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\begin{document}

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\section*{Sullivan model of a spherical fibration}

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

\hypertarget{rational_homotopy_theory}{}\paragraph*{{Rational homotopy theory}}\label{rational_homotopy_theory}

[[!include differential graded objects - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RationalEulerAndPontryaginClasses}{Rational Euler- and Pontryagin-class}\dotfill \pageref*{RationalEulerAndPontryaginClasses} \linebreak
\noindent\hyperlink{relation_to_rational_mapping_space_of_spheres}{Relation to rational mapping space of spheres}\dotfill \pageref*{relation_to_rational_mapping_space_of_spheres} \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}

A [[minimal model]] for a [[spherical fibration]] in [[rational homotopy theory]].

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

\hypertarget{RationalEulerAndPontryaginClasses}{}\subsubsection*{{Rational Euler- and Pontryagin-class}}\label{RationalEulerAndPontryaginClasses}

\begin{prop}
\label{SullivanModelForSphericalFibration}\hypertarget{SullivanModelForSphericalFibration}{}
Let $n \in \mathbb{N}$ be a [[natural number]], $n \geq 1$, let

\begin{equation}
\itexarray{
    S^n &\longrightarrow& E
    \\
    && \big\downarrow
    \\
    && X
  }
\label{SphericalFibration}\end{equation}
be a [[spherical fibration]] of [[topological spaces]] such that $X$ admits a [[Sullivan model]] $A_X \in dgcAlg$. Then a [[Sullivan model]] $A_E$ for the total space $E$ is of the following form:

\textbf{$n$ odd}

If $n = 2k+1$ is an [[odd number]], then

\begin{equation}
A_E 
  \;=\;
  A_X 
    \otimes 
  \mathbb{Q}\big[
    \omega_{2k+1}
  \big]
  /
  \big(
    d \omega_{2k+1} = c_{2k+2}
  \big)
\label{SullivanModelForOddDimensionalSphericalFibration}\end{equation}
for some

\begin{equation}
c_{2k+2} \in A_X
\label{RationalEulerClass}\end{equation}
being the rational [[Euler class]] of the [[spherical fibration]].

In particular, if $E = S(V)$ is the [[unit sphere bundle]] of a [[real vector bundle]] $V \to X$, then

\begin{displaymath}
[c_{2k}] = \chi
\end{displaymath}
is the [[Euler class]] of that vector bundle and $\omega_{2k+1}$ is a [[cochain]] on the [[unit sphere bundle]] $S(E)$ which on the [[fundamental class]] of any [[n-sphere|(2k+1)-sphere]] [[fiber]] evaluates to minus unity:

\begin{equation}
\left\langle
    \omega_{2k+1},
    \left[
      S^{2k+1}
    \right]
  \right\rangle
  \;=\;
  -1
  \,.
\label{FiberIntegrationOfOddGenerator}\end{equation}
$\backslash$linebreak

\textbf{$n$ even}

If $n = 2k$ is an [[even number]], then [[generalized the|the]] [[Sullivan model]] $A_E$ for a rank-$2k$ [[spherical fibration]] over some $X$ with [[Sullivan model]] $A_X$ is

\begin{equation}
A_E
  \;=\;
  A_X 
    \otimes 
  \mathbb{Q}
  \Big[
    \omega_{2k} , \omega_{4k-1}
  \Big]
  /
  \left(
    \itexarray{
      d 
      \,
      \omega_{2k}
      &=& 
      0
      \\
      d \omega_{4k-1} 
      & =& 
      -
      \omega_{2k} \wedge \omega_{2k}
      + 
      c_{4k} 
    }
  \right)
\label{FibS2kModel}\end{equation}
where

\begin{enumerate}%
\item the new generator $\omega_{2k}$ evaluates to unity on the [[fundamental classes]] of the [[n-sphere|2k-sphere]] [[fibers]] $S^{2k} \simeq E_x \hookrightarrow E$ over each point $x \in X$:

\begin{displaymath}
\big\langle  \omega_{2k}, [S^{2k}] \big\rangle
  \;=\;
  1
\end{displaymath}

\item $c_{4k} \in A_X$ is some element in the base algebra, which by \eqref{FibS2kModel} is closed and represents the rational [[cohomology class]] of the [[cup product|cup]] square of the class of $\omega_{2k}$:

\begin{displaymath}
\big[
    c_{4k}
  \big]
  \;=\;
  \big[ 
    \omega_{2k} 
  \big]^2
  \;\in\;
  H^{4k}
  \big( 
   X, \mathbb{Q}
  \big)
\end{displaymath}
and this class classifies the spherical fibration, rationally.



