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Pré-Publication, Document De Travail Année : 2019

Twisted Gromov and Lefschetz invariants associated with bundles

Gilberto Spano
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Résumé

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 $\mathcal{B}$.

Dates et versions

hal-02325313 , version 1 (22-10-2019)

Identifiants

Citer

Gilberto Spano. Twisted Gromov and Lefschetz invariants associated with bundles. 2019. ⟨hal-02325313⟩
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