiber sequences}\dotfill \pageref*{RelationToHomotopyFiberSequences} \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}

Let $\mathcal{A}$ be an [[abelian category]] and write $Ch_\bullet(\mathcal{A})$ for its [[category of chain complexes]]. Under forming [[chain homology]]

\begin{displaymath}
H_0 : Ch_\bullet(\mathcal{A}) \to \mathcal{A}
\end{displaymath}
in some (any) fixed degree, a [[homotopy fiber sequence]] in $Ch_\bullet(\mathcal{A})$ is sent to a [[long exact sequence]] in $\mathcal{A}$. This is the \emph{homology long exact sequence}.

Often this is considered specifically for the case that the fiber sequence in $Ch_\bullet(\mathcal{A})$ is that induced from a [[short exact sequence]] in $\mathcal{A}$. In this case the further map (that which makes the sequence ``long'') is called the [[connecting homomorphism]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

(\ldots{})

For the moment see still at \emph{[[fiber sequence]]}, for instance the section \emph{\href{fiber%20sequence#LongSequCoh}{long exact sequence in cohomology}} there.

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToHomotopyFiberSequences}{}\subsubsection*{{Relation to homotopy fiber sequences}}\label{RelationToHomotopyFiberSequences}

We discuss the relation of homology long exact sequences to [[homotopy cofiber sequences]] of chain complexes. Technical details corresponding to the following survey are at \emph{[[mapping cone]]} in the section \emph{\href{http://ncatlab.org/nlab/show/mapping+cone#HomologyExactSequencesAndFiberSequences}{Mapping cone -- Homology exact sequences and fiber sequences}}.

$\,$

While the notion of a [[short exact sequence]] of [[chain complexes]] is very useful for computations, it does not have invariant meaning if one considers chain complexes as objects in (abelian) [[homotopy theory]], where one takes into account [[chain homotopies]] between [[chain maps]] and takes [[equivalence]] of chain complexes not to be given by [[isomorphism]], but by [[quasi-isomorphism]].

For if a [[chain map]] $A_\bullet \to B_\bullet$ is the degreewise [[kernel]] of a chain map $B_\bullet \to C_\bullet$, then if $\hat A_\bullet \stackrel{\simeq}{\to} A_\bullet$ is a [[quasi-isomorphism]] (for instance a [[projective resolution]] of $A_\bullet$) then of course the composite chain map $\hat A_\bullet \to B_\bullet$ is in general far from being the degreewise kernel of $C_\bullet$. Hence the notion of degreewise kernels of chain maps and hence that of short exact sequences is not meaningful in the homotopy theory of chain complexes in $\mathcal{A}$ (for instance: not in the [[derived category]] of $\mathcal{A}$).

That short exact sequences of chain complexes nevertheless play an important role in [[homological algebra]] is due to what might be called a ``technical coincidence'':

\begin{prop}
\label{}\hypertarget{}{}
If $A_\bullet \to B_\bullet \to C_\bullet$ is a [[short exact sequence]] of [[chain complexes]], then the [[commuting square]]

\begin{displaymath}
\itexarray{
     A_\bullet &\to& 0
     \\
     \downarrow && \downarrow
     \\
     B_\bullet &\to& C_\bullet
  }
\end{displaymath}
is not only a [[pullback]] square in $Ch_\bullet(\mathcal{A})$, exhibiting $A_\bullet$ as the [[fiber]] of $B_\bullet \to C_\bullet$ over $0 \in C_\bullet$, it is in fact also a \emph{[[homotopy pullback]]}.

\end{prop}
This means it is [[universal property|universal]] not just among commuting such squares, but also among such squares which commute possibly only up to a [[chain homotopy]] $\phi$:

\begin{displaymath}
\itexarray{
     Q_\bullet &\to& 0
     \\
     \downarrow &\swArrow_{\phi}& \downarrow
     \\
     B_\bullet &\to& C_\bullet
  }
\end{displaymath}
and with morphisms between such squares being maps $A_\bullet \to A'_\bullet$ correspondingly with further chain homotopies filling all diagrams in sight.

\begin{proof}
This follows from using the basic property (see at \href{exact+sequence#Definition}{exact sequence -- Definition}) that in a short exact sequence $A_\bullet \to B_\bullet \to C_\bullet$ the morphism on the right is a degreewise [[surjection]] together with a basic result in the theory of [[model categories]] or in fact that of [[categories of fibrant objects]] which is discussed in detail at \emph{[[homotopy pullback]]} and also at \emph{[[factorization lemma]]}:

by the existence of the [[projective model structure on chain complexes]], we may regard every chain complex as a [[fibrant object]] and every degreewise surjection as a [[fibration]]. By the basic theorem discussed at \href{homotopy%20pullback#ConstructionsGeneral}{Homotopy pullback -- Properties -- General} these are sufficient conditions for the ordinary pullback as above to produce a chain complex that represents the homotopy-correct [[homotopy pullback]] (which, beware, is defined up ``weak chain homology equivalence'' only, hence up to [[zig-zags]] of [[quasi-isomorphism]]).

