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\begin{document}

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\section*{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{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_definition}{Basic definition}\dotfill \pageref*{basic_definition} \linebreak
\noindent\hyperlink{generalized_version}{Generalized version}\dotfill \pageref*{generalized_version} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{super_lie_algebra_structure}{Super Lie algebra structure}\dotfill \pageref*{super_lie_algebra_structure} \linebreak
\noindent\hyperlink{OfAnElementWithItself}{Of elements with themselves}\dotfill \pageref*{OfAnElementWithItself} \linebreak
\noindent\hyperlink{as_primary_homotopy_operations}{As primary homotopy operations}\dotfill \pageref*{as_primary_homotopy_operations} \linebreak
\noindent\hyperlink{in_terms_of_samelson_products}{In terms of Samelson products}\dotfill \pageref*{in_terms_of_samelson_products} \linebreak
\noindent\hyperlink{RelationToSullivanModels}{Relation to the Sullivan models}\dotfill \pageref*{RelationToSullivanModels} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{ReferencesInRationalHomotopyTheory}{In rational homotopy theory}\dotfill \pageref*{ReferencesInRationalHomotopyTheory} \linebreak


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

\hypertarget{basic_definition}{}\subsubsection*{{Basic definition}}\label{basic_definition}

\begin{defn}
\label{}\hypertarget{}{}
For $X$ a [[pointed space|pointed]] [[topological space]] ([[CW-complex]]), the \emph{Whitehead products} (\hyperlink{Whitehead41}{Whitehead 41, Section 3}) are the [[bilinear maps]] on the elements of the [[homotopy groups]] $\pi_\bullet(X)$ of $X$ of the form

\begin{equation}
[-,-]_{Wh}
  \;\colon\;
  \pi_{n_1}\big( X \big)
  \otimes_{\mathbb{Z}}
  \pi_{n_2}\big( X \big)
  \longrightarrow
  \pi_{n_1 + n_2 - 1}\big( X \big)
  \phantom{AA}
  \text{for all}
  \;
  n_i \in \mathbb{N}
  ;
  \;
  n_i \geq 1
  \,,
\label{WhiteheadProductMapInSingleDegrees}\end{equation}
given by sending any [[pair]] of [[homotopy classes]]

\begin{displaymath}
\big[
    S^{n_i} \overset{\phi_i}{\longrightarrow} X
  \big]
  \;\in\;
  \pi_{ n_i }
  \big(
    X
  \big)
\end{displaymath}
to the [[homotopy class]] of the top [[composition|composite]] in the [[diagram]]

\begin{displaymath}
\itexarray{
    S^{ n_1 + n_2 -1 }
    &
    \overset{      
      f_{n_1,n_2}
    }{
      \longrightarrow
    }
    &
    S^{n_1}
    \vee
    S^{n_2}
    &
    \overset{
      (\phi_1, \phi_2)
    }{\longrightarrow}
    &
    X
    \\
    \big\downarrow
    &
    (po)
    &
    \big\downarrow
    \\
    D^{ n_1 + n_2 }
    &\underset{}{\longrightarrow}&
    S^{n_1} \times S^{n_2}
  }
\end{displaymath}
where $f_{n_1, n_2}$ is [[generalized the|the]] [[attaching map]] exhibiting the [[product space]] $S^{n_1} \times S^{n_2}$ as the result of a [[cell attachment]] to the [[wedge sum]] $S^{n_1} \vee S^{n_2}$.

\end{defn}
In this form this appears for instance in \hyperlink{FelixHalperinThomas00}{Félix-Halperin-Thomas, p. 176 with p. 177}.

\hypertarget{generalized_version}{}\subsubsection*{{Generalized version}}\label{generalized_version}

There is also a \textbf{generalized Whitehead product} where we can take more general homotopy classes ([[continuous maps]] up to [[homotopy]]) $\alpha\in [S^\cdot Y,X]$ and $\beta\in [S^\cdot Z,X]$ to produce a class $[\alpha,\beta]_{Wh}\in[Y\star Z,X]$. Here $S^\cdot$ denotes the [[reduced suspension]] operation on pointed spaces and $\star$ denotes the [[join of spaces|join]] of CW-complexes. Notice that $pt\star Z = C^\cdot(Z)$ and the reduced [[cone]] of a point is $C^\cdot(pt)=S^1$. Thus for $Y=Z=pt$ the generalized Whitehead product reduced to the usual Whitehead product.

