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\section*{equivariant stable cohomotopy}

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

\hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology}

[[!include cohomology - contents]]

\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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{AsAlgebraicKTheoryOverTheFieldWithOneElement}{As equivariant algebraic K-theory over $\mathbb{F}_1$}\dotfill \pageref*{AsAlgebraicKTheoryOverTheFieldWithOneElement} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{equivariant_stable_}{Equivariant stable $\pi_3^{\mathbb{S}}$}\dotfill \pageref*{equivariant_stable_} \linebreak
\noindent\hyperlink{of_the_point_the_burnside_ring}{Of the point: The Burnside ring}\dotfill \pageref*{of_the_point_the_burnside_ring} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{relation_to_burnside_ring}{Relation to Burnside ring}\dotfill \pageref*{relation_to_burnside_ring} \linebreak
\noindent\hyperlink{relation_to_segalcarlsson_completion_theorem}{Relation to Segal-Carlsson completion theorem}\dotfill \pageref*{relation_to_segalcarlsson_completion_theorem} \linebreak
\noindent\hyperlink{as_equivariant_ktheory_over_the_field_with_one_element}{As equivariant K-theory over the field with one element}\dotfill \pageref*{as_equivariant_ktheory_over_the_field_with_one_element} \linebreak
\noindent\hyperlink{in_mbrane_charge_quantization}{In M-brane charge quantization}\dotfill \pageref*{in_mbrane_charge_quantization} \linebreak


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

The [[equivariant cohomology|equivariant]] [[generalized cohomology theory]] which is [[Brown representability theorem|represented]] by the [[equivariant sphere spectrum]] may also be called \emph{equivariant stable cohomotopy}, as it is the [[equivariant stable homotopy theory]] version of [[stable cohomotopy]], hence of [[cohomotopy]]. This is to be thought of as the first order [[Goodwillie-Taylor tower|Goodwillie approximation]] of plain (``unstable'') [[equivariant cohomotopy]].

Just as the plain [[sphere spectrum]] is a distinguished object of plain [[stable homotopy theory]], so the [[equivariant sphere spectrum]] is distinguished in [[equivariant stable homotopy theory]] and hence so is equivariant stable cohomotopy theory.

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

\hypertarget{AsAlgebraicKTheoryOverTheFieldWithOneElement}{}\subsubsection*{{As equivariant algebraic K-theory over $\mathbb{F}_1$}}\label{AsAlgebraicKTheoryOverTheFieldWithOneElement}

The following is known as the \emph{Barratt-Priddy-Quillen theorem}:

\begin{prop}
\label{StableCohomotopyIsKTheoryOfFinSet}\hypertarget{StableCohomotopyIsKTheoryOfFinSet}{}
\textbf{([[stable cohomotopy]] is K-theory of [[FinSet]])}

Let $\mathcal{C} =$ [[FinSet]] be a [[skeleton]] of the category of [[finite sets]], regarded as a [[permutative category]]. Then the [[K-theory of a permutative category|K-theory of this permutative category]]

\begin{displaymath}
K(FinSet) \;\simeq\; \mathbb{S}
\end{displaymath}
is represented by the [[sphere spectrum]], hence is stable cohomotopy.

\end{prop}
This is due to \hyperlink{BarrattPriddy72}{Barratt-Priddy 72} reproved in \hyperlink{Segal74}{Segal 74, Prop. 3.5}. See also \hyperlink{Priddy73}{Priddy 73}, \hyperlink{Glasman13}{Glasman 13}.

\begin{remark}
\label{StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}\hypertarget{StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement}{}
\textbf{([[stable cohomotopy]] as [[algebraic K-theory]] over the [[field with one element]])}

