%%% ==================================================================== %%% @LaTeX-file{ %%% filename = "Spinaction.tex", %%% author = "Jim Bryan'' %%% address = "Jim Bryan %%% 1917a Francisco St. %%% Berkeley, CA 94703 %%% USA", %%% telephone = "510-848-3399" %%% email = "jbryan@math.uci.edu" %%% keywords = "Spin action, Seiberg-Witten, 11/8-th's conjecture" %%% } %%% ==================================================================== \documentclass[12pt]{amsart} \pagestyle{headings} %%%% MSRI preprint series specs: \headheight=7pt \topmargin=14pt \textheight=574pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \address{M.S.R.I. \\ 1000 Centennial Dr.\\ Berkeley, CA 94720 } \email{jbryan@msri.org} \title[$\znums /2^{p}$ actions on spin 4-manifolds] {Seiberg-Witten Theory and $\znums /2^{p}$ actions\\on spin 4-manifolds} \author{Jim Bryan} \date{} \usepackage{amsmath,amsfonts} %\usepackage{amsthm} \usepackage{amscd} % Theorem environments %% \theoremstyle{plain} %% This is the default \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \newtheorem{lem}[thm]{Lemma} \newtheorem{prop}[thm]{Proposition} \newtheorem{ax}{Axiom} \theoremstyle{definition} \newtheorem{defn}{Definition}[section] \theoremstyle{remark} \newtheorem{rem}{Remark}[section] \newtheorem{example}{Example}[section] \newtheorem*{notation}{Notation} %\numberwithin{equation}{section} \newcommand{\thmref}[1]{Theorem~\ref{#1}} \newcommand{\secref}[1]{\S\ref{#1}} \newcommand{\lemref}[1]{Lemma~\ref{#1}} \newcommand{\szero}[1]{s^{-1}_{#1}(0)} \newcommand{\itsy} {\scriptscriptstyle} % for really small subscripts \newcommand{\cnums} {{\mathbf C}} % complex numbers \newcommand{\nnums} {{\mathbf N}} % natural numbers \newcommand{\rnums} {{\mathbf R}} % real numbers \newcommand{\znums} {{\mathbf Z}} % integers \newcommand{\qnums} {{\mathbf Q}} % rationals \newcommand{\cpbar} {\overline{\mathbf {CP}}^2} \newcommand{\dbar}{\bar\partial} \newcommand{\del}{\partial} \newcommand{\dirac}{\partial \hspace{-5pt}\raisebox{1.5pt}{/}\hspace{-1pt}} \newcommand{\ontoarrow}{\hspace{3pt}\to \hspace{-19pt}{\longrightarrow}} \newcommand{\Hom}{\operatorname{Hom}} \newcommand{\Ker}{\operatorname{Ker}} \newcommand{\End}{\operatorname{End}} \newcommand{\Tr}{\operatorname{tr}} \newcommand{\tr}{\operatorname{tr}} \newcommand{\Coker}{\operatorname{Coker}} \newcommand{\im}{\operatorname{Im}} \newcommand{\notexists}{\exists\hspace{-6pt}\raisebox{1.5pt}{/}} \newcommand{\res}[1]{\!\!\mid_{#1}} %restricted to \newcommand{\comp}{\hspace{-3pt}\stackrel{\scriptstyle \circ}{}\hspace{-3pt}} %composed with \newcommand{\ie}{{\em i.e. }} \newcommand{\eg}{{\em e.g. }} \newcommand{\point}{\text{pt.}} \newcommand{\til}[1]{\Tilde{#1}} \newcommand{\ttil}[1]{\Tilde{\Tilde{#1}}} \newcommand{\1}{{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}} \newcommand{\Zp}{\znums /2^{p}} \newcommand{\Zi}{\znums /2^{i}} \newcommand{\eleveneights}{``11/8-th's'' } \newcommand{\teneights}{``10/8-th's'' } \newcommand{\Hyper}{\left(\smallmatrix 0&1\\1&0\endsmallmatrix\right)} \newcommand{\embed}{\hookrightarrow} \newcommand{\CP}{\cnums \mathbf{P}^{2}} \newcommand{\pin}{\operatorname{Pin}{(2)}} \newcommand{\smargin}[1]{\marginpar{\tiny{#1}}} \newcommand{\rootminus}{\sqrt{-1}} \newcommand{\Ztwoq}{(\znums /2)^{q}} \begin{document} \begin{abstract} Furuta's \teneights theorem gives a bound on the magnitude of the signature of a smooth spin 4-manifold in terms of the second Betti number. We show that in the presence of a $\Zp $ action, his bound can be strengthened. As applications, we give new genus bounds on classes with divisibility and we give a classification of involutions on rational cohomology $K3$'s. We utilize the action of $Pin(2)\til{\times }\Zp $ on the Seiberg-Witten moduli space. Our techniques also provide a simplification of the proof of Furuta's theorem. \end{abstract} \thanks{Supported by M.S.R.I. postdoctoral fellowship.} \maketitle %\markboth{$\Zp $ actions on spin manifolds} %{$\Zp $ actions on spin manifolds} %\renewcommand{\sectionmark}[1]{} \section{Introduction}\label{sec: intro} In early 1995, Furuta \cite{Furuta} proved that if $X$ is a smooth, compact, connected spin 4-manifold with non-zero signature $\sigma (X)$, then \begin{equation}\label{eq: 108ths} \frac{5}{4}|\sigma (X)|+2 \leq b_{2}(X). \end{equation} This estimate has been dubbed the \teneights theorem in comparison with the \eleveneights conjecture which predicts the following bound: $$ \frac{11}{8}|\sigma (X)|\leq b_{2}(X). $$ The inequality (\ref{eq: 108ths}) follows by a surgery argument from the non-positive signature, $b_{1}(X)=0$ case: \begin{thm}[Furuta]\label{thm: Furuta's} Let $X$ be a smooth spin 4-manifold with $b_{1}(X)=0$ with non-positive signature. Let $k=-\sigma (X)/16$ and $m=b_{2}^{+}(X)$. Then, $$ 2k+1\leq m $$ if $m\neq 0$. \end{thm} The main result of this paper improves the above bound by $p$ under the assumption that $X$ has a $\Zp $ action satisfying some non-degeneracy conditions (the analogues of the condition $m\neq 0$ in the above theorem). A $\Zp $ action is called a {\em spin action} if the generator of the action $\tau :X\to X$ lifts to the spin bundle $\hat{\tau }:P_{Spin}\to P_{Spin}$. Such an action is of {\em even type} if $\hat{\tau }$ has order $2^{p}$ and is of {\em odd type} if $\hat{\tau }$ has order $2^{p+1}$. Using \cite{At-Bo}, it easy to determine if an action is spin and whether it is of even or odd type. We assume throughout that $X$ is a smooth spin 4-manifold with $b_{1}(X)=0$ and oriented so that the signature is non-positive. We will continue with the notation $k=-\sigma (X)/16$ and $m=b_{2}^{+}(X)$. Our main results are the following: \begin{thm}\label{thm: thmA} Suppose that $\tau:X\to X$ generates a smooth $\Zp $ action that is spin and of odd type. Let $X_{i}$ denote the quotient of $X$ by $\Zi \subset \Zp$. Then $$ 2k+1+p\leq m $$ if $m\neq 2k+b_{2}^{+}(X_{1})$ and $b_{2}^{+}(X_{i})\neq b_{2}^{+}(X_{j}) >0$ for $i\neq j$. \end{thm} For even actions, the easiest results to state are for involutions: \begin{thm}\label{thm: thmB} Suppose that $\sigma :X\to X$ is a smooth involution of even type. Then $$ 2k+2\leq m $$ if $m\neq b_{2}^{+}(X/\sigma )>0$. \end{thm} This is a special case of the following theorem: \begin{thm}\label{thm: thmD} Suppose that $\sigma _{1},\ldots,\sigma _{q}:X\to X$ are smooth involutions of even type generating an action of $\Ztwoq$. Then $$ 2k+1+q\leq m $$ if $m\neq b_{2}^{+}(X/g)$ for any non-trivial element $g\in \Ztwoq $ and $b_{2}^{+}(X/\Ztwoq )\neq 0$. \end{thm} \begin{rem}\label{rem: non-deg conditions can also be in terms of fixed set} The non-degeneracy conditions in Theorems \ref{thm: thmA}, \ref{thm: thmB}, and \ref{thm: thmD} can be restated in terms of the fixed point set although for our applications, the conditions on the quotient are more convenient. \end{rem} We also include the following \begin{thm}\label{thm: thmC} Suppose that $\tau:X\to X$ generates a smooth $\Zp $ action that is spin and of either type. Then $b_{2}^{+}(X/\tau )\neq 0$ if $k>0$. \end{thm} One can apply Theorem \ref{thm: thmA} to a cover of a four manifold branched along a smoothly embedded surface. The inequality can be used to get a bound on the genus of the embedded surface (see Theorem \ref{thm: genus bound in the general case}). One special case is the following: \begin{thm}\label{thm: genus bound using double cover} Let $M$ be a smooth, compact, oriented, simply connected 4-manifold (not necessarily spin) with $b_{2}^{+}(M)>1$ and let $\Sigma \embed M$ be a smooth embedding of a genus $g$ surface. Suppose that the homology class defined by $\Sigma $ is divisible by 2 and that $[\Sigma] /2\equiv w_{2}(M) \mod 2$. Then $$ g\geq \frac{5}{4}\left(\frac{[\Sigma] ^{2}}{4} -\sigma (M)\right)-b_{2}(M)+2. $$ \end{thm} This bound is typically weaker than the adjunction inequality from the Seiberg-Witten invariants (see \cite{M-S-T}), but the bound applies to manifolds even with zero Seiberg-Witten invariants. For examples where the bound is sharp and new we have (see also example \ref{example}): \begin{cor}\label{cor: gmin for ((4,4),6) and (2,6)} The minimal genus of an embedded surface in $\CP \# \CP $ representing the class $(6,2)$ is 10. The minimal genus of an embedded surface in $S^{2}\times S^{2} \# \CP $ representing the class $((4,4),6)$ is 19. \end{cor} \begin{proof} In each case the lower bound of Theorem \ref{thm: genus bound using double cover} is realized by the connected sum of smooth algebraic curves in each factor. \end{proof} As another application of the theorems we give a ``classification'' of involutions on rational cohomology $K3$'s. \begin{thm}\label{thm: classification of involutions on Q-K3's} Let $\sigma :X\to X$ be a spin involution of a rational cohomology $K3$ (\ie $b_{1}(X)=0$ and $Q_{X}\cong Q_{K3}$ ). If $\sigma $ is of even type then it has exactly 8 isolated fixed points and $b_{2}^{+}(X/\sigma )=3$; if $\sigma $ is of odd type then $b_{2}^{+}(X/\sigma )=1$. \end{thm} This theorem recovers as a special case a theorem of Donaldson concerning involutions on the $K3$ (\cite{D-K} Cor. 9.1.4) and is related to a theorem of Ruberman \cite{Ru}. We also remark that both possibilities in the theorem actually occur. The proof of Theorems \ref{thm: thmA}, \ref{thm: thmB}, \ref{thm: thmD}, and \ref{thm: thmC} uses Furuta's technique of ``finite dimensional approximation'' for the Seiberg-Witten moduli spaces to reduce the problem to algebraic topology. The main innovation of our technique is our approach to the equivariant $K$-theory. In particular, we do not need the Adam's operations in equivariant $K$-theory and can thus simplify that part of Furuta's proof. In section \ref{sec: S-W theory} we introduce the equations and use Furuta's technique to study the moduli space; in section \ref{sec: equiv K-theory} we use equivariant $K$-theory and representation theory to study the $G$-equivariant properties of the moduli space and prove the main theorems; section \ref{sec: applications} is devoted to applications, primarily genus bounds obtained by branched covers and our classification of involutions on rational cohomology $K3$'s. We would like to thank R. Cohen, B. Gompf, C. Gordon, D. Kotschick, P. Kronheimer, T. Lawson, N. C. Leung, G. Mat\'{\i}c, P. Oszvath, D. Ruberman, R. Stern, P. Teichner, and R. Wentworth for helpful conversations, and we would like to thank M.S.R.I. for providing an excellent research environment where this work was carried out. \section{Seiberg-Witten Theory}\label{sec: S-W theory} In this section we use Furuta's ``finite dimensional approximation'' technique to study the Seiberg-Witten moduli space in the presence of our $\Zp $ symmetry. The goal of the section is to use the Seiberg-Witten solutions to produce a certain $G$-equivariant map between spheres. This map will then be studied by algebraic topology in section \ref{sec: equiv K-theory} to produce a proof of the main theorems. Let $X$ be a smooth, compact, connected, spin 4-manifold with $b_{1}(X)=0$ and fix an orientation so that $\sigma (X)\leq 0$. By Rochlin's theorem, $\sigma (X) $ is divisible by 16 and so let $k=-\sigma (X)/16$ and let $m=b_{2}^{+}(X)$. Let $\tau :X\to X$ be an orientation preserving diffeomorphism generating a $\Zp $ action and fix an invariant Riemannian metric $g$. We assume that $\tau $ is a spin action. By definition this means that $\tau^{*}(\wp )-\wp $ is zero as an element of $H^{1}(X;\znums /2)$ where $\wp $ is the spin structure (the difference of two spin structures is naturally an element of $H^{1}(X;\znums /2)$). There is then a lift $\hat{\tau }$ of $\tau $ to the spin bundle: $$ \begin{CD} P_{\text{Spin}}&@>{\hat{\tau }}>>&P_{\text{Spin}}\\ @VVV& & @VVV\\ P_{\text{SO(4)}}&@>{d{\tau }}>>&P_{\text{SO(4)}}\\ @VVV& & @VVV\\ X& @>{\tau }>>& X. \end{CD} $$ The order of $\hat{\tau }$ is either $2^{p}$ or $2^{p+1}$ depending on whether $(\hat{\tau })^{2^{p}}$ is the identity or the non-trivial deck transformation of the double cover $P_{\text{Spin}}\to P_{\text{SO(4)}}$. We say that $\tau $ is of even type if $\hat{\tau }$ has order $2^{p}$ and of odd type if $\hat{\tau }$ has order $2^{p+1}$. A lemma of Atiyah and Bott makes it easy to determine the type of $\tau $: The $2^{p-1}$-th iteration of $\tau $ is an involution $\sigma $ of $X$ and will have a fixed point set consisting of manifolds of constant even dimension. If the fixed point set of $\sigma $ consists of points (or all of $X$), then $\tau $ is of even type; if the fixed point set is dimension 2, then $\tau $ is of odd type. In the case where $\sigma $ acts freely, then $\tau $ is of even or odd type depending on whether the quotient $X/\sigma $ is spin or not (c.f. \cite{At-Bo} or \cite{Br-Gok}). On a spin manifold, the Seiberg-Witten equations for the trivial spin$^{\cnums }$ structure take on a somewhat special form. In \cite{Furuta}, Furuta gives a concise and elegant description of the equations. We give a slightly different description to avoid repetition and to make the exposition more closely match standard notation. Let $S=S^{+}\oplus S^{-}$ denote the decomposition of the spinor bundle into the positive and negative spinor bundles. Clifford multiplication induces an isomorphism $$\rho :\Lambda^{*}_{\cnums }\to \End _{\cnums }(S) $$ between the bundle of complex valued forms and endomorphisms of $S$. Let $\dirac :\Gamma (S^{+})\to \Gamma (S^{-})$ be the Dirac operator. The Seiberg-Witten equations are for a pair $(a,\phi )\in \Omega ^{1}(X,\rootminus \rnums ) \times \Gamma (S^{+})$ and they are \begin{eqnarray*} \dirac \phi +\rho (a)\phi & =& 0,\\ \rho (d^{+}a)-\phi \otimes \phi ^{*}+\tfrac{1}{2}|\phi |^{2}\1 &=&0,\\ d^{*}a& =& 0. \end{eqnarray*} This system of equations is elliptic as written---the last equation defines a slice for the (based) gauge group of the usual equations. The image of $\rootminus \Lambda ^{2}_{+}$ under $\rho $ is the tracefree, hermitian endomorphisms of $S^{+}$ which we denote by $\rootminus \mathfrak{su}(S^{+})$. (We remark that by taking a slightly different gauge fixing condition, one could combine the last two equations in the single equation $\rho (d^{+}a+d^{*}a)=\phi \otimes \phi ^{*}.$) Following Furuta's notation, we regard the equations as the zero set of a map $$ D+Q:V\to W' $$ where $V$ is the $L_{2}^{4}$-completion of $\Gamma (\rootminus \Lambda ^{1}\oplus S^{+})$ and $W'$ is the $L^{3}_{2}$ completion of $\Gamma (S^{-}\oplus \rootminus \mathfrak{su}(S^{+})\oplus \rootminus \Lambda ^{0})$ and $D$ and $Q$ are the linear and quadratic pieces of the equations respectively, \ie \begin{eqnarray*} D(a,\phi )& =& (\dirac \phi ,\rho (d^{+}a),d^{*}a),\\ Q(a,\phi )& =& (\rho (a)\phi ,\phi \otimes \phi ^{*}-\tfrac{1}{2}|\phi |^{2}\1 ,0). \end{eqnarray*} The image of $D+Q$ is $L^{2}$-orthogonal to the constant functions in $\rootminus \Omega ^{0}\subset W'$. We define $W$ to be the orthogonal complement of the constant functions in $W'$ and consider $D+Q$ as defined on $W$: $$ D+Q:V\to W. $$ We now wish to determine the group of symmetries of the equations. As in Furuta, we will see that $\pin$ acts and, as one would expect, there are additional symmetries arising from $\tau $. The symmetry group turns out to depend on whether $\tau $ is of even or odd type. Define $\pin \subset \text{SU}(2)$ to be the centralizer of $S^{1}\subset \text{SU}(2)$. Regarding $\text{SU(2)}$ as the group of unit quaternions and taking $S^{1}$ to be elements of the form $e^{\rootminus \theta }$, $\pin $ then consists of elements of the form $e^{\rootminus \theta }$ or $e^{\rootminus \theta }J$ (our quaternions are spanned by $\langle 1,\rootminus ,J,\rootminus J \rangle$). We define the action of $\pin $ on $V$ and $W$ as follows: Since $S^{+}$ and $S^{-}$ are $\text{SU(2)}$ bundles, $\pin $ naturally acts on $\Gamma (S^{\pm })$ by multiplication on the left. $\znums /2$ acts on $\Gamma (\Lambda ^{*}_{\cnums })$ by multiplication by $\pm 1$ and this pulls back to an action of $\pin $ by the natural map $\pin \to \znums /2$. A calculation shows that this pullback also describes the induced action of $\pin $ on $\rootminus \mathfrak{su}(S^{+})$. Both $D$ and $Q$ are then seen to be $\pin $ equivariant maps. Note that the action of $S^{1}\subset \pin $ is the ordinary action of the constant gauge transformations. The isometry $\tau $ acts on $V$ and $W$ by pull back by $d\tau $ and $\hat{\tau }$; $D$ and $Q$ are equivariant with respect to this action. If $\tau $ is of even type, then it induces an action of $\Zp $ on $V$ and $W$; if $\tau $ is of odd type then it induces an action of $\znums /2^{p+1}$. In the even case the symmetry group is thus $$ G_{ev}\equiv \pin \times \znums /2^{p}. $$ In the odd case $\pin \times \znums /2^{p+1}$ acts but we see that the diagonal $\znums /2$ subgroup acts trivially: the $2^{p}$-th iteration of $\hat{\tau } $ acts on the spinors by the action induced by the non-trivial deck transformation; this is the same as the action of the constant gauge transformation $-1\subset S^{1}\subset \pin $. Thus the symmetry group for the odd case is $$ G_{odd}\equiv \frac{\pin \times \znums /2^{p+1}}{\znums /2}. $$ Throughout the sequel we write $G$ to mean $G_{odd}$ or $G_{ev}$. To state Furuta's finite dimensional approximation theorem we define $V_{\lambda }$ for any $\lambda \in \rnums $ to be the subspace of $V$ spanned by the eigenspaces of $D^{*}D$ with eigenvalues less than or equal to $\lambda $. Similarly, define $W_{\lambda }$ using $DD^{*}$ and write $p _{\lambda }$ for the $L^{2}$-orthogonal projection from $W$ to $W_{\lambda }$. Define $$ D_{\lambda }+Q_{\lambda }:V_{\lambda }\to W_{\lambda } $$ by the restriction of $D+p_{\lambda }Q$ to $V_{\lambda }$. Since the eigenspaces of $D^{*}D$ and $DD^{*}$ are $G$ invariant and $p_{\lambda }$ is a $G$-map, $D_{\lambda }+Q_{\lambda }$ is a $G$ equivariant map between finite dimensional $G$ representations and it is an approximation to $D+Q$ in the following sense: \begin{lem}[Lemma 3.4 of \cite{Furuta}]\label{lem: D+Qlambda-1(0) is compact} There exists an $R\in \rnums $ such that for all $\lambda >R$ the inverse image of zero $(D_{\lambda }+Q_{\lambda })^{-1}(0)$ is compact. \end{lem} The lemma allows Furuta to use $D_{\lambda }+Q_{\lambda }$ to construct a $G$-equivariant map on disks preserving boundaries $$ f_{\lambda }:(BV_{\lambda ,\cnums },SV_{\lambda ,\cnums })\to (BW_{\lambda ,\cnums },SW_{\lambda ,\cnums }) . $$ Here $V_{\lambda ,\cnums }=V_{\lambda }\otimes \cnums $, $BV_{\lambda ,\cnums }$ is homotopic to a ball in $V_{\lambda ,\cnums }$, and $SV_{\lambda ,\cnums }$ is the boundary of $BV_{\lambda ,\cnums }$ with similar definitions for $W_{\lambda ,\cnums }$, $BW_{\lambda ,\cnums }$, and $SW_{\lambda ,\cnums }$. The virtual $G$-representation $[V_{\lambda }]-[W_{\lambda }]\in R(G)$ is the $G$-index of $D$ and can be determined by the $G$ index theorem and is independent of $\lambda $. We discuss its computation in the next section. \section{Equivariant $K$-theory}\label{sec: equiv K-theory} In this section we use equivariant $K$-theory to deduce restrictions on the map $f_{\lambda }:BV_{\lambda ,\cnums }\to BW_{\lambda ,\cnums }$. Combining this with the index theorem determining $[V_{\lambda }]-[W_{\lambda }]\in R(G)$ we will prove the main theorems. Our $K$-theoretic techniques avoid Furuta's use of the equivariant Adam's operations and thus also provide a simplification in the proof of his \teneights theorem. \subsection{The representation ring of $G_{ev}$ and $G_{odd}$.}\label{subsec: R(G)} We write $R(\Gamma )$ for the complex representation ring of a compact Lie group $\Gamma $ and we write direct sum and tensor product of representations additively and multiplicatively respectively. The representation ring $R(\Zp )$ of $\Zp $ is isomorphic to the group ring $\znums (\Zp )$ and is generated by the standard one dimensional representation $\zeta$ . We write $1$ for the trivial representation, so for example $\zeta ^{2^{p}}=1$, and as a $\znums $ module, $R(\Zp )$ is generated by $1,\zeta ,\ldots,\zeta ^{2^{p}-1}.$ The group $\pin $ has one non-trivial one dimensional representation which we denote $\til{1}$ given by pulling back the non-trivial $\znums /2$ representation by the map $\pin \to \znums /2.$ It has a countable series of 2 dimensional irreducible representations $h_{1},h_{2},\ldots$. The representation $h_{1}$, which we sometimes write as $h$, is the restriction of the standard representation of $SU(2)$ to $\pin \subset SU(2)$. The representations $h_{i}$ can be obtained using the relation $h_{i}h_{j}=h_{i+j}+h_{|i-j|}$ where by convention $h_{0}$ denotes $1+\til{1}$. Note that $\til{1}\cdot h_{i}=h_{i}$. Let $\theta $ denote the standard 1 dimensional representation of $S^{1}$ so that $R(S^{1})\cong \znums [\theta ,\theta ^{-1}]$. It is easy to see that $h_{i}$ restricts to $\theta ^{i}+\theta ^{-i}$ as an $S^{1}$ representation. Since $G_{ev}=\pin \times \Zp $, the representation ring is just the tensor product $R(G_{ev})=R(\pin )\otimes R(\Zp )$. We can thus write a general element $\beta \in R(G_{ev})$ as $$ \beta =\beta _{0}(\zeta )1+\til{\beta }_{0}(\zeta )\til{1}+\sum_{i=1}^{\infty }\beta _{i}(\zeta )h_{i} $$ where $\beta _{0}$, $\til{\beta}_{0}$, $\beta _{1}$,\dots are degree $2^{p}-1$ polynomials in $\zeta $ and all but a finite number of the $\beta _{i}$'s are 0. The irreducible representations of $G_{odd}=(\pin \times \znums /2^{p+1})/(\znums /2)$ are the representations of $\pin \times \znums /2^{p+1}$ that are invariant under $\znums /2$. To avoid confusion we will use $\xi $ for the generator of $R(\znums /2^{p+1})$. The $\pin \times \znums /2^{p+1}$ representations $\xi ^{i}$ and $h_{i}$ are non-trivial restricted to $\znums /2$ if and only if $i$ is odd and the representation $\til{1}$ is trivial restricted to $\znums /2$. The subring $R(G_{odd})\subset R(\pin \times \znums /2^{p+1})$ is therefore generated by $1,\til{1},\xi ^{2},$ and $\xi ^{i}h_{j}$ where $i\equiv j \pmod 2$. We write a general element $\beta \in R(G_{odd})$ as $$ \beta =\beta _{0}(\xi )+\til{\beta }_{0}(\xi )+\sum_{i=1}^{\infty }\beta _{i}(\xi )h_{i} $$ where now $\beta _{0},\til{\beta }_{0},$ and $\beta _{2i}$ are {\em even} polynomials of degree $2^{p+1}-2$ and the $\beta _{2i+1}$'s are {\em odd} polynomials of degree $2^{p+1}-1$. In summary we have: \begin{thm}\label{thm: the relations of R(G)} The ring $R(G_{ev})$ is generated by $1,$ $\til{1},$ $h_{1},$ $h_{2},\ldots,$ and $\zeta $ with the relations $\zeta ^{2^{p}}=1$ and $h_{i}h_{j}=h_{i+j}+h_{|i-j|}$, where $h_{0}=1+\til{1}$. The ring $R(G_{odd})$ is generated by $1,$ $\til{1},$ $\xi ^{2},$ $h_{2},$ $h_{4},\ldots,$ and $\xi h_{1},$ $\xi h_{3},\ldots $ with the relations $\xi ^{2^{p+1}}=1$ and $h_{i}h_{j}=h_{i+j}+h_{|i-j|}$, where $h_{0}=1+\til{1}$. For $G=G_{odd}$ or $G_{ev}$ the restriction map $R(G)\to R(S^{1})\cong \znums [\theta ,\theta ^{-1}]$ is given by $\til{1}\mapsto 1,$ $ \xi \mapsto 1$ (or $\zeta\mapsto 1$), and $h_{i}\mapsto \theta ^{i}+\theta ^{-i}$. \end{thm} \subsection{The index of $D$.