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\begin{document}

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\section*{Sullivan model of loop space}

[[!redirects Sullivan model of free loop space]]

\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{construction}{Construction}\dotfill \pageref*{construction} \linebreak
\noindent\hyperlink{ConstructionForFreeLoopSpace}{For the free loop space}\dotfill \pageref*{ConstructionForFreeLoopSpace} \linebreak
\noindent\hyperlink{ConstructionForBasedLoopSpace}{For the based loop space}\dotfill \pageref*{ConstructionForBasedLoopSpace} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{homotopy_quotient_by__and_cyclic_homology}{Homotopy quotient by $S^1$ and cyclic homology}\dotfill \pageref*{homotopy_quotient_by__and_cyclic_homology} \linebreak
\noindent\hyperlink{RelationToHochschildHomology}{Relation to Hochschild homology and cyclic homology}\dotfill \pageref*{RelationToHochschildHomology} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{4SphereAndTwistedDeRham}{Free loop space of the 4-sphere and twisted de Rham cohomology}\dotfill \pageref*{4SphereAndTwistedDeRham} \linebreak
\noindent\hyperlink{free_loop_space_of_the_2sphere}{Free loop space of the 2-sphere}\dotfill \pageref*{free_loop_space_of_the_2sphere} \linebreak
\noindent\hyperlink{IteratedLoopSpacesOfNSpheres}{Iterated based loop spaces of $n$-spheres}\dotfill \pageref*{IteratedLoopSpacesOfNSpheres} \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}

In [[rational homotopy theory]], given a [[rational topological space]] modeled by a [[Sullivan model]] [[dg-algebra]], there is an explicit description of the Sullivan model of its [[loop space]], ([[free loop space]] or [[based loop space]]).

This is a special case of [[Sullivan models of mapping spaces]].

\hypertarget{construction}{}\subsection*{{Construction}}\label{construction}

\hypertarget{ConstructionForFreeLoopSpace}{}\subsubsection*{{For the free loop space}}\label{ConstructionForFreeLoopSpace}

\begin{prop}
\label{SullivanModelForTheFreeLoopSpace}\hypertarget{SullivanModelForTheFreeLoopSpace}{}
\textbf{([[Sullivan model]] for [[free loop space]])}

Let $(\wedge^\bullet V, d_X)$ be a [[semifree dg-algebra]] being a [[minimal Sullivan model]] of a [[rational space|rational]] [[simply connected space]] $X$. Then a Sullivan model for the [[free loop space]] $\mathcal{L} X$ is given by

\begin{displaymath}
(\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X})
  \,,
\end{displaymath}
where

\begin{itemize}%
\item $s V$ is the [[graded vector space]] obtained from $V$ by shifting degrees down by one: $deg(s v) = deg(v)-1$;


\item $d_{\mathcal{L}X}$ is defined on elements $v$ of $V$ by

\begin{displaymath}
d_{\mathcal{L}X} v \coloneqq d v
\end{displaymath}
and on elements $s v$ of $s V$ by

\begin{displaymath}
d_{\mathcal{L}X} s v \coloneqq - s ( d v )
  \,,
\end{displaymath}
where on the right $s \colon V \to s V$ is extended as a graded [[derivation]] $s \colon \wedge^2 V \to \wedge^\bullet (V \oplus s V)$.



\end{itemize}
\end{prop}
This is due to (\hyperlink{VigueSullivan76}{Vigu\'e{}-Sullivan 76}). Review includes (\hyperlink{FelixHalperinThomas00}{Felix-Halperin-Thomas 00, p. 206}, \hyperlink{Hess06}{Hess 06, example 2.5}, \hyperlink{FelixOpreaTanre08}{F\'e{}lix-Oprea-Tanre 08, theorem 5.11}).

\begin{remark}
\label{}\hypertarget{}{}
The formula in prop. \ref{SullivanModelForTheFreeLoopSpace} is the same as that for the [[Weil algebra]] of the [[L-infinity algebra]] of wich $(\wedge^\bullet V,d_X)$ is the [[Chevalley-Eilenberg algebra]], except that here $s$ shifts \emph{down} whereas for the Weil algebra it shifts \emph{up}.

