\documentclass[%
12pt,%
t,%
%trans%
]{beamer}

%
% Based on:
% doc/latex/beamer/solutions/generic-talks/generic-ornate-15min-45min.en.tex
% but considerably altered.
% Should compile on any ``reasonable'' TeX distribution (I used teTeX
% installed on a Mac via fink, also tested on teTeX on a Linux box.
% No idea what MikTeX would make of it).
%

%
% This is the version of the presentation that I would have given had
% I had more time to read the user guide in full and time to think
% carefully about how I wanted to implement my presentation.  The
% frames themselves are exactly as in the presentation - that is,
% there is no change in content - but they may look a little
% different.
%

%
% Specific things that are different:
%
% The overall theme:  I originally chose a very sparse look following
% the maxim ``less is more''.  Given that this was my first beamer
% presentation, I think that that was the correct decision; but having
% seen some others and gotten a feel for what is useful information I
% now think that I would have gone for a slightly more informative
% theme and so have changed it thus in this version.
%
% Transparencies:  I wanted to be able to produce a transparency
% version for two reasons.  Firstly, to print off transparencies to
% use in the event of a computer failure.  Secondly, to put a version
% on the web (putting it on the web as is seems a little overkill).
% However, I couldn't figure out how to make the ``Questions'' frame
% appear correct in all of its incarnations.  I have finally worked
% out how to do this and so this version, when given the 'trans'
% option, should produce a pdf suitable for either transparencies or
% for the web.
%
% Frametitles and labels: I wanted to be able to refer to the last
% slide of each frame for two reasons.  Firstly, I wanted some visual
% clue that it was the last slide and so pressing ``space'' (or
% whatever) would take me to the next frame; I chose to change the
% colour of the frame title but there are lots of things one could
% do.  Secondly, I wanted to label the last slide of each frame so
% that at the end of the talk if someone wanted to ask about something
% on a given frame, clicking the link on a ``navigation'' frame (at
% the end of the presentation) would take me to the last slide of the
% given frame - i.e. the one with everything revealed on it.  In the
% original version I did this by hand, counting the number of slides
% in each frame.  I have since figured out a way to do this slightly
% more elegantly.
%
% Spacing: I was keen to make sure that everything was pushed towards
% the top of the frame (without overcrowding) to increase visibility
% for those towards the back of the room (it wasn't sloped).  I wasn't
% aware of the [t] option to the beamer document class so used lots of
% \vfills.  Now that I've learn't of the [t] option those are mostly
% unnecessary.  It's still nice to space out sparse frames a bit,
% though.
%

%\includeonlyframes{current}

%
% When working on only one or a few frames, the above vastly speeds
% things up.
%


\mode<beamer>
{
%
% This load of themes are for the beamer mode; there's a different set
% for the 'trans' mode.
%

  \useoutertheme[right,height=25pt]{sidebar}

%
% Puts a sidebar on the right hand edge of the display showing the
% contents of the presentation, also marks where we are in the talk.
% This also allows control over the height of the headline.  As all my
% frame titles are short, I don't need much space in the headline.
%

  \usecolortheme{dolphin}
  \usecolortheme{rose}

%
% These two set the colour scheme for the document.
%

  \usefonttheme[stillsansserifsmall]{serif}

%
% Serif fonts look nice, but we concede that at small scales they're
% not so legible so we stick with sans serif for small fonts.
%

  \setbeamertemplate{navigation symbols}{}

%
% The talk is linear so we don't need that navigation bar at the
% bottom.
%
}

\mode<trans>{
  \usetheme{Pittsburgh}
}

%\setbeamercolor{frametitle}{fg=blue}

\usepackage{pxfonts}

\usepackage{amsmath}
\usepackage[arrow,matrix,curve]{xy}
\CompileMatrices


\DeclareMathOperator{\Topcat}{Top}
\DeclareMathOperator{\gab}{GAb}
\DeclareMathOperator{\Sq}{Sq}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\F}{\mathbb{F}}

\newtheorem{proposition}[theorem]{Proposition}

%
% When a single symbol is in math text surrounded by normal text it
% can get a little lost, such as: ``Let x be a number''.  To get round
% this problem, I put such mathematics in a slightly different
% colour.  I did this manually as I felt it best not to over use this
% feature.
%