Notice that for $F$ a [[field]], the [[K-theory of a permutative category]] of its [[category of modules]] $F Mod$ is its [[algebraic K-theory]] $K F$ (see \href{K-theory+of+a+permutative+category#OrdinaryAlgebraicKTheoryFromPermutativeCategoryOfProjectiveModules}{this example})

\begin{displaymath}
K F \;\simeq\; K(F Mod)
  \,.
\end{displaymath}
Now ([[pointed sets|pointed]]) [[finite sets]] may be regarded as the modules over the ``[[field with one element]]'' $\mathbb{F}_1$ (see \href{field+with+one+element#Modules}{there}):

\begin{displaymath}
\mathbb{F}_1 Mod
  \;=\;
  FinSet^{\ast/}
\end{displaymath}
If this is understood, example \ref{StableCohomotopyIsKTheoryOfFinSet} says that [[stable cohomotopy]] is the algebraic K-theory of the [[field with one element]]:

\begin{displaymath}
\mathbb{S} \;\simeq\; K \mathbb{F}_1
  \,.
\end{displaymath}
\end{remark}
This perspective is highlighted for instance in (\hyperlink{Deitmar06}{Deitmar 06, p. 2}, \hyperlink{Guillot06}{Guillot 06}, \hyperlink{Mahanta17}{Mahanta 17}, \hyperlink{DundasGoodwillieMcCarthy13}{Dundas-Goodwillie-McCarthy 13, II 1.2}, \hyperlink{MoravaSomeBackground}{Morava}, \hyperlink{ConnesConsani16}{Connes-Consani 16}).

The perspective that the [[K-theory]] $K \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[stable Cohomotopy]] has been highlighted in (\hyperlink{Deitmar06}{Deitmar 06, p. 2}, \hyperlink{Guillot06}{Guillot 06}, \hyperlink{Mahanta17}{Mahanta 17}, \hyperlink{DundasGoodwillieMcCarthy13}{Dundas-Goodwillie-McCarthy 13, II 1.2}, \hyperlink{MoravaSomeBackground}{Morava}, \hyperlink{ConnesConsani16}{Connes-Consani 16}). Generalized to [[equivariant stable homotopy theory]], the statement that [[equivariant K-theory]] $K_G \mathbb{F}_1$ over $\mathbb{F}_1$ should be [[equivariant stable Cohomotopy]] is discussed in \hyperlink{ChuLorscheidSanthanam10}{Chu-Lorscheid-Santhanam 10, 5.3}.

[[!include Segal completion -- table]]

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

\hypertarget{equivariant_stable_}{}\subsubsection*{{Equivariant stable $\pi_3^{\mathbb{S}}$}}\label{equivariant_stable_}

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

\hypertarget{of_the_point_the_burnside_ring}{}\subsubsection*{{Of the point: The Burnside ring}}\label{of_the_point_the_burnside_ring}

\begin{prop}
\label{BurnsideRingIsEquivariantStableCohomotopyOfPoint}\hypertarget{BurnsideRingIsEquivariantStableCohomotopyOfPoint}{}
\textbf{([[Burnside ring is equivariant stable cohomotopy of the point]])}

Let $G$ be a [[finite group]], then its [[Burnside ring]] $A(G)$ is [[isomorphism|isomorphic]] to the [[equivariant stable cohomotopy]] [[cohomology ring]] $\mathbb{S}_G(\ast)$ of the [[point]] in degree 0.

\begin{displaymath}
A(G) \overset{\simeq}{\longrightarrow} \mathbb{S}_G(\ast)
  \,.
\end{displaymath}
\end{prop}
This is due to \hyperlink{Segal71}{Segal 71}, a detailed proof is given by \hyperlink{tomDieck79}{tom Dieck 79, theorem 8.5.1}. See also \hyperlink{Lueck05}{Lück 05, theorem 1.13}, \hyperlink{tomDieckPetrie78}{tom Dieck-Petrie 78}.

[[!include Segal completion -- table]]

More explicitly, this means that the Burnside ring of a group $G$ is isomorphic to the [[colimit]]

\begin{displaymath}
A(G) \simeq \underset{\longrightarrow}{\lim}_V [S^V,S^V]_G
\end{displaymath}
over $G$-[[representations]] in a complete [[G-universe]], of $G$-[[homotopy classes]] of $G$-equivariant based [[continuous functions]] from the [[representation sphere]] $S^V$ to itself (\hyperlink{GreenleesMay95}{Greenlees-May 95, p. 8}).