\end{enumerate}
Moreover, if the [[spherical fibration]] $E \to X$ happens to be the [[unit sphere bundle]] $E = S(V)$ of a [[real vector bundle]] $V \to X$, then

\begin{enumerate}%
\item the class of $\omega_{2k}$ is $1/2$ the rationalized [[Euler class]] $\chi(\widehat V)$ of the corresponding (\ldots{}) rank [[reduction of the structure group|reduction]] $\widehat V$ of $V$:

\begin{displaymath}
\big[ \omega_{2k} \big]
  \;=\;
  \tfrac{1}{2}\chi\big( \widehat V \big)
  \;\in\;
  H^{2k}\big( X, \mathbb{Q} \big)
\end{displaymath}

\item the class of $c_{4k}$ is $1/4$th the rationalized $k$th [[Pontryagin class]] $p_k(V)$ of $V$:

\begin{displaymath}
\big[
    c_{4k}
  \big]
  \;=\;
  \tfrac{1}{4}
  p_k(V)
  \;\in\;
  H^{4k}\big( X, \mathbb{Q}\big)
  \,.
\end{displaymath}


\end{enumerate}
\end{prop}
This may be found as \hyperlink{FelixHalperinThomas00}{Félix-Halperin-Thomas 00, 15, Example 4, p. 202}, see also \hyperlink{FelixOpreaTanre16}{Félix-Oprea-Tanré 16, Prop. 2.3}. The fiber integral \eqref{FiberIntegrationOfOddGenerator} follows by \href{Euler+class#TrivializationOfEulerFormOnUnitSphereBundle}{this Prop.}.

\begin{remark}
\label{NonMinimalityOfSullivanModels}\hypertarget{NonMinimalityOfSullivanModels}{}
Beware that the Sullivan models for spherical fibrations in Prop. \ref{SullivanModelForSphericalFibration} are not in general \emph{minimal} Sullivan models.

For example over the [[classifying space]] $B SO(8)$ of [[SO(8)]] with indecomposable [[Euler class]] generator $\chi_8$ the equation $d \omega_7 = \chi_8$ \eqref{SullivanModelForOddDimensionalSphericalFibration} for the univeral 7-sperical fibration $S^7 \sslash SO(8) \to B SO(8)$ violates the Sullivan minimality condition (which requires that the right hand side is at least a binary wedge product of generators, or equivalently that the degree of the new generator $\omega_7$ is greater than that of any previous generators).

But the Sullivan models in Prop. \ref{SullivanModelForSphericalFibration} are \emph{relative} minimal models, relative to the Sullivan model for the base.

This means in particular that the new generators of these models reflect non-[[torsion subgroup|torsion]] [[relative homotopy groups]], but not in general non-torsion absolute homotopy groups.

\end{remark}
\hypertarget{relation_to_rational_mapping_space_of_spheres}{}\subsubsection*{{Relation to rational mapping space of spheres}}\label{relation_to_rational_mapping_space_of_spheres}

By general facts (see at [[∞-action]]) a spherical fibration as in \eqref{SphericalFibration} is classified by a map to the [[classifying space]] $B Aut(S^n)$ of the [[automorphism ∞-group]] $Aut(S^n) \hookrightarrow Maps(S^n, S^n)$ inside the [[mapping space]] from $S^n$ to itself, which is those [[connected components]] corresponding to [[degree of a continuous function|degree]] $\pm 1$

\begin{displaymath}
Aut(S^n) \;=\; Maps_{\pm 1}\big( S^n, S^n\big)
  \,.
\end{displaymath}
Hence the spherical fibration is given by the [[homotopy pullback]]

\begin{equation}
\itexarray{
    S^n 
      &\longrightarrow& 
    E
      &\longrightarrow& 
    S^{n}\sslash Aut(S^n)
    \\
    && 
    \big\downarrow
    &{}_{(pb)}&
    \big\downarrow
    \\
    && 
    X
    &\underset{c}{\longrightarrow}&
    B Aut(S^n)
  }
\label{SphericalFibration}\end{equation}
of the universal spherical fibration along a classifying map $c$.

