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\begin{document}

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\section*{Serre fibration}

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

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

\textbf{Serre fibration} $\Leftarrow$ [[Hurewicz fibration]] $\Rightarrow$ [[Dold fibration]] $\Leftarrow$ [[shrinkable map]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{closure_properties}{Closure properties}\dotfill \pageref*{closure_properties} \linebreak
\noindent\hyperlink{RelationToHurewiczFibrations}{Relation to Hurewicz fibrations}\dotfill \pageref*{RelationToHurewiczFibrations} \linebreak
\noindent\hyperlink{long_exact_sequences_of_homotopy_groups}{Long exact sequences of homotopy groups}\dotfill \pageref*{long_exact_sequences_of_homotopy_groups} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

\begin{defn}
\label{TopologicalGeneratingAcyclicCofibrations}\hypertarget{TopologicalGeneratingAcyclicCofibrations}{}
Write

\begin{displaymath}
J_{Top}
    \coloneqq
  \left\{
      D^n \stackrel{(id,\delta_0)}{\hookrightarrow} D^n \times I
  \right\}_{n \in \mathbb{N}}
  \;\subset Mor(Top)
\end{displaymath}
for the [[set]] of inclusions of the topological [[n-disks]], into their [[cylinder objects]] (the [[product topological space]] with the [[topological interval]]), along (for definiteness) the left endpoint inclusion.

\end{defn}
\begin{defn}
\label{SerreFibration}\hypertarget{SerreFibration}{}
A \textbf{Serre fibration} is a $J_{Top}$-[[injective morphism]], def. \ref{TopologicalGeneratingAcyclicCofibrations}, hence a [[continuous function]] $f \colon X \longrightarrow Y$ that has the [[right lifting property]] with respect to all inclusions of the form $(id,0) \colon D^n \hookrightarrow D^n \times I$ that include the standard topological [[n-disk]] into its standard [[cylinder object]].

\end{defn}
I.e. $f$ is a Serre fibration if for every [[commuting square]] of [[continuous functions]] of the form

\begin{displaymath}
\itexarray{
    D^n  &\longrightarrow& X
    \\
    {}^{\mathllap{(id,0)}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n \times I &\longrightarrow& Y
  }
\end{displaymath}
then there exists a continuous function $h \colon D^n \times I \to X$ such as to make a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    D^n  &\longrightarrow& X
    \\
    {}^{\mathllap{(id,0)}}\downarrow 
    &{}^h\nearrow& 
    \downarrow^{\mathrlap{f}}
    \\
    D^n \times I &\longrightarrow& Y
  }
\end{displaymath}
\begin{remark}
\label{}\hypertarget{}{}
The class of Serre fibrations serves as the class of abstract [[fibrations]] in the [[classical model structure on topological spaces]] (whence ``Serre-Quillen model structure'').

\end{remark}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{closure_properties}{}\subsubsection*{{Closure properties}}\label{closure_properties}

\begin{prop}
\label{SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes}\hypertarget{SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes}{}
A Serre fibration has the right lifting property against all [[retracts]] of $J_{Top}$-[[relative cell complexes]] (def. \ref{TopologicalGeneratingAcyclicCofibrations}).

\end{prop}
\begin{proof}
By general closure properties of [[projective and injective morphisms]], see there \href{injective+or+projective+morphism#ClosurePropertiesOfInjectiveAndProjectiveMorphisms}{this proposition} for details.

\end{proof}
\hypertarget{RelationToHurewiczFibrations}{}\subsubsection*{{Relation to Hurewicz fibrations}}\label{RelationToHurewiczFibrations}

\begin{remark}
\label{}\hypertarget{}{}
The condition in def. \ref{SerreFibration} is part of the condition on a [[Hurewicz fibration]], hence every [[Hurewicz fibration]] is in particular a Serre fibration.

\end{remark}
The converse is false:

\begin{example}
\label{SerreFibrationsWhichAreNotHurewiczFibrations}\hypertarget{SerreFibrationsWhichAreNotHurewiczFibrations}{}
\textbf{(Serre fibrations which are not Hurewicz fibrations)}

An example of a generalized [[covering space]] which is a Serre fibration but not a Hurewicz fibration is given by Jeremy Brazas \href{https://mathoverflow.net/a/241597/381}{here}.

\end{example}
But under some regularity condition it does becomes true:

\begin{prop}
\label{SerreFibrationsBetweenCWComplexesAreHurewiczFibrations}\hypertarget{SerreFibrationsBetweenCWComplexesAreHurewiczFibrations}{}
\textbf{(Serre fibrations of [[CW-complexes]] are [[Hurewicz fibrations]])}

In the [[convenient category of topological spaces|convenient category]] of [[compactly generated topological spaces|compactly generated]] [[weakly Hausdorff topological spaces]]) a Serre fibration in which the total space and base space are both [[CW complexes]] is a [[Hurewicz fibration]].

