Gromov Invariants and Symplectic Maps

TL;DR

This paper calculates Gromov invariants for symplectic mapping cylinders and their fiber sums, linking them to Lefschetz zeta functions and Alexander polynomials, thus producing new examples of exotic symplectic manifolds.

Contribution

It introduces a method to compute Gromov invariants of symplectic mapping cylinders using Lefschetz zeta functions and Alexander polynomials, leading to new exotic symplectic manifolds.

Findings

Computed Gromov invariants via Lefschetz zeta functions

Expressed invariants in terms of Alexander polynomials

Constructed new exotic symplectic manifolds

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.