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\begin{document}

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\section*{Adams operation}

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

\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{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{adams_conjecture}{Adams conjecture}\dotfill \pageref*{adams_conjecture} \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{ReferencesInRepresentationTheory}{In representation theory}\dotfill \pageref*{ReferencesInRepresentationTheory} \linebreak


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

Given a [[complex line bundle]] $L$ over a [[space]] $X$ its $k$th tensor power $L^{\otimes k}$ is another line bundle for any $k \in \mathbb{N}$. The line bundles define certain elements of [[topological K-theory]] group $K(X)$, and there is a unique operation $\psi^k : K(X) \to K(X)$, the \emph{$k$th Adams operation}, such that:

\begin{itemize}%
\item $\psi^k([L]) = [L^{\otimes k}]$ if $[L]$ is the $K$-theory class of any line bundle,


\item $\psi^k : K(X) \to K(X)$ is a group homomorphism,


\item $\psi^k$ is a natural transformation: any map $f: X \to Y$ induces a map $f^* : K(Y) \to K(X)$ on $K$-theory, and $\psi^k \circ f^* = f^* \circ \psi^k$.



\end{itemize}
More abstractly, Adams operations can be defined on any [[Lambda-ring]]. They are an example of [[power operations]].

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

The Adams operations have an explicit definition in terms of the [[Lambda-ring]] structure on [[topological K-theory]], this we state as def. \ref{DefinitionInTermsOfLambdaRing} below. While explicit, this definition may look contrived on first sight. But it turns out that it satisfies a list of properties, of which two simple ones already uniquely characterize the Adams operations. This is proposition \ref{TheBasicProperties} below.

\begin{prop}
\label{LambdaRingStructureOnKTheory}\hypertarget{LambdaRingStructureOnKTheory}{}
\textbf{(Lambda-ring structure on topological K-theory)}

Let $X$ be a [[compact topological space]] and write $K(X)$ for its [[topological K-theory]] [[ring]]. For $E$ a [[vector bundle]] over $X$, write $[E] \in K(X)$ for its class in K-theory. Given $E$, write

\begin{displaymath}
\lambda_t[E]
  \;\coloneqq\;
  \underoverset{k = 0}{\infty}{\sum}
  [\wedge^k_X E] t^k
  \;\;\in\;\;
  K(X)[ [t] ]
\end{displaymath}
for the [[formal power series]] with [[coefficients]] in the ring $K(X)$ being the K-theory classes of the skew-symmetrized [[tensor product of vector bundles]] of $E$ with itself.

Since the constant term of this power series is always the unit $[\wedge^0 E] = 1$, hence

\begin{displaymath}
\lambda_t[E] \in 1 + (t) \cdot K(X)[ [t] ]
\end{displaymath}
there exists a multiplicative inverse formal power series $\lambda_t[E]^{-1}$.

Then given the class of a [[virtual vector bundle]] $[E] - [F] \in K(X)$, define more generally

\begin{displaymath}
\lambda_t[[E- F]]
  \;\;\coloneqq\;\,
  \lambda_t[E] \cdot \lambda_t[F]^{-1}
  \;\;\in\;\;
  K(X)[ [t] ]
  \,.
\end{displaymath}
\end{prop}
\begin{prop}
\label{DefinitionInTermsOfLambdaRing}\hypertarget{DefinitionInTermsOfLambdaRing}{}
\textbf{(explicit definition of Adams operation)}

For $E$ a [[vector bundle]] over some [[topological space]] $X$, write

\begin{displaymath}
\psi^0(E) \coloneqq rank(E)
\end{displaymath}
for the [[bundle]] which over each [[connected component]] of $X$ is the [[trivial bundle|trivial]] vector bundle of the same [[rank of a vector bundle|rank]] as $E$ over that component.

Define a [[formal power series]] with [[coefficients]] in the K-theory ring $K(X)$ by

\begin{displaymath}
\begin{aligned}
    \psi_t(E)
    & \coloneqq
   \underoverset{\infty}{k = 0}{\sum}
   \psi^k(E) t^k
   \\
    & \coloneqq
    \psi^0(E)
      -
     t \frac{d}{d t} log \lambda_{-t}(E)
    \;\;\in\;\;
    K(X)[ [t] ]
  \end{aligned}
  \,,
\end{displaymath}
where $\lambda_t$ is the [[Lambda-ring]] operation from def. \ref{LambdaRingStructureOnKTheory}.

Here the [[derivative]] of the [[logarithm]] of formal power series stands for the usual expression in terms of the [[geometric series]]:

\begin{displaymath}
\begin{aligned}
    \frac{d}{d t} log \lambda_{-t}(E)
    & =
    \frac{1}{\lambda_{-t}(E)}
    \frac{d}{d t}
     \lambda_{-t}(E)
     \\
     & =
     \underoverset{\infty}{k = 0}{\sum}
     \left(
        1  - \lambda_{-t}(E)
     \right)^k
    \cdot
     \frac{d}{d t} \lambda_{-t}(E)
  \end{aligned} 
  \,.
\end{displaymath}
The \textbf{$k$th Adams operation} is the [[cohomology operation]] on [[topological K-theory]]

\begin{displaymath}
\psi^k \;\colon\; K(-) \longrightarrow K(-)
\end{displaymath}
which is the coefficient of $t^k$ in $\psi_t$.

