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\begin{document}

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\section*{co-binary Sullivan differential is Whitehead product}

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

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

[[!include homotopy - contents]]

\hypertarget{lie_theory}{}\paragraph*{{Lie theory}}\label{lie_theory}

[[!include infinity-Lie theory - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{notation_and_conventions}{Notation and conventions}\dotfill \pageref*{notation_and_conventions} \linebreak
\noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{HopfFibrations}{Hopf fibrations}\dotfill \pageref*{HopfFibrations} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

We discuss (Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} below) how the [[rationalization]] of the [[Whitehead product]] is the co-binary part of the [[Sullivan model|Sullivan differential]] in [[rational homotopy theory]].

\hypertarget{notation_and_conventions}{}\subsection*{{Notation and conventions}}\label{notation_and_conventions}

We make explicit some notation and normalization conventions that enter the statement.

In the following, for $W$ a $\mathbb{Z}$-[[graded module]], we write

\begin{displaymath}
W \wedge W
  \;\coloneqq\;
  Sym^2(W)
  \;\coloneqq\;
  \big(
    W \otimes W
  \big)
  / 
  \big(
   \alpha \otimes \beta 
     \sim
   (-1)^{ n_\alpha n_\beta } 
   \beta \otimes \alpha
  \big)
  \,,
\end{displaymath}
where on the right $\alpha, \beta \in W$ are elements of homogeneous degree $n_\alpha, n_\beta \in \mathbb{Z}$, respectively. The point is just to highlight that ``$(-)\wedge(-)$'' is not to imply here a degree shift of the generators (as it typically does in the usual notation for [[Grassmann algebras]]).

Let $X$ be a [[simply connected topological space]] with [[Sullivan model]]

\begin{equation}
CE( \mathfrak{l} X )
  \;=\;
  \big( 
    Sym^\bullet\big(V^\ast\big), d_X 
  \big)
\label{SullivanModelX}\end{equation}
for $V^\ast$ the [[graded vector space]] of generators, which is the $\mathbb{Q}$-linear [[dual space|dual]] [[graded vector space]] of the [[graded object|graded]] $\mathbb{Z}$-[[module]] (=[[graded abelian group]]) of [[homotopy groups]] of $X$:

\begin{displaymath}
V^\ast
  \;\coloneqq\;
  Hom_{Ab}\big(
    \pi_\bullet(X),
    \mathbb{Q}
  \big)
  \,.
\end{displaymath}
Declare the [[wedge product]] pairing to be given by

\begin{equation}
\itexarray{
    V^\ast
    \wedge
    V^\ast
    &\overset{\Phi}{\longrightarrow}&
    Hom_{Ab}
    \big( 
      \pi_\bullet(X) \wedge \pi_\bullet(X)
      ,
      \mathbb{Q}
    \big)
    \\
    (\alpha, \beta)
    &\mapsto&
    \Big(
      v \wedge w
      \;\mapsto\;
      (-1)^{ n_\alpha \cdot n_\beta }
      \alpha(v)\cdot \beta(w)
      +
      \beta(v)\cdot \alpha(w)
    \Big)
  }
\label{WedgeProductNormalization}\end{equation}
where $\alpha$, $\beta$ are assumed to be of homogeneous degree $n_\alpha, n_\beta \in \mathbb{N}$, respectively.

(Notice that the usual normalization factor of $1/2$ is \emph{not} included on the right. This normalization follows \hyperlink{AndrewsArkowitz78}{Andrews-Arkowitz 78, above Thm. 6.1}.)

Finally, write

\begin{equation}
[-]_2
  \;\colon\;
  Sym^\bullet\big(V^\ast\big)  
  \longrightarrow
  V^\ast \wedge V^\ast
\label{PojectionToBinary}\end{equation}
for the linear projection on quadratic polynomials in the graded [[symmetric algebra]].

\hypertarget{statement}{}\subsection*{{Statement}}\label{statement}

\begin{prop}
\label{CoBinarySullivanDifferentialIsWhiteheadProduct}\hypertarget{CoBinarySullivanDifferentialIsWhiteheadProduct}{}
\textbf{([[co-binary Sullivan differential is Whitehead product]])}

Let $X$ be a [[simply connected topological space]] of rational [[finite type]], so that it has a [[Sullivan model]] with Sullivan differential $d_X$ \eqref{SullivanModelX}.

Then the co-binary component \eqref{PojectionToBinary} of the Sullivan differential equals the $\mathbb{Q}$-[[linear dual map]] of the [[Whitehead product]] $[-,-]_X$ on the [[homotopy groups]] of $X$:

\begin{displaymath}
[d_X \alpha]_2
  \;=\;
  [-,-]_X^\ast
  \,.
\end{displaymath}
More explicitly, the following [[commuting diagram|diagram commutes]]:

\begin{displaymath}
\itexarray{
    V^\ast 
      &\overset{ [-]_2\circ d_X }{\longrightarrow}& 
    V^\ast \wedge V^\ast
    \\
    \big\downarrow^{ \mathrlap{=} }
    &&
    \big\downarrow^{ \mathrlap{\Phi} }
    \\
    Hom_{Ab}
    \big(
      \pi_\bullet(X),
      \mathbb{Q}
    \big)
    &
      \underset{
        Hom_{Ab}\big(
          [-,-]_X
          ,
          \;
          \mathbb{Q}
        \big)
      }{
        \longrightarrow
      }
    &
    Hom_{Ab}
    \big(
      \pi_\bullet(X)
        \wedge    
      \pi_\bullet(X),
      \;
      \mathbb{Q}
    \big)
  }
  \,,
\end{displaymath}
where the wedge product on the right is normalized as in \eqref{WedgeProductNormalization}.

