Torsions for manifolds with boundary and glueing formulas

We extend the definition of analytic and Reidemeister torsion for closed compact Riemannian manifolds, to compact Riemannian manifolds with boundary (M, ∂M), given a parallel flat bundle F of A - Hilbert modules of finite type and a decomposition of the boundary ∂M = ∂_M ∪∂₊M into disjoint components. When F is induced from the universal covering of M, and the fundamental group of M is infinite (cf. [BFKM]), these torsions are known as the L₂-analytic resp. L₂-Reidemeister torsions. If the system (M, ∂_M, ∂₊M,F) is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ principal covering where Γ is a residually finite group, we derive from work of LÜCK (cf. [L3]), that, the system (M, ∂_M, ∂₊M, F) is of determinant class (cf. Theorem 5.1 in Appendix A).