  The construction of invariants of three-dimensional manifolds with a
triangulated boundary, proposed earlier by the author for the case when the
boundary consists of not more than one connected component, is generalized to
any number of components. These invariants are based on the torsion of acyclic
complexes of geometric origin. The relevant tool for studying our invariants
turns out to be F.A. Berezin's calculus of anti-commuting variables; in
particular, they are used in the formulation of the main theorem of the paper,
concerning the composition of invariants under a gluing of manifolds. We show
that the theory obeys a natural modification of M. Atiyah's axioms for
anti-commuting variables.
