Topological K-theory first appeared in a 1961 paper by Atiyah and Hirzebruch; their paper adapted the work of Grothendieck on algebraic varieties to a topological setting. Since that time, topological K-theory (which we will henceforth simply call K-theory) has become a powerful and indespensible tool in topology, differential geometry, and index theory. The goal of this book is to provide a self-contained introduction to the subject.

This book is primarily aimed at beginning graduate students, but also for working mathematicians who know little or nothing about the subject and would like to learn something about it. The material in this book is suitable for a one semester course on K-theory; for this reason, I have included exercises at the end of each chapter. I have tried to keep the prerequisites for reading this book to a minimum; I will assume that the reader knows the following:

• Linear Algebra: Vector spaces, bases, linear transformations, similarity, trace, determinant.

• Abstract Algebra: Groups, rings, homomorphisms and isomorphisms, quotients, products.

• Topology: Metric spaces, completeness, compactness and connectedness, local compactness, continuous functions, quotient topology, subspace topology, partitions of unity.

To appreciate many of the motivating ideas and examples in K-theory, it is helpful, but not essential, for the reader to know the rudiments of differential topology, such as smooth manifolds, tangent bundles, differential forms, and de Rham cohomology. In Chapter 4, the theory of characteristic classes is developed in terms of differential forms and de Rham cohomology; for readers not familiar with these topics, I give a quick introduction at the beginning of that chapter.