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Generalized Symplectic Geometries and the Index of Families of Elliptic Problems
Liviu I. Nicolaescu, University of Michigan, Ann Arbor
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Memoirs of the American Mathematical Society
1997; 80 pp; softcover
Volume: 128
ISBN-10: 0-8218-0621-1
ISBN-13: 978-0-8218-0621-0
List Price: US$45 Individual Members: US$27
Institutional Members: US\$36
Order Code: MEMO/128/609

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.

• Algebraic preliminaries
• Topological preliminaries
• (p-q)-lagrangians and classifying spaces for K-theory
• Symplectic reductions
• Clifford Symmetric Fredholm operators
• Families of boundary value problems for Dirac operators
• Appendices
• References