\documentclass[12pt,color=dvipsnames]{beamer} % % This is the tex file of the presentation that I actually gave. Some % of the stuff in here is only here because it was in the sample that % I copied from; some other stuff is a basic hack that I put in to do % something complicated and which I couldn't see a simple way to do % otherwise. I didn't have overmuch time to learn the ins and outs of % beamer so had to take the 'quick and dirty' route a few times. Now % that I've had time to read the whole manual, I've figured out what I % would do differently. That's in a different file - as this is an % example I thought it best to put up what I actually did as well as % what I would have done. % % % Options to beamer: the default font size is 11pt; I chose to % increase this to 12pt to make everything that bit bigger and clearer % (and also to ensure that I didn't put too much on any one slide). % The color=dvipsnames option allows one to use more exotic colour % names such as 'WildStrawberry'. I didn't, in the end, but it's nice % to have the option. % \mode { % % Sets up the look of the thing. I chose Pittsburgh as it seemed a % fairly sparse theme. I also used the sidebar outer theme to give me % more control over the size of the frame titles (as I was using % simple one-line titles I didn't need all the space usually % allocated). The 0pt on the width effectively turns off the sidebar % itself. That seemed to muck around with the colours a bit so the % colour theme 'lily' restores things to normal again. As I was using % a larger font size, I figured that I could get away with serif % lettering but made sure that any small font was in the more legible % sans-serif (see the beamer user manual section on fonts). % \usetheme{Pittsburgh} \useoutertheme[width=0pt,height=25pt]{sidebar} \usecolortheme{lily} \usefonttheme[stillsansserifsmall]{serif} } % % Despite using the colour theme 'lily', the frametitle colour wasn't % what it originally was. % \setbeamercolor{frametitle}{fg=blue} % % Apart from the pxfonts line, the next block was all in the original % template that I copied from. I prefer pxfonts to Computer Modern; % they're in the standard teTeX distribution so should be on most TeX % installations. If LaTeX complains about fonts when compiling this % document, try commenting out the pxfonts line. Then go and get % yourself some decent TeX fonts. % \usepackage[english]{babel} \usepackage[latin1]{inputenc} \usepackage{pxfonts} \usepackage[T1]{fontenc} % % Standard stuff now; amsthm is loaded by default with beamer. % \usepackage{amsmath} \usepackage[arrow,matrix,curve]{xy} % % The argument of a 'mode' command is only active in the specified % mode. I was thinking of producing a handout based on the slides, % but didn't in the end. If I had done so, this would have made it % clear what was a slide and what wasn't (say, if I'd done them 2-up) % by making the background slightly grey. % \mode{\beamertemplatesolidbackgroundcolor{black!5}} % % More standard stuff % \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} % % Sometimes - and only sometimes - math text in normal text can get a % little lost; generally in situations like: ``Let n be an integer.'' % So I defined a new colour to make this stand out a little. I % implemented this manually rather than changing all math text (to do % that, see the beamer user guide) as I only wanted to change the % colour where I thought it really necessary. % \definecolor{math}{named}{violet} % % I wanted a visual clue that I was on the last slide of a frame. I % chose to change the colour of the frame title. This colour is the % colour to change to. % \definecolor{last}{named}{purple} \bibliographystyle{halpha} \title{Delooping Moravian Maps} \subtitle{Stable and Unstable Operations in the Morava K--theories\\ % arXiv: math.AT/0605471\\[20pt] 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} % % This is the most complicated frame. Slides of this frame appear at % two later places. Here we get slides 1 and 2, the second time we % get slide 3, then last of all we get 4 to 7. In actual fact, slides % 2, 3, and 4 are identical but making them actually the same makes % LaTeX complain