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\begin{document}

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\section*{homotopy groups of spheres}

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

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

[[!include homotopy - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable Homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{homotopy_groups_of_spheres}{}\section*{{Homotopy groups of spheres}}\label{homotopy_groups_of_spheres}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Tables}{Tables}\dotfill \pageref*{Tables} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{serre_finiteness_theorem}{Serre finiteness theorem}\dotfill \pageref*{serre_finiteness_theorem} \linebreak
\noindent\hyperlink{nishida_nilpotence_theorem}{Nishida nilpotence theorem}\dotfill \pageref*{nishida_nilpotence_theorem} \linebreak
\noindent\hyperlink{RelationToFramedBordismRing}{Relation to the framed bordism ring}\dotfill \pageref*{RelationToFramedBordismRing} \linebreak
\noindent\hyperlink{jhomomorphism_and_adams_einvariant}{J-homomorphism and Adams e-invariant}\dotfill \pageref*{jhomomorphism_and_adams_einvariant} \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{image_of_the_jhomomorphism}{Image of the J-homomorphism}\dotfill \pageref*{image_of_the_jhomomorphism} \linebreak
\noindent\hyperlink{formalization_in_homotopy_type_theory}{Formalization in homotopy type theory}\dotfill \pageref*{formalization_in_homotopy_type_theory} \linebreak



\vspace{.5em} \hrule \vspace{.5em}
\begin{quote}%
My initial inclination was to call this book \emph{[[The Music of the Spheres]]}, but I was dissuaded from doing so by my diligent publisher, who is ever mindful of the sensibilities of librarians. (\hyperlink{Ravenel86}{D. Ravenel 86, preface})

With all due respect to anyone interested in them, the stable homotopy groups of spheres are a mess. ([[Stable homotopy and generalised homology|J. F. Adams 74, p 204]])


\end{quote}

\vspace{.5em} \hrule \vspace{.5em}
\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The [[homotopy groups]] of [[spheres]] $\pi_{n+k}(S^n)$ are the [[homotopy classes]] of maps $S^{n+k} \longrightarrow S^n$

\begin{displaymath}
\pi_{n+k}(S^n) \coloneqq [S^{n+k}, S^n]
  \,.
\end{displaymath}
For fixed $k$, the [[colimit]] over $n$ with respect to the [[suspension]] homomorphism

\begin{displaymath}
\pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})
\end{displaymath}
over all $\pi_{n+k}(S^n)$ (called the $k$-[[stem]]) is called the \emph{[[stable homotopy groups]] of spheres} (also: the ``stable $k$-[[stem]]'')

\begin{displaymath}
\pi_k^S = \coloneqq \underset{\longrightarrow}{\lim}_n \pi_{n+k}(S^n)  
 \,.
\end{displaymath}
In fact, by the [[Freudenthal suspension theorem]], the value of the $\pi_{n+k}(S^n)$ stabilizes for $n \gt k+1$ (depend only on $k$ in this range), whence the name.

The [[stable homotopy groups]] of sphere are equivalently the [[homotopy groups of a spectrum]] for the [[sphere spectrum]] $\mathbb{S}$

\begin{displaymath}
\pi_k^S = \pi_k(\mathbb{S})
  \,.
\end{displaymath}
The stable homotopy groups of spheres are notorious for their immense computational richness. Many of the tools of [[algebraic topology]] and [[stable homotopy theory]] were devised to compute more and more of the stable stems. This notably include the [[Adams spectral sequence]], the [[Adams-Novikov spectral sequence]].

\hypertarget{Tables}{}\subsection*{{Tables}}\label{Tables}

The first few stable homotopy groups of the [[sphere spectrum]] $\mathbb{S}$ are

\begin{tabular}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l}
$k =$&0&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&$\cdots$\\
\hline 
$\pi_k(\mathbb{S}) =$&$\mathbb{Z}$&$\mathbb{Z}/2$&$\mathbb{Z}/2$&$\mathbb{Z}/{24}$&$0$&$0$&$\mathbb{Z}/2$&$\mathbb{Z}/{240}$&$(\mathbb{Z}/2)^2$&$(\mathbb{Z}/2)^3$&$\mathbb{Z}/6$&$\mathbb{Z}/{504}$&$0$&$\mathbb{Z}/3$&$(\mathbb{Z}/2)^2$&$\mathbb{Z}/{480} \oplus \mathbb{Z}/2$&$\cdots$\\
\end{tabular}

The following tables show the [[p-primary group|p-primary decomposition]] of these and the following stable homotopy groups.