\end{remark}
\hypertarget{ConstructionForBasedLoopSpace}{}\subsubsection*{{For the based loop space}}\label{ConstructionForBasedLoopSpace}

For $X$ a [[pointed topological space]] and for the [[circle]] $S^1$ regarded as [[pointed topological space|pointed]] by any base point $\ast \to S^1$ there is the following [[homotopy fiber sequence]] which exhibits the [[based loop space]] as the [[homotopy fiber]] of the [[evaluation map]] out of the [[free loop space]]:

\begin{displaymath}
\Omega X 
  \overset{fib(ev_\ast)}{\longrightarrow}
  \mathcal{L}X 
  \overset{ ev_\ast }{\longrightarrow}
  X
  \,.
\end{displaymath}
With the dgc-algebra model from Prop. \ref{SullivanModelForTheFreeLoopSpace} for $\mathcal{L}X$ it follows that the dgc-algebra model for the based loop space is the [[homotopy cofiber]] [[dgc-algebra]] $(\wedge^\bullet( s V ), d_{\Omega X})$ in

\begin{displaymath}
(\wedge^\bullet( s V ), d_{\Omega X})
  \overset{ cofib\big( (ev_\ast)^\ast \big) }{\longleftarrow}
  (\wedge^\bullet( V \oplus s V ), d_{\mathcal{L}X})
  \overset{ (ev_\ast)^\ast }{\longleftarrow}
  (\wedge\bullet V, d_X)
  \,.
\end{displaymath}
This the inclusion on the right is manifestly a [[relative Sullivan algebra]], its [[homotopy cofiber]] is represented by the ordinary cofiber, which is readily read off:

\begin{prop}
\label{SullivanModelForBasedLoopSpace}\hypertarget{SullivanModelForBasedLoopSpace}{}
\textbf{([[Sullivan model]] for [[based loop space]])}

For $X$ a [[connected topological space|cnnected]] and [[simply connected topological space]] with [[Sullivan model]] $(\wedge\bullet V, d_X)$, the Sullivan model $(\wedge^\bullet( s V ), d_{\Omega X})$ of its [[based loop space]] $\Omega X$ is the dgc-algebra obtained from $(\wedge\bullet V, d_X)$ by shifting down all generators in degree by 1, and by keeping only the co-unary componend of the [[differential]].

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

\hypertarget{homotopy_quotient_by__and_cyclic_homology}{}\subsubsection*{{Homotopy quotient by $S^1$ and cyclic homology}}\label{homotopy_quotient_by__and_cyclic_homology}

\begin{prop}
\label{ModelForS1quotient}\hypertarget{ModelForS1quotient}{}
Given a [[Sullivan model]] $(\wedge^\bullet (V \oplus s V), d_{\mathcal{L}X})$ for a [[free loop space]] as in prop. \ref{SullivanModelForTheFreeLoopSpace}, then a Sullivan model for the [[cyclic loop space]], i.e. for the [[homotopy quotient]] $\mathcal{L} X // S^1$ with respect to the canonical [[circle group]] action that rotates loops (i.e. for the [[Borel construction]] $\mathcal{L}X \times_{S^1} E S^1$) is given by

\begin{displaymath}
(\wedge^\bullet(  V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1})
\end{displaymath}
where

\begin{itemize}%
\item $\omega_2$ is in degree 2;


\item $d_{\mathcal{L}X/S^1}$ is defined on generators $w \in V\oplus s V$ by

\begin{displaymath}
d_{\mathcal{L}X/S^1} w
    \;\coloneqq\;
  d_{\mathcal{L}X} w + \omega_2 \wedge s w
  \,.
\end{displaymath}