about multiple references and makes the labelling go % wrong. Thus the best way is to ensure that at least the first slide % of any new appearance has not been seen before, even if it is % identical to another. % % % This also shows the recolouring of the frame title. This occurs on % frames 2,3, and 7 (which are the last slides of each occurrence). % I also put labels on the last slides of each occurrence. This was % so that at the end of the talk I could jump back to any given frame % and be sure of landing on the last slide of that frame (i.e. with % all the junk displayed). This worked as one would hope: the % 'qnsorig' jumped back to this occurrence of the frame, 'qnsreprise' % to the second, and 'qnsconclude' to the third. % \begin{frame}<1-2>[label=qns] \frametitle{{\only<2,3,7>{\color{last}}Questions}} \label<2>{qnsorig} \label<3>{qnsreprise} \label<7>{qnsconclude} \uncover<5->{ {\color{blue}For the Morava K--theories:} } \vfill \begin{enumerate} \item When is an unstable cohomology operation a component of a stable one? \vfill \uncover<6->{ {\color{blue} If it can be delooped once.} } \vfill \pause \item If we have a component of a stable operation, can we construct the other components? \vfill \uncover<7->{ {\color{blue} Yes; easily, using the periodicity.} } \end{enumerate} \vfill \end{frame} \begin{frame} \frametitle{{\only<3>{\color{last}}Preliminaries}} \label<3>{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} \vfill \end{frame} \begin{frame} \frametitle{{\only<4>{\color{last}}Forgetfulness}} \label<4>{forget} Three views of \(E^*(-)\): \pause \vfill \begin{itemize} \item One functor, \(E^*(-)\), into graded abelian groups. \pause \vfill \item A family of functors, \(\{E^k(-)\}\), into abelian groups. \pause \vfill \item A family of functors, \(\{E^k(-)\}\), into sets. \end{itemize} \vfill \end{frame} \begin{frame} \frametitle{{\only<2>{\color{last}}Operations}} \label<2>{nat} An \alert{operation} is a natural transformation between functors. \pause \vfill \(F, G : \mathcal{C} \to \mathcal{D}\) contravariant. \(\nu : F \to G\) is: for every \(\mathcal{C}\)--object \(X\), \(\nu_X : F(X) \to G(X)\) such that: % \[ \xymatrix{ F(X) \ar[r]^{\nu_X} \ar@{}[rd]|{\circlearrowleft} & % G(X) \\ % F(Y) \ar[r]^{\nu_Y} \ar[u]^{F(f)} & % G(Y) \ar[u]_{G(f)} } \] \vfill \end{frame} \begin{frame} \frametitle{{\only<5>{\color{last}}Operations}} \label<5>{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 \[ \mathcal{S}^h \to \mathcal{A}_k^{k+h} \subseteq \mathcal{U}_k^{k+h} \] \vfill \end{frame} \begin{frame} \frametitle{{\only<5>{\color{last}}Examples}} \label<5>{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} \vfill \end{frame} \againframe<3>{qns} \begin{frame} \frametitle{{\only<2>{\color{last}}Looping}} \label<2>{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: % \[ (\Omega r)_X : E^{k-1}(X) \cong E^k(\Sigma X) \xrightarrow{r_{\Sigma X}} E^l(\Sigma X) \cong E^{l-1}(X) \] \vfill \end{frame} \begin{frame} \frametitle{{\only<6>{\color{last}}Looping}} \label<6>{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. \vfill \end{frame} % % This is one of the more complicated slides. This has an xypic % diagram with elements revealed piece-by-piece. Unfortunately, an xy % diagram doesn't work well with the \uncover command so one has to use % the \only version. This means that the diagram jumps around as more % bits are revealed. The solution that I found was to use pitprops % and their horizontal versions to ensure that the diagram remained % where it was put. Essentially, the pitprops were just big enough % that any new bits did not require any reformating of the diagram. % Try making all the '0pt's to '1pt's in the arguments of the \rule % commands in the diagram. % \begin{frame} \frametitle{{\only<7>{\color{last}}Example: K--theory}} \label<7>{exk} K--theory: \uncover<4->{\(2\)--periodic.