The horizontal index is the degree $n$ of the stable homotopy group $\pi_n$. The appearance of a string of $k$ connected dots vertically above index $n$ means that there is a [[direct sum|direct summand]] [[primary group]] of [[order of a group|order]] $p^k$. The bottom rows in each case are given by the [[image of the J-homomorphism]].

See at \emph{\href{fundamental+theorem+of+finitely+generated+abelian+groups#GraphicalRepresentation}{fundamental theorem of finitely generated abelian grouops -- Graphical representation}} for details on the notation used in these table, and see example \ref{InterpretTable} below for illustration.

(the following graphics are taken from \hyperlink{Hatcher}{Hatcher}, based on (\hyperlink{Ravenel86}{Ravenel 86})

\textbf{$p = 2$-primary component}



\textbf{$p = 3$-primary component}



\textbf{$p = 5$-primary component}



\begin{example}
\label{InterpretTable}\hypertarget{InterpretTable}{}
The [[finite abelian group]] $\pi_3(\mathbb{S}) \simeq \mathbb{Z}_{24}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_8 \oplus \mathbb{Z}_3$. Here $8 = 2^3$ corresponds to the three dots above $n = 3$ in the first table, and $3 = 3^1$ to the single dot over $n = 3$ in the second.

The [[finite abelian group]] $\pi_7(\mathbb{S}) \simeq \mathbb{Z}_{240}$ decomposes into [[primary groups]] as $\simeq \mathbb{Z}_{16} \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_5$. Here $16 = 2^4$ corresponds to the four dots above $n = 7$ in the first table, and $3 = 3^1$ to the single dot over $n = 7$ in the second and $5 = 5^1$ to the single dot over $n = 7$ in the third table.

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

\hypertarget{serre_finiteness_theorem}{}\subsubsection*{{Serre finiteness theorem}}\label{serre_finiteness_theorem}

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{([[Serre finiteness theorem]])}

The [[homotopy group]] $\pi_{n+k}(S^k)$ is a [[finite group]] except

\begin{enumerate}%
\item for $n = 0$ in which case $\pi_k(S^k) = \mathbb{Z}$;


\item $k = 2m$ and $n = 2m -1$ in which case

\begin{displaymath}
\pi_{4m - 1}(S^{2m}) \simeq \mathbb{Z} \oplus F_m
\end{displaymath}
for $F_m$ a [[finite group]].



\end{enumerate}
\end{theorem}
(\hyperlink{Serre53}{Serre 53})

\hypertarget{nishida_nilpotence_theorem}{}\subsubsection*{{Nishida nilpotence theorem}}\label{nishida_nilpotence_theorem}

The [[Nishida nilpotence theorem]]\ldots{}

\hypertarget{RelationToFramedBordismRing}{}\subsubsection*{{Relation to the framed bordism ring}}\label{RelationToFramedBordismRing}

By [[Thom's theorem]], for any [[(B,f)-structure]] $\mathcal{B}$, there is an [[isomorphism]] (of [[commutative rings]])

\begin{displaymath}
\Omega^{\mathcal{B}}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(M\mathcal{B})
\end{displaymath}
from the [[cobordism ring]] of manifolds with stable normal $\mathcal{B}$-structure to the [[homotopy groups of a spectrum|homotopy groups]] of the universal $\mathcal{B}$-[[Thom spectrum]].

Now for $\mathcal{B} = Fr$ [[framing]] structure, then

\begin{displaymath}
M Fr \simeq \mathbb{S}
\end{displaymath}
is equivalently the [[sphere spectrum]]. Hence in this case [[Thom's theorem]] states that there is an isomorphism

\begin{displaymath}
\Omega^{fr}_\bullet
  \overset{\simeq}{\longrightarrow}
  \pi_\bullet(\mathbb{S})
\end{displaymath}
between the framed cobordism ring and the stable homotopy groups of spheres.