\end{itemize}
Moreover, the canonical sequence of morphisms of [[dg-algebras]]

\begin{displaymath}
(\wedge \omega_2, d = 0)
    \longrightarrow
  (\wedge^\bullet(  V\oplus s V \oplus \langle \omega_2\rangle ), d_{\mathcal{L}X/S^1})
    \longrightarrow
  (\wedge^\bullet(  V\oplus s V ), d_{\mathcal{L}X})
\end{displaymath}
is a model for the rationalization of the [[homotopy fiber sequence]]

\begin{displaymath}
\mathcal{L}X 
    \longrightarrow
  \mathcal{L}X / / S^1
    \longrightarrow
  B S^1
\end{displaymath}
which exhibits the [[infinity-action]] (by the discussion there) of $S^1$ on $\mathcal{L}X$.

\end{prop}
This is due to (\hyperlink{VigueBurghelea85}{Vigu\'e{}-Burghelea 85, theorem A}).

\hypertarget{RelationToHochschildHomology}{}\subsubsection*{{Relation to Hochschild homology and cyclic homology}}\label{RelationToHochschildHomology}

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

The [[ordinary cohomology]] $H^\bullet$ of its [[free loop space]] is the [[Hochschild homology]] $HH_\bullet$ of its [[singular cohomology|singular chains]] $C^\bullet(X)$:

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
\end{displaymath}
Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the cyclic loop space $\mathcal{L}X/^h S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
\end{displaymath}
(\hyperlink{Loday11}{Loday 11})

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then by prop. \ref{SullivanModelForTheFreeLoopSpace} and prop. \ref{ModelForS1quotient}, and the general statements at \emph{[[rational homotopy theory]]}, the [[cochain cohomology]] of the above [[minimal Sullivan models]] for $\mathcal{L}X$ and $\mathcal{l}X/S^1$ compute the rational [[Hochschild homology]] and [[cyclic homology]] of (the cochains on) $X$, respectively.

In the special case that the [[topological space]] $X$ carries the structure of a [[smooth manifold]], then the singular cochains on $X$ are equivalent to the [[dgc-algebra]] of [[differential forms]] (the [[de Rham algebra]]) and hence in this case the statement becomes that

\begin{displaymath}
H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
\begin{displaymath}
H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
\end{displaymath}
This is known as \emph{[[Jones' theorem]]} (\hyperlink{Jones87}{Jones 87})

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at \emph{\href{Hochschild+cohomology#JonesTheorem}{Hochschild cohomology -- Jones' theorem}}.

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

\hypertarget{4SphereAndTwistedDeRham}{}\subsubsection*{{Free loop space of the 4-sphere and twisted de Rham cohomology}}\label{4SphereAndTwistedDeRham}

We discuss the Sullivan model for the free and cyclic loop space of the [[4-sphere]]. This may also be thought of as the [[cocycle space]] for [[rational Cohomotopy|rational 4-Cohomotopy]], see \hyperlink{FiorenzaSatiSchreiber16}{FSS16, Section 3}.

\begin{example}
\label{}\hypertarget{}{}
Let $X = S^4$ be the [[4-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model

\begin{displaymath}
(\wedge^\bullet \langle g_4, g_7 \rangle, d)
\end{displaymath}
with

\begin{displaymath}
d g_4 = 0\,,\;\;\;\; d g_7 = -\tfrac{1}{2} g_4 \wedge g_4
  \,.
\end{displaymath}
Hence prop. \ref{SullivanModelForTheFreeLoopSpace} gives for the rationalization of $\mathcal{L}S^4$ the model

\begin{displaymath}
( \wedge^\bullet \langle \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4} )
\end{displaymath}
with

\begin{displaymath}
\begin{aligned}
    d_{\mathcal{L}S^4} h_3 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_4 & = 0
    \\
    d_{\mathcal{L}S^4} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4
    \\
  \end{aligned}
\end{displaymath}
and prop. \ref{ModelForS1quotient} gives for the rationalization of $\mathcal{L}S^4 / / S^1$ the model

\begin{displaymath}
( \wedge^\bullet \langle \omega_2, \omega_4, \omega_6, h_3, h_7 \rangle  , d_{\mathcal{L}S^4 / / S^1} )
\end{displaymath}
with

