  In this paper, we introduce the notion of Reidemeister torsion for
quasi-isomorphisms of based chain complexes over a field. We call a chain map a
quasi-isomorphism if its induced homomorphism between homology is an
isomorphism. Our notion of torsion generalizes the torsion of acyclic based
chain complexes, and is a chain homotopy invariant on the collection of all
quasi-isomorphisms from a based chain complex to another. It shares nice
properties with torsion of acyclic based chain complexes, like multiplicativity
and duality. We will further generalize our torsion to quasi-isomorphisms
between free chain complexes over a ring under some mild condition. We
anticipate that the study of torsion of quasi-isomorphisms will be fruitful in
many directions, and in particular, in the study of links in 3-manifolds.