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

\hypertarget{super_lie_algebra_structure}{}\subsubsection*{{Super Lie algebra structure}}\label{super_lie_algebra_structure}

If one assigns degree $n-1$ to the $n$th [[homotopy group]] $\pi_n$, then the degree-wise Whitehead products \eqref{WhiteheadProductMapInSingleDegrees} organize into a single degree-0 bilinear pairing on the [[graded abelian group]] which is the [[direct sum]] of all the [[homotopy groups]]:

\begin{equation}
\itexarray{
    \pi_{\bullet + 1}(X) 
    =
    &
    \pi_1(X)
    &\oplus&
    \pi_2(X)
    &\oplus&
    \pi_3(X)
    &\oplus&
    \cdots
    \\
    deg =
    &
    0
    &&
    1
    &&
    2
  }
\label{DegreeShiftedHomotopyGroups}\end{equation}
This unified Whitehead product is \emph{graded} skew-symmetric in that for $\phi_i \in \pi_{n_i}\big( X \big)$ it satisfies

\begin{displaymath}
\big[
    \phi_1,
    \,
    \phi_2
  \big]_{Wh}
  \;=\;
  -
  (-1)^{ (n_1 - 1)(n_2-1) }
  \big[
    \phi_1,
    \,
    \phi_2
  \big]_{Wh}
\end{displaymath}
and it satisfies the corresponding graded [[Jacobi identity]] (\hyperlink{Hilton55}{Hilton 55, Theorem B}).

This makes the Whitehead bracket the [[Lie bracket]] of a [[super Lie algebra]] [[structure]] on $\pi_{\bullet-1}(X)$ \eqref{DegreeShiftedHomotopyGroups}, over the [[ring]] of [[integers]] (sometimes called, in this context, a \emph{graded quasi-Lie algebra}, see \hyperlink{OfAnElementWithItself}{below}).

\hypertarget{OfAnElementWithItself}{}\subsubsection*{{Of elements with themselves}}\label{OfAnElementWithItself}

Beware that the skew-symmetry of [[Lie algebras]] over the [[integers]], as opposed to over a [[field]] of [[characteristic zero]], implies for any element $\phi$ of [[even number|even]] homogeneous degree -- hence here for elements of [[homotopy groups]] in [[odd number|odd]] degree -- only that the bracket with itself vanishes after multiplication by 2

\begin{displaymath}
[\phi,\phi]_{Wh} = - [\phi,\phi]_{Wh}
  \phantom{AA}
  \text{hence equivalently}
  \phantom{AA}
  2 \cdot [\phi,\phi]_{Wh} = 0
\end{displaymath}
but not necessarily that $[\phi,\phi]_{Wh} = 0$ by itself -- since [[multiplication]] by 2 is not an [[isomorphism]] over the [[integers]].

But this means that the Whitehead bracket of any even-degree [[element]] with itself -- hence of any [[element]] of a [[homotopy group]] in [[odd number|odd]] degree -- has [[order of a group element|order]] at most 2, hence is in the 2-[[torsion subgroup]] of the respective [[homotopy group]].

\hypertarget{as_primary_homotopy_operations}{}\subsubsection*{{As primary homotopy operations}}\label{as_primary_homotopy_operations}

The Whitehead products form one of the [[primary homotopy operations]].

In fact, together with [[composition operations]] and [[fundamental group]]-[[actions]] they [[generators and relations|generate]] all such operations.

This is related to the definition of [[Pi-algebras]].

\hypertarget{in_terms_of_samelson_products}{}\subsubsection*{{In terms of Samelson products}}\label{in_terms_of_samelson_products}

In the context of [[simplicial groups]], representing [[connected]] [[homotopy types]], there is a formula for the Whitehead product in terms of a [[Samelson product]], which in turn is derived from a [[shuffle]] product which is a sort of non-commutative version of the [[Eilenberg-Zilber map]]. These simplicial formulae come from an analysis of the structure of the [[product of simplices]].

(This formula for the Whitehead product is due to [[Dan Kan]] and can be found in the old survey article of [[Ed Curtis]]. The proof that it works was never published.)