\begin{displaymath}
\begin{aligned}
    d_{\mathcal{L}S^4 / / S^1} h_3 & = 0
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_2 & = 0
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_4 & = h_3 \wedge \omega_2 
    \\
    d_{\mathcal{L}S^4 / / S^1} \omega_6 & = h_3 \wedge \omega_4
    \\
    d_{\mathcal{L}S^4 / / S^1} h_7 & = -\tfrac{1}{2} \omega_4 \wedge \omega_4 + \omega_2 \wedge \omega_6
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
\begin{prop}
\label{}\hypertarget{}{}
Let $\hat \mathfrak{g} \to \mathfrak{g}$ be a [[central extension|central]] [[Lie algebra extension]] by $\mathbb{R}$ of a finite dimensional Lie algebra $\mathfrak{g}$, and let $\mathfrak{g} \longrightarrow b \mathbb{R}$ be the corresponding [[L-∞ algebra cohomology|L-∞ 2-cocycle]] with coefficients in the [[line Lie n-algebra|line Lie 2-algebra]] $b \mathbb{R}$, hence ([[schreiber:The brane bouquet|FSS 13, prop. 3.5]]) so that there is a [[homotopy fiber sequence]] of [[L-∞ algebras]]

\begin{displaymath}
\hat \mathfrak{g}
    \longrightarrow
  \mathfrak{g}
    \overset{\omega_2}{\longrightarrow}
  b \mathbb{R}
\end{displaymath}
which is dually modeled by

\begin{displaymath}
CE(\hat \mathfrak{g})
    =
  ( \wedge^\bullet ( \mathfrak{g}^\ast \oplus \langle e \rangle ), d_{\hat \mathfrak{g}}|_{\mathfrak{g}^\ast} = d_{\mathfrak{g}},\;  d_{\hat \mathfrak{g}} e = \omega_2)
  \,.
\end{displaymath}
For $X$ a space with [[Sullivan model]] $(A_X,d_X)$ write $\mathfrak{l}(X)$ for the corresponding [[L-∞ algebra]], i.e. for the $L_\infty$-algebra whose [[Chevalley-Eilenberg algebra]] is $(A_X,d_X)$:

\begin{displaymath}
CE(\mathfrak{l}X) = (A_X,d_X)
  \,.
\end{displaymath}
Then there is an [[isomorphism]] of [[hom-sets]]

\begin{displaymath}
Hom_{L_\infty Alg}( \hat \mathfrak{g}, \mathfrak{l}(S^4) )
  \;\simeq\;
  Hom_{L_\infty Alg/b \mathbb{R}}( \mathfrak{g}, \mathfrak{l}( \mathcal{L}S^4 / S^1 ) )
  \,,
\end{displaymath}
with $\mathfrak{l}(S^4)$ from prop. \ref{SullivanModelForTheFreeLoopSpace} and $\mathfrak{l}(\mathcal{L}S^4 //S^1)$ from prop. \ref{ModelForS1quotient}, where on the right we have homs in the [[slice category|slice]] over the [[line Lie n-algebra|line Lie 2-algebra]], via prop. \ref{ModelForS1quotient}.

Moreover, this isomorphism takes

\begin{displaymath}
\hat \mathfrak{g}
    \overset{(g_4, g_7)}{\longrightarrow}
  \mathfrak{l}(S^4)
\end{displaymath}
to

\begin{displaymath}
\itexarray{
    \mathfrak{g} 
      &&
      \overset{(\omega_2,\omega_4, \omega_6, h_3,h_7)}{\longrightarrow}
      &&
    \mathfrak{l}( \mathcal{L}X / S^1 )
    \\
    & 
    {}_{\mathllap{\omega_2}}\searrow 
      && 
    \swarrow_{\mathrlap{\omega_2}}
    \\
    && b \mathbb{R}
  }
  \,,
\end{displaymath}
where

\begin{displaymath}
\omega_4 = g_4 - h_3 \wedge e
  \;\,,
  \;\;\;
  h_7 = g_7 + \omega_6 \wedge e
\end{displaymath}
with $e$ being the central generator in $CE(\hat \mathfrak{g})$ from above, and where the equations take place in $\wedge^\bullet \hat \mathfrak{g}^\ast$ with the defining inclusion $\wedge^\bullet \mathfrak{g}^\ast \hookrightarrow \wedge^\bullet \mathfrak{g}^\ast$ understood.