\definecolor{math}{named}{violet}

%
% The next load of commands are for putting stuff at the last slide of
% a frame.  What I wanted to do was put a visual signal to say that I
% was at the last slide (I chose changing the colour of the title) and
% put a label on the last slide so that at the end of the talk, going
% back to that slide via a link takes one to the last slide.  The same
% idea could be used for making something appear on the last slide,
% but I would advise against it - there are potential problems with
% refering to the last slide as it can change between compilations.
% In fact, if using this 'last slide' feature I would turn it off
% whilst writing the presentation and only turn it on again when the
% presentation was finished - this will ensure that the 'last slide'
% of each frame isn't going to change again.  The '\lastoff' command
% turns off all this junk; comment it out to turn it back on.
%

%
% The basic idea is to define a counter which is set to the number of
% slides in the frame.  On a `normal' frame, this is enough: all we
% need to do is use this counter as an overlay specification to get
% something on the last slide.  However, on frames that are going to
% be repeated or ones that have more slides defined than used, we need
% to offset the number of slides.  Thus we have a second counter which
% allows for an offset.
%

%
% It would be ideal to have the 'number of slides' counter defined at
% the start of each frame, however I'm not going to start mucking
% around with the \frame command!  Therefore, I've defined the command
% '\last' which sets the 'last' counter to the number of slides in the
% frame.  This could either be called once at the start of a frame, or
% integrally in a given command.  I've chosen the latter route as it
% involves less commands when invoking this stuff.  The 'lastoffset'
% can then be set or added to on a frame-by-frame (or even
% slide-by-slide) basis.  It is reset to zero at the start of each
% frame.  The 'last' counter is also reset to zero at the start of
% each frame just for safety.
%

%
% Using this stuff is simple.  To make something appear on the last
% frame, just type:
%
% \last%
% \uncover<\value{last}>{This appears on the last frame}
%
% However, I would be very wary of using the last counter with stuff
% like 'uncover' or 'only'; labels and colours are about all I would
% use it for.
%
% Trying to do something on the last slide only makes sense for the
% beamer mode.  In trans mode we turn off the colour change and put
% the label on the first slide.

\newcounter{last}[framenumber]
\newcounter{lastoffset}[framenumber]

\newcommand{\last}{%
\setcounter{last}{1}%
\addtocounter{last}{\insertframeendpage}%
\addtocounter{last}{-\insertframestartpage}%
\addtocounter{last}{\value{lastoffset}}%
}

\newcommand{\lastlabel}[1]{%
\last%
\label<\value{last}| trans:1>{#1}%
}

\definecolor{last}{named}{white}

%
% This was purple in the original version, but with the altered colour
% theme it seemed better to put it in white.
%

\newcommand{\lastframetitle}[1]{%
\last%
\frametitle{\color<\value{last}| trans:0>{last}#1}
}

\newcommand{\lastoff}{%
\renewcommand{\last}{%
\setcounter{last}{1}%
}}

%\lastoff

\bibliographystyle{halpha}

\title{Delooping Moravian Maps}

\subtitle{Stable and Unstable Operations\\in the Morava K--theories\\
%
arXiv: math.AT/0605471\\[16pt]
21st British Topology Meeting
}

\author[Stacey \and Whitehouse]%
{Andrew Stacey\inst{1} \and Sarah Whitehouse\inst{2}}

\institute[Sheffield]%
{
  University of Sheffield \\
\inst{2}Partially and \inst{1}fully supported by the EPSRC, grant no.:
  GR/S76823/01%
}

\date{12th September 2006}


\begin{document}

\begin{frame}
\label{title}
  \titlepage
\end{frame}

\section{The Questions}

%
% This first proper frame is probably the most complicated as parts of
% it will be repeated twice later in the talk.  The '<1-2>' after the
% '\begin{frame}' indicates that only the first two slides of this
% frame will be shown here.  The other slides will appear later via
% the '\againframe' command.
%
% Right, this bit is NOT OBVIOUS AT ALL!
% It took a while to figure out how to make this frame work in the
% 'trans' mode (a mode which I would recommend for making a backup OHP
% version and for making a web version).  Just putting the option
% 'trans' into the documentclass produced the /final/ version of this
% frame at all occurences (i.e. the version with the answers for
% Morava K-theories which should only be shown at the end).  What I
% wanted was for the first two occurences to just have the questions
% with the answers appearing at the end.
%
% I tried just about every possible combination to make this work; but
% what I finally worked out (and which is not clear in the
% otherwise-excellent user guide) is that the different modes have
% different counts for the slides.  Once I'd figured this out the rest
% was child's play.
%
% The first occurence of this frame is slides 1 and 2 for <beamer>
% mode but only slide 1 for <trans>.  The second is slide 3 for
% <beamer> but slide 2 for <trans>.  The third is then slides 4 to 7
% for <beamer> but slide 3 for <trans>.  This makes it crucial to get
% the overlay specifications right!
%


