For closed manifolds, there is a highly elaborated theory of number-valued invariants, attached to the underlying manifold, structures and differential operators. On open manifolds, nearly all of this fails, with the exception of some special classes. The goal of this monograph is to establish for open manifolds, structures and differential operators an applicable theory of number-valued relative invariants. This is of great use in the theory of moduli spaces for nonlinear partial differential equations and mathematical physics. The book is self-contained: in particular, it contains an outline of the necessary tools from nonlinear Sobolev analysis.
Absolute Invariants for Open Manifolds and Bundles: Absolute Characteristic Numbers; Index Theorems for Open Manifolds; Non-linear Sobolev Structures; Generalized Dirac Operators: Generalized Dirac Operators, Their Heat Kernel and Spectral Properties; Duhamels Principle, Trace Class Conditions and Scattering Theory; Trace Class Properties; Relative Index Theory: Relative Index Theorems; The Spectral Shift Function and the Scattering Index; Relative Zeta Functions, Eta Functions, Determinants, Partition Functions of QFT and Torsion.