\hypertarget{RelationToSullivanModels}{}\subsubsection*{{Relation to the Sullivan models}}\label{RelationToSullivanModels}

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]]. First we make explicit some notation and normalization conventions that enter this 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]].

Then:

\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}, following \hyperlink{DeligneGriffithsMorganSullivan75}{Deligne-Griffiths-Morgan-Sullivan 75})

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

\begin{example}
\label{WhiteheadProductCorrespondingToComplexHopfFibration}\hypertarget{WhiteheadProductCorrespondingToComplexHopfFibration}{}
\textbf{([[Whitehead product]] corresponding to [[complex Hopf fibration]])}

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 $h_{\mathbb{C}} \in \pi_3\big(  S^2 \big)$ (represented by the [[complex Hopf fibration]])



\end{enumerate}
Then the Whitehead product satisfies

\begin{displaymath}
\big[
    id_{S^2},
    \; 
    id_{S^2}
  \big]
  \;=\;
  2 \cdot h_{\mathbb{C}}
  \,.
\end{displaymath}
\end{example}
$\backslash$linebreak

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

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

The concept is originally due to

\begin{itemize}%
\item [[J. H. C. Whitehead]], Section 3 of \emph{On Adding Relations to Homotopy Groups}, Annals of Mathematics Second Series, Vol. 42, No. 2 (Apr., 1941), pp. 409-428 (\href{https://www.jstor.org/stable/1968907}{jstor:1968907})

\end{itemize}
with early discussion in

\begin{itemize}%
\item [[Peter Hilton]], [[J. H. C. Whitehead]], \emph{Note on the Whitehead Product}, Annals of Mathematics Second Series, Vol. 58, No. 3 (Nov., 1953), pp. 429-442 (\href{https://www.jstor.org/stable/1969746}{jstor:1969746})


\item [[Peter Hilton]], \emph{On the homotopy groups of unions of spheres}, J. London Math. Soc., 1955, 30, 154–172 ([[Hilton55.pdf:file]])



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia \emph{\href{http://en.wikipedia.org/wiki/Whitehead_product}{Whitehead product}}

\end{itemize}
Discussion of Whitehead products specifically of [[homotopy groups of spheres]]:

\begin{itemize}%
\item [[Ioan Mackenzie James]], \emph{On the Suspension Triad}, Annals of Mathematics Second Series, Vol. 63, No. 2 (Mar., 1956), pp. 191-247 (\href{https://www.jstor.org/stable/1969607}{arXiv:1969607})


\item [[Ioan Mackenzie James]], \emph{On the Suspension Sequence}, Annals of Mathematics Second Series, Vol. 65, No. 1 (Jan., 1957), pp. 74-107 (\href{https://www.jstor.org/stable/1969666}{arXiv:1969666})



\end{itemize}
Discussion of Whitehead products in [[homotopy type theory]]:

\begin{itemize}%
\item [[Guillaume Brunerie]], section 3.3 of \emph{On the homotopy groups of spheres in homotopy type theory} (\href{https://arxiv.org/abs/1606.05916}{arXiv:1606.05916})

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

Discussion of Whitehead products in [[rational homotopy theory]] ([[the co-binary Sullivan differential is the dual Whitehead product]]):

\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})


\item Christopher Allday, \emph{Rational Whitehead products and a spectral sequence of Quillen}, Pacific J. Math. Volume 46, Number 2 (1973), 313-323 (\href{https://projecteuclid.org/euclid.pjm/1102946308}{euclid:1102946308})


\item Christopher Allday, \emph{Rational Whitehead product and a spectral sequence of Quillen, II}, Houston Journal of Mathematics, Volume 3, No. 3, 1977 (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.434.8821&rep=rep1&type=pdf}{pdf})


\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})


\item Peter Andrews, [[Martin Arkowitz]], \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})


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


\item Francisco Belchí, [[Urtzi Buijs]], José M. Moreno-Fernández, Aniceto Murillo, \emph{Higher order Whitehead products and $L_\infty$ structures on the homology of a DGL}, Linear Algebra and its Applications, Volume 520 (2017), pages 16-31 (\href{https://arxiv.org/abs/1604.01478}{arXiv:1604.01478})


\item Takahito Naito, \emph{A model for the Whitehead product in rational mapping spaces} (\href{https://arxiv.org/abs/1106.4080}{arXiv:1106.4080})



\end{itemize}
[[!redirects Whitehead products]]



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