\begin{frame}<1-2| trans:1>[label=qns]
%
% Because this frame is repeated and not all is shown at once, the
% 'last' counter rarely refers to the actual last slide (it does so
% only on the first time around).  This is why we have the
% 'lastoffset' counter so that we can shift stuff around as needed.
%
\temporal<3>{}{%
\setcounter{lastoffset}{2}%
}{%
\setcounter{lastoffset}{3}%
}%
\lastframetitle{Questions \uncover<5->{and Answers}}
%
% Okay, I admit it.  I have made /one/ change to the given
% presentation.  The words ``and Answers''
%
\label<2| trans:1>{qnsorig}
\label<3| trans:2>{qnsreprise}
\label<7| trans:3>{qnsconclude}

%
% These labels refer to the /last/ slide of each of the repetitions of
% this frame.  The referencing works as one might hope: the
% 'qnsreprise' label refers to the first continuation of this frame
% and a hyperlink to that label jumps to that continuation and not
% back to this point.
%
% We can't use '\lastlabel' in this situation as we want a different
% name for the label in each of the three situations.  Also we needed
% slightly more complicated overlay specifications to do with the
% 'trans' option.
%

\uncover<5-| trans:3>{
{\color{blue}For the Morava K--theories:}
}

\vspace{10pt}

  \begin{enumerate}
  \item When is an unstable cohomology operation a component of a
stable one?
\vspace{10pt}


\uncover<6-| trans:3>{
{\color{blue} If it can be delooped once.}
}
\vspace{10pt}

\pause
  \item If we have a component of a stable operation, can we construct
the other components?
\vspace{10pt}


\uncover<7-| trans:3>{
{\color{blue} Yes;  easily, using the periodicity.}
}

  \end{enumerate}
\end{frame}


%
% The table of contents frame was not in the original presentation.  I
% put it here, after the questions, since the ``Questions'' frame acts
% a bit like an abstract and it seems silly to explain the outline
% without an idea of what the talk is actually about.
%

\begin{frame}
\frametitle{Outline}
\tableofcontents[hideallsubsections]
\end{frame}



\section{Preliminaries}

\subsection{Cohomology Theories}

\begin{frame}
\lastframetitle{Preliminaries}

\lastlabel{prelim}

Let \(E^*(-)\) be a \alert{graded, generalised cohomology
  theory}. \pause

\vspace{20pt}

Contravariant functor \(E^*(-) : \Topcat \to \gab\)

\begin{itemize}
\item \(X\) topological space \(\xymatrix@1{{} \ar@{~>}[r] & {}}\)
\(E^*(X)\), graded abelian group.

\item \(f : X \to Y\) continuous \(\xymatrix@1{{} \ar@{~>}[r] & {}}\)
\(f^* : E^*(Y) \to E^*(X)\) of graded abelian groups (degree zero), with
 \((f g)^* = g^* f^*\).

\pause

\item \(E^*(-)\) \alert{intertwines suspensions}: \\
\(E^k(\Sigma X) \cong (\Sigma E^*(X))^k = E^{k-1}(X)\), natural in \(X\).
\end{itemize}

\end{frame}


\begin{frame}
\lastframetitle{Forgetfulness}
\lastlabel{forget}

Three views of \(E^*(-)\):

\pause
\vspace{20pt}

\begin{itemize}
\item One functor, \(E^*(-)\), into graded abelian groups. \pause
\vspace{20pt}

\item A family of functors, \(\{E^k(-)\}\), into abelian
groups. \pause
\vspace{20pt}

\item A family of functors, \(\{E^k(-)\}\), into sets.
\end{itemize}

\end{frame}

\subsection{Operations}

\begin{frame}
\lastframetitle{Operations}
\lastlabel{nat}

An \alert{operation} is a natural transformation between
functors. \pause


\(F, G : \mathcal{C} \to \mathcal{D}\) contravariant.

