Topological K–theory, the first generalized cohomology theory to be studied thoroughly, was introduced around 1960 by Atiyah and Hirzebruch, based on the Periodicity Theorem of Bott proved just a few years earlier. In some respects K–theory is more elementary than classical homology and cohomology, and it is also more powerful for certain purposes. Some of the best-known applications of algebraic topology in the twentieth century, such as the theorem of Bott and Milnor that there are no division algebras after the Cayley octonions, or Adams’ theorem determining the maximum number of linearly independent tangent vector fields on a sphere of arbitrary dimension, have relatively elementary proofs using K–theory, much simpler than the original proofs using ordinary homology and cohomology.

The first portion of this book takes these theorems as its goals, with an exposition that should be accessible to bright undergraduates familiar with standard material in undergraduate courses in linear algebra, abstract algebra, and topology. Later chapters of the book assume more, approximately the contents of a standard graduate course in algebraic topology. A concrete goal of the later chapters is to tell the full story on the stable J–homomorphism, which gives the first level of depth in the stable homotopy groups of spheres. Along the way various other topics related to vector bundles that are of interest independent of K–theory are also developed, such as the characteristic classes associated to the names Stiefel and Whitney, Chern, and Pontryagin.