Torsion Invariants of Combed 3-Manifolds with Boundary Pattern and Legendrian Links

We extend Turaev’s definition of torsion invariants of 3-dimensional manifolds equipped with non-singular vector fields, by allowing (suitable) tangency circles to the boundary, and manifolds with non-zero Euler characteristic. We show that these invariants apply in particular to (the exterior of) Legendrian links in contact 3-manifolds. Our approach uses a combinatorial encoding of vector fields, based on standard spines. In this paper we extend this encoding from closed manifolds to manifolds with boundary. Mathematics Subject Classification (1991): 57N10 (primary), 57Q10, 57R25 (secondary). Reidemeister torsion is a classical yet very vital topic in 3-dimensional topology, and it was recently used in a variety of important developments. To mention a few, torsion is a fundamental ingredient of the Casson-Walker-Lescop invariants (see e.g. [12]), and more generally of the perturbative approach to quantum invariants (see e.g. [11]). Relations have been pointed out between torsion and hyperbolic geometry [20]. Turaev’s torsion of non-singular vector fields on 3-manifolds [22] has been recognized to have deep connections with some 3-dimensional versions of the Seiberg-Witten invariants [16], [23]. It is also worth recalling that vector fields (and framings), have also been used by Kuperberg [10] to construct new invariants of different nature, based on Hopf algebras more general than those employed for quantum invariants. (There are reasons to speculate that also Kuperberg’s invariants should have a torsion content and relations with Turaev’s work, but we do not insist on this.) In this paper, using (actually, improving) our theory of branched standard spines [2] and building on [22], we extend Turaev’s definition of torsion by allowing vector fields to have (appropriate) tangency circles to the boundary. Moreover, we do not require the manifolds to have zero Euler characteristic (an assumption which is at the base of Turaev’s theory). To be precise, we accept any compact oriented manifold with (possibly empty) boundary, and non-singular vector fields with the only restriction ∗The second named author gratefully acknowledges financial support by GNSAGA-CNR