\documentclass[12pt]{amsart} \headheight=8pt \topmargin=0pt \textheight=634pt \textwidth=432pt \oddsidemargin=18pt \evensidemargin=18pt \usepackage{epsf} \begin{document} \title{Manifolds not containing Gompf nuclei} \author{Andr\'{a}s I. Stipsicz} \thanks{The author was supported by the Magyary Zolt\'an Foundation and by OTKA. Research at MSRI is supported in part by NSF grant DMS-9022140.} \address{Department of Analysis,\\ ELTE TTK, Budapest, Hungary} \email{stipsicz@cs.elte.hu} \newcommand{\cp}{{\Bbb {CP}}} \newcommand{\bfz}{{\Bbb {Z}}} %************************************************** \newtheorem{fact}{Fact}[section] \newtheorem{lemma}[fact]{Lemma} \newtheorem{theorem}[fact]{Theorem} \newtheorem{definition}[fact]{Definition} \newtheorem{remark}[fact]{Remark} \newtheorem{proposition}[fact]{Proposition} %************************************************** \newenvironment{prooff}{\medskip \par \noindent {\it Proof}\ }{\hfill $\square$ \medskip \par} \def\sqr#1#2{{\vcenter{\hrule height.#2pt \hbox{\vrule width.#2pt height#1pt \kern#1pt \vrule width.#2pt}\hrule height.#2pt}}} \def\square{\mathchoice\sqr67\sqr67\sqr{2.1}6\sqr{1.5}6} \def\pf#1{\medskip \par \noindent {\it #1.}\ } \def\endpf{\hfill $\square$ \medskip \par} \def\demo#1{\medskip \par \noindent {\it #1.}\ } \def\enddemo{\medskip \par} \def\qed{~\hfill$\square$} \font\sevenrm=cmr7 \font\tenmsb=msbm10 at 11pt \font\sevenmsb=msbm7 at 9pt \font\fivemsb=msbm5 at 7pt \newfam\msbfam \textfont\msbfam=\tenmsb \scriptfont\msbfam=\sevenmsb \scriptscriptfont\msbfam=\fivemsb \def\Bbb#1{\fam\msbfam\relax#1} \def\complex{{\Bbb C}} \def\IF{{\Bbb F}} \def\que{{\Bbb Q}} \def\pee{{\Bbb P}} \def\real{{\Bbb R}} \def\nat{{\Bbb N}} \def\zed{{\Bbb Z}} \def\C{{\cal C}} \def\O{{\cal O}} \def\cS{{\cal S}} \def\simtimes{\,\tilde\times\,} %{\ \buildrel \sim\over\times\ } \def\chix{{\raise.5ex\hbox{$\chi$}}} \def\frac#1#2{{\textstyle{#1\over#2}}} \def\cpn{{\complex {\Bbb {P}}^n}} \def\cpone{{\complex {\Bbb {P}}^1}} \def\cptwo{{\complex {\Bbb {P}}^2}} \def\ocptwo{\overline{\complex {\Bbb {P}}}^2} %\def\num{{\,\#\,}} \def\blk{$\underline{\qquad}$} \def\ep{\varepsilon} \def\ov{\overline} \begin{abstract} In this note we show that there are 4-manifolds not containing Gompf nucleus $N_2$; in this way we answer Problem 4.98 of Kirby's problem list (see \cite{K}) in the negative. \end{abstract} \maketitle \section{Introduction} \label{elso} This note is devoted to answer a question in Kirby's problem list \cite{K} asking whether every simply connected smooth 4-manifold with $b^+\geq 3$ contains a Gompf nucleus $N_2$ (Problem 4.98 in \cite{K}). We prove that by doing logarithmic transformations on three linearly independent tori in the {\em K3\/}-surface we get a 4-manifold not contaning $N_2$. To make our statements more precise we need a few definitions. The hypersurface $X=\lbrace [z_0:z_1:z_2:z_3]\in \cp ^3\ \vert \ \sum _{i=0} ^3 z_i ^4=0\rbrace \subset \cp ^3$ is a simply connected, smooth 4-manifold with $c_1(X)=0$, hence it is a {\em K3-surface\/}. It is known that all simply connected complex surfaces with vanishing first Chern class are diffeomorphic, consequently from the differential topological point of view $X$ is {\em the K3\/}-surface. The complex surface $X$ admits a holomorphic fibration $\pi \colon X\to \cp ^1$ such that the generic fiber is a smooth elliptic curve --- a 2-dimensional torus. Such fibrations are called {\em elliptic