(No relationship between the covering map and the CW structures is required.)

\end{prop}
This is due to (\hyperlink{SteinbergerWest84}{Steinberger-West 84}) with the corrected proof due to (\hyperlink{Cauty92}{Cauty 92}) (pointers via [[Peter May]] \href{https://mathoverflow.net/a/241611/381}{here}). Theorem-page at [[a Serre fibration between CW-complexes is a Hurewicz fibration]].

\hypertarget{long_exact_sequences_of_homotopy_groups}{}\subsubsection*{{Long exact sequences of homotopy groups}}\label{long_exact_sequences_of_homotopy_groups}

Since Serre fibrations are the abstract fibrations in the Serre-[[classical model structure on topological spaces]], the following statement follows from general [[model category]] theory. But it may also be seen by direct inspection, as follows.

\begin{lemma}
\label{CylinderOverCWComplexIsJTopRelativeCellComplex}\hypertarget{CylinderOverCWComplexIsJTopRelativeCellComplex}{}
For $X$ a finite [[CW-complex]], then its inclusion $X \overset{(id, \delta_0)}{\longrightarrow} X \times I$ into its standard [[cylinder object|cylinder]] is a $J_{Top}$-[[relative cell complex]] (def. \ref{TopologicalGeneratingAcyclicCofibrations}).

\end{lemma}
\begin{proof}
First erect a cylinder over all 0-cells

\begin{displaymath}
\itexarray{
     \underset{x \in X_0}{\coprod} D^0 &\longrightarrow& X
     \\
     \downarrow &(po)& \downarrow
     \\
     \underset{x\in X_0}{\coprod} D^1 &\longrightarrow& Y_1
  }
  \,.
\end{displaymath}
Assume then that the cylinder over all $n$-cells of $X$ has been erected using attachment from $J_{Top}$. Then the union of any $(n+1)$-cell $\sigma$ of $X$ with the cylinder over its boundary is homeomorphic to $D^{n+1}$ and is like the cylinder over the cell ``with end and interior removed''. Hence via attaching along $D^{n+1} \to D^{n+1}\times I$ the cylinder over $\sigma$ is erected.

\end{proof}
\begin{prop}
\label{SerreFibrationGivesExactSequenceOfHomotopyGroups}\hypertarget{SerreFibrationGivesExactSequenceOfHomotopyGroups}{}
Let $f\colon X \longrightarrow Y$ be a [[Serre fibration]], def. \ref{SerreFibration}, let $y \colon \ast \to Y$ be any point and write

\begin{displaymath}
F_y \overset{\iota}{\hookrightarrow} X \overset{f}{\longrightarrow} Y
\end{displaymath}
for the [[fiber]] inclusion over that point. Then for every choice $x \colon \ast \to X$ of lift of the point $y$ through $f$, the induced sequence of [[homotopy groups]]

\begin{displaymath}
\pi_{\bullet}(F_y, x)
    \overset{\iota_\ast}{\longrightarrow}
  \pi_\bullet(X, x)
    \overset{f_\ast}{\longrightarrow}
  \pi_\bullet(Y)
\end{displaymath}
is [[exact sequence|exact]], in that the [[kernel]] of $f_\ast$ is canonically identified with the [[image]] of $\iota_\ast$:

\begin{displaymath}
ker(f_\ast) \simeq im(\iota_\ast)
  \,.
\end{displaymath}
\end{prop}
\begin{proof}
It is clear that the image of $\iota_\ast$ is in the kernel of $f_\ast$ (every sphere in $F_y\hookrightarrow X$ becomes constant on $y$, hence contractible, when sent forward to $Y$).

For the converse, let $[\alpha]\in \pi_{\bullet}(X,x)$ be represented by some $\alpha \colon S^{n-1} \to X$. Assume that $[\alpha]$ is in the kernel of $f_\ast$. This means equivalently that $\alpha$ fits into a [[commuting diagram]] of the form

\begin{displaymath}
\itexarray{
    S^{n-1} &\overset{\alpha}{\longrightarrow}& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n &\overset{\kappa}{\longrightarrow}& Y
  }
  \,,
\end{displaymath}
where $\kappa$ is the contracting homotopy witnessing that $f_\ast[\alpha] = 0$.

