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\begin{document}

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\section*{power operation}

\begin{quote}%
under construction


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

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

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

[[!include cohomology - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{SteenrodPowerOperations}{Steenrod squares and Steenrod power operations}\dotfill \pageref*{SteenrodPowerOperations} \linebreak
\noindent\hyperlink{kudoarakidyerlashof_operations}{Kudo-Araki-Dyer-Lashof operations}\dotfill \pageref*{kudoarakidyerlashof_operations} \linebreak
\noindent\hyperlink{adams_operations}{Adams operations}\dotfill \pageref*{adams_operations} \linebreak
\noindent\hyperlink{OnK1LocalKUAlgebras}{On $K(1)$-local $KU$-algebras}\dotfill \pageref*{OnK1LocalKUAlgebras} \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}

Power operations are [[cohomology operations]] in [[multiplicative cohomology theory]] which are higher-degree analogs of [[cup product]]-squares [[symmetric algebra|symmetrized]] in the right [[homotopy theory|homotopy-theoretic sense]].

\begin{remark}
\label{AsPDerivations}\hypertarget{AsPDerivations}{}
At least to some extent, power operations may be understood as the [[higher algebra]]-generalization of the ordinary $p$-power map $(-)^p$ on a [[commutative ring]], the one that appears in the definition of [[Fermat quotients]], [[p-derivations]] and [[Frobenius morphisms]].

See for instance \hyperlink{Lurie}{Lurie, from remark 2.2.7 on}) for relation to the [[Frobenius homomorphism]] and see the example \hyperlink{OnK1LocalKUAlgebras}{below}. See (\hyperlink{Guillot06}{Guillot 06}, \hyperlink{MoravaSanthanam}{Morava-Santhanam 12}) for further discussion and speculation in this direction.

\end{remark}
For $E$ an [[E-∞ ring]] and $X$ a [[topological space]] ([[∞-groupoid]], [[homotopy type]]), a map $a\;\colon\;X \to E$ is a [[cocycle]] in the [[cohomology]] of $X$ with [[coefficients]] in $E$.

The $n$-th [[cup product]] power of this $a$ is the composite

\begin{displaymath}
a^n \;\colon\; X^{\times n} \stackrel{(a,\cdots,a)}{\longrightarrow} E^{\times n} \stackrel{\mu}{\longrightarrow} E
  \,,
\end{displaymath}
where the second map is the product operation in $E$. Since this is by assumption commutative up to coherent higher homotopy, this map factors through the [[homotopy quotient]] by the [[∞-action]] of the [[symmetric group]] $\Sigma_n$

\begin{displaymath}
a^n \;\colon\; X \times \ast //\Sigma_n \longrightarrow   X^n//\Sigma_n \longrightarrow E
  \,.
\end{displaymath}
The cohomology class of this $E$-cocycle on $X \times B \Sigma_n$ is the $n$-th (symmetric) power of $a$.

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

\hypertarget{SteenrodPowerOperations}{}\subsubsection*{{Steenrod squares and Steenrod power operations}}\label{SteenrodPowerOperations}

On [[ordinary cohomology]] over a [[topological space]], the power operations are the [[Steenrod operations]];

Specifically for $n = 2$ and $E = H \mathbb{Z}_2$ then the second (symmetric) power of $a \in H(X,\mathbb{Z}_2)$ is an element in $H^\bullet(\mathbb{R}P^\infty \times X, \mathbb{Z}_2) \simeq H^\bullet(X,\mathbb{Z}_2)[x]$ and the [[coefficients]] of this [[polynomial]] in $x$ are the [[Steenrod operations]] on $a$.

For $p \gt 2$ there are the Steenrod power operations (e.g. \hyperlink{Rognes12}{Rognes 12, around theorem 3.3}).

\hypertarget{kudoarakidyerlashof_operations}{}\subsubsection*{{Kudo-Araki-Dyer-Lashof operations}}\label{kudoarakidyerlashof_operations}

On an [[infinite loop space]] the power operations are the [[Kudo-Araki-Dyer-Lashof operations]]

\hypertarget{adams_operations}{}\subsubsection*{{Adams operations}}\label{adams_operations}

In the context of [[complex K-theory]] power operations are the [[Adams operations]].

\hypertarget{OnK1LocalKUAlgebras}{}\subsubsection*{{On $K(1)$-local $KU$-algebras}}\label{OnK1LocalKUAlgebras}

\begin{quote}%
From \href{http://mathoverflow.net/a/178627/381}{this MO comment} by [[Akhil Mathew]]:


\end{quote}
Let $R$ be a [[K(n)-local stable homotopy theory|K(1)-local]] [[E-∞ ring]] under ([[p-completion|p-adic]]) [[complex K-theory]] [[KU]]. Then there exists a basic power operation $\theta: \pi_0 R \to \pi_0 R$ (see \hyperlink{Hopkins}{Hopkins}) such that :

\begin{itemize}%
\item $\psi(x) \stackrel{\mathrm{def}}{=} x^p + p \theta(x)$ defines a ring homomorphism from $\pi_0 R \to \pi_0 R$.


