Let \((X,\omega)\) be a closed symplectic manifold, and let \(f\colon X\longrightarrow X\) be a symplectomorphism. Form the symplectic mapping torus \(X_f:=X\times{\mathbb R}\times{\mathbb S}^1/{\mathbb Z},\) where the \({\mathbb Z}\)-action is generated by \((x,s,\theta)\longmapsto(f(x),s+1,\theta).\) In this paper, the authors compute the Gromov invariants of \(X_f,\) by expressing them in terms of the Lefschetz zeta-function of \(f.\) More precisely, for each given class \(A\in H_2(X_f),\) they assemble some of the Ruan-Tian invariants (which count perturbed holomorphic curves) into a “partial Gromov series” \(\text{ Gr}^A(X_f)\): a power series in a variable \(t_A\) whose coefficients count, in a non-trivial fashion, the perturbed holomorphic curves representing multiples of \(A.\) Then, they define the full degree zero Gromov series \(\text{Gr}(X_f)\) to be the product of the \(\text{ Gr}^A(X_f)\) over all primitive classes \(A,\) which is invariant under diffeomorphisms and isotopies of \(\omega,\) and thus it gives an invariant \(\text{Gr}\colon \{\text{deformation classes}\}\to \mathbb{Z}\) defined in all dimensions. The map \(f\) induces a map \(f_{*k}\) on \(H_k(X;{\mathbb{Q}}).\) Since \(X_f\) fibers over the torus \(T^2\) with fiber \(X\) when \(\det(I-f_{*1})=\pm 1\) there is a well-defined section class \(T\in H_2(X_f;{\mathbb Z}).\) The authors computes the Gromov invariants of the multiples of this section class. In fact, they prove the following theorem.
Theorem. If \(\det(I-f_{*1})=\pm 1,\) the partial Gromov series of \(X_f\) for the section class \(T\) is given by the Lefschetz zeta-function of \(f,\) \(\zeta_f,\) in the variable \(t=t_T:\) \[ \text{ Gr}^T(X_f)=\zeta_f(t)={{\displaystyle{\prod_{k odd} \det(I-tf_{*k})}}\over{\displaystyle{\prod_{k even} \det(I-tf_{*k})}}}. \] The authors give further applications in computing the Gromov invariants of fiber sums of \(X_f\) with other symplectic manifolds. (Such fiber sums are defined by \(Z(f)=Z\sharp_{F=T}X_f,\) where \((Z,\omega')\) is a symplectic 4-manifold, \(F\subset Z\) is a symplectically embedded torus with trivial normal bundle and, \(T\subset X_f\) being represented by a symplectically embedded torus, \(F=T\) means the identification of \(T\) with \(F.\) When the construction is carefully done then \(Z(f)\) is a symplectic manifold.) They finally provide a computational way for distinguishing a large set of interesting non-Kähler symplectic manifolds. In dimension \(4\) this gives a symplectic construction of the “exotic” elliptic surfaces of Fintushel and Stern, and in higher dimensions it gives many examples of manifolds that are diffeomorphic but not “equivalent” as symplectic manifolds.