Torsions for manifolds with boundary and glueing formulas

We extend the definition of analytic and Reidemeister torsion from closed compact Riemannian manifolds to compact Riemannian manifolds with boundary $(M, \partial M)$, given a flat bundle $\Cal F$ of $\Cal A$-Hilbert modules of finite type and a decomposition of the boundary $\partial M =\partial_- M \cup \partial_+ M$ into disjoint components. In particular we extend the $L-2$ analytic and Reidemeister torsions to compact manifolds with boundary. If the system $(M,\partial_-M, \partial_+M, \Cal F)$ is of determinant class we compute the quotient of the analytic and the Reidemeister torsion and prove glueing formulas for both of them. In particular we answer positively Conjecture 7.6 in [LL]