}\label{subsec: index of D} The virtual representation $[V_{\lambda ,\cnums }]-[W_{\lambda ,\cnums }]\in R(G)$ is the same as $\operatorname{Ind}(D)=[\Ker D]-[\Coker D]$. Furuta determines $\operatorname{Ind}(D)$ as a $\pin $ representation; denoting the restriction map $r:R(G)\to R(\pin )$, Furuta shows $$ r(\operatorname{Ind}(D))=2kh-m\til{1} $$ where $k=-\sigma(X)/16$ and $m=b^{+}_{2}(X)$. Thus $\operatorname{Ind}(D)=sh-t\til{1}$ where $s$ and $t$ are polynomials in $\xi $ or $\zeta $ such that $s(1)=2k$ and $t(1)=m$. In the case of $G=G_{ev}$ we write $$ \begin{array}{ccc} s(\zeta )=\sum_{i=1}^{2^{p}}s_{i}\zeta ^{i}&\text{and}&t(\zeta )=\sum_{i=1}^{2^{p}}t_{i}\zeta ^{i} \end{array} $$ and for $G=G_{odd}$ we have $$ \begin{array}{ccc} s(\xi )=\sum_{i=1}^{2^{p}}s_{i}\xi ^{2i-1}&\text{and}&t(\xi )=\sum_{i=1}^{2^{p}}t_{i}\xi ^{2i} . \end{array} $$ To compute the coefficients $t_{i}$ and $s_{i}$ one can use the $\Zp $-index theorem. We will only need information about the $t_{i} $ coefficients. From the definition of the operator $D$ we see that the polynomial $\sum_{i=1}^{2^{p}}t_{i}\zeta ^{i}$ is the honest $\Zp $ representation $$\Coker ((d^{*},d^{+}):\Omega ^{1}\to \Omega ^{0}_{\bot}\oplus \Omega ^{2}_{+})$$ where $\Omega ^{0}_{\bot}$ denotes the $L^{2}$-orthogonal complement of the constant functions (this is true for both the odd and even cases). The constant coefficient, $t_{2^{p}}$ is the dimension of the invariant part of $H^{2}_{+}(X)$ which is just $b_{2}^{+}(X/\tau )$. More generally, $b_{2}^{+}$ of the quotients of $X$ by subgroups of $\Zp $ are given by the various sums of the $t_{i}$'s. The dimension of the subspace of $\sum_{i=1}^{2^{p}}t_{i}\xi ^{i}$ invariant under $\znums /2^{j}\subset \znums /2^{p}$ is given by $$ \sum_{i\equiv 0 \bmod 2^{j}}t_{i}=b_{2}^{+}(X_{j}) $$ where $X_{j}$ is the quotient of $X$ by $\znums /2^{j}$ and $X_{0}=X$ by convention. We summarize the discussion in the following \begin{thm}\label{thm: index of D} The index of $D$ is given by $$ [V_{\lambda ,\cnums }]-[W_{\lambda ,\cnums }]=sh-t\til{1}\in R(G) $$ where if $G=G_{ev}$ then $$ \begin{array}{ccc} s(\zeta )=\sum_{i=1}^{2^{p}}s_{i}\zeta ^{i}&\text{and}&t(\zeta )=\sum_{i=1}^{2^{p}}t_{i}\zeta ^{i} \end{array} $$ and if $G=G_{odd}$ then $$ \begin{array}{ccc} s(\xi )=\sum_{i=1}^{2^{p}}s_{i}\xi ^{2i-1}&\text{and}&t(\xi )=\sum_{i=1}^{2^{p}}t_{i}\xi ^{2i} . \end{array} $$ For either $G_{ev} $ or $G_{odd}$ we have \begin{eqnarray*} \sum_{i=1}^{2^{p}}s_{i}& =& 2k\\ \sum_{i\equiv 0\bmod 2^{j}}t_{i}&=&b^{+}_{2}(X_{j}). \end{eqnarray*} \end{thm} \subsection{The Thom isomorphism and a character formula for the $K$-theoretic degree.}\label{subsec: Thom iso and char formula} The Thom isomorphism theorem in equivariant $K$-theory for a general compact Lie group is a deep theorem proved using elliptic operators \cite{At-elliptic} . The subsequent character formula of this section uses only elementary properties of the Bott class. We follow tom Dieck \cite{tD} pgs. 254--255 for this discussion. Let $V$ and $W$ be complex $\Gamma $ representations for some compact Lie group $\Gamma $. Let $BV$ and $BW$ denote balls in $V$ and $W$ and let $f:BV\to BW$ be a $\Gamma $-map preserving the boundaries $SV$ and $SW$. $K_{\Gamma }(V)$ is by definition $K_{\Gamma }(BV,SV)$ and by the equivariant Thom isomorphism theorem, $K_{\Gamma }(V)$ is a free $R(\Gamma )$ module with generator the Bott class $\lambda (V)$. Applying the $K$-theory functor to $f$ we get a map $$ f^{*}:K_{\Gamma }(W)\to K_{\Gamma }(V) $$ which defines a unique element $\alpha _{f}\in R(\Gamma )$ by the equation $f^{*}(\lambda (W))=\alpha _{f}\cdot \lambda (V)$. The element $\alpha _{f}$ is called the {\em $K$-theoretic degree} of $f$. Let $V_{g}$ and $W_{g}$ denote the subspaces of $V$ and $W$ fixed by an element $g\in \Gamma $ and let $V_{g}^{\bot}$ and $W_{g}^{\bot}$ be the orthogonal complements. Let $f^{g}:V_{g}\to W_{g}$ be the restriction of $f$ (well defined because of equivariance) and let $d(f^{g})$ denote the ordinary topological degree of $f^{g}$ (by definition, $d(f^{g})=0$ if $\dim V_{g}\neq \dim W_{g}$). For any $\beta \in R(\Gamma )$ let $\lambda _{-1}\beta $ denote the alternating sum $\sum_{}(-1)^{i}\lambda ^{i}\beta $ of exterior powers. T. tom Dieck proves a character formula for the degree $\alpha _{f}$: \begin{thm}\label{thm: char formula} Let $f:BV\to BW$ be a $\Gamma $-map preserving boundaries and let $\alpha _{f}\in R(\Gamma )$ be the $K$-theory degree. Then $$ \Tr _{g}(\alpha _{f})=d(f^{g})\Tr _{g}(\lambda _{-1}(W_{g}^{\bot}-V_{g}^{\bot})) $$ where $\Tr _{g}$ is the trace of the action of an element $g\in \Gamma $. \end{thm} This formula will be especially useful in the case where $\dim W_{g}\neq \dim V_{g}$ so that $d(f^{g})=0$. We also recall here that $\lambda _{-1}(\sum_{i}a_{i}r_{i})=\prod_{i}(\lambda _{-1}r_{i})^{a_{i}}$ and that for a one dimensional representation $r$, we have $\lambda _{-1}r=(1-r)$. A two dimensional representation such as $h$ has $\lambda _{-1}h=(1-h+\Lambda ^{2} h)$. In this case, since $h$ comes from an $SU(2)$ representation, $\Lambda ^{2}h=\det h=1$ so $\lambda _{-1}h=(2-h)$. All the proofs in the following subsections proceed by using the character formula to examine the $K$-theory degree $\alpha _{f_{\lambda }}$ of the map $f_{\lambda }:BV_{\lambda ,\cnums }\to BW_{\lambda ,\cnums }$ coming from the Seiberg-Witten equations. We will abbreviate $\alpha _{f_{\lambda }}$ as just $\alpha $ and $V_{\lambda ,\cnums }$ and $W_{\lambda, \cnums }$ as just $V$ and $W$ and we will use the following elements of $G$. Let $\phi \in S^{1}\subset \pin \subset G$ be an element generating a dense subgroup of $S^{1}$; let $\eta \in \Zp $ and $\nu \in \znums /2^{p+1}$ be generators and recall that there is the element $J\in \pin $ coming from the quaternions. Note that the action of $J$ on $h$ has two invariant subspaces on which $J$ acts by multiplication with $\rootminus $ and $-\rootminus $. \subsection{Proof of Furuta's theorem (Theorem \ref{thm: Furuta's})} Consider $\alpha =\alpha _{f_{\lambda }}\in R(\pin )$; it has the form $$ \alpha =\alpha _{0}+\til{\alpha }_{0}\til{1}+\sum_{i=1}^{\infty }\alpha _{i}h_{i} . $$ Since $\phi $ acts non-trivially on $h$ and trivially on $\til{1}$, we have that $\dim V_{\phi }\neq \dim W_{\phi }$ as long as $m>0$. The character formula then gives $$ \Tr _{\phi }(\alpha )=0=\alpha _{0}+\til{\alpha }_{0}+\sum_{i=1}^{\infty }\alpha _{i}(\phi ^{i}+\phi ^{-i}) $$ so $\alpha _{0}=-\til{\alpha }_{0}$ and $\alpha _{i}=0$ for $i\geq 1$. Since $J$ acts non-trivially on both $h$ and $\til{1}$, $\dim V_{J}=\dim