Elliptic Theory on Singular Manifolds (Hardcover)

I Singular Manifolds and Differential Operators

GEOMETRY OF SINGULARITIES
Preliminaries
Manifolds with conical singularities
Manifolds with edges
ELLIPTIC OPERATORS ON SINGULAR MANIFOLDS
Operators on manifolds with conical singularities
Operators on manifolds with edges
Examples of elliptic edge operators

II Analytical Tools

PSEUDODIFFERENTIAL OPERATORS
Preliminary remarks
Classical theory
Operators in sections of Hilbert bundles
Operators on singular manifolds
Ellipticity and finiteness theorems
Index theorems on smooth closed manifolds
LOCALIZATION (SURGERY) IN ELLIPTIC THEORY
The index locality principle
Localization in index theory on smooth manifolds
Surgery for the index of elliptic operators on singular manifolds
Relative index formulas on manifolds with isolated singularities

III Topological Problems

INDEX THEORY
Statement of the problem
Invariants of interior symbol and symmetries
Invariants of the edge symbol
Index theorems
Index on manifolds with isolated singularities
Supplement. Classification of elliptic symbols with symmetry and K-theory
Supplement. Proof of Proposition 5.16
ELLIPTIC EDGE PROBLEMS
Morphisms
The obstruction to ellipticity
A formula for the obstruction in topological terms
Examples. Obstructions for geometric operators

IV Applications and Related Topics

FOURIER INTEGRAL OPERATORS ON SINGULAR MANIFOLDS
Homogeneous canonical (contact) transformations
Definition of Fourier integral operators
Properties of Fourier integral operators
The index of elliptic Fourier integral operators
Application to quantized contact transformations
Example
RELATIVE ELLIPTIC THEORY
Analytic aspects of relative elliptic theory
Topological aspects of relative elliptic theory
INDEX OF GEOMETRIC OPERATORS ON MANIFOLDS WITH CYLINDRICAL ENDS
Operators on manifolds with cylindrical ends
Index formulas
HOMOTOPY CLASSIFICATION OF ELLIPTIC OPERATORS
The homotopy classification problem
Classification on smooth manifolds
Atiyah-de Rham duality
Abstract elliptic operators and analytic K-homology
Classification on singular manifolds
Some applications
LEFSCHETZ FORMULAS
Main result
Proof of the theorem
Contributions of conical points as sums of residues
Supplement. The Lefschetz number
Supplement. The Sternin-Shatalov method

APPENDICES
Spectral Flow
Eta Invariants
Index of Parameter-Dependent Elliptic Families

Bibliographic Remarks
Bibliography
Index