**Memoirs of the American Mathematical Society**

1997;
80 pp;
Softcover

MSC: Primary 58; 47; 55;
Secondary 15; 16

Print ISBN: 978-0-8218-0621-0

Product Code: MEMO/128/609

List Price: $45.00

Individual Member Price: $27.00

**Electronic ISBN: 978-1-4704-0194-8
Product Code: MEMO/128/609.E**

List Price: $45.00

Individual Member Price: $27.00

# Generalized Symplectic Geometries and the Index of Families of Elliptic Problems

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*Liviu I. Nicolaescu*

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).

#### Readership

Graduate students and research mathematicians interested in index theory; topologists and gauge theorists.

#### Table of Contents

# Table of Contents

## Generalized Symplectic Geometries and the Index of Families of Elliptic Problems

- Contents vii8 free
- 1 Algebraic preliminaries 114 free
- 2 Topological preliminaries 518
- 3 (p,q)-lagrangians and classifying spaces for K-theory 1225
- 4 Symplectic reductions 2437
- 5 Clifford symmetric Fredholm operators 3548
- 6 Families of boundary value problems for Dirac operators 4659
- A. Gap convergence of linear operators 6477
- B. Gap continuity of families of BVP's for Dirac operators 6881
- C. Pseudodifferential Grassmanians and BVP's for Dirac operators 7184
- D. The proof of Proposition 6.1 7487
- References 7891