fibrations\/}. It can be assumed that $\pi $ has a singular fiber homeomorphic to the 2-dimensional sphere $S^2$ --- such fibers are called {\em cusp fibers\/}. The fibration also has a section $\sigma \colon \cp ^1\to X$, the image of which is a sphere $S\subset X$ with square $[S]^2=-2$. The {\em Gompf nucleus\/} $N_2$ is by definition the tubular neighborhood of the union of a cusp fiber and a section in $X$. The manifold $N_2$ admits a handle decomposition with one 0-handle and two 2-handles, where these two 2-handles are attached to $S^3=\partial (0-handle)$ according to the Kirby diagram shown in Figure~\ref{egyes}. \begin{figure} \centerline{ \epsfbox{elso.eps}} \caption{Kirby picture for $N_2$} \label{egyes} \end{figure} It can be shown that the {\em K3\/}-surface $X$ contains three disjoint copies of $N_2$. If a 4-manifold $M^4$ contains a 2-dimensional torus $T$ with square 0, then we can perform a {\em logarithmic transformation\/} on $T$: deleting the tubular neighborhood of $T$ (which is diffeomorphic to $D^2\times T^2$) and regluing it via a diffeomorphism $\varphi \colon \partial (M\setminus D^2\times T^2 )\to \partial (D^2\times T^2)$ we get a new manifold $M_{\varphi}$. It turnes out that if $T$ is the fiber in a Gompf nucleus $N_2\subset M$, then the diffeomorphism type of $M_{\varphi}$ will depend only on one nonnegative number $p$ associated to $\varphi$. This number is called the {\em multiplicity\/} of the logarithmic transformation. For more about elliptic surfaces, nuclei and logarithmic transformations see \cite{G}, \cite{FS1} and \cite{GS}. Now perform logarithmic transformations of multiplicity 2 on the three tori contained by the three disjoint nuclei in the {\em K3\/}-surface $X$. The resulting manifold will be denoted by $X_{2,2,2}$. \begin{theorem} The 4-manifold $X_{2,2,2}$ does not contain a Gompf nucleus $N_2$. \label{elsotetel} \end{theorem} \begin{remark} {\rm One of the most interesting questions in 4-manifold theory is whether all 4-manifolds are of simple type or not. (For the definition of simple type see Section~\ref{masodik}.) It is known that if $M^4$ contains a homologically essential torus with square 0, then $M$ is of simple type. There is no example of a 4-manifold with $b^+\geq 3$ and not containing a torus with square 0. One can ask whether every 4-manifold contains a cusp neeighborhood (a tubular neighborhood of a cusp fiber) --- or even a Gompf nucleus $N_2$. The above theorem shows an example of a manifold which contains no $N_2$; $X_{2,2,2}$ is still of simple type, however.} \end{remark} One can modify the question by trying to find more general nuclei $N_n$ in 4-manifolds. The manifold $N_n$ is described by the Kirby diagram shown in Figure~\ref{kettes} --- it can be defined alternatively as the tubular neighborhhod of the union of a cups fiber and a section in a simply connected elliptic surface (admitting a section) with Euler characteristic $12n$ (see \cite{G}). \begin{figure}[b] \centerline{ \epsfbox{kettes.eps}} \caption{Kirby picture for $N_n$} \label{kettes} \end{figure} \begin{theorem} For every $n$ there is a manifold $Y_n$ such that $Y_n$ does not contain $N_n$. \label{masodiktetel} \end{theorem} A related question would be to find a 4-manifold $M$ with the property that $M$ does not contain $N_2(p,q)$; here $N_2(p,q)$ denotes the 4-manifold with boundary we get by performing two logarithmic transformations (of