On Lens Spaces and Their Symplectic Fillings

The standard contact structure $\xi_0$ on the three–sphere $S^3$ is invariant under the action of $\mathbb Z/p\mathbb Z$ yielding the lens space $L(p,q)$, therefore every lens space carries a natural quotient contact structure $\overline\xi_0$. A theorem of Eliashberg and McDuff classifies the symplectic fillings of ($L(p,1),{\overline\xi_0})$ up to diffeomorphism. Here we announce a generalization of that result to every lens space. In particular, we give an explicit handlebody decomposition of every symplectic filling of $(L(p,q),{\overline\xi_0})$ for every $p$ and $q$. Our results imply: <br >(a)</br > There exist infinitely many lens spaces $L(p,q)$ with $q\not= 1$ such that the contact 3–manifold $(L(p,q),{\overline\xi_0})$ admits only one symplectic filling up to blowups and diffeomorphisms. <br >(b)</br > For any natural number $N$, there exist infinitely many lens spaces $L(p,q)$ such that $(L(p,q),{\overline\xi_0})$ admits more than $N$ symplectic fillings up to blowups and diffeomorphisms.

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Published 1 January 2004