\end{prop}
(\hyperlink{AndrewsArkowitz78}{Andrews-Arkowitz 78, Thm. 6.1}, see also \hyperlink{FelixHalperinThomas00}{Félix-Halperin-Thomas 00, Prop. 13.16})

$\backslash$linebreak

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

\hypertarget{HopfFibrations}{}\subsubsection*{{Hopf fibrations}}\label{HopfFibrations}

For $X = S^2$ the [[2-sphere]], consider the following two [[elements]] of its [[homotopy groups]] ([[homotopy groups of spheres|of spheres]], as it were):

\begin{enumerate}%
\item $id_{S^2} \in \pi_2\big( S^2 \big)$ (represented by the [[identity function]] $S^2 \to S^2$)


\item $h_{\mathbb{C}} \in \pi_3\big(  S^3 \big)$ (represented by the [[complex Hopf fibration]])



\end{enumerate}
Then the Whitehead product satisfies

\begin{equation}
\big[
    id_{S^2},
    \; 
    id_{S^2}
  \big]
  \;=\;
  2 \cdot h_{\mathbb{C}}
  \,.
\label{WhiteheadProductOf2Sphere}\end{equation}
(by \href{Whitehead+product#WhiteheadProductCorrespondingToComplexHopfFibration}{this Example}).

Now let

\begin{displaymath}
vol_{S^2},\; vol_{S^3}
  \;\in\;
  CE\big( \mathfrak{l}S^2 \big)
\end{displaymath}
be the two generators of the [[rational n-sphere|Sullivan model of the 2-sphere]], \emph{normalized} such that they correspond to the [[volume forms]] of the [[2-sphere]] and (after [[pullback of differential forms|pullback]] along the [[complex Hopf fibration]] $h_{\mathbb{C}}$) of the [[3-sphere]], respectively.

This means that the [[Sullivan model|Sullivan differential]] is

\begin{equation}
d_{S^2} vol_{S^3}
  \;=\;
  c \cdot 
  vol_{S^2} \wedge vol_{S^2}
\label{GenericDifferentialForSullivanModelOf2Sphere}\end{equation}
for some [[rational number]] $c \in \mathbb{Q}$.

Notice that with the normalization in \eqref{WedgeProductNormalization} we have

\begin{displaymath}
\begin{aligned}
  \Phi(vol_{S^2}, vol_{S^2})\big(  id_{S^2} \wedge id_{S^2} \big)
  & =
  (-1)^{2 \cdot 2}
  vol_{S^2}\big( id_{S^2}\big)
  \cdot
  vol_{S^2}\big( id_{S^2}\big)
  +  
  vol_{S^2}\big( id_{S^2}\big)
  \cdot
  vol_{S^2}\big( id_{S^2}\big)
  \\
  & =
  2
  \end{aligned}
\end{displaymath}
Therefore Prop. \ref{CoBinarySullivanDifferentialIsWhiteheadProduct} gives

\begin{displaymath}
\itexarray{
    \big\{
      vol_{S^3}
    \big\}
    &\overset{[-]_2\circ d_{S^2}}{\longrightarrow}&
    c' \cdot ( vol_{S^2} \otimes vol_{S^2} )
    \\
    \big\downarrow
    &&
    \big\downarrow^{\mathrlap{\Phi}}
    \\
    \big\{
     h_{\mathbb{H}}
    \big\}
    &\overset{ [-,-]_X^\ast }{\longrightarrow}&
    \big\{
      id_{S^2} \wedge id_{S^2}
      \mapsto 
      2
    \big\}
  }
\end{displaymath}
where in the bottom row we used the Whitehead product \eqref{WhiteheadProductOf2Sphere}.

Hence $c = 1$:

\begin{displaymath}
d_{S^2} vol_3 
  \;=\; 
  - vol_{S^2} \wedge vol_{S^2} 
  \,.
\end{displaymath}
See also \hyperlink{FelixHalperinThomas00}{Félix-Halperin-Thomas 00, Example 1 on p. 178}.

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

Under the general relation between the Sullivan model and the original Quillen model of [[rational homotopy theory]], the statement comes from

\begin{itemize}%
\item [[Daniel Quillen]], section I.5 of \emph{Rational Homotopy Theory}, Annals of Mathematics Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (\href{https://www.jstor.org/stable/1970725}{jstor:1970725})

\end{itemize}
It is made fully explicit in

\begin{itemize}%
\item Peter Andrews, [[Martin Arkowitz]], Theorem 6.1 \emph{Sullivan's Minimal Models and Higher Order Whitehead Products}, Canadian Journal of Mathematics, 30(5), 961-982, 1978 (\href{https://doi.org/10.4153/CJM-1978-083-6}{doi:10.4153/CJM-1978-083-6})

\end{itemize}
where the result is attributed to

\begin{itemize}%
\item [[Pierre Deligne]], [[Phillip Griffiths]], [[John Morgan]], [[Dennis Sullivan]], \emph{Real homotopy theory of Kähler manifolds}, Invent Math (1975) 29: 245 (\href{https://doi.org/10.1007/BF01389853}{doi:10.1007/BF01389853})

\end{itemize}
which however just touches on it in passing.

Textbook accounts:

\begin{itemize}%
\item [[Yves Félix]], [[Steve Halperin]], J.C. Thomas, Prop. 13.16 in \emph{Rational Homotopy Theory}, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000.

\end{itemize}
[[!redirects the co-binary Sullivan differential is the dual Whitehead product]] [[!redirects the co-binary Sullivan differential is the dual of the Whitehead product]]



\end{document}