multiplicity $p$ and $q$) along the fiber of the Gompf nucleus $N_2$. Using a recent construction of Fintushel and Stern \cite{FS3} such $M$ can be found, see \cite{SSz}. {\em Acknowledgement}: The author would like to thank MSRI for their hospitality and support, and Zolt\'an Szab\'o for many helpful conversations. \section{Basic classes} \label{masodik} In studying differential topological properties of smooth 4-manifolds, {\em Seiberg-Witten basic classes\/} turn out to be very powerful tools. For the definition of these objects see \cite{A}, \cite{M} or \cite{GS}, here we restrict ourselves to a very short outline. Assume that $M^4$ is a smooth, oriented, simply connected, closed 4-manifold with $b^+_M \geq 3$. A cohomology class $K\in H^2 (M;\bfz )$ with $K\equiv w_2(M)\ (mod\ 2)$ uniquely determines a spin$^c$ structure on $M$, and for such a structure a certain pair of partial differential equations (the so-called Seiberg-Witten equations) can be described --- involving a choice of metric on $M$, a coupled Dirac operator and a perturbation 2-form. By a delicate "counting argument" of the solutions of these equations a number $SW_M(K)$ can be associated to the cohomology class $K$. It turnes out that this number (up to sign) is a smooth invariant of the manifold $M$, more precisely \begin{theorem} If $f\colon M'\to M$ is an orientation preserving diffeomorphism then $SW_M(K)=\pm SW _{M'}(f^{\ast}K)$. Moreover, for a fixed 4-manifold $M$ there are only finitely many classes $K$ with $SW_M(K)\neq 0$, and $SW_M(-K)=\pm SW_M(K)$.\qed \end{theorem} \begin{definition} {\rm The cohomology class $K\in H^2(M;\bfz )$ is called a {\em Seiberg-Witten basic class\/} if $SW_M(K)\neq 0$. The 4-manifold $M$ is of simple type if $SW_M(K)\neq 0$ implies that $K^2=3\sigma (M)+\chi (M)$; here $\sigma (M)$ and $\chi (M)$ stand for the signature and Euler characteristic of $M$.} \end{definition} The most important relation between the smooth topology of a 4-manifold $M$ and its basic classes $\lbrace K_i \ \vert \ i=1,...,n\rbrace $ is shown by the {\em generalized adjunction formula\/}: \begin{theorem} {\rm {(Kronheimer-Mrowka)}} If $\Sigma ^2\subset M^4$ is a smooth, connected 2-dimen\-sional surface of genus $g(\Sigma )$, $[\Sigma ]\neq 0$ and $[\Sigma ]^2\geq 0$, then for every basic class $K$ we have $$ 2g(\Sigma )-2 \geq [\Sigma ]^2+\vert K([\Sigma ])\vert .$$ \qed \end{theorem} \section{Proofs of Theorems~\ref{elsotetel} and \ref{masodiktetel}} \label{harmadik} Assume that $T_1,T_2$ and $T_3$ are three tori in the {\em K3\/}-surface lying in three disjoint Gompf nuclei. Perform logarithmic transformations of multiplicity 2 on each $T_i$. The basic classes of the resulting manifold $X_{2,2,2}$ are determined by Fintushel and Stern \cite{FS2}. \begin{proposition} The basic classes of $X_{2,2,2}$ are the Poincar\'e duals of the homology classes $\pm [{{T_1}\over 2}] \pm [{{T_2}\over 2}] \pm [{{T_3}\over 2}]$.