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\begin{document}

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\section*{Hopf invariant}

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

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

[[!include homotopy - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{generic_values}{Generic values}\dotfill \pageref*{generic_values} \linebreak
\noindent\hyperlink{hopf_invariant_one}{Hopf invariant one}\dotfill \pageref*{hopf_invariant_one} \linebreak
\noindent\hyperlink{via_sullivan_models}{Via Sullivan models}\dotfill \pageref*{via_sullivan_models} \linebreak
\noindent\hyperlink{whitehead_integral_formula}{Whitehead integral formula}\dotfill \pageref*{whitehead_integral_formula} \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{ReferencesWhiteheadIntegralFormula}{Whitehead's integral formula}\dotfill \pageref*{ReferencesWhiteheadIntegralFormula} \linebreak


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

For $n \in \mathbb{N}$ with $n \gt 1$, consider [[continuous functions]] between [[spheres]] of the form

\begin{displaymath}
\phi \;\colon\; S^{2n-1} \longrightarrow S^n
  \,.
\end{displaymath}
The [[homotopy cofiber]] of $\phi$

\begin{displaymath}
cofib(\phi) \simeq S^n \underset{S^{2n-1}}{\cup} D^{2n}
\end{displaymath}
has [[ordinary cohomology]]

\begin{displaymath}
H^k(cofib(\phi), \mathbb{Z})
  \simeq
  \left\{
    \itexarray{  
      \mathbb{Z} & for\; k = n, 2n;
      \\
      0 &  otherwise
    }
  \right.
  \,.
\end{displaymath}
Hence for $\alpha, \beta$ generators of the cohomology groups in degree $n$ and $2n$ (unique up to choice of sign), respectively, there exists an [[integer]] $h(\phi)$ which expresses the [[cup product]] square of $\alpha$ as a multiple of $\beta$:

\begin{displaymath}
\alpha \cup \alpha = h(\phi) \cdot \beta 
  \,.
\end{displaymath}
This integer $h(\phi) \in \mathbb{Z}$ is called the \emph{Hopf invariant} of $\phi$ (e.g. \hyperlink{MosherTangora86}{Mosher-Tangora 86, p. 33}).

It depends on the choices made only up to sign. In particular it has a well-defined image $[h(\phi)] \in \mathbb{F}_2 = \mathbb{Z}/2\mathbb{Z}$, and as such it is the [[Steenrod square]]

\begin{displaymath}
[h(\phi)] \cdot (-)
    \;\colon\; 
  \mathbb{F}_2
    \simeq
  H^n(cofib(\phi), \mathbb{F}_2) 
   \stackrel{Sq^n}{\longrightarrow}
  H^{2n}(cofib(\phi), \mathbb{F}_2) 
   \simeq
  \mathbb{F}_2
  \,.
\end{displaymath}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{generic_values}{}\subsubsection*{{Generic values}}\label{generic_values}

For $n$ odd, the Hopf invariant necessarily vanishes. For $n$ even however, then there is a homomorphism

\begin{displaymath}
\pi_{2n-1}(S^n) \longrightarrow  \mathbb{Z}
\end{displaymath}
whose [[image]] contains at least the even integers.

\hypertarget{hopf_invariant_one}{}\subsubsection*{{Hopf invariant one}}\label{hopf_invariant_one}

Hence a famous open question in the 1950s was for which maps $\phi$ one has Hopf invariant one, $h(\phi) = 1$.

The \emph{[[Hopf invariant one theorem]]} (\href{Hopf+invariant+one#Adams60}{Adams60}) states that the only maps of Hopf invariant one, $h(\phi) = 1$, are the [[Hopf constructions]] on the four real [[normed division algebras]]:

\begin{itemize}%
\item the [[real Hopf fibration]];


\item the [[complex Hopf fibration]];


\item the [[quaternionic Hopf fibration]];


\item the [[octonionic Hopf fibration]].



