Gromov Invariants and Symplectic Maps

Abstract

Given a symplectomorphism f of a symplectic manifold X, one can form the `symplectic mapping cylinder' $X_f=(X\times R\times S^1)/Z$ where the Z action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$ with other symplectic manifolds. This is done by expressing the Gromov invariants in terms of the Lefschetz zeta function of f and, in special cases, in terms of the Alexander polynomials of knots. The result is a large set of interesting non-Kahler symplectic manifolds with computational ways of distinguishing them. In particular, this gives a simple symplectic construction of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern and of related `exotic' symplectic 6-manifolds.

27 pages, AMS-LATEX, 2 figures

References in corpus (2)

• Virtual moduli cycles and Gromov-Witten invariants of general symplectic manifolds
• The Gromov Invariants of Ruan-Tian and Taubes