  In this paper, we begin constructing a new finite-dimensional topological
quantum field theory (TQFT) for three-manifolds, based on group PSL(2,C) and
its action on a complex variable by fractional-linear transformations, by
providing its key ingredient -- a new type of chain complexes. As these
complexes happen to be acyclic often enough, we make use of their torsion to
construct different versions of manifold invariants. In particular, we show how
to construct a large set of invariants for a manifold with boundary, analogous
to the set of invariants based on Euclidean geometric values and used in a
paper by one of the authors for constructing a "Euclidean" TQFT. We show on
examples that our invariants are highly nontrivial.