} \pause \[ \xymatrix{ K^{-2}(X) \only<3->{\ar[d]^{\Omega^2 r}} & % K^{-1}(X) \only<3->{\ar[d]^{\Omega r}} & % K^0(X) \ar[d]^{r} & % K^1(X) \only<6->{\ar[d]^{\Omega (\beta^{-1} r \beta)}} & % K^2(X) \ar@{}[d]^{\rule{25pt}{0pt}} \ar@{}[ll]_{\rule{0pt}{30pt}} \only<5->{\ar[d]^{\beta^{-1} r \beta}} \only<4->{\ar@(ul,ur)[ll]_{\beta}} \\ % K^{-2}(X) & % K^{-1}(X) & % K^0(X) \ar@{}[rr]_{\rule{0pt}{30pt}} \only<4->{\ar@(dr,dl)[rr]_{\beta^{-1}}} & % K^1(X) & % K^2(X) % } \] \uncover<7->{ \[ \text{Question: } r = \Omega^2(\beta^{-1} r \beta) ? \] } \vfill \end{frame} \begin{frame} \frametitle{{\only<5>{\color{last}}Adams Operations}} \label<5>{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 \(k = -1\)) } \vfill \end{frame} \begin{frame} \frametitle{{\only<4>{\color{last}}Coefficients}} \label<4>{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} \vfill \end{frame} \begin{frame} \frametitle{{\only<6>{\color{last}}Mod \(p\)}} \label<6>{modp} \alert{Warning:} \(K(X; \F_p) \ne K(X)/(p)\). \pause \[ p = 3, \; k = 11 \] \[ \xymatrix{ K^0(X;\F_p) \ar[d]^{\Psi^{11}} & % K^2(X;\F_p) \only<3->{\ar[d]^{\frac1{11} \Psi^{11} \only<4->{=2 \Psi^{11}}}} & % K^4(X;\F_p) \ar@{}[d]^{\rule{45pt}{0pt}} \only<5->{\ar[d]^{\frac2{11} \Psi^{11} \only<6->{=\Psi^{11}}}} \\ % K^0(X;\F_p) & % K^2(X;\F_p) & % K^4(X;\F_p) } \] \vfill \end{frame} \begin{frame} \frametitle{{\only<5>{\color{last}}Answers}} \label<5>{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} \vfill \end{frame} \begin{frame} \frametitle{{\only<2>{\color{last}}All Operations}} \label<2>{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} \vfill \end{frame} \begin{frame} \frametitle{{\only<5>{\color{last}}Reconstruction}} \label<5>{recon} Start with a component of a stable operation. \[ \xymatrix{ \only<3->{K^i(X;\F_p)} \only<4->{\ar[d]^{\Omega^j r_k}} & % K^k(X;\F_p) \ar[d]^{r_k} & % \only<2->{K^l(X;\F_p)} \ar@{}[ll]_{\rule{0pt}{30pt}} \only<3->{\ar@(ul,ur)[ll]_{\beta^{m(p-1)}}} \only<2->{\ar[d]_{r_l }^{\only<5->{=\beta^{-m(p-1)}(\Omega^j r_k) \beta^{m(p-1)}}}} \ar@{}[d]^{\rule{100pt}{0pt}} \\ % \only<3->{K^{i+h}(X;\F_p)} \ar@{}[rr]_{\rule{0pt}{30pt}} \only<3->{\ar@(dr,dl)[rr]_{\beta^{-m(p-1)}}} & % K^{k+h}(X;\F_p) & % \only<2->{K^{l+h}(X;\F_p)} \\ % \rule{70pt}{0pt} & % \rule{70pt}{0pt} & % \rule{70pt}{0pt} } \] \vfill \end{frame} \begin{frame} \frametitle{{\only<3>{\color{last}}Recognition Improvement}} \label<3>{recog} \vfill \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 \vfill \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 \vfill That is, if \(r\) deloops \alert{once} then it deloops \alert{as many times as we like}. \vfill \end{frame} \begin{frame} \frametitle{{\only<4>{\color{last}}Morava K--theories}} \label<4>{morava} For each prime \(p\), a sequence of cohomology theories \(\{K(n)^*(-)\}\) -- the {\color{red}chr}{\color{orange}oma}{\color{yellow}tic}{\color{green}\ fi}{\color{blue}ltr}{\color{purple}ati}{\color{violet}on.} \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} \vfill \end{frame} \begin{frame} \frametitle{{\only<2>{\color{last}}Recognition and Reconstruction}} \label<2>{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} \vfill \end{frame} \begin{frame} \frametitle{{\only<3>{\color{last}}Notes}} \label<3>{notes} \vfill \begin{enumerate} \item The periodicity has changed to reflect the periodicity of the cohomology theory. \pause \vfill \item The periodicity is always that of the cohomology theory. (Compare with \(K^*(-;\F_p)\)) \pause \vfill \item The ``delooping'' condition has not changed: if we can deloop \alert{once} we can deloop as many times as we like. \end{enumerate} \vfill \end{frame} \begin{frame} \frametitle{{\only<4>{\color{last}}Remarks}} \label<4>{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} \vfill \end{frame} \againframe<4->{qns} \begin{frame} \frametitle{Navigation} \begin{columns} \begin{column}{6cm} \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}{6cm} \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} \vfill \end{frame} \end{document} % % This bibilography didn't appear in the final presentation, but I % would have included it in any handout or online version. I created % it by doing a normal bibliography and then including the .bbl file % and manually formating it to fit in the frames. % % The original frame was thus: % %\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} % \begin{frame} \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} \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}