The rational homotopy type of these [[connected components]] of the [[mapping space]] are given by [[Sullivan models of mapping spaces]]:

\begin{prop}
\label{RationalTypeOfMappingSpaceSnToSn}\hypertarget{RationalTypeOfMappingSpaceSnToSn}{}
Let $n \in \mathbb{N}$ be a [[natural number]] and $fcolon S^n \to S^n$ a [[continuous function]] from the [[n-sphere]] to itself. Then the [[connected component]] $Maps_f\big( S^n, S^n\big)$ of the [[mapping space]] which contains this map has the following [[rational homotopy theory|rational]] [[homotopy type]]:

\begin{equation}
Maps_f\big( S^n, S^n\big)
  \;\simeq_{\mathbb{Q}}\;
  \left\{
    \itexarray{
      S^n \times S^{n-1} &\vert& n\,\text{even}\,, deg(f) = 0
      \\
      S^{2n-1} &\vert& n \, \text{even}\,, deg(f) \neq 0
      \\
      S^n &\vert& n\, \text{odd}
    }
  \right.
\label{RationalHomotopyTypeOfMappingSpacesSnToSn}\end{equation}
where $deg(f)$ is the [[degree of a continuous function|degree]] of $f$.

\end{prop}
(\hyperlink{MollerRaussen85}{Møller-Raussen 85, Example 2.5}, \hyperlink{CohenVoronov05}{Cohen-Voronov 05, Lemma 5.3.5})

\begin{remark}
\label{}\hypertarget{}{}
Here Prop. \ref{RationalTypeOfMappingSpaceSnToSn} and Prop. \ref{SullivanModelForSphericalFibration} are two aspects of the same situation:

For $n = 2k+1$ an [[odd number]] the rational [[Euler class]] \eqref{RationalEulerClass} of the spherical fibration is the class of the rational classifying map to the shift of $S^{2k+1}$ in \eqref{RationalHomotopyTypeOfMappingSpacesSnToSn};

for $n = 2k$ an [[even number]] the rational [[Pontryagin class]] \eqref{RationalPontryaginClass} of the spherical fibration is the class of the rational classifying map to the shift of $S^{4k-1}$ in \eqref{RationalHomotopyTypeOfMappingSpacesSnToSn}.

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

[[!include Sullivan models -- examples]]

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

\begin{itemize}%
\item [[Yves Félix]], [[Steve Halperin]], J.C. Thomas, \emph{Rational Homotopy Theory}, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.


\item [[Jesper Møller]], [[Martin Raussen]], \emph{Rational Homotopy of Spaces of Maps Into Spheres and Complex Projective Spaces}, Transactions of the American Mathematical Society Vol. 292, No. 2 (Dec., 1985), pp. 721-732 (\href{https://www.jstor.org/stable/2000242}{jstor:2000242})


\item [[Ralph Cohen]], [[Alexander Voronov]], \emph{Notes on string topology} (\href{https://arxiv.org/abs/math/0503625}{arXiv:math/0503625})


\item [[Yves Félix]], John Oprea, [[Daniel Tanré]], Prop. 2.3 in \emph{Lie-model for Thom spaces of tangent bundles}, Proc. Amer. Math. Soc. 144 (2016), 1829-1840 (\href{http://www.ams.org/journals/proc/2016-144-04/S0002-9939-2015-12829-8/S0002-9939-2015-12829-8.pdf}{pdf}, \href{https://doi.org/10.1090/proc/12829}{doi:10.1090/proc/12829})



\end{itemize}
[[!redirects Sullivan mode of spherical fibrations]]

[[!redirects Sullivan models of spherical fibrations]]



\end{document}