\(\nu : F \to G\) is:

 for every \(\mathcal{C}\)--object \(X\),
\(\nu_X : F(X) \to G(X)\) such that:
%
\[
  \xymatrix{
    F(X) \ar[r]^{\nu_X} \ar@{}[rd]|{\circlearrowleft} & 
%
    G(X) \\
%
    F(Y) \ar[r]^{\nu_Y} \ar[u]^{F(f)} &
%
    G(Y) \ar[u]_{G(f)}
}
\]

\end{frame}

\begin{frame}
\lastframetitle{Operations}
\lastlabel{ops}

There are \alert{three} types of operation: \pause

\begin{itemize}
\item {\color{blue}Stable:} \(r : E^*(-) \to E^*(-)\) of graded
abelian groups, respecting suspension.
\hfill
\(\mathcal{S}^h\)\pause

\vspace{20pt}

\item {\color{blue} Additive:} \(r : E^k(-) \to E^l(-)\) of abelian
groups.
\hfill
\(\mathcal{A}_k^l\) \pause

\vspace{20pt}

\item {\color{blue} Unstable:} \(r : E^k(-) \to E^l(-)\) of sets.
\hfill
\(\mathcal{U}_k^l\)
\end{itemize}

\pause

\[
\mathcal{S}^h \to \mathcal{A}_k^{k+h} \subseteq \mathcal{U}_k^{k+h}
\]

\end{frame}

\begin{frame}
\lastframetitle{Examples}
\lastlabel{exop}

\begin{itemize}
\item Coefficient operations on \(E^*(-)\): \(n (x) = n
x\); \pause

\item Multiplication operations on \(H^*(-)\): \(x \mapsto x^k\) \pause

\item Steenrod squares on \(H^*(-;\F_2)\). \pause

\item Bott periodicity in K--theory: \(\beta : K^{k+2}(X)
\xrightarrow{\cong} K^k(X)\); \pause

\item Adams operations in K--theory: for \(k \in \Z\), \(\Psi^k :
K^0(X) \to K^0(X)\).

\(\Psi^k(L) = L^{\otimes k}\), \(\Psi^k(V \oplus W) = \Psi^k(V) \oplus
\Psi^k(W)\).
\end{itemize}

\end{frame}

%
% This is the first repetition; only it isn't quite a repetition: it
% is a continuation of the ``Questions'' frame, picking up at slide 3
% (which just happens to be a copy of slide 2).
%

\againframe<3| trans:2>{qns}

\section{Reconstructing Stable Operations}
\subsection{Looping}

\begin{frame}
\lastframetitle{Looping}
\lastlabel{loop}

Consider an unstable operation:
%
\[
  r : E^k(-) \to E^l(-)
\]

\pause

Define a new operation:
%
\[
  \Omega r : E^{k-1}(-) \to E^{l-1}(-)
\]
%
by:
%
\[
  (\Omega r)_X : E^{k-1}(X) \cong E^k(\Sigma X) \xrightarrow{r_{\Sigma
      X}} E^l(\Sigma X) \cong E^{l-1}(X)
\]

\end{frame}

\begin{frame}
\lastframetitle{Looping}
\lastlabel{loopprop}

\begin{proposition}
\begin{enumerate}
\item {\color{math}\(\Omega r\)} is an unstable (additive) operation;

\pause

\item If {\color{math}\(r\)} is the \(k\)th component of a stable operation, \\
{\color{math}\((-1)^{l-k} \Omega r\)} is the \((k-1)\)th component.
\end{enumerate}
\end{proposition}

\pause

\alert<3>{Lower} components are easy.

\pause

\alert<4>{Higher} components are the hard part.

\pause

That is, to \alert<5>{deloop} the operation \(r\).

\pause

Mild help: often have a uniqueness theorem.