W_{J}=0$ so $d(f^{J})=1$ and the character formula says \begin{eqnarray*} \Tr _{J}(\alpha )& =& \Tr _{J}(\lambda _{-1}(m\til{1}-2kh))\\ & =& \Tr _{J}((1-\til{1})^{m}(2-h)^{-2k})\\ & =& 2^{m-2k} \end{eqnarray*} using $\Tr _{J}h=0$ and $\Tr _{J}\til{1}=-1$. On the other hand $\Tr _{J}(\alpha )=\Tr _{J}(\alpha _{0}(1-\til{1}))=2\alpha _{0}$ so the degree is $$ \alpha =2^{m-2k-1}(1-\til{1}) $$ and we can conclude $2k+1\leq m$. \qed The proofs of Theorems \ref{thm: thmA}, \ref{thm: thmB}, and \ref{thm: thmD} are a generalization of the above proof. \subsection{Proof of Theorem \ref{thm: thmB}} In this case the group is $G=G_{ev}=\pin \times \znums /2$ and $$ V-W=(s_{1}\zeta +s_{2})h-(t_{1}\zeta +t_{2})\til{1} $$ where $t_{1}+t_{2}=m$ and $s_{1}+s_{2}=2k$. The hypothesis $m\neq b_{2}^{+}(X/\sigma )>0$ translates to $t_{1}+t_{2}\neq t_{2}>0$ which is equivalent to both $t_{1}$ and $t_{2}$ being non-zero. Both $\phi $ and $\phi \nu $ act non-trivially on $h$ and trivially on $\til{1}$ so (since $t_{2}\neq 0$) we have $d(f^{\phi })=d(f^{\phi \nu })=0$ so that $\Tr _{\phi }(\alpha )=\Tr _{\phi \nu }(\alpha )=0$. The general form of $\alpha $ is $$ \alpha =\alpha _{0}(\zeta )+\til{\alpha }_{0}(\zeta )\til{1} +\sum_{i=1}^{\infty }\alpha _{i}(\zeta )h_{i} $$ so the conditions $\Tr _{\phi }(\alpha )=\Tr _{\phi \nu }(\alpha )=0$ imply that \begin{eqnarray*} \alpha _{0}(\pm 1)+\til{\alpha }_{0}(\pm 1)& =& 0\\ \alpha _{i}(\pm 1)& =& 0 \end{eqnarray*} since $\alpha _{0},$ $\til{\alpha }_{0}$, and $\alpha _{i}$ are degree 1 in $\zeta $ we see that $\alpha _{0}=-\til{\alpha }_{0}$ and $\alpha _{i}=0$ so $$ \alpha =(\alpha _{0}^{1}\zeta +\alpha _{0}^{2})(1-\til{1}). $$ Now $J\nu $ acts trivially on $\zeta \til{1}$ and non-trivially on $\zeta h$ and $h$ so since $t_{1}\neq 0$, $d(f^{J\nu })=0$. Thus we get $$ \Tr _{J\nu }(\alpha )=0=(-\alpha _{0}^{1}+\alpha _{0}^{2})\cdot 2 $$ so that $\alpha _{0}^{1}=\alpha _{0}^{2}$. Finally, as before, $J$ acts non-trivially on $W$ and $V$ so that $d(f^{J})=1$ and $\Tr _{J}(\alpha )=2^{m-2k}$ so it must be the case that $$ \alpha = 2^{m-2k-2}(1+\zeta )(1-\til{1}) $$ and thus $2k+2\leq m$.\qed \subsection{Proof of Theorem \ref{thm: thmD}} We generalize the preceding proof. Now $G=\pin \times \Ztwoq $ and we write $\zeta _{i}$ for the non-trivial representation of the $i$-th copy of $\znums /2$. The index $\operatorname{Ind}(D)$ is given by $$ V-W=s(\zeta _{1},\ldots,\zeta _{q})h-t(\zeta _{1},\ldots,\zeta _{q})\til{1} $$ where $s$ and $t$ are polynomial functions in the variables $\zeta _{1},\ldots,\zeta _{q}$ of multi-degree $(1,\ldots,1)$. The hypothesis $m\neq b_{2}^{+}(X/g)$ then implies that the representation $t(\zeta _{1},\ldots,\zeta _{q})$ contains a summand on which $g$ acts as $-1$. Since $J$ acts on $\til{1} $ by $-1$ we see that the representation $t(\zeta _{1},\ldots,\zeta _{q})\til{1}$ has a positive dimensional subspace fixed by $Jg$ for every non-trivial $g\in \Ztwoq $. On the other hand, $Jg$ always acts non-trivially on $s(\zeta _{1},\ldots,\zeta _{q})h$ so the character formula gives us $$ \Tr _{Jg}(\alpha )=0. $$ Since $b_{2}^{+}(X/\Ztwoq )\neq 0$, the coefficient of the trivial representation in $t(\zeta _{1},\ldots,\zeta _{q})$ is non-zero and so $\phi g$ always fixes a non-trivial subspace of $t\til{1}$. On the other hand, $\phi g$ always acts non-trivially on $sh$ so again the character formula shows that $$ \Tr _{\phi g}(\alpha )=0. $$ The general form of $\alpha $ is $$ \alpha =\alpha _{0}+\til{\alpha }_{0}\til{1}+\sum_{i=1}^{\infty }\alpha _{i}h_{i} $$ where $\alpha _{0}$, $\til{\alpha }_{0}$, and $\alpha _{i}$ are polynomial functions in $\zeta _{1},\ldots,\zeta _{q}$ of multi-degree $(1,\ldots,1)$. Now for any polynomial function $\beta (x_{1},\ldots,x_{q})$ of multi-degree $(1,\ldots,1)$, if we know that $\beta ((-1)^{n_{1}},\ldots,(-1)^{n_{q}})=0$ for any arrangement of the signs, then $\beta \equiv 0$. Thus the formula $\Tr _{\phi g}(\alpha )=0$ for all $g\in \Ztwoq $ implies that $\alpha _{0}+\til{\alpha }_{0}\equiv 0$ and $\alpha _{i}\equiv 0$. We can therefore write $$ \alpha =\alpha _{0}(\zeta _{1},\ldots,\zeta _{q})(1-\til{1}). $$ Since $J$ acts non-trivially on both $U$ and $V$ we can compute as before: $$\Tr _{J}(\alpha )=\Tr _{J}(\lambda _{-1}(W-V))=2^{m-2k}.$$ This equation, along with the $2^{q}-1$ equations $\Tr _{Jg}(\alpha )=0$ for $g\neq 1\in \Ztwoq $ give $2^{q} $ independent conditions on $\alpha _{0}$ which determine it uniquely. It must be the following: $$ \alpha =2^{m-2k-1-q}(1+\zeta _{1})(1+\zeta _{2})\cdots (1+\zeta _{q})(1-\til{1}) $$ and thus $2k+1+q\leq m$.\qed \subsection{Proof of Theorem \ref{thm: thmA}} Recall that $$ [V]-[W]=s(\xi )h-t(\xi )\til{1}\in R(G_{odd}) $$ with $s(\xi )=\sum_{i=1}^{2^{p}}s_{i}\xi ^{2i-1}$ and $t(\xi )=\sum_{i=1}^{2^{p}}t_{i}\xi ^{2i}$. The $K$-theory degree $\alpha =\alpha _{f_{\lambda }}$ of $f_{\lambda }$ has the form $$ \alpha =\alpha _{0}(\xi )+\til{\alpha }_{0}(\xi )+\sum_{i=1}^{\infty }\alpha _{i}(\xi )h_{i} . $$ We compute the following characters of $\alpha $: \begin{lem}\label{lem: chars of alpha} $\Tr _{\phi \eta ^{j}}(\alpha )=0$ for $j=1,\ldots,2^{p}, \Tr _{J\eta ^{j}}(\alpha )=0$ for $j=1,\ldots,2^{p}-1$, and $\Tr _{J}(\alpha )=2^{m-2k}$. \end{lem} {\em Proof:} We introduce the notation $(V)_{g}=\dim V_{g}$. By the character formula, $\Tr _{\phi \eta ^{j}}(\alpha )=0$ if $$ (V)_{\phi \eta^{j}}-(W)_{\phi \eta ^{j}}=(s(\xi )h-t(\xi )\til{1})_{\phi \eta ^{j}}\neq 0. $$ Since $\phi \eta ^{j}$ acts non-trivially on every $\xi ^{i}h$ and $\phi \eta ^{j} $ acts trivially on $\til{1}$, we have $$ (V)_{\phi \eta ^{j}}-(W)_{\phi \eta ^{j}}\leq -t_{2^{p}}. $$ By the hypothesis in Theorem \ref{thm: thmA}, $t_{2^{p}}=b_{2}^{+}(X/\tau )>0$ so $\Tr _{\phi \eta ^{j}}(\alpha )=0$. To show $\Tr _{J\eta ^{j}}(\alpha )=0$ for $j=1,\ldots,2^{p}-1$ we need to show $(V)_{J\eta ^{j}}-(W)_{J\eta ^{j}}\neq 0. $ The 2 dimensional representation $h$ decomposes into two complex lines on which $J$ acts as $\rootminus $ and $-\rootminus $, so $$ (\xi ^{2i-1}h)_{J\eta ^{j}}=\begin{cases} 1& \text{if $\eta ^{j} $ acts on $\xi ^{2i-1}$ by $\pm \rootminus $}\\ 0& \text{otherwise.