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

[[!include flavours of cohomotopy -- table]]

\begin{itemize}%
\item [[Burnside ring]]


\item [[Segal conjecture]], [[Sullivan conjecture]]



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

\hypertarget{relation_to_burnside_ring}{}\subsubsection*{{Relation to Burnside ring}}\label{relation_to_burnside_ring}

Relation to [[Burnside ring]]:

\begin{itemize}%
\item [[Graeme Segal]], \emph{Equivariant stable homotopy theory}, In Actes du Congr\`e{}s International des Math \'e{}maticiens (Nice, 1970), Tome 2 , pages 59–63. Gauthier-Villars, Paris, 1971 ([[SegalEquivariantStableHomotopyTheory.pdf:file]])


\item [[Tammo tom Dieck]], T. Petrie, \emph{Geometric modules over the Burnside ring}, Invent. Math. 47 (1978) 273-287 (\href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf}{pdf})


\item [[Tammo tom Dieck]], \emph{[[Transformation Groups and Representation Theory]]}, Springer 1979



\end{itemize}
\hypertarget{relation_to_segalcarlsson_completion_theorem}{}\subsubsection*{{Relation to Segal-Carlsson completion theorem}}\label{relation_to_segalcarlsson_completion_theorem}

Relation to [[Segal-Carlsson completion theorem]]:

\begin{itemize}%
\item Czes Kosniowski, \emph{Equivariant cohomology and Stable Cohomotopy}, Math. Ann. 210, 83-104 (1974) (\href{https://doi.org/10.1007/BF01360033}{doi:10.1007/BF01360033} \href{https://link.springer.com/content/pdf/10.1007/BF01360033.pdf}{pdf})


\item Erkki Laitinen, \emph{On the Burnside ring and stable cohomotopy of a finite group}, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (\href{https://www.jstor.org/stable/24491306}{jstor:24491306}, [[Laitinen79.pdf:file]])


\item [[Gunnar Carlsson]], \emph{Equivariant Stable Homotopy and Segal's Burnside Ring Conjecture}, Annals of Mathematics Second Series, Vol. 120, No. 2 (Sep., 1984), pp. 189-224 (\href{https://www.jstor.org/stable/2006940}{jstor:2006940}, \href{https://web.math.rochester.edu/people/faculty/doug/otherpapers/carlsson.pdf}{pdf})


\item [[Wolfgang Lück]], \emph{The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups}, Pure Appl. Math. Q. 1 (2005), no. 3, Special Issue: In memory of Armand Borel. Part 2, 479--541 (\href{https://arxiv.org/abs/math/0504051}{arXiv:math/0504051})


\item Noe Barcenas, \emph{Nonlinearity, Proper Actions and Equivariant Stable Cohomotopy} (\href{https://arxiv.org/abs/1302.1712}{arXiv:1302.1712})



\end{itemize}
A lift of [[Seiberg-Witten invariants]] to classes in [[circle group]]-equivariant stable cohomotopy is discussed in

\begin{itemize}%
\item Stefan Bauer, Mikio Furuta \emph{A stable cohomotopy refinement of Seiberg-Witten invariants: I} (\href{http://arxiv.org/abs/math/0204340}{arXiv:math/0204340})


\item Stefan Bauer, \emph{A stable cohomotopy refinement of Seiberg-Witten invariants: II} (\href{http://arxiv.org/abs/math/0204267}{arXiv:math/0204267})


\item Christian Okonek, Andrei Teleman, \emph{Cohomotopy Invariants and the Universal Cohomotopy Invariant Jump Formula}, J. Math. Sci. Univ. Tokyo 15 (2008), 325-409 (\href{http://www.matmor.unam.mx/~barcenas/nonlinearity.pdf}{pdf})



\end{itemize}
\hypertarget{as_equivariant_ktheory_over_the_field_with_one_element}{}\subsubsection*{{As equivariant K-theory over the field with one element}}\label{as_equivariant_ktheory_over_the_field_with_one_element}

The identification of stable cohomotopy with the [[K-theory of a permutative category|K-theory of the permutative category]] of [[finite set]] is due to

\begin{itemize}%
\item [[Michael Barratt]], [[Stewart Priddy]], \emph{On the homology of non-connected monoids and their associated groups}, Commentarii Mathematici Helvetici, December 1972, Volume 47, Issue 1, pp 1–14 (\href{https://doi.org/10.1007/BF02566785}{doi:10.1007/BF02566785})


\item [[Graeme Segal]], \emph{Categories and cohomology theories}, Topology vol 13, pp. 293-312, 1974 (\href{http://ncatlab.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf}{pdf})



