in the late 1920's the relentless march ofideas and discoveries had carried physics to a generally acceptedrelativistic theory of the electron. the physicist p.a.m. dirac,however, was dissatisfied with the prevailing ideas and, somewhatin isolation, sought for a better formulation. by 1928 he succeededin finding a theory which accorded with his own ideas and also fitmost of the established principles of the time. ultimately thistheory proved to be one of the great intellectual achievements ofthe period. it was particularly remarkable for the internal beautyof its mathematical structure which not only clarified muchpreviously mysterious phenomena but also predicted in a compellingway the existence of an electron-like particle of negative energy.indeed such particles were subsequently found to exist and ourunderstanding of nature was transformed.
because of its compelling beauty and physical significance it isperhaps not surprising that the ideas at the heart of dirac'stheory have also been discovered to play a role of great importancein modern mathematics, particularly in the interrelations betweentopology, geometry and analysis. a great part of this newunderstanding comes from the work of m. atiyah and i. singer. it istheir work and its implications which form the focus of thisbook.
preface
acknowledgments
introduction
chapter ⅰ clifford algebras, spin groups and their
representations
1. clifford algebras
2. the groups pin and spin
3. the algebras cln, and clr,s
4. the classification
5. representations
6. lie algebra structures
7. some direct applications to geometry
8. some further applications to the theory of lie groups
9. k-theory and the atiyah-bott-shapiro construction
10. kr-theory and the (1,1)-periodicity theorem
preface
acknowledgments
introduction
chapter ⅰ clifford algebras, spin groups and their
representations
1. clifford algebras
2. the groups pin and spin
3. the algebras cln, and clr,s
4. the classification
5. representations
6. lie algebra structures
7. some direct applications to geometry
8. some further applications to the theory of lie groups
9. k-theory and the atiyah-bott-shapiro construction
10. kr-theory and the (1,1)-periodicity theorem
chapter ⅱ spin geometry and the dirac operators
1. spin structures on vector bundles
2. spin manifolds and spin cobordism
3. clifford and spinor bundles
.4. connections on spinor bundles
5. the dirac operators
6. the fundamental elliptic operators
7. clk-linear dirac operators
8. vanishing theorems and some applications
chapter ⅲ index theorems
1. differential operators
2. sobolev spaces and sobolev theorems
3. pseudodifferential operators
4. elliptic operators and parametrices
5. fundamental results for elliptic operators
6. the heat kernel and the index
7. the topological invariance of the index
8. the index of a family of elliptic operators
9. the g-index
10. the clifford index
11. multiplicative sequences and the chern character
12. thom isomorphisms and the chern character defect
13. the atiyah-singer index theorem
14. fixed-point formulas for elliptic operators
15. the index theorem for families
16. families of real operators and the clk-index theorem
17. remarks on heat and supersymmetry
chapter ⅳ applications in geometry and topology
1. lntegrality theorems
2. immersions of manifolds and the vector field problem
3. group actions on manifolds
4. compact manifolds of positive scalar curvature
5. positive scalar curvature and the fundamental group
6. complete manifolds of positive scalar curvature
7. the topology of the space of positive scalar curvaturemetrics
8. clifford multiplication and kiihler manifolds
9. pure spinors, complex structures, and twistors
10. reduced holonomy and calibrations
11. spinor cohomology and complex manifolds with vanishing firstchern class
12. the positive mass conjecture in general relativity
appendix a principal g-bundles
appendix b classifying spaces and characteristic classes
appendix c orientation classes and thom isomorphisms ink-theory
appendix d spine-manifolds
bibliography
index
notation index
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