Absolute torsion and eta-invariant

In a recent joint work with V. Turaev [6], we defined a new concept of combinatorial torsion which we called absolute torsion. Compared with the classical Reidemeister torsion, it has the advantage of having a well-determined sign. Also, the absolute torsion is defined for arbitrary orientable flat vector bundles, and not only for unimodular ones, as is classical Reidemeister torsion. In this paper I show that the sign behavior of the absolute torsion, under a continuous deformation of the flat bundle, is determined by the eta-invariant and the Pontrjagin classes. This result has a twofold significance. Firstly, it justifies the definition of the absolute torsion by establishing a relation to the well-known geometric invariants of manifolds. Viewed differently, the result of this paper allows to express (partially) the eta-invariant, which is defined using analytic tools, in terms of the absolute torsion, having a purely topological definition. The result may find applications in studying the spectral flow by methods of combinatorial topology.