\end{prop}
This is observed in (\hyperlink{FiorenzaSatiSchreiber16}{Fiorenza-Sati-Schreiber 16}, \hyperlink{FSS16b}{FSS 16b}), where it serves to formalize, on the level of [[rational homotopy theory]], the [[double dimensional reduction]] of [[M-branes]] in [[M-theory]] to [[D-branes]] in [[type IIA string theory]] (for the case that $\mathfrak{g}$ is type IIA [[super Minkowski spacetime]] $\mathbb{R}^{9,1\vert \mathbf{16}+ \overline{\mathbf{16}}}$ and $\hat \mathfrak{g}$ is 11d [[super Minkowski spacetime]] $\mathbb{R}^{10,1\vert \mathbf{32}}$, and the cocycles are those of [[schreiber:The brane bouquet]]).

\begin{proof}
By the fact that the underlying graded algebras are free, and since $e$ is a generator of odd degree, the given decomposition for $\omega_4$ and $h_7$ is unique.

Hence it is sufficient to observe that under this decomposition the defining equations

\begin{displaymath}
d g_4 = 0
  \,,\;\;\;
  d g_{7} = -\tfrac{1}{2} g_4 \wedge g_4
\end{displaymath}
for the $\mathfrak{l}S^4$-valued cocycle on $\hat \mathfrak{g}$ turn into the equations for a $\mathfrak{l} ( \mathcal{L}S^4 / S^1 )$-valued cocycle on $\mathfrak{g}$. This is straightforward:

\begin{displaymath}
\begin{aligned}
    & d_{\hat \mathfrak{g}} ( \omega_4 + h_3 \wedge e ) = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} (\omega_4 - h_3 \wedge \omega_2) = 0
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_{\mathfrak{g}} \omega_4 = h_3 \wedge \omega_2
    \;\;\; and \;\;\;
    d_{\mathfrak{g}} h_3 = 0
  \end{aligned}
\end{displaymath}
as well as

\begin{displaymath}
\begin{aligned}
    & d_{\hat \mathfrak{g}} ( h_7 - \omega_6 \wedge e ) 
      = -\tfrac{1}{2}( \omega_4 + h_3 \wedge e ) \wedge (\omega_4 + h_3\wedge e)
   \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 - \omega_6 \wedge \omega_2
    =
    -\tfrac{1}{2}\omega_4 \wedge \omega_4
    \;\;\; and \;\;\;
    - d_\mathfrak{g} \omega_6 = - h_3 \wedge \omega_4 
    \\
    \Leftrightarrow \;\;\;\;
    &
    d_\mathfrak{g} h_7 = -\tfrac{1}{2}\omega_4 \wedge \omega_4  + \omega_6 \wedge \omega_2 
    \;\;\; and \;\;\;
    d_\mathfrak{g} h_6 = h_3 \wedge \omega_4
  \end{aligned}
\end{displaymath}
\end{proof}
\hypertarget{free_loop_space_of_the_2sphere}{}\subsubsection*{{Free loop space of the 2-sphere}}\label{free_loop_space_of_the_2sphere}

\begin{example}
\label{}\hypertarget{}{}
Let $X = S^2$ be the [[2-sphere]]. The corresponding [[rational n-sphere]] has minimal Sullivan model

\begin{displaymath}
(\wedge^\bullet \langle g_3, g_2 \rangle, d)
\end{displaymath}
with