\end{itemize}
see also

\begin{itemize}%
\item [[Stewart Priddy]], \emph{Transfer, symmetric groups, and stable homotopy theory}, in \emph{Higher K-Theories}, Springer, Berlin, Heidelberg, 1973. 244-255 (\href{https://link.springer.com/content/pdf/10.1007/BFb0067060.pdf}{pdf})


\item [[Saul Glasman]], \emph{The multiplicative Barratt-Priddy-Quillen theorem and beyond}, talk 2013 (\href{http://math.mit.edu/~sglasman/bpq-beamer.pdf}{pdf})



\end{itemize}
The resulting interpretation of stable cohomotopy as [[algebraic K-theory]] over the [[field with one element]] is amplified in the following texts:

\begin{itemize}%
\item [[Bjørn Dundas]], [[Thomas Goodwillie]], [[Randy McCarthy]], chapter II, section 1.2 of \emph{The local structure of algebraic K-theory}, Springer 2013 (\href{http://math.mit.edu/~nrozen/juvitop/dundas.pdf}{pdf})


\item [[Anton Deitmar]], \emph{Remarks on zeta functions and K-theory over $\mathbb{F}_1$} (\href{https://arxiv.org/abs/math/0605429}{arXiv:math/0605429})


\item [[Pierre Guillot]], \emph{Adams operations in cohomotopy} (\href{https://arxiv.org/abs/math/0612327}{arXiv:0612327})


\item [[Snigdhayan Mahanta]], \emph{G-theory of $\mathbb{F}_1$-algebras I: the equivariant Nishida problem}, J. Homotopy Relat. Struct. 12 (4), 901-930, 2017 (\href{https://arxiv.org/abs/1110.6001}{arXiv:1110.6001})


\item Chenghao Chu, [[Oliver Lorscheid]], Rekha Santhanam, \emph{Sheaves and K-theory for $\mathbb{F}_1$-schemes}, Advances in Mathematics, Volume 229, Issue 4, 1 March 2012, Pages 2239-2286 (\href{https://arxiv.org/abs/1010.2896}{arxiv:1010.2896})



\end{itemize}
see also

\begin{itemize}%
\item [[Jack Morava]], \emph{Some background on Manin's theorem $K(\mathbb{F}_1) \sim \mathbb{S}$} (\href{http://www.alainconnes.org/docs/Morava.pdf}{pdf}, [[MoravaSomeBackground.pdf:file]])


\item [[Alain Connes]], [[Caterina Consani]], \emph{Absolute algebra and Segal's Gamma sets}, Journal of Number Theory 162 (2016): 518-551 (\href{https://arxiv.org/abs/1502.05585}{arXiv:1502.05585})


\item John D. Berman, p. 92 of \emph{Categorified algebra and equivariant homotopy theory}, PhD thesis 2018 (\href{http://www.people.virginia.edu/~jdb8pc/Thesis.pdf}{pdf})



\end{itemize}
\hypertarget{in_mbrane_charge_quantization}{}\subsubsection*{{In M-brane charge quantization}}\label{in_mbrane_charge_quantization}

Discussion for [[M-brane]] physics:

\begin{itemize}%
\item [[John Huerta]], [[Hisham Sati]], [[Urs Schreiber]], \emph{[[schreiber:Equivariant homotopy and super M-branes|Real ADE-equivariant (co)homotopy and Super M-branes]]} (\href{https://arxiv.org/abs/1805.05987}{arXiv:1805.05987})

([[equivariant rational homotopy theory|rational]] [[equivariant cohomotopy]])


\item [[nLab:Hisham Sati]], [[nLab:Urs Schreiber]], \emph{[[schreiber:Equivariant Cohomotopy implies orientifold tadpole cancellation]]} (\href{https://arxiv.org/abs/1909.12277}{arXiv:1909.12277})

(implying [[RR-field tadpole cancellation]])



\end{itemize}
[[!redirects equivariant stable Cohomotopy]]

[[!redirects equivariant stable cohomotopy theory]] [[!redirects equivariant stable cohomotopy theories]]

[[!redirects equivariant stable Cohomotopy theory]] [[!redirects equivariant stable Cohomotopy theories]]

[[!redirects stable equivariant cohomotopy]] [[!redirects stable equivariant Cohomotopy]]



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