For discussion of computation of $\pi_\bullet(\mathbb{S})$ this way, see for instance (\hyperlink{WangXu10}{Wang-Xu 10, section 2}) and (\hyperlink{Putnam}{Putnam}).

For instance

\begin{itemize}%
\item $\Omega^{fr}_0 = \mathbb{Z}$ because there are two $k$-framings on a single point, corresponding to $\pi_0(O(k)) \simeq \mathbb{Z}_2$, the negative of a point with one framing is the point with the other framing, and so under disjoint union, the framed points form the group of integers;


\item $\Omega^{fr}_1 = \mathbb{Z}_2$ because the only compact connected 1-manifold is the circle, there are two framings on the circle, corresponding to $\pi_1(O(k)) \simeq \mathbb{Z}_2$ and they are their own negatives.



\end{itemize}
\hypertarget{jhomomorphism_and_adams_einvariant}{}\subsubsection*{{J-homomorphism and Adams e-invariant}}\label{jhomomorphism_and_adams_einvariant}

The following characterizes the [[image]] of the [[J-homomorphism]]

\begin{displaymath}
J \;\colon\; \pi_\bullet(O) \longrightarrow \pi_\bullet(\mathbb{S})
\end{displaymath}
from the [[homotopy groups]] of the [[stable orthogonal group]] to the stable homotopy groups of spheres. This was first conjectured in (\hyperlink{Adams66}{Adams 66}) (since called the \emph{Adams conjecture}) and then proven in (\hyperlink{Quillen71}{Quillen 71}).

\begin{remark}
\label{}\hypertarget{}{}
By the discussion at \emph{\href{orthogonal%20group#HomotopyGroups}{orthogonal group -- homotopy groups}} we have that the [[homotopy groups]] of the [[stable orthogonal group]] are

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
$\pi_n(O)$&$\mathbb{Z}_2$&$\mathbb{Z}_2$&0&$\mathbb{Z}$&0&0&0&$\mathbb{Z}$\\
\end{tabular}

Because all groups appearing here and in the following are [[cyclic groups]], we instead write down the [[order of a group|order]]

\begin{tabular}{l|l|l|l|l|l|l|l|l}
$n\;mod\; 8$&0&1&2&3&4&5&6&7\\
\hline 
${\vert\pi_n(O)\vert}$&2&2&1&$\infty$&1&1&1&$\infty$\\
\end{tabular}

\end{remark}
\begin{theorem}
\label{}\hypertarget{}{}
The [[stable homotopy groups of spheres]] $\pi_n(\mathbb{S})$ are the [[direct sum]] of the ([[cyclic group|cyclic]]) [[image]] of the [[J-homomorphism]], and the [[kernel]] of the [[Adams e-invariant]].

Moreover,

\begin{itemize}%
\item for $n = 0 \;mod \;$ and $n = 1 \;mod \; 8$ and $n$ positive the J-homomorphism is [[injection|injective]], hence its image is $\mathbb{Z}_2$,


\item for $n = 3\; mod\; 8$ and $n = 7 \; mod \; 8$ hence for $n = 4 k -1$, the [[order of a group|order]] of the image is equal to the [[denominator]] of $B_{2k}/4k$, where $B_{2k}$ is the [[Bernoulli number]]


\item for all other cases the image is necessarily zero.