\begin{displaymath}
d g_2 = 0\,,\;\;\;\; d g_3 = -\tfrac{1}{2} g_2 \wedge g_2
  \,.
\end{displaymath}
Hence prop. \ref{SullivanModelForTheFreeLoopSpace} gives for the rationalization of $\mathcal{L}S^2$ the model

\begin{displaymath}
( \wedge^\bullet \langle \omega^A_2, \omega^B_2, h_1, h_3 \rangle  , d_{\mathcal{L}S^2} )
\end{displaymath}
with

\begin{displaymath}
\begin{aligned}
    d_{\mathcal{L}S^2} h_1 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^A_2 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A
    \\
    d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2
  \end{aligned}
\end{displaymath}
and prop. \ref{ModelForS1quotient} gives for the rationalization of $\mathcal{L}S^2 / / S^1$ the model

\begin{displaymath}
( \wedge^\bullet \langle \omega^A_2, \omega^B_2, \omega^C_2 h_1, h_3 , d_{\mathcal{L}S^2 / / S^1} )
\end{displaymath}
with

\begin{displaymath}
\begin{aligned}
    d_{\mathcal{L}S^2} h_1 & = 0
    \\
    d_{\mathcal{L}S^2} \omega^A_2 & = \omega^C_2 \wedge h_1
    \\
    d_{\mathcal{L}S^2} \omega^B_2 & = h_1 \wedge \omega_2^A
    \\
    d_{\mathcal{L}S^2} \omega^C_2 & = 0
    \\
    d_{\mathcal{L}S^2} h_3 & = -\tfrac{1}{2} \omega^A_2 \wedge \omega^A_2 + \omega^C_2 \wedge \omega^B_2
  \end{aligned}
  \,.
\end{displaymath}
\end{example}
$\backslash$linebreak

\hypertarget{IteratedLoopSpacesOfNSpheres}{}\subsubsection*{{Iterated based loop spaces of $n$-spheres}}\label{IteratedLoopSpacesOfNSpheres}

By iterating the Sullivan model construction for the [[based loop space]] from Prop. \ref{SullivanModelForBasedLoopSpace} and using the [[Sullivan models of n-spheres]] we have that:

\begin{prop}
\label{SullivanModelsOfMapsFromSkToSnFornLargerk}\hypertarget{SullivanModelsOfMapsFromSkToSnFornLargerk}{}
\textbf{([[Sullivan models]] for [[iterated loop spaces]] of [[n-spheres]])}

The [[Sullivan model]] of the $k$-fold [[iterated based loop space]] $\Omega^k S^n$ of the [[n-sphere]] for $k \lt n$ is

\begin{displaymath}
CE\mathfrak{l}
  \big(
    \Omega^k S^n
  \big)
  \;=\;
  \left\{
  \itexarray{
    \left( 
      \itexarray{
        d\,\omega_{n-k} & = 0
      }
    \right)
    &\vert&
    n \;\text{is odd}
   \\
    \left( 
      \itexarray{
        d\,\omega_{n-k} & = 0
        \\
        d\,\omega_{2n-1-k} & = 0
      }
    \right)
    &\vert&
    n \;\text{is even}
  }
  \right.
  \phantom{AAAA}
  \text{for}\; k \lt n
  \,.
\end{displaymath}
\end{prop}
(see also \hyperlink{KallelSjerve99}{Kallel-Sjerve 99, Prop. 4.10})

For the edge case $\Omega^D S^D$ the above formula does not apply, since $\Omega^{D-1} S^D$ is not [[simply connected topological space|simply connected]] (its [[fundamental group]] is $\pi_1\big( \Omega^{D-1}S^D \big) = \pi_0 \big(\Omega^D S^D\big) = \pi_D(S^D) = \mathbb{Z}$, the 0th [[stable homotopy group of spheres]]).

