AT-theory was introduced by A. Grothendieck in his formulation of the
Riemann- Roch theorem (cf. Borel and Serre [2]). For each projective
algebraic variety, Grothendieck constructed a group from the category of
coherent algebraic sheaves, and showed that it had many nice properties.
Atiyah and Hirzebruch [3] con- sidered a topological analog defined
for any compact space X, a group K{X) constructed from the category of
vector bundles on X. It is this ''topological J^-theory" that this book
will study. Topological ^-theory has become an important tool in
topology. Using- theory, Adams and Atiyah were able to give a simple
proof that the only spheres which can be provided with //-space
structures are S^, S^ and S'^. Moreover, it is possible to derive a
substantial part of stable homotopy theory from A^-theory (cf. J. F.
Adams [2]). Further applications to analysis and algebra are found in
the work of Atiyah-Singer [2], Bass [1], Quillen [1], and others.
A key factor in these applications is Bott periodicity (Bott [2]). The
purpose of this book is to provide advanced students and mathematicians
in other fields with the fundamental material in this subject. In
addition, several applications of the type described above are included.
In general we have tried to make this book self-contained, beginning
with elementary concepts wherever possible; however, we assume that the
reader is familiar with the basic definitions of homotopy theory:
homotopy classes of maps and homotopy groups (cf.