\end{itemize}
\end{theorem}
[[!include image of J -- table]]

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

\begin{itemize}%
\item [[EHP spectral sequence]]


\item [[Adams–Novikov spectral sequence]]


\item [[Music of the Spheres]]



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

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

Introductions and surveys include

\begin{itemize}%
\item Alex Wright, \emph{Homotopy groups of spheres: A very basic introduction} (\href{http://www-personal.umich.edu/~alexmw/HomotopyGroupsOfSoheres.pdf}{pdf})


\item [[Guozhen Wang]], Zhouli Xu \emph{A survey of computations of homotopy groups of Spheres and Cobordisms}, 2010 (\href{http://math.mit.edu/~guozhen/homotopy%20groups.pdf}{pdf})


\item [[Andrew Putman]], \emph{Homotopy groups of spheres and low-dimensional topology} (\href{http://www.math.rice.edu/~andyp/notes/HomotopySpheresLowDimTop.pdf}{pdf})


\item [[Allen Hatcher]], \emph{Pictures of stable homotopy groups of spheres} (\href{http://www.math.cornell.edu/~hatcher/stemfigs/stems.html}{html})


\item [[Mark Mahowald]], [[Doug Ravenel]], \emph{Towards a Global Understanding of the Homotopy Groups of Spheres} (\href{http://www.math.rochester.edu/people/faculty/doug/mypapers/global.pdf}{pdf})


\item [[Haynes Miller]], [[Doug Ravenel]], \emph{Mark Mahowald's work on the homotopy groups of spheres} (\href{http://www-math.mit.edu/~hrm/ksem/miller-ravenel.pdf}{pdf})


\item [[eom]], \emph{\href{http://www.encyclopediaofmath.org/index.php/Spheres,_homotopy_groups_of_the}{Spheres, homotopy groups of the}}


\item [[Doug Ravenel]], \emph{[[Complex cobordism and stable homotopy groups of spheres]]}, since 1986 (\href{http://www.math.rochester.edu/people/faculty/doug/mu.html}{web})


\item [[Stanley Kochmann]], \emph{Stable Homotopy Groups of Spheres -- A Computer-Assisted Approach}, Lecture Notes in Mathematics, 1990


\item [[Stanley Kochmann]], section 5 of of \emph{[[Bordism, Stable Homotopy and Adams Spectral Sequences]]}, AMS 1996



\end{itemize}
A tabulation of stable homotopy groups of spheres is in

\begin{itemize}%
\item [[Doug Ravenel]], Appendix 3 of \emph{[[Complex cobordism and stable homotopy groups of spheres]]} (\href{http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA3.pdf}{pdf})

\end{itemize}
Original articles on basic properties include

\begin{itemize}%
\item [[Jean-Pierre Serre]] \emph{Groupes d'homotopie et classes de groupes abelien}, Ann. of Math. 58 (1953), 258--294.

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Homotopy_groups_of_spheres}{Homotopy groups of spheres}}


\item MO, \emph{\href{http://mathoverflow.net/questions/31004/computational-complexity-of-computing-homotopy-groups-of-spheres}{Computational complexity of computing homotopy groups of spheres}}



\end{itemize}
\hypertarget{image_of_the_jhomomorphism}{}\subsubsection*{{Image of the J-homomorphism}}\label{image_of_the_jhomomorphism}

Discussion of the [[image]] of the [[J-homomorphism]] is due to

\begin{itemize}%
\item [[John Adams]], \emph{On the groups $J(X)$ IV}, Topology 5: 21,(1966) \emph{Correction}, Topology 7 (3): 331 (1968)


\item [[Daniel Quillen]], \emph{The Adams conjecture}, Topology. an International Journal of Mathematics 10: 67--80 (1971)



\end{itemize}
\hypertarget{formalization_in_homotopy_type_theory}{}\subsubsection*{{Formalization in homotopy type theory}}\label{formalization_in_homotopy_type_theory}

For formalization in [[homotopy type theory]] see

\begin{itemize}%
\item [[homotopytypetheory:HomePage|HoTT wiki]], \emph{[[homotopytypetheory:homotopy groups of spheres]]}


\item [[UF-IAS-2012]], \emph{\href{http://uf-ias-2012.wikispaces.com/HomotopyGroupsOfSpheres}{HomotopyGroupsOfSpheres}}


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



\end{itemize}
[[!redirects homotopy group of a sphere]] [[!redirects homotopy groups of a sphere]] [[!redirects homotopy groups of spheres]]

[[!redirects stable homotopy group of spheres]] [[!redirects stable homotopy groups of spheres]]



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