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\begin{document}

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\section*{rational n-sphere}

\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{sullivan_models}{Sullivan models}\dotfill \pageref*{sullivan_models} \linebreak
\noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak
\noindent\hyperlink{hopf_invariant}{Hopf invariant}\dotfill \pageref*{hopf_invariant} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak


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

The \emph{rational $n$-sphere} is the [[rationalization]] of the [[n-sphere]].

As a [[topological space]] this is a [[rational topological space]] closest to the $n$-sphere, in the sense of [[rational homotopy theory]].

\hypertarget{sullivan_models}{}\subsection*{{Sullivan models}}\label{sullivan_models}

The minimal [[Sullivan model]] of a [[sphere]] $S^{2k+1}$ of odd [[dimension]] is the dg-algebra with a single generator $\omega_{2k+1}$ in degre $2k+1$ and vanishing differential

\begin{displaymath}
d \omega_{2k+1} = 0
  \,.
\end{displaymath}
The minimal [[Sullivan model]] of a sphere $S^{2k}$ of even dimension, for $k \geq 1$. is the dg-algeba with a generator $\omega_{2k}$ in degree $2k$ and another generator $\omega_{4k-1}$ in degree $4k+1$ with the differential defined by

\begin{displaymath}
d \omega_{2k}= 0
\end{displaymath}
\begin{displaymath}
d \omega_{4k+1} = \omega_{2k}\wedge \omega_{2k}
  \,.
\end{displaymath}
One may understand this form the central fact of [[rational homotopy theory]] (the proposition \href{rational+homotopy+theory#HomotopyGroupsFromSullivanGenerators}{here}):

An $n$-sphere has rational cohomology concentrated in degree $n$. Hence its minimal Sullivan model needs at least one closed generator in that degree. In the odd dimensional case one such is already sufficient, since the wedge square of that generator vanishes and hence produces no higher degree cohomology classes. But in the even degree case the wedge square $\omega_{2k}\wedge \omega_{2k}$ needs to be canceled in cohomology. That is accomplished by the second generator $\omega_{4k-1}$.

Again by \href{rational+homotopy+theory#HomotopyGroupsFromSullivanGenerators}{that proposition}, this now implies that the rational [[homotopy groups of spheres]] are concentrated, in degree $2k+1$ for the odd $(2k+1)$-dimensional spheres, and in degrees $2k$ and $4k-1$ in for the even $2k$-dimensional spheres. (These are the non-torsion [[homotopy groups of spheres]] appearing in the [[Serre finiteness theorem]].)

For instance the [[4-sphere]] has rational homotopy in degree 4 and 7. The one in degree 7 being represented by the [[quaternionic Hopf fibration]].

Hence, odd dimensional $n$-spheres are [[rational homotopy equivalence|rationally homotopy equivalent]] to [[Eilenberg-MacLane spaces]] $K(\mathbb{Z},n)$, while even-dimensional spheres are not.

\hypertarget{applications}{}\subsection*{{Applications}}\label{applications}

\hypertarget{hopf_invariant}{}\subsubsection*{{Hopf invariant}}\label{hopf_invariant}

\begin{prop}
\label{RecognitionFromSullivanModels}\hypertarget{RecognitionFromSullivanModels}{}
By standard results in [[rational homotopy theory]], every [[continuous function]]

\begin{displaymath}
S^{4k-1} \overset{\phi}{\longrightarrow} S^{2k}
\end{displaymath}
corresponds to a unique [[dgc-algebra]] [[homomorphism]]

\begin{displaymath}
CE
  \big( 
    \mathfrak{l}S^{4k-1}
  \big)
  \overset{ CE(\mathfrak{l}\phi) }{\longleftarrow}
  CE
  \big( 
    \mathfrak{l}S^{2k}
  \big)
\end{displaymath}
between [[Sullivan models]] [[rational n-sphere|of n-spheres]].

The unique free [[coefficient]] of this homomorphism $CE(\mathfrak{l}\phi)$ is the [[Hopf invariant]] $HI(\phi)$ of $\phi$:

$\backslash$begin\{center\} $\backslash$begin\{imagefromfile\} ``file\_name'': ``HopfInvariantFromSullivanModels.jpg'', ``width'': 530 $\backslash$end\{imagefromfile\} $\backslash$end\{center\}

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

\begin{itemize}%
\item [[Serre finiteness theorem]]


\item The [[non-abelian cohomology|non-abelian]] [[generalized cohomology theory]] [[representable functor|represented]] by [[rational n-spheres]] is \emph{[[rational Cohomotopy cohomology theory]]}.



\end{itemize}
[[!include Sullivan models -- examples]]

[[!redirects rational n-spheres]]

[[!redirects rational sphere]] [[!redirects rational spheres]]

[[!redirects Sullivan model of n-spheres]] [[!redirects Sullivan models of n-spheres]]

[[!redirects Sullivan model of a sphere]] [[!redirects Sullivan models of spheres]]

[[!redirects Sullivan model of an n-sphere]] [[!redirects Sullivan model of n-spheres]]



\end{document}