\end{frame}

\subsection{Example: K--theory}

%
% The next frame uses an xypic diagram and reveals bits of it on each
% slide.  I found that putting an arrow in an \uncover command didn't
% work as it was always there (xypic presumably ignores colours) so
% the only way to successively reveal parts of the diagram was to put
% them in an \only command; however that causes the diagram to shift
% around a bit as more text is put in.  The simplest solution that I
% found was to put 'pitprops' in appropriate places to ensure that the
% diagram stayed in the same place on each slide.  To see what I mean,
% change all the '0pt's to '1pt's in the \rule commands below; several
% black lines will appear in the diagram which project just beyond the
% longest, highest, or lowest bit of text.
%


\begin{frame}
\lastframetitle{Example: K--theory}
\lastlabel{exk}

K--theory: \uncover<4->{\(2\)--periodic.}

\pause

\[
  \xymatrix@C-5pt{
    K^{-2}(X) \only<3->{\ar[d]^{\Omega^2 r}} &
%
    K^{-1}(X) \only<3->{\ar[d]^{\Omega r}} &
%
    K^0(X) \ar[d]^{r} &
%
    K^1(X) \only<6->{\ar[d]^{\Omega (\beta^{-1} r \beta)}} &
%
    K^2(X)
    \ar@{}[d]^{\rule{25pt}{0pt}}
    \ar@{}[ll]_{\rule{0pt}{30pt}}
    \only<5->{\ar[d]^{\beta^{-1} r \beta}}
    \only<4->{\ar@(ul,ur)[ll]_{\beta}} \\
%
    K^{-2}(X) &
%
    K^{-1}(X) &
%
    K^0(X)
    \ar@{}[rr]_{\rule{0pt}{30pt}}
    \only<4->{\ar@(dr,dl)[rr]_{\beta^{-1}}} &
%
    K^1(X) &
%
    K^2(X)
%
}
\]

\uncover<7->{
\[
\text{Question: }  r = \Omega^2(\beta^{-1} r \beta) ?
\]
}


\end{frame}



\begin{frame}
\lastframetitle{Adams Operations}
\lastlabel{exadam}

\[
  \Omega^2(\beta^{-1} \Psi^k \beta) = k \Psi^k
\]

\pause

\[
  \xymatrix{
    K^{-2}(X) \only<3->{\ar[d]^{k \Psi^k}} &
%
    K^0(X) \ar[d]^{\Psi^k} &
%
    K^2(X)
    \ar@{}[d]^{\rule{18pt}{0pt}}
    \only<4->{\ar[d]^{\frac1k \Psi^k}} \\
%
    K^{-2}(X) &
%
    K^0(X) &
%
    K^2(X)
%
}
\]

\uncover<5->{
\(\frac1k \Psi^k\) not an operation on \(K^0(-)\) \\
%
(unless \(k = 1\) or \(k = -1\))
}



\end{frame}

\subsection{Coefficients}

\begin{frame}
\lastframetitle{Coefficients}
\lastlabel{coeff}

How to divide by \(k\): introduce coefficients.

\(R\) a commutative, unital ring\\
%
\(K(-;R)\) K--theory with coefficients in \(R\).

\pause

\begin{examples}
\begin{enumerate}
\item \(R = \Q\)\pause, but \(K^*(-;\Q) \cong H^\pm(-; \Q)\)
\pause

\item \(R = \Z_{(p)}\) retains \(p\)--typical information

\(\Psi^k\) is stable if \(p \not\mid k\)\\
 (see work of Clarke, Crossley, and Whitehouse)

\end{enumerate}
\end{examples}

\end{frame}

\section{K--theory Mod p}

\subsection{Adams Operations}

\begin{frame}
\lastframetitle{Mod \(p\)}
\lastlabel{modp}

\alert{Warning:} \(K(X; \F_p) \ne K(X)/(p)\).

\pause

\[
p = 3, \; k = 11
\]

\[
  \xymatrix{
    K^0(X;\F_p) \ar[d]^{\Psi^{11}} &
%
    K^2(X;\F_p) \only<3->{\ar[d]^{\frac1{11} \Psi^{11} \only<4->{=2
          \Psi^{11}}}} &
%
    K^4(X;\F_p) \ar@{}[d]^{\rule{45pt}{0pt}}
    \only<5->{\ar[d]^{\frac2{11} \Psi^{11}
        \only<6->{=\Psi^{11}}}} \\
%
    K^0(X;\F_p) &
%
    K^2(X;\F_p) &
%
    K^4(X;\F_p)
  }
\]

\end{frame}

\begin{frame}
\lastframetitle{Answers}
\lastlabel{ans}

\(\Psi^{11}\) repeats with period \(4 \pause = 2(3 - 1)\)

\pause

In \(K^*(-; \F_p)\), for \(p \not\mid k\), \(\Psi^k\)
  repeats with period \(2(p-1)\) (Fermat)