\item $\theta$ satisfies all the identities needed to make $\psi$ a ring-homomorphism after ``division by p.'' For instance $\psi(x+y) = \psi(x) + \psi(y)$ implies that

\begin{displaymath}
\theta(x+y) = \theta(x) + \theta(y) + \frac{x^p - y^p - (x+y)^p}{p}
  \,,
\end{displaymath}
where the last term is an integral polynomial in $x,y$ and is interpreted as such.



\end{itemize}
(see also \hyperlink{Rezk09}{Rezk 09, example 1.3})

This is a ``$\theta$-algebra.''/[[p-derivation]] as in remark \ref{AsPDerivations} above.

Note in particular that $\psi$ is a lift of the [[Frobenius homomorphism]]. There are generalizations of $\psi, \theta$ at higher [[chromatic levels]], too, and there is a modular interpretation of the resulting algebraic structure in (\hyperlink{Rezk09}{Rezk 09}).

By (\hyperlink{Strickland98}{Strickland 98}) we have that if $G$ is the [[formal group]] associated to a [[Morava E-theory]], then Frobenius lifts (twhich corresponds to degree $p^k$ subgroups of $G$) are classified by maps into $E^0(B \Sigma_{p^r})/I_{t r}$ where $I_{t r}$ is the transfer ideal. So, for example, the map $\psi$ above corresponds to a universal map $KU^0 \to KU^0(B \Sigma_p)/I_{t r} \simeq KU^0$.

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

\begin{itemize}%
\item [[Adams operation]]


\item [[logarithmic cohomology operation]]


\item [[spectral symmetric algebra]]



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

The basic idea is nicely described in

\begin{itemize}%
\item [[Charles Rezk]], \href{http://mathoverflow.net/a/6384/381}{MO comment} Nov 09

\end{itemize}
(from which some of the above text is adapted).

More technical surveys include

\begin{itemize}%
\item [[Charles Rezk]], \emph{Lectures on power operations} (\href{http://www.math.uiuc.edu/~rezk/power-operation-lectures.dvi}{dvi}) (2006)


\item [[Charles Rezk]], \emph{Power operations in Morava E-theory -- a survey} (2009) (\href{http://www.math.uiuc.edu/~rezk/midwest-2009-power-ops.pdf}{pdf})


\item [[Charles Rezk]], \emph{Isogenies, power operations, and homotopy theory}, article (\href{http://www.math.uiuc.edu/~rezk/rezk-icm-talk-posted.pdf}{pdf}) and talk at ICM 2014 (\href{http://www.math.uiuc.edu/~rezk/rezk-icm-2014-slides.pdf}{pdf})



\end{itemize}
Lecture notes on the Steenrod squares and power operations include

\begin{itemize}%
\item [[John Rognes]], section 3 of \emph{The Adams spectral sequence}, 2012 (\href{http://folk.uio.no/rognes/papers/notes.050612.pdf}{pdf})

\end{itemize}
The original articles are

\begin{itemize}%
\item [[Charles Rezk]], \emph{The units of a ring spectrum and a logarithmic cohomology operation}, J. Amer. Math. Soc. 19 (2006), 969-1014 (\href{http://arxiv.org/abs/math/0407022}{arXiv:math/0407022})


\item [[Charles Rezk]], \emph{Power operations for Morava E-theory of height 2 at the prime 2} (\href{http://arxiv.org/abs/0812.1320}{arXiv:0812.1320})


\item [[Charles Rezk]], \emph{The congruence criterion for power operations in Morava E-theory}, Homology, Homotopy and Applications, 11(2), 327-379 (\href{http://arxiv.org/abs/0902.2499}{arXiv:0902.2499})



\end{itemize}
More discussion in the generality of [[E-infinity arithmetic geometry]] is in

\begin{itemize}%
\item [[Jacob Lurie]], \emph{[[Rational and p-adic Homotopy Theory]]}

\end{itemize}
Discussion for $K(1)$-local $E_\infty$-rings is in

\begin{itemize}%
\item [[Michael Hopkins]], \emph{$K(1)$-local $E_\infty$-Ring spectra} (\href{http://www.math.rochester.edu/people/faculty/doug/otherpapers/knlocal.pdf}{pdf})

\end{itemize}
and discussion of power operations in [[Morava E-theory]] is in

\begin{itemize}%
\item [[Matthew Ando]], \emph{Isogenies of formal group laws and power operations in the cohomology theories $E_n$}, Duke Math. J. Volume 79, Number 2 (1995), 423-485 (\href{http://projecteuclid.org/euclid.dmj/1077285158}{Euclid})


\item [[Neil Strickland]], \emph{Morava E-theory of symmetric groups} (\href{http://arxiv.org/abs/math/9801125}{arXiv:math/9801125})



\end{itemize}
Comments on the analogy between power operations in homotopy theory and [[Lambda ring]] structure in [[Borger's absolute geometry]] are in

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


\item [[Jack Morava]], Rakha Santhanam, \emph{Power operations and Absolute geometry}, 2012 (\href{http://www.lemiller.net/media/slidesconf/AbsolutePower.pdf}{pdf})



\end{itemize}
[[!redirects power operations]]



\end{document}