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\section*{equivariant sphere spectrum}

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

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

[[!include stable homotopy theory - contents]]

\hypertarget{representation_theory}{}\paragraph*{{Representation theory}}\label{representation_theory}

[[!include representation theory - 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{EquivariantHomotopyGroups}{Equivariant homotopy groups}\dotfill \pageref*{EquivariantHomotopyGroups} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{Z2equivariance}{$\mathbb{Z}_2$-equivariance}\dotfill \pageref*{Z2equivariance} \linebreak
\noindent\hyperlink{cyclic_equivariance}{Cyclic ($G_D \hookrightarrow SO(2)$-)equivariance}\dotfill \pageref*{cyclic_equivariance} \linebreak
\noindent\hyperlink{equivariance_2}{$G_{ADE} \hookrightarrow SO(3)$-equivariance}\dotfill \pageref*{equivariance_2} \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{equivariance_3}{$\mathbb{Z}/2$-equivariance}\dotfill \pageref*{equivariance_3} \linebreak
\noindent\hyperlink{equivariance_4}{$\mathbb{Z}/4$-equivariance}\dotfill \pageref*{equivariance_4} \linebreak


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

The [[sphere spectrum]] in ([[global equivariant stable homotopy theory|global]]) [[equivariant stable homotopy theory]].

Its [[RO(G)-grading|RO(G)-graded]] [[homotopy groups]] are the equivariant version of the [[stable homotopy groups of spheres]].

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

The $G$-equivariant sphere spectrum is the [[equivariant suspension spectrum]] of the [[0-sphere]] $S^0 = \ast_+$

\begin{displaymath}
\mathbb{S} = \Sigma^\infty_G S^0
\end{displaymath}
for $S^0$ regarded as equipped with the (necessarily) trivial $G$-action. It follows that for $V$ an orthogonal linear $G$-representation then in [[RO(G)-degree]] $V$ the equivariant sphere spectrum is the corresponding [[representation sphere]] $\mathbb{S}(V) \simeq S^V$.

(e.g. \hyperlink{Schwede15}{Schwede 15, example 2.10})

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

\hypertarget{EquivariantHomotopyGroups}{}\subsubsection*{{Equivariant homotopy groups}}\label{EquivariantHomotopyGroups}

Just as for the plain [[sphere spectrum]], the [[equivariant homotopy groups]] of the equivariant sphere spectrum in ordinary [[integer]] degrees $n$ are all [[torsion subgroup|torsion]], except at $n = 0$:

\begin{displaymath}
\pi_n^H(\mathbb{S})\otimes \mathbb{Q} =
  \left\{
    \itexarray{
       \cdots & for \; n = 0
       \\
       0 & otherwise
    }
  \right.
\end{displaymath}
(\hyperlink{GreenleesMay95}{Greenlees-May 95, prop. A.3})

But in some [[RO(G)-degrees]] there may appear further non-torsion groups, see the \hyperlink{Examples}{examples} below.

In degree 0, the [[tom Dieck splitting]] applied to the [[equivariant suspension spectrum]] $\mathbb{S} = \Sigma^\infty_G S^0$ gives that $\pi_0^G(\mathbb{S})$ is the [[free abelian group]] on the set of [[conjugacy classes]] of [[subgroups]] of $G$:

\begin{equation}
\pi_0^G(\mathbb{S})
  \simeq
  \underset{[H \subset G]}{\oplus} \pi_0^{W_G H}(\Sigma_+^\infty E (W_G H))
  \simeq
  \mathbb{Z}[conjugacy\;classes\;of\;subgroups]
\label{pi_0ofEquivariantSphere}\end{equation}
(e.g. \hyperlink{Schwede15}{Schwede 15, p. 64})

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

\hypertarget{Z2equivariance}{}\subsubsection*{{$\mathbb{Z}_2$-equivariance}}\label{Z2equivariance}

Consider $G= \mathbb{Z}_2$ the [[cyclic group of order 2]] and write $\pi^S_{p,q}$ for the homotopy group in [[RO(G)-grading|RO(G)-degree]] given by the representation on $\mathbb{R}^{p+q}$ with $\mathbb{Z}_2$ acts by [[reflection]] on the first $p$ coordinates, and trivially on the remaining $q$ coordinates:

