Geometric torsions and an Atiyah-style topological field theory

We generalize the construction of invariants of three-dimensional manifolds with a triangulated boundary that we previously proposed for the case where the boundary consists of not more than one connected component to any number of components. These invariants are based on the torsion of acyclic complexes of geometric origin. An adequate tool for studying such invariants turns out to be Berezin's calculus of anticommuting variables; in particular, they are used to formulate our main theorem, concerning the composition of invariants under a gluing of manifolds. We show that the theory satisfies a natural modification of Atiyah's axioms for anticommuting variables.