\pause

\begin{proposition}
In \(K^*(-; \F_p)\):
 \begin{enumerate}
\item {\color{math}\(\Psi^k\)} is stable if and only if {\color{math}\(p \not\mid k\)};
\pause
\item If {\color{math}\(\Psi^k\)} is stable the (even) components are blocks of:
%
\[
    (\Psi^k, k^{p-2} \Psi^k, k^{p-3} \Psi^k, k^{p-4} \Psi^k, \dotsc,
    k\Psi^k).
\]
 \end{enumerate}
\end{proposition}

\end{frame}

\subsection{Mod p Reconstruction}

\begin{frame}
\lastframetitle{All Operations}
\lastlabel{allop}

\begin{theorem}[S -- Whitehouse]
The components of a stable operation on \(K^*(-;\F_p)\) repeat with
periodicity \(2(p-1)\).
\end{theorem}

\pause

\begin{corollary}
If {\color{math}\(\Omega^{2(p-1)}(\beta^{-(p-1)} r \beta^{p-1}) = r\)}
then {\color{math}\(r\)} is a component of a stable operation.
\end{corollary}

\end{frame}

\begin{frame}
\lastframetitle{Reconstruction}
\lastlabel{recon}

Start with a component of a stable operation.


\[
  \xymatrixnocompile@C-32pt{
    \only<3->{K^i(X;\F_p)} \only<4->{\ar[d]^{\Omega^j r_k}} &
%
    K^k(X;\F_p) \ar[d]^{r_k} &
%
    \only<2->{K^l(X;\F_p)} \ar@{}[ll]_{\rule{0pt}{30pt}} \only<3->{\ar@(ul,ur)[ll]_{\beta^{m(p-1)}}}
    \only<2->{\ar[d]_{r_l }^{\only<5->{=\beta^{-m(p-1)}(\Omega^j r_k) \beta^{m(p-1)}}}}
    \ar@{}[d]^{\rule{100pt}{0pt}} \\
%
    \only<3->{K^{i+h}(X;\F_p)} \ar@{}[rr]_{\rule{0pt}{30pt}}
    \only<3->{\ar@(dr,dl)[rr]_{\beta^{-m(p-1)}}} &
%
    K^{k+h}(X;\F_p) &
%
    \only<2->{K^{l+h}(X;\F_p)} \\
%
    \rule{70pt}{0pt} &
%
    \rule{70pt}{0pt} &
%
    \rule{70pt}{0pt}
  }
\]

\end{frame}


\subsection{Mod p Recognition}


\begin{frame}
\lastframetitle{Recognition Improvement}
\lastlabel{recog}


\begin{corollary}
If {\color{math}\(\Omega^{2(p-1)}(\beta^{-(p-1)} r \beta^{p-1}) = r\)}
then {\color{math}\(r\)} is a component of a stable operation.
\end{corollary}

\pause


\begin{theorem}[S -- Whitehouse]
Let {\color{math}\(r\)} be an unstable operation on \(K^*(-;\F_p)\).
Then {\color{math}\(r\)} is a component of a stable operation if (and
only if) there is an unstable operation {\color{math}\(s\)} with
{\color{math}\(r = \Omega s\)}.
\end{theorem}

\pause


That is, if \(r\) deloops \alert{once} then it deloops \alert{as many
  times as we like}.


\end{frame}

\section{Morava K--theories}

\subsection{Recognition and Reconstruction}

\begin{frame}
\lastframetitle{Morava K--theories}
\lastlabel{morava}

For each prime \(p\), a sequence of cohomology theories
\(\{K(n)^*(-)\}\) -- the
\colorbox{black}{%
{\color{red}c}%
{\color{red!80!yellow}h}%
{\color{red!60!yellow}r}%
{\color{red!40!yellow}o}%
{\color{red!20!yellow}m}%
{\color{yellow}a}%
{\color{yellow!80!green}t}%
{\color{yellow!60!green}i}%
{\color{yellow!40!green}c}%
{\color{yellow!20!green}\ f}%
{\color{green}i}%
{\color{green!80!blue}l}%
{\color{green!60!blue}t}%
{\color{green!40!blue}r}%
{\color{green!20!blue}a}%
{\color{blue}t}%
{\color{blue!75!violet}i}%
{\color{blue!50!violet}o}%
{\color{blue!25!violet}n}%
{\color{violet}.}%
}

