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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{stable cohomotopy} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{AsAlgebraicKTheoryOverTheFieldWithOneElement}{As algebraic K-theory over $\mathbb{F}_1$}\dotfill \pageref*{AsAlgebraicKTheoryOverTheFieldWithOneElement} \linebreak \noindent\hyperlink{kahnpriddy_theorem}{Kahn-Priddy theorem}\dotfill \pageref*{kahnpriddy_theorem} \linebreak \noindent\hyperlink{boardman_homomorphisms}{Boardman homomorphisms}\dotfill \pageref*{boardman_homomorphisms} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The [[generalized cohomology theory]] which is [[Brown representability theorem|represented]] by the [[sphere spectrum]] is also called \emph{stable cohomotopy}, as it is the [[stable homotopy theory]] version of [[cohomotopy]]. Equivalenty, it is the cohomolgical [[duality|dual]] concept to [[stable homotopy homology theory]]. By the [[Pontryagin-Thom theorem]] this is equivalently [[framed manifold|framed]] [[cobordism cohomology theory]]. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{AsAlgebraicKTheoryOverTheFieldWithOneElement}{}\subsubsection*{{As 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{kahnpriddy_theorem}{}\subsubsection*{{Kahn-Priddy theorem}}\label{kahnpriddy_theorem} The [[Kahn-Priddy theorem]] characterizes a comparison map between stable cohomotopy and [[cohomology]] with [[coefficients]] in the infinite [[real projective space]] $\mathbb{R}P^\infty \simeq B \mathbb{Z}/2$. \hypertarget{boardman_homomorphisms}{}\subsubsection*{{Boardman homomorphisms}}\label{boardman_homomorphisms} Consider the [[unit]] morphism \begin{displaymath} \mathbb{S} \longrightarrow H \mathbb{Z} \end{displaymath} from the [[sphere spectrum]] to the [[Eilenberg-MacLane spectrum]] of the [[integers]]. For any [[topological space]]/[[spectrum]] postcomposition with this morphism induces [[Boardman homomorphisms]] of [[cohomology groups]] (in fact of [[commutative rings]]) \begin{equation} b^n \;\colon\; \pi^n(X) \longrightarrow H^n(X, \mathbb{Z}) \label{BoardmandCohomotopyToOrdinaryCohomology}\end{equation} from the [[stable cohomotopy]] of $X$ in degree $n$ to its [[ordinary cohomology]] in degree $n$. \begin{prop} \label{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}\hypertarget{BoundsOnCoKernelOfBoardmandFromStableCohomotopyToOrdinaryCohomology}{} \textbf{(bounds on ([[cokernel|co-]])[[kernel]] of [[Boardman homomorphism]] from [[stable cohomotopy]] to [[integral cohomology]])} If $X$ is a [[CW-spectrum]] which \begin{enumerate}% \item is $(m-1)$-[[n-connected object of an (infinity,1)-topos|(m-1)-connected]] \item of dimension $d \in \mathbb{N}$ \end{enumerate} then \begin{enumerate}% \item the [[kernel]] of the [[Boardman homomorphism]] $b^n$ \eqref{BoardmandCohomotopyToOrdinaryCohomology} for \begin{displaymath} m \leq n\leq d -1 \end{displaymath} is a $\overline{\rho}_{d-n}$-[[torsion subgroup|torsion group]]: \begin{displaymath} \overline{\rho}_{d-n} ker(b^n) \;\simeq\; 0 \end{displaymath} \item the [[cokernel]] of the [[Boardman homomorphism]] $b^n$ \eqref{BoardmandCohomotopyToOrdinaryCohomology} for \begin{displaymath} m \leq n \leq d - 2 \end{displaymath} is a $\overline{\rho}_{d-n-1}$-[[torsion subgroup|torsion group]]: \begin{displaymath} \overline{\rho}_{d-n-1} coker(b^n) \;\simeq\; 0 \end{displaymath} \end{enumerate} where \begin{displaymath} \overline{\rho}_{i} \;\coloneqq\; \left\{ \itexarray{ 1 &\vert& i\leq 1 \\ \underoverset{j = 1}{i}{\prod} exp\left( \pi_j\left( \mathbb{S}\right) \right) &\vert& \text{otherwise} } \right. \end{displaymath} is the [[multiplication|product]] of the [[exponent of a group|exponents]] of the [[stable homotopy groups of spheres]] in [[positive number|positive]] degree $\leq i$. \end{prop} (\hyperlink{Arlettaz04}{Arlettaz 04, theorem 1.2}) \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} [[!include flavours of cohomotopy -- table]] \begin{itemize}% \item [[Boardman homomorphism]] \item [[Spanier-Whitehead duality]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The basic concept appears in \begin{itemize}% \item [[Frank Adams]], part III, section 6 of \emph{[[Stable homotopy and generalised homology]]}, 1974 \end{itemize} 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} The [[Kahn-Priddy theorem]] is due to \begin{itemize}% \item [[Daniel Kahn]], [[Stewart Priddy]], \emph{Applications of the transfer to stable homotopy theory}, Bull. Amer. Math. Soc. Volume 78, Number 6 (1972), 981-987 (\href{https://projecteuclid.org/euclid.bams/1183534135}{Euclid}) \end{itemize} The relation to [[β-rings]] is discussed in \begin{itemize}% \item E. Vallejo, \emph{Polynomial operations from Burnside rings to representation functors}, J. Pure Appl. Algebra, 65 (1990), pp. 163–190. \item E. Vallejo, \emph{Polynomial operations on stable cohomotopy}, Manuscripta Math., 67 (1990), pp. 345–365 \item E. Vallejo, \emph{The free β-ring on one generator, J. Pure Appl. Algebra, 86 (1993), pp. 95–108.} \item \hyperlink{Guillot06}{Guillot 06} \end{itemize} see also \begin{itemize}% \item [[Jack Morava]], Rekha Santhanam, \emph{Power operations and absolute geometry} (\href{http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf}{pdf}) \end{itemize} Discussion of [[Boardman homomorphisms]] from stable cohomotopy is in \begin{itemize}% \item Dominique Arlettaz, \emph{The generalized Boardman homomorphisms}, Central European Journal of Mathematics March 2004, Volume 2, Issue 1, pp 50-56 \end{itemize} A lift of [[Seiberg-Witten invariants]] to classes in [[circle group]]-[[equivariant stable cohomotopy]] is discussed in \begin{itemize}% \item \emph{A stable cohomotopy refinement of Seiberg-Witten invariants: I} (\href{http://arxiv.org/abs/math/0204340}{arXiv:math/0204340}) \item \emph{A stable cohomotopy refinement of Seiberg-Witten invariants: II} (\href{http://arxiv.org/abs/math/0204267}{arXiv:math/0204267}) \end{itemize} [[!redirects stable Cohomotopy]] [[!redirects stable cohomotopy theory]] [[!redirects stable Cohomotopy theory]] [[!redirects stable cohomotopy cohomology theory]] [[!redirects stable Cohomotopy cohomology theory]] \end{document}