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\section*{Hopf construction}

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

\hypertarget{bundles}{}\paragraph*{{Bundles}}\label{bundles}

[[!include bundles - contents]]

\hypertarget{topology}{}\paragraph*{{Topology}}\label{topology}

[[!include topology - contents]]

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

[[!include homotopy - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{HomotopyFiber}{Homotopy fiber}\dotfill \pageref*{HomotopyFiber} \linebreak
\noindent\hyperlink{relation_to_hopf_invariant}{Relation to Hopf invariant}\dotfill \pageref*{relation_to_hopf_invariant} \linebreak
\noindent\hyperlink{RealizationAsAQuasiFibration}{Realization as a quasi-fibration}\dotfill \pageref*{RealizationAsAQuasiFibration} \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}

Let $X$ be an [[H-space]]. The \emph{Hopf construction} (\hyperlink{Hopf35}{Hopf 35}) on $X$ is a [[fibration]]

\begin{displaymath}
X \hookrightarrow X\ast X \to \Sigma X
\end{displaymath}
whose [[fiber]] is $X$, whose base space is the [[suspension]] of $X$, and whose total space is the [[join of topological spaces|join]] of $X$ with itself. (\hyperlink{Stasheff70}{Stasheff 70, chapter 1}).

Specialized to $X$ the [[sphere]] of [[dimension]] [[0-sphere|0]], [[1-sphere|1]], [[3-sphere|3]], or [[7-sphere|7]], the Hopf construction yields the \emph{\hyperlink{HopfFibrations}{Hopf fibrations}}. (And by the [[Hopf invariant one theorem]] these are the only dimensions for in which spheres are H-spaces.)

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

\begin{defn}
\label{SuspensionJoin}\hypertarget{SuspensionJoin}{}
Write $I \coloneqq [0,1]$ for the unit interval, regarded as a [[topological space]].

Let $X,Y$ be [[topological spaces]].

\begin{enumerate}%
\item The [[suspension]] $\Sigma X$ is the [[quotient space]]

\begin{displaymath}
\Sigma X \coloneqq (X \times I)_{/\sim}
\end{displaymath}
by the [[equivalence relation]] given by

\begin{displaymath}
(x_1,0) \sim (x_2,0) \;\,,\;\;  (x_1, 1) \sim (x_2, 1) \;\;\; \forall x_1,x_2 \in X
\end{displaymath}

\item The [[join of topological spaces|join]] $X \ast Y$ is the [[quotient space]]

\begin{displaymath}
X \ast Y \coloneqq (X \times I \times Y)_{/\sim}
\end{displaymath}
by the [[equivalence relation]]

\begin{displaymath}
(x, 0, y_1) \simeq (x,0,y_2) \;\;,\;\; (x_1,1,y) \sim (x_2, 1, y)
  \,.
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{defn}
\label{HopfConstruction}\hypertarget{HopfConstruction}{}
Given a [[continuous function]] of the form

\begin{displaymath}
f \colon X \times Y \longrightarrow Z
\end{displaymath}
its \emph{Hopf construction} is the continuous function

\begin{displaymath}
H_f \colon X \ast Y \longrightarrow \Sigma Z
\end{displaymath}
out of the [[join of topological spaces|join]] into the [[suspension]], given in the coordinates of def. \ref{SuspensionJoin} by

\begin{displaymath}
H_f \colon (x,t,y) \mapsto (f(x,y), t)
  \,.
\end{displaymath}
\end{defn}
\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{HomotopyFiber}{}\subsubsection*{{Homotopy fiber}}\label{HomotopyFiber}

If $X$ is an ``grouplike H-space'', in that it is a topological [[magma]] such that left multiplication acts by [[weak homotopy equivalences]], then the [[homotopy fiber]] of the Hopf construction $X \ast X \to \Sigma X$ over any point is [[weak homotopy equivalence|weakly homotopy equivalent]] to $X$ (\href{https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=76088#Comment_76088}{here}).

Beware that it may not generally be true that the ordinary [[fibers]] of the Hopf construction are [[weak homotopy equivalence|weakly homotopy equivalent]] to the [[homotopy fibers]], see also the discussion of [[quasifibrations]] \hyperlink{RealizationAsAQuasiFibration}{below}. But in the classical examples it Happens to be the case, see at [[Hopf fibration]].