\pause

\begin{itemize}
\item \(K(0)^*(-) = H^*(-;\Q)\) \pause
\item \(K(1)^*(-)\) summand of \(K^*(-;\F_p)\) \pause
\item \(K(n)^*(-)\):
%
\begin{itemize}
\item is periodic, period \(2(p^n-1)\)
\item has coefficients \(\F_p[v_n, {v_n}^{-1}]\), \(\lvert v_n \rvert
= -2(p^n-1)\)
\item has K\"unneth formula and duality
\end{itemize}
\end{itemize}

\end{frame}

\begin{frame}
\lastframetitle{Recognition and Reconstruction}
\lastlabel{recrec}

\begin{theorem}[S -- Whitehouse]
\begin{enumerate}
\item The components of a stable operation in \(K(n)^*(-)\) repeat
with periodicity \(2(p^n-1)\);

\pause

\item If {\color{math}\(r\)} is an unstable operation such that there
is another unstable operation {\color{math}\(s\)} with
{\color{math}\(r = \Omega s\)} then {\color{math}\(r\)} is a component
of a (unique) stable operation.
\end{enumerate}
\end{theorem}

\end{frame}

\subsection{Comments}

\begin{frame}
\lastframetitle{Notes}
\lastlabel{notes}



\begin{enumerate}
\item The periodicity has changed to reflect the periodicity of the
cohomology theory. \pause



\item The periodicity is always that of the cohomology theory.

(Compare with \(K^*(-;\F_p)\))
 \pause


\item The ``delooping'' condition has not changed: if we can deloop
\alert{once} we can deloop as many times as we like.
\end{enumerate}


\end{frame}

\begin{frame}
\lastframetitle{Remarks}
\lastlabel{remarks}

\begin{enumerate}
\item Projection \(P : \mathcal{U}_k^l \to \mathcal{U}_k^l\) via:
%
\[
  P r = \Omega^{2(p^n-1)} ({v_n}^{-1} r v_n)
\]
%
such that \(r\) is a component of a stable operation if and only if
\(r = P r\).

\pause

\item Closely linked to the Bousfield--Kuhn functor.

\pause

\item Reconstruction is easy using periodicity.

\pause

\item Proof is a straightforward analysis of the \(p\)--series of the
formal group law.
\end{enumerate}

\end{frame}

\subsection{Answers}

\againframe<4-| trans:3>{qns}

%
% The point of the next frame was so that at the end of the talk I
% could quickly get back to any frame that I liked, a sort of reverse
% table of contents.  A bit pointless in the 'trans' mode so we don't
% display it then.
%

\section<beamer>*{Navigation}

\begin{frame}<trans:0>
\frametitle{Navigation}

\begin{columns}
\begin{column}{.5\textwidth}
\hyperlink{title}{\beamergotobutton{Title Page}}\\
\hyperlink{qnsorig}{\beamergotobutton{Questions}}\\
\hyperlink{prelim}{\beamergotobutton{Preliminaries}}\\
\hyperlink{forget}{\beamergotobutton{Forgetfulness}}\\
\hyperlink{nat}{\beamergotobutton{Operations (natural transformations)}}\\
\hyperlink{ops}{\beamergotobutton{Operations}}\\
\hyperlink{exop}{\beamergotobutton{Examples of Operations}}\\
\hyperlink{qnsreprise}{\beamergotobutton{Questions (reprise)}}\\
\hyperlink{loop}{\beamergotobutton{Looping an Operation}}\\
\hyperlink{loopprop}{\beamergotobutton{Properties of Looping}}\\
\hyperlink{exk}{\beamergotobutton{Example: K--theory}}\\
\hyperlink{exadam}{\beamergotobutton{Example: Adams Operations}}
\end{column}
\begin{column}{.5\textwidth}
\hyperlink{coeff}{\beamergotobutton{Coefficients}}\\
\hyperlink{modp}{\beamergotobutton{Mod \(p\)}}\\
\hyperlink{ans}{\beamergotobutton{Answers (Adams Operations)}}\\
\hyperlink{allop}{\beamergotobutton{All Operations}}\\
\hyperlink{recon}{\beamergotobutton{Reconstruction}}\\
\hyperlink{recog}{\beamergotobutton{Recognition}}\\
\hyperlink{morava}{\beamergotobutton{Morava K--theories}}\\
\hyperlink{recrec}{\beamergotobutton{Reconstruction and Recognition}}\\
\hyperlink{notes}{\beamergotobutton{Notes}}\\
\hyperlink{remarks}{\beamergotobutton{Remarks}}\\
\hyperlink{qnsconclude}{\beamergotobutton{Questions (conclusion)}}
\end{column}
\end{columns}



