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Introduction

In the previous chapter we built a ring-valued functor \(K(X)\) by group-completing the monoid of vector bundles. It is only one group, whereas a cohomology theory needs a group in every degree. In Chapter 15 we saw how to produce these from a representing space. If \(\widetilde{K}(X) \cong [X, Z]_*\text{,}\) then \([X, \Omega^n Z]_* \cong [\Sigma^n X, Z]_*\) is the natural candidate for \(\widetilde{K}^{-n}(X)\text{.}\) This chapter carries out that plan in three steps.
  1. Vector bundles of rank \(k\) are classified by maps into the infinite Grassmannian \(BU(k) = \mathrm{Gr}_k(\mathbb{C}^\infty)\text{.}\) Brown’s theorem gives a classifying space abstractly, and we identify it with the Grassmannian.
  2. Since \(\widetilde{K}\) only sees bundles up to stable equivalence (Proposition 16.22), we let \(k \to \infty\text{.}\) The resulting space \(BU\) satisfies \(\widetilde{K}(X) \cong [X, \mathbb{Z} \times BU]_*\) for finite CW complexes \(X\text{.}\)
  3. Looping gives the negative degrees. Bott periodicity, \(\Omega^2(\mathbb{Z} \times BU) \simeq \mathbb{Z} \times BU\text{,}\) shows that they repeat with period \(2\text{,}\) and this lets us define the positive degrees too. The result is a \(2\)-periodic cohomology theory represented by an \(\Omega\)-spectrum.
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