自旋几何

prefaceacknowledgmentsintroductionchapter 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 theoremchapter 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 applicationschapter 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 supersymmetrychapter 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　bibliographyindexnotation index

　　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.