orsions for Manifolds with Boundary and Glueing Formulas

We extend the definition of analytic and Reidemeister torsion for closed compact iemannian manifolds, to compact Riemannian manifolds with boundary ( M , δ M ), given a parallel at bundle F of A ‐Hilbert modules of finite type and a decomposition of the boundary δ M = ‐ M ⊔ δ+ 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 L 2 ‐analytic resp. 2 ‐Reidemeister torsions. If the system ( M, δ‐M, δ+M, F ) is of determinant class we compute he quotient of the analytic and the Reidemeister torsion and prove gluing formulas for both of hem. In particular we answer positively Conjecture 7.6 in [LL]. If F is induced from a Γ‐principal overing where Γ is a residually finite group, we derive from work of Lück (cf. [L3]), that the system ( M, δ‐M, δ+M, F ) is of determinant class (cf. Theorem 5.1 in Appendix A).