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Introduction

Topological K-theory is the most important example of a generalized cohomology theory other than ordinary cohomology. Unlike the theories of Chapter 15, which were produced abstractly, it starts from geometric objects, namely vector bundles. Isomorphism classes of vector bundles on $X$ already behave like a cohomology theory in two respects. They pull back along maps, and homotopic maps induce the same pullback. What they lack is subtraction, since direct sum makes them only a commutative monoid. In this chapter we set up vector bundles, prove that pullback is homotopy invariant, and then force subtraction by group completion. The result is an abelian group $K(X)\text{.}$ In the next chapter we show that $K(X)$ is the degree-$0$ part of a cohomology theory.

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