} \end{cases} $$ $\eta ^{j}$ acts as $\pm \rootminus $ on $\xi ^{2i-1}$ if and only if $j$ and $i$ satisfy $$ (2i-1)j\equiv \pm 2^{p-1}\bmod 2^{p+1}. $$ If $j=2^{p-1} $ then the condition is satisfied for every $i$; if $j\neq 2^{p-1}$ the condition is never satisfied (divide both sides by the highest power of 2 that divides $j$ to get an odd number on one side and an even on the other). Thus $$ (s(\xi )h)_{J\eta ^{j}}=\begin{cases} 2k& \text{if $j=2^{p-1}$,}\\ 0& \text{if $j\neq 2^{p-1}$}. \end{cases} $$ Now $J\eta ^{j}$ acts on $\til{1}$ by $-1$ and so $$ (\xi ^{2i}\til{1})_{J\eta ^{j}}=\begin{cases} 1& \text{if $\eta ^{j}$ acts on $\xi ^{2i}$ by $-1$ and}\\ 0& \text{otherwise.} \end{cases} $$ Thus we see that \begin{eqnarray*} (t(\xi )\til{1})_{J\eta ^{j}}& =&\sum_{2ij\equiv 2^{p}\bmod 2^{p+1}} t_{i}\\ & =& \sum_{ij\equiv 2^{p-1}\bmod 2^{p}}t_i. \end{eqnarray*} We want to see that the non-degeneracy conditions of the theorem imply $(V-W)_{J\eta ^{j}}\neq 0$. The conditions are equivalent to $$ 00$ means that $s(\zeta )=\sum_{i=1}^{2^{p}}s_{i}\zeta ^{i}$ has at least one positive coefficient. From Theorem \ref{thm: Furuta's}, $k>0$ also implies $m>0$. We will prove that $b_{2}^{+}(X/\tau )\neq 0$ by contradiction. Suppose $b_{2}^{+}(X/\tau )=0$ so that the constant coefficient of $t(\zeta )$ is 0, \ie $t_{2^{p}}=0$. Since $m>0$ we have $(W-V)_{\phi }\neq 0$ so $\Tr _{\phi }(\alpha )=0$ and since $t_{2^{p}}=0$, we know that $\phi \nu $ acts non-trivially on all of $V$ and $W$ so \begin{eqnarray*} \Tr _{\phi \nu }(\alpha )&=&\lambda _{-1}(W-V)\\ &=&\frac{(1-\nu )^{t_{1}}(1-\nu ^{2})^{t_{2}}\cdots (1-\nu ^{2^{p}-1})^{t_{2^{p}-1}}}{(1-\nu \phi )^{s_{1}}(1-\nu \phi ^{-1})^{s_{1}}\cdots }. \end{eqnarray*} Since at least one of the $s_{i}$'s is positive, the above expression has arbitrarily high powers of $\phi $ in it. On the other hand \begin{eqnarray*} \Tr _{\phi \nu }(\alpha )&=&\Tr _{\phi \nu }\left(\alpha _{0}(\zeta )+\til{\alpha }_{0}(\zeta )\til{1}+\sum_{i=1}^{\infty }\alpha _{i}(\zeta )h_{i} \right)\\ & =&\alpha _{0}(\nu )+\til{\alpha }_{0}(\nu )+\sum_{i=1}^{\infty }\alpha _{i}(\nu )(\phi ^{i}+\phi ^{-i}) \end{eqnarray*} has only finitely many non-zero $\alpha _{i}$ terms, which is our contradiction. \qed \section{Applications}\label{sec: applications} \subsection{Genus bounds.}\label{subsec: genus bounds} Our original motivation for this work is our application to genus bounds. We refer the reader to our paper \cite{Br-Gok} for details of the set up. Let $M$ be a smooth 4-manifold (not necessarily spin) and let $\Sigma \embed M$ be a smoothly embedded surface representing a homology class $a\in H_{2}(M;\znums )$. A basic question in 4-manifold topology asks for the minimal genus $g_{min}(a)$ of a smoothly embedded surface representing a given class $a$. In order to determine $g_{min}(a)$ one needs good lower bounds on the genus of embedded surfaces and constructions realizing those bounds. The basic ideas we use go back to Hsiang-Szczarba \cite{Hs-Sz}, Rochlin \cite{Ro}, and Kotschick-Mat\'{\i}c \cite{Ko-Ma} (see also T. Lawson \cite{Lawson}). Suppose the class $a$ is divisible by some number $d$, then one can construct the $d$-fold branched cover $X\to M$ branched along $\Sigma $. Under favorable hypotheses, $X$ will be spin and the covering transformation is a spin action. The signature and Euler characteristic of $X$ can be computed in terms of $\Sigma \cdot \Sigma $ and $g(\Sigma )$ and so the bounds of Theorems \ref{thm: Furuta's} and \ref{thm: thmA} give genus bounds. The most straightforward implementation of this idea is for double branched covers and results in Theorem \ref{thm: genus bound using double cover}. We will prove the general result coming from $2^{p}$ covers in this section. \begin{prop}\label{prop: branched cover is spin} Let $\Sigma \embed M$ be a smoothly embedded surface of genus $g$ in a smooth, simply-connected, compact, oriented 4-manifold. Suppose that $[\Sigma ]\in H_{2}(M)$ has the property that $2^{p}|[\Sigma ]$ and $[\Sigma ]/2^{p}\equiv w_{2}(M)\bmod 2$. Then there is a spin 4-manifold $X$ with a spin $\znums /2^{p}$ action $\tau :X\to X$ of odd type such that $X/\tau =M$. Furthermore, $k=-\sigma (X)/16$ and $m=b_{2}^{+}(X)$ are given by: \begin{eqnarray*} k&=&\frac{1}{16}\left(-2^{p}\sigma (M)+\frac{4^{p}-1}{3\cdot 2^{p}}[\Sigma ]\cdot [\Sigma ] \right),\\ m&=&2^{p}b_{2}^{+}(M)+(2^{p}-1)g-\frac{4^{p}-1}{6\cdot 2^{p}}[\Sigma ]\cdot [\Sigma ]. \end{eqnarray*} \end{prop} {\em Proof:} This is in \cite{Br-Gok}; a sketch of the proof is the following. The condition $2^{p}|[\Sigma ]$ allows one to construct the $2^{p}$-fold branched cover $X\to M$ branched along $\Sigma $. One can compute $w_{2}(X)$ using a formula of Brand \cite{Brand} and the condition of $[\Sigma ]/2^{p}\equiv w_{2}(M)\bmod 2$ guarantees $w_{2}(X)$ vanishes. $\pi _{1}(X)$ is finite (see \cite{Ro}, \cite{Ko-Ma}) so $b_{1}(X)=b_{3}(X)=0$ and so $m$ and $k$ can be determined by $e(X)$ and $\sigma (X)$ which can be computed by the $G$-index theorem. The covering action $\tau :X\to X$ is automatically spin because $H_{1}(X;\znums /2)=0$ (see \cite{Ko-Ma}) and the action is of odd type since the fixed point set is two dimensional. \qed Here is our general genus bound (Theorem \ref{thm: genus bound using double cover} is a special case): \begin{thm}\label{thm: genus bound in the general case} Let $\Sigma \embed M$ be a smoothly embedded surface of genus $g$ in a smooth, simply-connected, compact, oriented 4-manifold. Suppose that $[\Sigma ]\in H_{2}(M)$ has the property that $2^{p}|[\Sigma ]$ and $[\Sigma ]/2^{p}\equiv w_{2}(M)\bmod 2$. Suppose $b_{2}^{+}(M)>1$ and $g\neq [\Sigma ]^{2}(1+2^{2i-2p+1})/6-b_{2}^{+}(M)$ for $i=1,\ldots,p-1 $. Then \begin{equation}\label{eqn: genus bound} g\geq \frac{1}{2^{p}-1}\left[\frac{5}{4}\left(\frac{4^{p}-1}{6\cdot 2^{p}}[\Sigma ]^{2}-2^{p-1}\sigma (M) \right)+1+p-2^{p-1}b_{2}(M) \right]. \end{equation} \end{thm} \begin{rem} The conditions on $b_{2}^{+}(M)$ and $g$ are so that the non-degeneracy conditions of Theorem \ref{thm: thmA} are met. The condition on $g$ is not present for $p=1$. Although it is aesthetically unpleasing to have restrictions on $g$ in the hypothesis, in practice the conditions are easy to dispense with (see example \ref{example} ). Even without the conditions one still gets an inequality coming from Furuta's theorem which is the same as the bound (\ref{eqn: genus bound}) but without the $p$ term in the brackets. Our theorem thus improves the Furuta bound by $p/(2^p-1)$. Probably the best applications occur in the case $p=1$ (as in Theorem \ref{thm: genus bound using double cover} where we improve the Furuta bound by 1). However, we will give examples where Theorem \ref{thm: genus bound in the general case} provides sharp bounds unobtainable from the Furuta bound. \end{rem} \begin{example}\label{example} Let $M=\# _{N}{\bf CP}^{2}$ with $N>1$ and consider $\Sigma \embed M$ representing the class $(4,\ldots,4)$. We can apply Theorem \ref{thm: genus bound in the general case} with $p=2$. It tells us $$ g\geq \tfrac{1}{3}(8N+3) $$ if $g\neq 3N$. When $N\geq 6$ the condition on $g$ is no condition and for $N=2,\ldots,5$ the theorem gives $g\geq 3N$ since if $g=3N-1$ it would contradict $g\geq\tfrac{1}{3}(8N+3)$. Since the connected sum of the algebraic representative in each factor has genus $3N$, the theorem gives a sharp bound for $N=2,\ldots,5$. \end{example} {\em Proof of Theorem \ref{thm: genus bound in the general case} :} Let $X\to M$ be the $2^{p}$-th branched cover and, to match the notation of Theorem \ref{thm: thmA}, let $X_{i}$ be the $2^{p-i}$ branched cover. Then $X_{0}=X$, $X_{p}=M$ and $X_{i}$ is the quotient of $X$ by $\znums /2^{i}\subset \Zp $. Let $m_{i}=b^{+}_{2}(X_{i})$ which is computable by essentially the same formula as in Proposition \ref{prop: branched cover is spin} $$ m_{i}=2^{p-i}m_{p}+(2^{p-i}-1)g -\frac{4^{p-i}-1}{6\cdot 2^{p-i}}[\Sigma ]^{2}. $$ Note that $m_{0}\geq m_{1}\geq\cdots \geq m_{p}>1$. The inequality of the theorem is equivalent to $2k+1+p\leq m_{0}$ so the only way the theorem can fail is if $2k+\delta =m_{0}$ for $\delta \in \{1,\ldots,p \}$ {\em and} $m_{0}-2k=m_{1}$ or $m_{i}=m_{i-1}$ for $i=2,\ldots,p$. The hypothesis $g\neq [\Sigma ]^{2}(1+2^{2i-2p+1})/6-b_{2}^{+}(M)$ is equivalent to $m_{i}\neq m_{i+1}$ so we know that $m_{1}>m_{2}>\cdots >m_{p}=b_{2}^{+}(M)$. This implies $m_{1}\geq p-1+b_{2}^{+}(M)$ which, combined with the hypothesis $b_{2}^{+}(M)>1$ gives $m_{1}>p$. Then we see that we cannot have $2k+\delta =m_{0}$ and $m_{0}-2k=m_{1}$ since that would imply $\delta >p$.\qed \subsection{Involutions on rational cohomology $K3$'s.}\label{subsec: involutions on K3} The proof of Theorem \ref{thm: classification of involutions on Q-K3's} follows readily from our main theorems (Theorems \ref{thm: thmA}, \ref{thm: thmB}, and \ref{thm: thmC}). Suppose $X$ is a rational cohomology $K3$ so that $H^{*}(X;\qnums)\cong H^{*}(K3;\qnums)$ and suppose $\tau :X\to X$ is a spin involution. First suppose that $\tau $ is of odd type. Since the inequality of Theorem \ref{thm: thmA} is violated, one of the non-degeneracy conditions must also be violated and so $b_{2}^{+}(X/\tau )$ is either 1 or 0. Theorem \ref{thm: thmC} shows that $b_{2}^{+}(X/\tau )$ is not 0 so it must be 1. This can easily occur, for example $K3$ is a double branched cover over ${\bf CP}^{2}$, $S^{2}\times S^{2}$, and $E(1)\cong {\bf CP}^{2}\# 9\cpbar $ and it has a free involution covering the Enriques surface. Now suppose that $\tau $ is even so it must violate the non-degeneracy conditions of Theorem \ref{thm: thmB}. In conjunction with Theorem \ref{thm: thmC} we see that $b_{2}^{+}(X/\tau )$ must be 3. Since the fixed point set is at most points, the $G$-signature theorem tells us that $$ \sigma (X/\tau )=\tfrac{1}{2}\sigma (X)=-8. $$ If $N$ is then the number of fixed points, then the Lefschetz formula tells us the Euler characteristic $$ \chi (X/\tau )=\tfrac{1}{2}(\chi (X) +N)=12+N/2. $$ Since $b_{1}(X/\tau )=0$ and we know $b_{2}^{+}(X/\tau )=3$, we can solve the above equations to get $b_{2}^{-}(X/\tau )=11$ and $N=8$. It was pointed out to us by P. Kronheimer that this does occur. The construction is as follows. Let $\til{Y}$ be the $K3$ surface. There are 8 disjoint $-2$ spheres $S_{1},\ldots,S_{8}$ such that $S=\sum_{i=1}^{8}S_{i}$ is divisible by 2. Let $\til{X}$ be the double branched cover of $\til{Y}$ branched along $S$. The preimage of $S$ in $\til{X}$ is then 8 disjoint $(-1)$ spheres which we can blow down to obtain $X$, a smooth manifold with an involution fixing 8 points covering $Y$, the orbifold obtained by collapsing the $S_{i}$'s in $\til{Y}$: $$ \begin{CD} \til{X}@>>>X\\ @VVV@VVV\\ \til{Y}@>>>Y. \end{CD} $$ We will show that $X$ is a rational cohomology $K3$ (with a little more work, one can see that $X\cong K3$) and $b_{2}^{+}(Y)=3$ so this is the example we seek. From the signature theorem and Lefschetz formula we compute that $\chi (\til{X})=2\chi (\til{Y})-\chi (S)=32$ and $\sigma (\til{X})=2\sigma (\til{Y})-S^{2}=24$ and so, after blowing down, $\chi (X)=24$ and $\sigma (X)=-16$. Since the preimage of $S$ is characteristic in $\til{X}$ we know that $X$ is spin and since $b_{1}(\til{X})=b_{1}(X)=0$, $X$ is a rational cohomology $K3$. \begin{rem} It is easy to construct a {\em homeomorphism} $\tau :X\to X $ generating an involution with $b_{2}^{+}(X/\tau )=2$ (for example) but our classification implies this will not be smoothable, \ie there is no smooth structure on $X$ so that this $\tau $ is a diffeomorphism. \end{rem} \bibliography{biblio} \bibliographystyle{plain} \end{document} \endinput