\end{frame}

%
% A ``Further Reading'' section is a bit pointless in the talk, but
% can be useful in an online or handout edition.  The way to do it is
% to start with a bibtex-like bibliography, such as:
%
%\begin{frame}
%\frametitle{Further Reading}
%
%\nocite{math.AT/0605471,jb4,jbdjww,ww,tknspt,drww,nk2,ab2,fcmcsw,fcmcsw2}
%
%
%\bibliography{arxiv,articles,books,misc}
%\end{frame}
%
% Then import the bbl file into the tex file and remove (or comment
% out) the above.
%
% The <overlay specification> on these frames ensures that they won't
% appear in the actual presentation but only in the online version.
%

\mode<trans>{%
\section{Further Reading}%
}

\begin{frame}<beamer:0| trans:1>
\frametitle{Further Reading}

{\tiny

\begin{thebibliography}{CCW05}

\bibitem[BJW95]{jbdjww}
J.~Michael Boardman, David~Copeland Johnson, and W.~Stephen Wilson.
\newblock Unstable operations in generalized cohomology.
\newblock In {\em Handbook of algebraic topology}, pages 687--828.
  North-Holland, Amsterdam, 1995.

\bibitem[Boa95]{jb4}
J.~Michael Boardman.
\newblock Stable operations in generalized cohomology.
\newblock In {\em Handbook of algebraic topology}, pages 585--686.
  North-Holland, Amsterdam, 1995.

\bibitem[Bou87]{ab2}
A.~K. Bousfield.
\newblock Uniqueness of infinite deloopings for {$K$}-theoretic spaces.
\newblock {\em Pacific J. Math.}, 129(1):1--31, 1987.

\bibitem[CCW01]{fcmcsw2}
Francis Clarke, M.~D. Crossley, and Sarah Whitehouse.
\newblock Bases for cooperations in {$K$}-theory.
\newblock {\em $K$-Theory}, 23(3):237--250, 2001.

\bibitem[CCW05]{fcmcsw}
Francis Clarke, Martin Crossley, and Sarah Whitehouse.
\newblock Algebras of operations in {$K$}-theory.
\newblock {\em Topology}, 44(1):151--174, 2005.
\end{thebibliography}
}
\end{frame}

\begin{frame}<beamer:0| trans:1>
\frametitle{Further Reading}

{\tiny


\begin{thebibliography}{CCW05}

\bibitem[KST96]{tknspt}
Takuji Kashiwabara, Neil Strickland, and Paul Turner.
\newblock The {M}orava {$K$}-theory {H}opf ring for {$BP$}.
\newblock In {\em Algebraic topology: new trends in localization and
  periodicity (Sant Feliu de Gu\'\i xols, 1994)}, volume 136 of {\em Progr.
  Math.}, pages 209--222. Birkh\"auser, Basel, 1996.

\bibitem[Kuh89]{nk2}
Nicholas~J. Kuhn.
\newblock Morava {$K$}-theories and infinite loop spaces.
\newblock In {\em Algebraic topology (Arcata, CA, 1986)}, volume 1370 of {\em
  Lecture Notes in Math.}, pages 243--257. Springer, Berlin, 1989.

\bibitem[RW77]{drww}
Douglas~C. Ravenel and W.~Stephen Wilson.
\newblock The {H}opf ring for complex cobordism.
\newblock {\em J. Pure Appl. Algebra}, 9(3):241--280, 1976/77.

\bibitem[SW]{math.AT/0605471}
Andrew Stacey and Sarah Whitehouse.
\newblock {Stable and Unstable Operations in mod p Cohomology Theories},
  arXiv:math.AT/0605471.

\bibitem[Wil84]{ww}
W.~Stephen Wilson.
\newblock The {H}opf ring for {M}orava {$K$}-theory.
\newblock {\em Publ. Res. Inst. Math. Sci.}, 20(5):1025--1036, 1984.

\end{thebibliography}
}
\end{frame}
\end{document}