Now since $x$ is a lift of $y$, there exists a [[left homotopy]]

\begin{displaymath}
\eta  \;\colon\; \kappa \Rightarrow const_y
\end{displaymath}
as follows:

\begin{displaymath}
\itexarray{
    && S^{n-1} &\overset{\alpha}{\longrightarrow}& X
    \\
    && {}^{\mathllap{\iota_n}}\downarrow && \downarrow^{\mathrlap{f}}
    \\
    && D^n &\overset{\kappa}{\longrightarrow}& Y
    \\
    && \downarrow^{\mathrlap{(id,1)}} && \downarrow^{\mathrlap{id}}
    \\
    D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y
    \\
    \downarrow && &&  \downarrow
    \\
    \ast && \overset{y}{\longrightarrow} && Y
  }
\end{displaymath}
(for instance: regard $D^n$ as embedded in $\mathbb{R}^n$ such that $0 \in \mathbb{R}^n$ is identified with the basepoint on the boundary of $D^n$ and set $\eta(\vec v,t) \coloneqq \kappa(t \vec v)$).

The [[pasting]] of the top two squares that have appeared this way is equivalent to the following commuting square

\begin{displaymath}
\itexarray{
    S^{n-1} &\longrightarrow& &\overset{\alpha}{\longrightarrow}& X
    \\
    {}^{\mathllap{(id,1)}}\downarrow 
    &&
    && \downarrow^{\mathrlap{f}}
    \\
    S^{n-1} \times I 
      &\overset{(\iota_n, id)}{\longrightarrow}& 
    D^n \times I 
      &\overset{\eta}{\longrightarrow}& 
    Y
  }
  \,.
\end{displaymath}
Because $f$ is a [[Serre fibration]] and by lemma \ref{CylinderOverCWComplexIsJTopRelativeCellComplex} and prop. \ref{SerreFibrationHasRightLiftingAgainstJTopRelativeCellComplexes}, this has a [[lift]]

\begin{displaymath}
\tilde \eta \;\colon\; S^{n-1} \times I \longrightarrow X
  \,.
\end{displaymath}
Notice that $\tilde \eta$ is a basepoint preserving [[left homotopy]] from $\alpha = \tilde \eta|_1$ to some $\alpha' \coloneqq \tilde \eta|_0$. Being homotopic, they represent the same element of $\pi_{n-1}(X,x)$:

\begin{displaymath}
[\alpha'] = [\alpha]
  \,.
\end{displaymath}
But the new representative $\alpha'$ has the special property that its image in $Y$ is not just trivializable, but trivialized: combining $\tilde \eta$ with the previous diagram shows that it sits in the following commuting diagram

\begin{displaymath}
\itexarray{
    \alpha' \colon & S^{n-1} &\overset{(id,0)}{\longrightarrow}& S^{n-1}\times I &\overset{\tilde \eta}{\longrightarrow}& X
    \\
    & \downarrow^{\iota_n} && \downarrow^{\mathrlap{(\iota_n,id)}} && \downarrow^{\mathrlap{f}}
    \\
    & D^n &\overset{(id,0)}{\longrightarrow}& D^n \times I &\overset{\eta}{\longrightarrow}& Y
    \\
    & \downarrow && &&  \downarrow
    \\
    & \ast && \overset{y}{\longrightarrow} && Y
  }
  \,.
\end{displaymath}
The commutativity of the outer square says that $f_\ast \alpha'$ is constant, hence that $\alpha'$ is entirely contained in the fiber $F_y$. Said more abstractly, the [[universal property]] of [[fibers]] gives that $\alpha'$ factors through $F_y\overset{\iota}{\hookrightarrow} X$, hence that $[\alpha'] = [\alpha]$ is in the image of $\iota_\ast$.

\end{proof}
(\ldots{})

[[long exact sequence of homotopy groups]]

(\ldots{})

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

Every locally trivial topological [[fiber bundle]] is a Serre fibration.

In particular,

\begin{example}
\label{CoveringSpaceIsSerreFibration}\hypertarget{CoveringSpaceIsSerreFibration}{}
\textbf{([[covering space]] projection is Serre fibration)}

Every [[covering space]] projection is a Serre fibration, in fact a [[Hurewicz fibration]] (by \href{covering+space#HomotopyLiftingPropertyOfCoveringSpaces}{this prop.}).

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

\begin{itemize}%
\item [[Serre spectral sequence]]

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

\begin{itemize}%
\item M. Steinberger and J. West, \emph{Covering homotopy properties of maps between CW complexes or ANRs}, Proc. Amer. Math. Soc. 92 (1984), 573-577.


\item R. Cauty, \emph{Sur les ouverts des CW-complexes et les fibr\'e{}s de Serre}, Colloquy Math. 63 (1992), 1--7



\end{itemize}
[[!redirects Serre fibrations]]



\end{document}