0 M ar 1 99 7 Gromov Invariants and Symplectic Maps

where the Z action is generated by (x, s, θ) 7→ (f(x), s + 1, θ). In this paper we compute the Gromov invariants of the manifolds Xf and of fiber sums of the Xf 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-Kähler 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.