\hypertarget{relation_to_hopf_invariant}{}\subsubsection*{{Relation to Hopf invariant}}\label{relation_to_hopf_invariant}

Consider $X = S^{n-1}$ a [[sphere]]

\begin{prop}
\label{HopfInvariantFromProductOfDegrees}\hypertarget{HopfInvariantFromProductOfDegrees}{}
Given a [[continuous function]]

\begin{displaymath}
f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}
\end{displaymath}
the [[degree of a continuous function|degrees]]

\begin{displaymath}
\alpha \coloneqq deg(f(x,-)) \;\;\; \beta \coloneqq deg(f(-,x))
\end{displaymath}
are independent of the choice of $x \in S^{n-1}$. The [[Hopf invariant]] $h$ of the Hopf construction $H_f$ of $f$, def. \ref{HopfConstruction}, is the product of these two:

\begin{displaymath}
h(H_f) = \alpha \beta
  \,.
\end{displaymath}
\end{prop}
(\hyperlink{MosherTangora}{Mosher-Tangora, exercises to section 4, page 38})

\hypertarget{RealizationAsAQuasiFibration}{}\subsubsection*{{Realization as a quasi-fibration}}\label{RealizationAsAQuasiFibration}

Beware that \hyperlink{Stasheff70}{Stasheff 70, theorem 1.2} claims that Sugawara claimed that the Hopf construction for any [[CW-complex|CW]] H-space is necessarily a [[quasifibration]]. But it seems (\href{https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=75986#Comment_75986}{here}) that Sugawara never actually claimed this and also (\href{https://nforum.ncatlab.org/discussion/3391/hopf-fibration/?Focus=75985#Comment_75985}{here}) that it is not actually the case.

A different but homotopy-equivalent realization of the Hopf construction, which over grouplike H-spaces is guaranteed to be a [[quasifibration]], is maybe given in \hyperlink{DoldLashof59}{Dold-Lashof 59}, see also \hyperlink{Stasheff70}{Stasheff 70, theorem 1.4}.

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

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

When $X$ is a [[sphere]] that is an $H$-space, namely, one of the [[groups]] $S^0 = \mathbb{Z}/2$ the [[group of order 2]], $S^1 = U(1)$ the [[circle group]], the [[3-sphere]] [[special unitary group]] $S^3 = SU(2)$; or the [[7-sphere]] $S^7$ with its [[Moufang loop]] structure, then the Hopf construction produces the four \emph{[[Hopf fibrations]]}:

\begin{enumerate}%
\item $S^0 \hookrightarrow S^1 \to S^1$ -- [[real Hopf fibration]]
\item $S^1 \hookrightarrow S^3 \to S^2$ -- [[complex Hopf fibration]]
\item $S^3 \hookrightarrow S^7 \to S^4$ -- [[quaternionic Hopf fibration]]
\item $S^7 \hookrightarrow S^{15} \to S^8$ -- [[octonionic Hopf fibration]]

\end{enumerate}
In detail, let $A \in \{\mathbb{R}, \mathbb{C}, \mathbb{H}, \mathbb{O}\}$ be one of the real [[normed division algebras]] and write

\begin{displaymath}
n \coloneqq dim_{\mathbb{R}}(A) \in \{1,2,4,8\}
\end{displaymath}
for its [[dimension]] as a real vector space. Then the $S^{n-1}$-sphere may be identified with the subspace of unit norm elements in $A$:

\begin{displaymath}
S^{n-1} \simeq
  \left\{
    x \in A
    \,;
    {\vert x\vert}^2 = 1
  \right\}
  \,.
\end{displaymath}
Consider then the pairing map

\begin{displaymath}
f \colon S^{n-1}\times S^{n-1} \longrightarrow S^{n-1}
\end{displaymath}
which is the restriction to these unit norm elements of the product in $A$:

\begin{displaymath}
f \colon (x,y) \mapsto x \cdot y
\end{displaymath}
This is well defined by the very property that for [[normed division algebras]] the norm is multiplicative.

Accordingly, the [[join of topological spaces|join]] of two such spheres is naturally parameterized as follows

\begin{displaymath}
S^{n-1}\ast S^{n-1}
  =
  (S^{n-1}\times I \times S^{n-1})_{/\sim}
  \simeq
  \left\{
    (x,t,y)
    \,,
    {\vert x \vert}^2 = 2t
    \,,\;
    {\vert y\vert}^2 = 2 - 2t
  \right\}
\end{displaymath}
which makes manifest that

\begin{displaymath}
S^{n-1} \ast S^{n-1} \simeq S^{2n-1}
\end{displaymath}
Similarly, the [[suspension]] is parameterized by

\begin{displaymath}
\Sigma S^{n-1}
  =
  (S^{n-1}\times I)_{/\sim}
  \simeq
  \left\{
    (z,t)
    \,,\;
    {\vert z \vert}^2 + (1 - 2t)^2   = 1
  \right\}
\end{displaymath}
where we take $I = [0,1]$ and $t \in I$. This makes manifest that

\begin{displaymath}
\Sigma S^{n-1} \simeq S^n
  \,.
\end{displaymath}
Moreover, in this parameterization the Hopf construction, def. \ref{HopfConstruction}, which is given by

\begin{displaymath}
(x,y) \mapsto x \cdot \overline{y}
\end{displaymath}
manifestly gives the Hopf fibration map.

