Relative index theory, determinants and torsion for open manifolds - Eichhorn, Jürgen 1942- (World Scientific Pub. Co., 2009)
Includes bibliographical references (pages 331-337) and index
Introduction -- I. Absolute invariants for open manifolds and bundles. 1. Absolute characteristic numbers. 2. Index theorems for open manifolds -- II. Non-linear Sobolev structures. 1. Clifford bundles, generalized Dirac operators and associated Sobolev spaces. 2. Uniform structures of metric spaces. 3. Completed manifolds of maps. 4. Uniform structures of manifolds and Clifford bundles. 5. The classification problem, new (co- )homologies and relative characteristic numbers -- III. The heat kernel of generalized Dirac operators. 1. Invariance properties of the spectrum and the heat kernel. 2. Duhamel's principle, scattering theory and trace class conditions -- IV. Trace class properties. 1. Variation of the Clifford connection. 2. Variation of the Clifford structure. 3. Additional topological perturbations -- V. Relative index theory. 1. Relative index theorems, the spectral shift function and the scattering index -- VI. Relative [symbol]-functions, [symbol]-functions, determinants and torsion. 1. Pairs of asymptotic expansions. 2. Relative [symbol]-functions. 3. Relative determinants and QFT. 4. Relative analytic torsion. 5. Relative [symbol]-invariants. 6. Examples and applications -- VII. Scattering theory for manifolds with injectivity radius zero. 1. Uniform structures defined by decay functions. 2. The injectivity radius and weighted Sobolev spaces. 3. Mapping properties of e[symbol]. 4. Proof of the trace class property -- References -- List of notations -- Index
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
Keywords: MATHEMATICS / Geometry / Differential; Manifolds (Mathematics); Index theory (Mathematics); Torsionstheorie; Determinante; Indextheorie; Mannigfaltigkeit