\end{itemize}
\hypertarget{via_sullivan_models}{}\subsubsection*{{Via Sullivan models}}\label{via_sullivan_models}

\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{whitehead_integral_formula}{}\subsubsection*{{Whitehead integral formula}}\label{whitehead_integral_formula}

See at \emph{[[Whitehead integral formula]]} and see the references \hyperlink{ReferencesWhiteheadIntegralFormula}{below}

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

\begin{itemize}%
\item [[Whitehead integral formula]]

\begin{itemize}%
\item [[functional cup product]]


\item [[Hopf-Wess-Zumino term]]



\end{itemize}

\item [[Hopf degree theorem]]


\item [[EHP spectral sequence]]



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

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

\begin{itemize}%
\item [[Robert Mosher]], [[Martin Tangora]], p. 33 of \emph{Cohomology operations and applications in homotopy theory}, Harper \& Row 1986 (\href{https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf}{pdf})


\item [[Dale Husemöller]], chapter 15 of \emph{Fibre Bundles}, Graduate Texts in Mathematics 20, Springer New York (1966)


\item [[John Michael Boardman]], B. Steer, \emph{On Hopf Invariants} (\href{http://www.maths.ed.ac.uk/~aar/papers/boarstee.pdf}{pdf})


\item Michael Crabb, [[Andrew Ranicki]], \emph{The geometric Hopf invariant} (\href{http://www.maths.ed.ac.uk/~aar/slides/hopfbeam.pdf}{pdf})



\end{itemize}
See also:

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Hopf_invariant}{Hopf invariant}}

\end{itemize}
\hypertarget{ReferencesWhiteheadIntegralFormula}{}\subsubsection*{{Whitehead's integral formula}}\label{ReferencesWhiteheadIntegralFormula}

Discussion via [[differential forms]]/[[rational homotopy theory]] (see also at [[functional cup product]]):

\begin{itemize}%
\item [[J. H. C. Whitehead]], \emph{An expression of Hopf's invariant as an integral}, Proc. Nat. Acad. Sci. USA 33 (1947), 117–123 (\href{https://www.jstor.org/stable/87688}{jstor:87688})


\item [[Hassler Whitney]], Section 31 in \emph{Geometric Integration Theory}, 1957 (\href{https://press.princeton.edu/titles/3151.html}{pup:3151})


\item [[André Haefliger]], p. 3 of \emph{Whitehead products and differential forms}, In: P.A. Schweitzer (ed.), \emph{Differential Topology, Foliations and Gelfand-Fuks Cohomology}, Lecture Notes in Mathematics, vol 652. Springer 1978 (\href{https://doi.org/10.1007/BFb0063500}{doi:10.1007/BFb0063500})


\item [[Raoul Bott]], [[Loring Tu]], Prop. 17.22 in \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 Lee Rudolph, \emph{Whitehead's Integral Formula, Isolated Critical Points, and the Enhancement of the Milnor Number}, Pure and Applied Mathematics Quarterly Volume 6, Number 2, 2010 (\href{https://arxiv.org/abs/0912.4974}{arXiv:0912.4974})


\item [[Phillip Griffiths]], [[John Morgan]], Section 14.5 of \emph{Rational Homotopy Theory and Differential Forms}, Progress in Mathematics Volume 16, Birkhauser (1981, 2013) (\href{https://doi.org/10.1007/978-1-4614-8468-4}{doi:10.1007/978-1-4614-8468-4})


\item [[Dev Sinha]], [[Ben Walter]], \emph{Lie coalgebras and rational homotopy theory II: Hopf invariants}, Trans. Amer. Math. Soc. 365 (2013), 861-883 (\href{https://arxiv.org/abs/0809.5084}{arXiv:0809.5084}, \href{https://doi.org/10.1090/S0002-9947-2012-05654-6}{doi:10.1090/S0002-9947-2012-05654-6})


\item [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Twisted Cohomotopy implies M5 WZ term level quantization]]}, 2019 (\href{https://arxiv.org/abs/1906.07417}{arXiv:1906.07417})



\end{itemize}
[[!redirects Hopf invariants]]



\end{document}