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\begin{document}

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\section*{spherical fibration}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - 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{in_components}{In components}\dotfill \pageref*{in_components} \linebreak
\noindent\hyperlink{classifying_space}{Classifying space}\dotfill \pageref*{classifying_space} \linebreak
\noindent\hyperlink{as_module_bundles}{As $(\infty,1)$-module bundles}\dotfill \pageref*{as_module_bundles} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{adams_conjecture}{Adams conjecture}\dotfill \pageref*{adams_conjecture} \linebreak
\noindent\hyperlink{gysin_sequence}{Gysin sequence}\dotfill \pageref*{gysin_sequence} \linebreak
\noindent\hyperlink{rational_homotopy_type}{Rational homotopy type}\dotfill \pageref*{rational_homotopy_type} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{in_rational_homotopy_theory}{In rational homotopy theory}\dotfill \pageref*{in_rational_homotopy_theory} \linebreak


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

A \emph{spherical fibration} is a [[fiber bundle]] of [[spheres]] of some [[dimension]] (a [[sphere fiber bundle]]). Typically this is considered in [[homotopy theory]] where one considers [[fibrations]] whose [[fibers]] have the [[homotopy type]] of [[spheres]]; and this in turn is often considered in [[stable homotopy theory]] after [[stabilization]] (hence up to tensoring with trivial spherical fibrations) which makes spherical fibrations models for [[(∞,1)-module bundles]] for the [[sphere spectrum]] regarded as an [[E-∞ ring]].

Every [[real vector bundle]] becomes a spherical fibration in the sense of [[homotopy theory]] upon removing its [[zero section]] and this construction induces a map from vector bundles and in fact from [[topological K-theory]] to spherical fibrations, called the \emph{[[J-homomorphism]]}.

This is closely related to the [[Thom space]]/[[Thom spectrum]] construction for vector bundles.

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

\hypertarget{in_components}{}\subsubsection*{{In components}}\label{in_components}

For $X$ (the [[homotopy type]] of) a [[topological space]], a \emph{spherical fibration over it} is a [[fibration]] $E \to X$ such that each [[fiber]] has the [[homotopy type]] of a [[sphere]].

Given two spherical fibrations $E_1, E_2 \to X$, there is their fiberwise [[smash product]] $E_1 \wedge_X E_2 \to X$.

For $n \in \mathbb{N}$, write $\epsilon^n \colon X \times S^n \to X$ for the trivial sphere bundle of fiber dimension $n$. Two spherical fibrations $E_1, E_2 \to X$ are \emph{stably fiberwise equivalent} if there exists $n_1, n_2 \in \mathbb{N}$ such that there is a map

\begin{displaymath}
E_1 \wedge_X \epsilon^{n_1}
  \longrightarrow
  E_2 \wedge_X \epsilon^{n_2}
\end{displaymath}
over $X$ which is fiberwise a [[weak homotopy equivalence]].

One consider the [[abelian group]]

\begin{displaymath}
Sph(X) \in Ab
\end{displaymath}
to be the [[Grothendieck group]] of stable fiberwise equivalence classes of spherical fibrations, under fiberwise smash product.

\hypertarget{classifying_space}{}\subsubsection*{{Classifying space}}\label{classifying_space}

There is an associative [[H-space]], $G_n$, of homotopy equivalences of the $(n-1)$-sphere with composition. Then $B G_n$ acts as the [[classifying space]] for spherical fibrations with spherical fibre $S^{n-1}$ (\hyperlink{Stasheff63}{Stasheff 63}).

There is an inclusion of the [[orthogonal group]] $O(n)$ into $G_n$.

Suspension gives a map $G_n \to G_{n+1}$ whose limit is denoted $G$. Then $B G$ classifies stable spherical fibrations.

\hypertarget{as_module_bundles}{}\subsubsection*{{As $(\infty,1)$-module bundles}}\label{as_module_bundles}

(\ldots{})

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

\hypertarget{adams_conjecture}{}\subsubsection*{{Adams conjecture}}\label{adams_conjecture}

The [[Adams conjecture]] (a theorem) characterizes certain spherical fibrations in the image of the [[J-homomorphism]] as trivial.

\hypertarget{gysin_sequence}{}\subsubsection*{{Gysin sequence}}\label{gysin_sequence}

The [[long exact sequence in cohomology]] induced by a spherical fibration is called a \emph{[[Gysin sequence]]}.

\hypertarget{rational_homotopy_type}{}\subsubsection*{{Rational homotopy type}}\label{rational_homotopy_type}

See \emph{[[Sullivan model of a spherical fibration]]}.

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

\begin{itemize}%
\item [[sphere spectrum]]


\item [[Thom spectrum]]


\item [[twisted cohomotopy]]



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

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

An original reference is

\begin{itemize}%
\item [[Albrecht Dold]], [[Richard Lashof]], \emph{Principal quasifibrations and fibre homotopy equivalence of bundles}, 1958 (\href{http://www.maths.ed.ac.uk/~aar/papers/doldlashof.pdf}{pdf})

\end{itemize}
Treatment of the classifying space for spherical fibrations is in

\begin{itemize}%
\item [[James Stasheff]], \emph{A classification theorem for fibre spaces}, Topology Volume 2, Issue 3, October 1963, Pages 239-246.

\end{itemize}
Reviews include

\begin{itemize}%
\item [[Raoul Bott]], [[Loring Tu]], Chapter 11 of \emph{[[Differential Forms in Algebraic Topology]]}, Graduate Texts in Mathematics 82, Springer 1982 (\href{https://doi.org/10.1007/BFb0063500}{doi:10.1007/BFb0063500})


\item Howard Marcum, Duane Randall, \emph{The homotopy Thom class of a spherical fibration}, Proceedings of the AMS, volume 80, number 2 (\href{http://www.maths.ed.ac.uk/~aar/papers/marcum1.pdf}{pdf})


\item Per Holm, Jon Reed, section 7 of \emph{Structure theory of manifolds}, Seminar notes 1971\href{http://www.mn.uio.no/math/tjenester/kunnskap/kompendier/seminar-holm.pdf}{pdf}


\item Oliver Straser, Nena R\"o{}ttgens, \emph{Spivak normal fibrations} (\href{http://www.map.mpim-bonn.mpg.de/images/b/be/Regensburg2012Talk5.pdf}{pdf})


\item S. Husseini, \emph{Spherical fibrations} (\href{http://www.maths.ed.ac.uk/~aar/papers/husseini2.pdf}{pdf})



\end{itemize}
\hypertarget{in_rational_homotopy_theory}{}\subsubsection*{{In rational homotopy theory}}\label{in_rational_homotopy_theory}

Discussion in [[rational homotopy theory]] (for more see at \emph{[[Sullivan model of a spherical fibration]]}):

\begin{itemize}%
\item [[Yves Félix]], [[Steve Halperin]] and J.C. Thomas, p. 202 of \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 spherical fibrations]]

[[!redirects sphere bundle]] [[!redirects sphere bundles]]



\end{document}