Notice that it is again the multiplicativity of the norm in division algebras which makes this work: if ${\vert x \vert}^2 = 2t$ and ${\vert y\vert}^2 = 2 - 2t$ then it follows that

\begin{displaymath}
\begin{aligned}
     {\vert x \cdot \overline{y}\vert}^2 + (1- 2t)^2
     & = 
     {\vert x \vert}^2 {\vert y \vert}^2
     + (1-2t)^2
     \\
     & = 2t (2-2t) + (1 - 2t)^2 
     \\
     & = 1
  \end{aligned}
  \,,
\end{displaymath}
hence that indeed we have a well-defined map like so:

\begin{displaymath}
\itexarray{
      S^7 & \longrightarrow & S^4
      \\
     \left\{
       (x,t,y)
        \,,
       {\vert x \vert}^2 = 2t
        \,,\;
        {\vert y\vert}^2 = 2 - 2t
     \right\}
     &\stackrel{{(x,y) \mapsto z \coloneqq x \cdot \overline{y}}\atop{t \mapsto t}}{\longrightarrow}&
    \left\{
       (z,t)
       \,,\;
       {\vert z \vert}^2 + (1 - 2t)^2   = 1
    \right\}  
  }
  \,.
\end{displaymath}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The original sources are

\begin{itemize}%
\item [[Heinz Hopf]], \emph{\"U{}ber die Abbildungen von Sph\"a{}ren auf Sph\"a{}ren niedrigerer Dimension}, Fund. Math. 25: 427--440 (1935) (\href{https://eudml.org/doc/212801}{Euclid})


\item [[George Whitehead]], \emph{On the homotopy groups of spheres and rotation groups}, Annals of Mathematics. Second Series 43 (4): 634--640, (1942) (\href{http://www.jstor.org/stable/1968956}{JSTOR})


\item [[Albrecht Dold]], [[Richard Lashof]], \emph{Principal quasifibrations and fibre homotopy equivalence of bundles}, Illinois J. Math. Volume 3, Issue 2 (1959), 285-305 (\href{https://projecteuclid.org/euclid.ijm/1255455128}{euclid:1255455128})



\end{itemize}
Review inclides

\begin{itemize}%
\item [[Jim Stasheff]], chapter 1 in \emph{H-Spaces from a Homotopy point of view}, Lecture Notes in Mathematics Volume 161 1970

\end{itemize}
Textbook accounts include

\begin{itemize}%
\item [[Robert Mosher]], [[Martin Tangora]], p. 38 of \emph{Cohomology Operations and Application in Homotopy Theory}, Harper and Row (1968) (\href{http://math.jhu.edu/~anakade1/notes/Dec%202014/Mosher%20and%20Tangora%20Notes/mosher-tangora.pdf}{pdf})


\item Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, section 10.6 of \emph{Algebraic topology from a homotopical viewpoint}, Springer (2002) (\href{http://tocs.ulb.tu-darmstadt.de/106999419.pdf}{toc pdf})



\end{itemize}
See also

\begin{itemize}%
\item Guillermo Moreno, \emph{Hopf construction map in higher dimensions} (\href{https://arxiv.org/abs/math/0404172}{arXiv:math/0404172})


\item Feza Guersey, Chia-Hsiung Tze, (4b.2) in \emph{On the role of Division, Jordan and Related algebras in Particle Physics}



\end{itemize}
Discussion of the situation in [[parameterized homotopy theory]] includes

\begin{itemize}%
\item A. L. Cook, M.C. Crabb, \emph{Fiberwise Hopf structures on sphere bundles}, J. London Math. Soc. (2) 48 (1993) 365-384 (\href{http://www.maths.ed.ac.uk/~aar/papers/crabbcook.pdf}{pdf})


\item Kouyemon Iriye, \emph{Equivariant Hopf structures on a sphere}, J. Math. Kyoto Univ. Volume 35, Number 3 (1995), 403-412 (\href{http://projecteuclid.org/euclid.kjm/1250518704}{Euclid})



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Hopf_construction}{Hopf construction}}

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

[[!redirects Hopf map]] [[!redirects Hopf maps]]



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