The following groups contain $\mathbb{Z}$-summands:

\begin{itemize}%
\item $\pi^S_{0,0} = \mathbb{Z} \times \mathbb{Z}$ (by the [[equivariant Hopf degree theorem]], \href{Hopf+degree+theorem#EquivariantCohomotopyOfRepresentationSphereOfSignRepresentationInThatDegree}{this Example})


\item $\pi^S_{1,0}$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 8.7, p. 25}) generated by the [[complex Hopf fibration]] $\hat \eta$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, p. 24})


\item $\pi^S_{2,0}$ generated by $\hat \eta^2$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 9.6, p. 27})


\item $\pi^S_{3,0}$ generated by $\hat \eta^3$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 10.11, p. 31})


\item $\pi^S_{4,0}$ generated by $\hat \eta^4$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 11.3, p. 32})


\item $\pi^S_{5,0}$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 12.4, p. 34})


\item $\pi^S_{6,0}$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 13.12, p. 38})


\item $\pi^S_{7,0}$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 14.18, p. 45})


\item $\pi^S_{8,0}$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, theorem 15.26, p. 54})



\end{itemize}
In addition we have

\begin{itemize}%
\item $\pi^S_{2,1} \simeq \mathbb{Z}/24$ generated by the [[quaternionic Hopf fibration]] $\hat \nu$ (\hyperlink{ArakiIriye82}{Araki-Iriye 82, prop. 10.1, p. 28 with theorem 10.11, p. 31})

\end{itemize}
 In summary and more generally we have the following $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres in low bidegree:

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
The table shows the $\mathbb{Z}/2$-equivariant stable homotopy groups of spheres $\pi^S_{p,q}$ with $p+q$ increasing horizontally to the right, and $p$ increasing vertically upwards. The origin is the double-circled $\pi^S_{0,0} = \mathbb{Z}^2$. The [[complex Hopf fibration]] $\hat\eta$ generates $\pi^S_{1,0} = \mathbb{Z}$, and the [[quaternionic Hopf fibration]] generates $\pi^S_{2,1} = \mathbb{Z}/24$

graphics grabbed from \hyperlink{Dugger08}{Dugger 08}, based on \hyperlink{ArakiIriye82}{Araki-Iriye 82}

(beware that \hyperlink{Dugger08}{Dugger 08} uses a different bi-degree labeling convention: the $(p,q)$ here is $(p+q,p)$ in \hyperlink{Dugger08}{Dugger 08}, matching the coordinates of the above table)


\end{quote}
\begin{tabular}{l|l|l|l|l|l}
&$n_{sgn} + n$&$0$&$1$&$2$&$3$\\
\hline 
$n_{sgn}$&$\pi^{st}_{n_{sgn} + n}$&&&&\\
$2$&&$0$&$0$&$\mathbb{Z} = \langle (h_{\mathbb{C}})^2\rangle$&$\mathbb{Z}_{24} = \langle h_{\mathbb{H}}\rangle$\\
$1$&&$0$&$\mathbb{Z} = \langle h_{\mathbb{C}}\rangle$&$\pi_1^{st} \oplus \mathbb{Z}_2$&$\pi^{st}_2 \oplus \mathbb{Z}_2$\\
$0$&&$\pi_0^{st} \oplus \mathbb{Z}$&$\pi_1^{st} \oplus (\mathbb{Z}_2)^2$&$\pi_2^{st} \oplus (\mathbb{Z}_2)^2$&$\pi_3^{st} \oplus \mathbb{Z}_8 \oplus \mathbb{Z}_{24}$\\
\end{tabular}

\hypertarget{cyclic_equivariance}{}\subsubsection*{{Cyclic ($G_D \hookrightarrow SO(2)$-)equivariance}}\label{cyclic_equivariance}

The global equivariant sphere spectrum for all the [[cyclic groups]] over the [[circle group]] is canonically a [[cyclotomic spectrum]] and as such is the [[tensor unit]] in the [[monoidal (infinity,1)-category]] of [[cyclotomic spectra]] (see there).

\hypertarget{equivariance_2}{}\subsubsection*{{$G_{ADE} \hookrightarrow SO(3)$-equivariance}}\label{equivariance_2}

See at \emph{\href{quaternionic+Hopf+fibration#ClassInEquivariantStableHomotopyTheory}{quaternionic Hopf fibration -- Class in equivariant stable homotopy theory}}

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

\begin{itemize}%
\item The [[cohomology theory]] [[Brown representability theorem|represented]] by the equivariant sphere spectrum is \emph{[[equivariant stable cohomotopy]]}.

