Reidemeister torsion in generalized Morse theory

In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological invariant. It is hoped that this will provide a model for possible generalizations to Floer theory. In two papers with Yi-Jen Lee [HL1, HL2], we defined a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. We consider the flow dual to the 1-form via an auxiliary metric. Our invariant, which we call I, multiplies the algebraic Reidemeister torsion of the Novikov complex, which counts flow lines between critical points, by a zeta function which counts closed orbits of the flow. For a closed 1-form in a rational cohomology class, i.e. d of a circle-valued function, we proved in the above papers that I equals a form of topological Reidemeister torsion due to Turaev. This implies a posteriori that I is invariant under homotopy of the circle-valued function and the auxiliary metric. In this paper we reprove these results using an opposite approach: we first prove a priori that I is a topological invariant, depending only on the cohomology class of the closed 1-form. We then deduce that I agrees with Turaev torsion, by using invariance to reduce to the easier case of an exact 1-form. This approach has two advantages. First, it works for closed 1-forms in an arbitrary cohomology class, thus extending the results of our previous papers. Second, and perhaps more importantly, the proof of invariance here should provide a model for the possible construction of torsion in Floer theory.