\end{proof}
Equivalently, we have the formally dual result, proved using instead the existence of the [[injective model structure on chain complexes]]:

\begin{prop}
\label{}\hypertarget{}{}
If $A_\bullet \to B_\bullet \to C_\bullet$ is a [[short exact sequence]] of [[chain complexes]], then the [[commuting square]]

\begin{displaymath}
\itexarray{
     A_\bullet &\to& 0
     \\
     \downarrow && \downarrow
     \\
     B_\bullet &\to& C_\bullet
  }
\end{displaymath}
is not only a [[pushout]] square in $Ch_\bullet(\mathcal{A})$, exhibiting $C_\bullet$ as the [[cofiber]] of $A_\bullet \to B_\bullet$ over $0 \in C_\bullet$, it is in fact also a \emph{[[homotopy pushout]]}.

\end{prop}
But a central difference between [[fibers]]/[[cofibers]] on the one hand and [[homotopy fibers]]/[[homotopy cofibers]] on the other is that while the (co)fiber of a (co)fiber is necessarily trivial, the homotopy (co)fiber of a homotopy (co)fiber is in general far from trivial: it is instead the [[looping]] $\Omega(-)$ or [[suspension]] $\Sigma(-)$ of the codomain/domain of the original morphism: by the [[pasting law]] for homotopy pullbacks the [[pasting]] composite of successive [[homotopy cofibers]] of a given morphism $f : A_\bullet \to B_\bullet$ looks like this:

\begin{displaymath}
\itexarray{
     A_\bullet &\stackrel{f}{\to}& B_\bullet &\to& 0
     \\
     \downarrow &\swArrow_{\mathrlap{\phi}}& \downarrow &\swArrow& \downarrow
     \\
     0 &\to& cone(f) &\to& A[1]_{\bullet} &\stackrel{}{\to}& 0
     \\
     && \downarrow &\swArrow& \downarrow^{\mathrlap{f[1]}}  &\swArrow& \downarrow
     \\
     && 0 &\to& B[1] &\to& cone(f)[1]_\bullet &\to& \cdots
     \\
     && && \downarrow && \downarrow &\ddots&
     \\
     && && \vdots && && 
  }
\end{displaymath}
here

\begin{itemize}%
\item $cone(f)$ is a specific representative of the [[homotopy cofiber]] of $f$ called the \emph{[[mapping cone]]} of $f$, whose construction comes with an explicit [[chain homotopy]] $\phi$ as indicated, hence $cone(f)$ is homology-equivalence to $C_\bullet$ above, but is in general a ``bigger'' model of the homotopy cofiber;


\item $A[1]$ etc. is the [[suspension of a chain complex]] of $A$, hence the same chain complex but pushed up in degree by one.



\end{itemize}
This is discussed in detail at \emph{[[mapping cone]]}, see the section \emph{\href{mapping%20cone#InChainComplexes}{mapping cone - for chain complexes}}.

In conclusion we get from every morphim of chain complexes a long \textbf{[[homotopy cofiber sequence]]}

\begin{displaymath}
\cdots 
  \to 
     A_\bullet \stackrel{f}{\to}B_\bullet
  \stackrel{}{\to} 
  cone(f)
   \stackrel{}{\to}
  A[1]_\bullet
   \stackrel{f[1]}{\to}
  B[1]_\bullet
   \stackrel{}{\to}   
  cone(f)[1]_\bullet
   \to 
  \cdots
  \,.
\end{displaymath}
And applying the [[chain homology]] functor to this yields the long exact sequence in chain homology which is traditionally said to be associated to the short exact sequence $A_\bullet \to B_\bullet \to C_\bullet$.

In conclusion this means that it is not really the passage to homology groups which ``makes a short exact sequence become long''. It's rather that passing to homology groups is a shadow of passing to chain complexes regarded up to quasi-isomorphism, and \emph{this} is what makes every short exact sequence be realized as but a special presentation of a stage in a long [[homotopy fiber sequence]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Gysin sequence]]


\item [[Serre long exact sequence]]


\item [[long exact sequence of homotopy groups]]



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

Lecture notes include

\begin{itemize}%
\item Robert Ash, \emph{The long exact homology sequence and applications} (\href{http://www.math.uiuc.edu/~r-ash/Algebra/Supplement.pdf}{pdf})

\end{itemize}
[[!redirects long exact sequence in cohomology]] [[!redirects long exact sequences in homology]] [[!redirects long exact sequences in cohomology]]

[[!redirects homology exact sequence]] [[!redirects homology exact sequences]] [[!redirects cohomology exact sequence]] [[!redirects cohomology exact sequences]]

[[!redirects homology long exact sequence]] [[!redirects homology long exact sequences]] [[!redirects cohomology long exact sequence]] [[!redirects cohomology long exact sequences]]



\end{document}