\end{prop}
\begin{prop}
\label{TheBasicProperties}\hypertarget{TheBasicProperties}{}
\textbf{(basic properties and characterization of Adams operations)}

The Adams operations

\begin{displaymath}
\psi^k \;\colon\; K(X) \longrightarrow K(X)
\end{displaymath}
have the following properties, for all elements $x,y \in K(X)$ and $k, l \in \mathbb{N}$ and $p \; \text{prime}$:

\begin{enumerate}%
\item $\psi^k(x + y) = \psi^k(x) + \psi^k(y)$

($\psi^k$ is a [[natural transformation|natural]] [[abelian group]] [[homomorphism]])


\item $x \,\text{a line} \;\;\Rightarrow\;\; \psi^k(x) = x^k$

(applied to a class $x \coloneqq [L]$ represented by a [[line bundle]] $L$, $\psi^k$ is the $k$th [[tensor power]])


\item $\psi^k(x \cdot y) = \psi^k(x) \cdot \psi^k(y)$

($\psi^k$ is in fact a [[natural transformation|natural]] [[ring]] [[homomorphism]])


\item $\psi^k(\psi^l(x)) = \psi^{k l}(x)$


\item $\psi^p(x) = x^p \, \text{mod}\, p$


\item if $x \in \tilde K(S^{2n})$ ([[reduced cohomology]]) then

$x \in \tilde K(S^{2n}) \hookrightarrow K(S^{2n}) \;\;\Rightarrow\;\;\psi^k(x) = k^n \cdot x$.



\end{enumerate}
Moreover, the first two of these already uniquely characterize the Adams operations.

\end{prop}
e.g. \hyperlink{Wirthmuller12}{Wirthmuller 12, section 11}

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

\hypertarget{adams_conjecture}{}\subsubsection*{{Adams conjecture}}\label{adams_conjecture}

The [[Adams conjecture]] (a [[theorem]]) says that for all $k \in \mathbb{N}$ and $V \in K(X)$ there is $n \in \mathbb{N}$ such that the [[spherical fibration]] assigned to the [[K-theory]] class $k^n (\psi^k(V)-V)$ under the [[J-homomorphism]] is trivial, hence that

\begin{displaymath}
J
  \left(
     k^n \left(
       \psi^k(V) - V
     \right)
  \right)
  = 0
  \,.
\end{displaymath}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Lambda-ring]]


\item [[Steenrod square]]


\item [[Steenrod operation]]


\item [[power operation]]



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

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

\begin{itemize}%
\item [[Klaus Wirthmüller]], section 11 of \emph{Vector bundles and K-theory}, 2012 (\href{ftp://www.mathematik.uni-kl.de/pub/scripts/wirthm/Top/vbkt_skript.pdf}{pdf})


\item [[Allen Hatcher]], section 2.3 of \emph{Vector bundles and K-theory} (\href{https://www.math.cornell.edu/~hatcher/VBKT/VBpage.html}{web})


\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Adams_operation}{Adams operation}}


\item [[Jacob Lurie]], remark 2 in: \emph{[[Chromatic Homotopy Theory]]}, Lecture series 2010, Lecture 35 \emph{The image of $J$} (\href{http://www.math.harvard.edu/~lurie/252xnotes/Lecture35.pdf}{pdf})



\end{itemize}
\hypertarget{ReferencesInRepresentationTheory}{}\subsubsection*{{In representation theory}}\label{ReferencesInRepresentationTheory}

Adams operations on the [[representation ring]] (the [[equivariant K-theory]] of the point) are discussed in

\begin{itemize}%
\item [[Tammo tom Dieck]], section 3.5 of \emph{[[Transformation Groups and Representation Theory]]}, Lecture Notes in Mathematics 766 Springer 1979


\item Robert Boltje, \emph{A characterization of Adams operations on representation rings}, 2001 (\href{https://boltje.math.ucsc.edu/publications/p01v.pdf}{pdf})


\item [[Tammo tom Dieck]], section 6.2 of \emph{Representation theory}, 2009 (\href{http://www.uni-math.gwdg.de/tammo/rep.pdf}{pdf})


\item [[Michael Boardman]], \emph{Adams operations on Group representations}, 2007 (\href{http://www.math.jhu.edu/~jmb/note/adamrept.pdf}{pdf})


\item Ehud Meir, Markus Szymik, \emph{Adams operations and symmetries of representation categories} (\href{https://arxiv.org/abs/1704.03389}{arXiv:1704.03389})



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



\end{document}