But:

\begin{example}
\label{RationalModelsForBasedMappingSpaceSDToSD}\hypertarget{RationalModelsForBasedMappingSpaceSDToSD}{}
The rational model for $\Omega^D S^D$ follows from \href{Sullivan+model+of+mapping+space#RationalHomotopyTypeOfMapsNSphereToNsphere}{this Prop.} by realizing the pointed mapping space as the [[homotopy fiber]] of the [[evaluation map]] from the free mapping space:

\begin{displaymath}
\itexarray{
    \mathllap{
      \Omega^D S^D
      \simeq
    \;}
    Maps^{\ast/\!}\big( S^D, S^D\big)
    \\
    \big\downarrow^{\mathrlap{fib(ev_\ast)}}
    \\
    Maps(S^D, S^D)
    \\
    \big\downarrow^{\mathrlap{ev_\ast}}
    \\
    S^D
  }
\end{displaymath}
This yields for instance the following examples.

In odd dimensions:

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}3, S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} $\backslash$ast $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}3, S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$ard{\tt \symbol{94}}-\{ ($\backslash$mathrm\{id\}\_\{S{\tt \symbol{94}}3\})\_\{n $\backslash$in $\backslash$mathbb\{N\}\} \} $\backslash$ S{\tt \symbol{94}}3 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}3 $\backslash$end\{xymatrix\}

In even dimensions:

(In the following $h_{\mathbb{K}}$ denotes the [[Hopf fibration]] of the [[division algebra]] $\mathbb{K}$, hence $h_{\mathbb{C}}$ denotes the [[complex Hopf fibration]] and $h_{\mathbb{H}}$ the [[quaternionic Hopf fibration]].)

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}2, S{\tt \symbol{94}}2 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}1 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}2, S{\tt \symbol{94}}2 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$big( S{\tt \symbol{94}}2 $\backslash$times S{\tt \symbol{94}}1 $\backslash$big) $\backslash$sqcup $\backslash$big( $\backslash$underset\{ n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$big( p\_1, (h\_\{$\backslash$mathbb\{C\}\})\_\{n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{N\}\} $\backslash$big) \} $\backslash$ S{\tt \symbol{94}}2 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}2 $\backslash$end\{xymatrix\}

$\backslash$begin\{xymatrix\} $\backslash$mathrm\{Maps\}{\tt \symbol{94}}\{$\backslash$ast/!\} $\backslash$big( S{\tt \symbol{94}}4, S{\tt \symbol{94}}4 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{fib\}\_\{($\backslash$mathrm\{ev\}\_$\backslash$ast)\} \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$underset\{ n $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}3 $\backslash$ar@\{{\tt \symbol{94}}\{(\}-{\tt \symbol{62}}\}d $\backslash$ $\backslash$mathrm\{Maps\} $\backslash$big( S{\tt \symbol{94}}4, S{\tt \symbol{94}}4 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$mathrm\{ev\}\_$\backslash$ast \} $\backslash$ar@\{\}r|-\{ $\backslash$simeq\_\{$\backslash$mathbb\{Q\}\} \} \& $\backslash$big( S{\tt \symbol{94}}4 $\backslash$times S{\tt \symbol{94}}3 $\backslash$big) $\backslash$sqcup $\backslash$big( $\backslash$underset\{ n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{Z\} \}\{$\backslash$sqcup\} S{\tt \symbol{94}}7 $\backslash$big) $\backslash$ard{\tt \symbol{94}}-\{ $\backslash$big( p\_1, (h\_\{$\backslash$mathbb\{H\}\})\_\{n $\backslash$neq 0 $\backslash$in $\backslash$mathbb\{N\}\} $\backslash$big) \} $\backslash$ S{\tt \symbol{94}}4 $\backslash$ar@\{=\}r \& S{\tt \symbol{94}}4 $\backslash$end\{xymatrix\}

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

\begin{itemize}%
\item [[Hochschild homology]]


\item [[cyclic homology]]


\item [[rational model of mapping spaces]]


\item [[rational Cohomotopy]]



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

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

The original result is due to

\begin{itemize}%
\item [[Micheline Vigué-Poirrier]], [[Dennis Sullivan]], \emph{The homology theory of the closed geodesic problem}, J. Differential Geometry 11 (1976) 633-644.