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

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

General lecture notes include

\begin{itemize}%
\item [[Stefan Schwede]], example 2.10 in \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}

\end{itemize}
Discussion in [[rational equivariant stable homotopy theory]] includes

\begin{itemize}%
\item [[John Greenlees]], [[Peter May]], appendix A of \emph{Generalized Tate cohomology}, Mem. Amer. Math. Soc. 113 (1995) no 543 (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/GM-Tate-543.pdf}{pdf})

\end{itemize}
The sphere spectrum in [[global equivariant homotopy theory]] is discussed in

\begin{itemize}%
\item [[John Rognes]], \emph{Galois extensions of structured ring spectra} (\href{http://arxiv.org/abs/math/0502183}{arXiv:math/0502183})


\item Markus Hausmann, Dominik Ostermayr, \emph{Filtrations of global equivariant K-theory} (\href{http://arxiv.org/abs/1510.04011}{arXiv:1510.04011})



\end{itemize}
\hypertarget{equivariance_3}{}\subsubsection*{{$\mathbb{Z}/2$-equivariance}}\label{equivariance_3}

Discussion of $G$-equivariant homotopy groups for $G = \mathbb{Z}/2$ is in

\begin{itemize}%
\item [[Peter Landweber]], \emph{On Equivariant Maps Between Spheres with Involutions}, Annals of Mathematics Second Series, Vol. 89, No. 1 (Jan., 1969), pp. 125-137 (\href{http://www.jstor.org/stable/1970812}{jstor})


\item Sh\^o{}r\^o{} Araki, Kouyemon Iriye, \emph{Equivariant stable homotopy groups of spheres with involutions. I}, Osaka J. Math. Volume 19, Number 1 (1982), 1-55. (\href{http://projecteuclid.org/euclid.ojm/1200774828}{Euclid:1200774828})


\item Kouyemon Iriye, \emph{Equivariant stable homotopy groups of spheres with involutions. II}, Osaka J. Math. Volume 19, Number 4 (1982), 733-743 (\href{https://projecteuclid.org/euclid.ojm/1200775536}{euclid:1200775536})


\item [[Daniel Dugger]], [[Daniel Isaksen]], \emph{$\mathbb{Z}/2$-equivariant and R-motivic stable stems}, Proceedings of the American Mathematical Society 145.8 (2017): 3617-3627 (\href{https://arxiv.org/abs/1603.09305}{arXiv:1603.09305})



\end{itemize}
with exposition in

\begin{itemize}%
\item [[Daniel Dugger]], \emph{Motivic stable homotopy groups of spheres}, talk at \href{http://sma.epfl.ch/~hessbell/arolla/}{Third Arolla Conference on Algebraic Topology} 2008 (\href{http://sma.epfl.ch/~hessbell/arolla/slides12/Dugger.pdf}{pdf}, [[DuggerMotivicStableHomotopy08.pdf:file]])

\end{itemize}
\hypertarget{equivariance_4}{}\subsubsection*{{$\mathbb{Z}/4$-equivariance}}\label{equivariance_4}

Discussion for $G = \mathbb{Z}/4$ is in

\begin{itemize}%
\item M. C. Crabb, \emph{Periodicty in $\mathbb{Z}/4$ equivariant stable homotopy theory}, in [[Mark Mahowald]] and Stewart Priddy (eds.) \emph{Algebraic Topology}, Proceedings of the International Conference held March 21-24 (1988) Contemporary Mathematics, volume 96, 1989 (\href{http://www.ams.org/books/conm/096/conm096.pdf}{pdf}, \href{http://www.ams.org/books/conm/096/}{publisher page})

\end{itemize}
General background includes

\begin{itemize}%
\item [[Stefan Schwede]], \emph{[[Lectures on Equivariant Stable Homotopy Theory]]}, 2015 (\href{http://www.math.uni-bonn.de/people/schwede/equivariant.pdf}{pdf})

\end{itemize}
[[!redirects equivariant sphere spectra]]

[[!redirects global equivariant sphere spectrum]] [[!redirects global sphere spectrum]]



\end{document}