Generalized symplectic geometries and the index of families of elliptic problems
About this Title
Liviu I. Nicolaescu
Publication: Memoirs of the American Mathematical Society
Publication Year 1997: Volume 128, Number 609
ISBNs: 978-0-8218-0621-0 (print); 978-1-4704-0194-8 (online)
MathSciNet review: 1388897
MSC: Primary 58G12; Secondary 19K56, 55N15, 55P10, 58F05, 58G20
In this book, an index theorem is proved for arbitrary families of elliptic boundary value problems for Dirac operators and a surgery formula for the index of a family of Dirac operators on a closed manifold. Also obtained is a very general result on the cobordism invariance of the index of a family.
All results are established by first symplectically rephrasing the problems and then using a generalized symplectic reduction technique. This provides a unified approach to all possible parameter spaces and all possible symmetries of a Dirac operator (eight symmetries in the real case and two in the complex case).
Graduate students and research mathematicians interested in index theory; topologists and gauge theorists.
Table of Contents
- 1. Algebraic preliminaries
- 2. Topological preliminaries
- 3. $(p,q)$-lagrangians and classifying spaces for $K$-theory
- 4. Symplectic reductions
- 5. Clifford symmetric Fredholm operators
- 6. Families of boundary value problems for Dirac operators
- Appendix A. Gap convergence of linear operators
- Appendix B. Gap continuity of families of BVP’s for Dirac operators
- Appendix C. Pseudodifferential Grassmanians and BVP’s for Dirac operators
- Appendix D. The proof of Proposition 6.1