\item [[Micheline Vigué-Poirrier]], Dan Burghelea, \emph{A model for cyclic homology and algebraic K-theory of 1-connected topological spaces}, J. Differential Geom. Volume 22, Number 2 (1985), 243-253 (\href{https://projecteuclid.org/euclid.jdg/1214439821}{Euclid})



\end{itemize}
Review is in

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


\item [[Kathryn Hess]], example 2.5 of \emph{Rational homotopy theory: a brief introduction} (\href{http://arxiv.org/abs/math.AT/0604626}{math.AT/0604626})


\item [[Yves Félix]], John Oprea, [[Daniel Tanre]], \emph{Algebraic models in geometry}, Oxford graduate texts in mathematics 17 (2008)


\item A. Yu. Onishchenko and Th. Yu. Popelensky, \emph{Rational cohomology of the free loop space of a simply connected 4-manifold}, J. Fixed Point Theory Appl. 12 (2012) 69--9 (\href{http://link.springer.com/article/10.1007/s11784-013-0100-0}{DOI 10.1007/s11784-013-0100-0})


\item [[Luc Menichi]], \emph{Sullivan models and free loop space}, A short introduction to Sullivan models, with the Sullivan model of a free loop space and the detailed proof of Vigu\'e{}-Sullivan theorem on the Betti numbers of free loop space. Workshop on free loop space \`a{} Strasbourg, November 2008 (scanned notes \href{http://math.univ-angers.fr/perso/lmenichi/Sullivan_models_free_loop_space.pdf}{pdf})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]]: \emph{[[schreiber:Rational sphere valued supercocycles in M-theory|Rational sphere valued supercocycles in M-theory and type IIA string theory]]}, Journal of Geometry and Physics, Volume 114, Pages 91-108 (2017) (\href{https://arxiv.org/abs/1606.03206}{arXiv:1606.03206})


\item [[nLab:Domenico Fiorenza]], [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:T-Duality from super Lie n-algebra cocycles for super p-branes]]} (\href{https://arxiv.org/abs/1611.06536}{arXiv:1611.06536})



\end{itemize}
General background on [[Hochschild homology]] and [[cyclic homology]] is in

\begin{itemize}%
\item [[John D.S. Jones]], \emph{Cyclic homology and equivariant homology}, Invent. Math. \textbf{87}, 403-423 (1987) (\href{https://math.berkeley.edu/~nadler/jones.pdf}{pdf})


\item [[Jean-Louis Loday]], \emph{Cyclic homology}, Grundlehren Math.Wiss. \textbf{301}, Springer (1998)


\item [[Jean-Louis Loday]], \emph{Free loop space and homology} (\href{https://arxiv.org/abs/1110.0405}{arXiv:1110.0405})



\end{itemize}
The case of [[iterated based loop spaces]] of [[n-spheres]] is discussed also in

\begin{itemize}%
\item [[Sadok Kallel]], [[Denis Sjerve]], \emph{On Brace Products and the Structure of Fibrations with Section}, 1999 (\href{https://www.math.ubc.ca/~sjer/brace.pdf}{pdf}, [[KallelSjerv99.pdf:file]])

\end{itemize}
[[!redirects Sullivan models of free loop space]] [[!redirects Sullivan model of free loop spaces]]

[[!redirects Sullivan model of the free loop space]] [[!redirects Sullivan models of the free loop space]]

[[!redirects Sullivan model of a free loop space]] [[!redirects Sullivan models of a free loop space]]

[[!redirects Sullivan models of free loop spaces]] [[!redirects Sullivan model for free loop space]] [[!redirects Sullivan models for free loop space]] [[!redirects Sullivan model for free loop spaces]] [[!redirects Sullivan models for free loop spaces]]

[[!redirects Sullivan model of based loop space]] [[!redirects Sullivan models of based loop spaces]]

[[!redirects Sullivan model for based loop space]] [[!redirects Sullivan models for based loop spaces]]



\end{document}