Complex K-theory

One of the most classical cohomology theories is complex K-theory, established as part of algebraic topology by Atiyah and Hirzebruch (1959), building on work of Grothendieck. The starting point is the set of isomorphism classes of complex vector bundles over a space \(X\). This set admits an operation, given by direct sum. The complex K-theory group \(K^0(X)\) of \(X\) is defined as the group completion of this commutative monoid, formally adding inverses to all vector bundles. Its central property is Bott periodicity: the theory repeats with period \(2\).

Complex K-theory is represented by a commutative ring spectrum \(\KU \). The construction of this spectrum mirrors the definition of \(K^0(X)\) via group completion. First, one can refine the set of isomorphism classes of vector bundles to an anima \(\Vect (X)^{\simeq }\), which takes into account the topology on the sets of isomorphisms between vector bundles. The direct sum operation turns this into a coherently commutative monoid in \(\An \). Applying the homotopy-theoretic group completion operation of Chapter 5, we obtain a commutative group in \(\An \), which under the recognition principle corresponds to a connective spectrum \(\ku \). The 2-periodic K-theory spectrum \(\KU \) is then obtained by inverting a certain Bott class \(\beta \in \pi _2(\ku )\).

We recall complex vector bundles and their homotopy invariance in Section 9.1, and identify the infinite Grassmannian as their classifying anima in Section 9.2. Section 9.3 defines the groups \(K^0(X)\), proves that they are represented by \(\Z \times BU\), and states Bott periodicity. Section 9.4 then carries out the spectrum-level construction of \(\ku \) and \(\KU \) sketched above. The multiplicative structure comes from the tensor product of vector bundles; making it coherent requires the higher algebra of Part II, and is carried out in Section 19.7. We close by rationalizing \(\KU \) and deducing that the Chern character is a rational isomorphism.

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