Twisted Gromov and Lefschetz invariants associated with bundles

Abstract

Given a closed symplectic 4-manifold $(X,\omega)$, we define a twisted version of the Gromov-Taubes invariants for $(X,\omega)$, where the twisting coefficients are induced by the choice of a surface bundle over $X$. Given a fibered 3-manifold $Y$, we similarly construct twisted Lefschetz zeta functions associated with surface bundles: we prove that these are essentially equivalent to the Jiang's Lefschetz zeta functions of $Y$, twisted by the representations of $\pi_1(Y)$ that are induced by monodromy homomorphisms of surface bundles over $Y$. This leads to an interpretation of the corresponding twisted Reidemeister torsions of $Y$ in terms of products of "local" commutative Reidemeister torsions. Finally we relate the two invariants by proving that, for any fixed closed surface bundle $\mathcal{B}$ over $Y$, the corresponding twisted Lefschetz zeta function coincides with the Gromov-Taubes invariant of $S^1 \times Y$ twisted by the bundle over $S^1 \times Y$ naturally induced by $\mathcal{B}$.