This book offers a systematic and comprehensive presentation of the
concepts of a spin manifold, spinor fields, Dirac operators, and
A-genera, which, over the last two decades, have come to play a
significant role in many areas of modern mathematics. Since the deeper
applications of these ideas require various general forms of the
Atiyah-Singer Index Theorem, the theorems and their proofs, together
with all prerequisite material, are examined here in detail. The
exposition is richly embroidered with examples and applications to a
wide spectrum of problems in differential geometry, topology, and
mathematical physics. The authors consistently use Clifford algebras and
their representations in this exposition. Clifford multiplication and
Dirac operator identities are even used in place of the standard tensor
calculus. This unique approach unifies all the standard elliptic
operators in geometry and brings fresh insights into curvature
calculations. The fundamental relationships of Clifford modules to such
topics as the theory of Lie groups, K-theory, KR-theory, and Bott
Periodicity also receive careful consideration. A special feature of
this book is the development of the theory of Cl-linear elliptic
operators and the associated index theorem, which connects certain
subtle spin-corbordism invariants to classical questions in geometry and
has led to some of the most profound relations known between the
curvature and topology of manifolds.
