Electronic ISBN:  9781470401948 
Product Code:  MEMO/128/609.E 
List Price:  $45.00 
MAA Member Price:  $40.50 
AMS Member Price:  $27.00 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 128; 1997; 80 ppMSC: Primary 58; 47; 55; Secondary 15; 16;
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).ReadershipGraduate students and research mathematicians interested in index theory; topologists and gauge theorists.

Table of Contents

Chapters

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


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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.

Chapters

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