\qed \end{proposition} \begin{prooff} {\em of Theorem~\ref{elsotetel}:\/} Assume that $N_2\subset X_{2,2,2}$. The homology class of the fiber and the section in $N_2$ will be denoted by $f$ and $s$ respectively; note that $f$ and $g=f+s$ have square 0 and can be represented by tori. Consequently (by the generalized adjunction formula) a basic class $K$ of $X_{2,2,2}$ evaluates trivially on $f$ and $g$. Now $f\cdot ([{{T_i}\over 2}]+ [{{T_j}\over 2}]+ [{{T_k}\over 2}])=0$ and $f\cdot (-[{{T_i}\over 2}]+ [{{T_j}\over 2}]+ [{{T_k}\over 2}])0$ ($\lbrace i,j,k\rbrace =\lbrace 1,2,3\rbrace$) implies that $f\cdot [{{T_i}\over 2}]=0$, similarly $g\cdot [{{T_i}\over 2}] =0$ for $i=1,2,3$. Since the complement of the three disjoint nuclei in $X$ have negative definite intersection form, the above equalities imply that $f=\sum _{i=1} ^3 \alpha _i [T_i]$ and similarly $g=\sum _{i=1} ^3 \beta _j [T_j]$. These two latter equations, however, give a contradiction, since $f\cdot g=1$ but $(\sum _{i=1} ^3 \alpha _i [T_i])\cdot (\sum _{i=1} ^3 \beta _j [T_j])=0$. Consequently $N_2$ does not embed in $X_{2,2,2}$. \end{prooff} \begin{prooff} {\em of Theorem~\ref{masodiktetel}:\/} Perform logarithmic transsformations of multiplicity $2n$ on $T_1,T_2$ and $T_3$ as above; the resulting 4-manifold $X_{2n,2n,2n}$ will be denoted by $Y_n$. The basic classes of $Y_n$ are determined in \cite{FS2}: these are the Poincar\'e duals of the homology classes $\alpha [{{T_1}\over {2n}}]+\beta [{{T_2}\over {2n}}]+ \gamma [{{T_3}\over {2n}}]$ where $\alpha ,\beta ,\gamma \in \lbrace \pm (2j-1) \ \vert \ j=1,...,n\rbrace $. Assume now that $N_n\subset Y_n$. As before, $f$ and $s$ will denote the homology classes of the fiber and the section in $N_n$ respectively. Note that $f^2=0$ and $s^2=-n$. The class $f$ can be represented by a torus, while $g=s+nf$ can be represented by a surface of genus $n$; moreover $g^2=n$. The same argument as in the proof of Theorem~\ref{elsotetel} gives that $f=\sum _{i=1} ^3 \alpha _i [T_i]$. To show that every basic class of $Y_n$ evaluates trivially on $g$ involves a little trick: take the basic classes $\pm [{{T_1}\over 2}]\pm [{{T_2}\over 2}]\pm [{{T_3}\over 2}]$ and evaluate them on $g$. If one of them, say $K$, evaluates nontrivially on $g$, then for $L=(2n-1)K$ (which is also a basic class) we have $\vert L(g)\vert \geq 2n-1$, but then $L$ and $g$ would violate the generalized adjunction formula. Consequently $(\pm [{{T_1}\over 2}]\pm [{{T_2}\over 2}]\pm [{{T_3}\over 2}])\cdot g=0$ implying that $g\cdot [T_i]=0$ $(i=1,2,3)$. These latter equations show that $g=\sum _{j=1} ^3 \beta _j [T_j]$, and we have the same contradiction as before. \end{prooff} \begin{remark} {\rm Note that the proof given above also shows that no $N_k$ with $k\leq n$ embeds in $Y_n=X_{2n,2n,2n}$.} \end{remark} \begin{thebibliography}{SSz1} \bibitem[A]{A} S. Akbulut, {\em Lectures on Seiberg-Witten invariants}, Turkish J. Math. {\bf20} (1996), 95--119. \bibitem[FS1]{FS1} R. Fintushel, R. Stern {\em Surgery in Cusp Neighborhoods and the Geography of Irreducible 4-Manifolds}, to appear \bibitem[FS2]{FS2} R. Fintushel and R. Stern, {\em Rational blowdown of smooth 4-manifolds}, preprint \bibitem[FS3]{FS3} R. Fintushel and R. Stern, {\em Knots, links and 4-manifolds}, MSRI preprint \bibitem[G]{G} R. Gompf, {\em Nuclei of elliptic surfaces}, Topology {\bf30} (1991), 479--511. \bibitem[GS]{GS} R. Gompf and A. Stipsicz, {\em An introduction to 4-manifolds and Kirby Calculus}, book in preparation \bibitem[K]{K} R. Kirby, {\em Problems in low-dimensional topology}, in Geometric Topology (W. Kazez ed.) AMS/IP 1997. \bibitem[M]{M} J. Morgan, {\em The Seiberg-Witten equations and applications to the topology of smooth four-manifolds}, Math. Notes {\bf44} Princeton University Press, Princeton NJ, 1996. \bibitem[SSz]{SSz} A. Stipsicz and Z. Szab\'o, in preparation \end{thebibliography} \end{document}