We now pass from vector bundles to the classical complex K-theory groups. For a compact Hausdorff space, these are obtained by formally adjoining additive inverses to vector bundles. We will prove that \(K^0\) is represented by the space \(\Z \times BU\), define the Bott class, and state the classical Bott periodicity theorem.
Definition 9.3.1 (Complex K-theory). Let \(X\) be a compact Hausdorff space. We define the complex K-theory group \(K^0(X)\) as the Grothendieck group of the abelian monoid \(\pi _0\Vect (X)\): \[ K^0(X) := (\pi _0\Vect (X))^{\grp }. \] In other words, \(K^0(X)\) consists of equivalence classes of formal differences \(E- E'\) of complex vector bundles over \(X\), where we say that \(E_1 - E_1' = E_2 - E_2'\) whenever there is an isomorphism \(E_1 \oplus E_2' \oplus E_3 \cong E_1' \oplus E_2\oplus E_3\) of vector bundles over \(X\) for some \(E_3 \in \pi _0\Vect (X)\).
Example 9.3.2. For \(X = \pt \), a vector bundle is simply a complex vector space, which up to isomorphism is determined by its dimension. We thus see that the abelian monoid \(\pi _0\Vect (\pt )\) is isomorphic to \(\N \), and hence \(K^0(\pt ) \cong \Z \).
The abelian group \(K^0(X)\) is contravariantly functorial in \(X\): given a continuous map \(f\colon X \to Y\), pullback along \(f\) determines a map \[ f^*\colon \pi _0\Vect (Y) \to \pi _0\Vect (X), \] which induces a map \(f^*\colon K^0(Y) \to K^0(X)\) on Grothendieck groups.
Tensor product distributes over direct sum, so it makes \(\pi _0\Vect (X)\) into a commutative semiring. Its multiplication extends uniquely to a commutative ring structure on \(K^0(X)\), with unit the class of the trivial line bundle. Pullback maps are morphisms of rings. The details are left to Chapterexercise 9.3.
Corollary 9.3.3. Every homotopy equivalence \(f\colon X \to Y\) between compact Hausdorff spaces induces an isomorphism \[ f^*\colon K^0(Y) \xrightarrow {\cong } K^0(X) \] of abelian groups.
Proof. This follows from Theorem 9.1.4 by applying group completion. □
Definition 9.3.4 (Reduced K-theory). Let \(X\) be a compact Hausdorff space. We define the abelian group \(\widetilde {K}^0(X)\) as the cokernel of the map \(\Z \cong K^0(\pt ) \to K^0(X)\). This is called the reduced K-theory group of \(X\).
When \(X\) has a basepoint \(x\), this is also equivalent to the kernel of \(K^0(X) \to K^0(\{x\}) \cong \Z \).
Example 9.3.5 (Complex K-theory of the circle and the two-sphere). There are isomorphisms \[ K^0(S^1)\cong \Z , \qquad K^0(S^2)\cong \Z \times \Z , \] and hence \(\widetilde K^0(S^1)=0\) and \(\widetilde K^0(S^2)\cong \Z \). Identifying \(S^2\) with \(\CP ^1\), the class \([H]-[\ul {\C }]\) of the tautological line bundle generates \(\widetilde K^0(S^2)\).
Proof. The calculation for \(S^1\) follows from Exercise 9.1.6. For \(S^2\), the clutching classification of Exercise 9.1.7 identifies a positive-rank vector bundle with its rank and the winding number of the determinant of its clutching function. These invariants are additive under direct sum, so group completion gives \(K^0(S^2)\cong \Z \times \Z \). The second factor is the reduced group, and the clutching function of \(H\) has winding number \(1\) up to sign. □
We will use one more standard fact about vector bundles. It is the geometric reason that virtual bundles can be represented by honest bundles after adding enough trivial summands.
Theorem 9.3.6 (Stable complements, Hatcher (2003), Proposition 1.4). Let \(X\) be a compact Hausdorff space. For every complex vector bundle \(E \to X\) there exists another vector bundle \(E' \to X\) such that the direct sum \(E \oplus E'\) is isomorphic to a trivial vector bundle \(\ul {\C }^N := X \times \C ^{N}\) for some \(N \in \N \).
Proof sketch. Choose finitely many local trivializations and a subordinate partition of unity. Multiplying the local coordinate functions by the partition-of-unity functions embeds \(E\) into a trivial bundle \(X \times \C ^N\). After choosing a Hermitian inner product on this trivial bundle, the orthogonal complements of the fibers of \(E\) assemble to a vector bundle \(E'\) with \(E \oplus E' \cong \ul {\C }^N\). The details are the usual compactness and partition-of-unity argument; see Hatcher for the full proof. □
Corollary 9.3.7. There is a surjective morphism of monoids \(\pi _0\Vect (X) \to \widetilde {K}^0(X)\).
Proof. Represent a class in \(\widetilde K^0(X)\) by a difference \([E]-[F]\). Choose a vector bundle \(F'\) such that \(F\oplus F'\cong \ul {\C }^N\) by Theorem 9.3.6. Modulo classes of trivial bundles, the given class is then represented by the honest vector bundle \(E\oplus F'\). □
9.3.1 Representability of \(K^0\)
We now prove the classical representability theorem for \(K^0\) using the Grassmannian classification theorem and stable complements.
Choose a linear isometric isomorphism \(\C ^\infty \oplus \C ^\infty \cong \C ^\infty \). Direct sum induces, up to homotopy, a multiplication on \(\Z \times BU\): the \(\Z \)-coordinates are added, while on the \(BU\)-coordinate we use the block-sum maps \[ \Gr _n(\C ^\infty )\times \Gr _m(\C ^\infty ) \longrightarrow \Gr _{n+m}(\C ^\infty \oplus \C ^\infty ) \cong \Gr _{n+m}(\C ^\infty ), \] passing to the colimit. The proposition below identifies the resulting operation on \([X,\Z \times BU]\) with direct sum in \(K^0(X)\), and hence shows in particular that it is independent of the choices and defines an abelian group structure.
Proposition 9.3.8. Let \(X\) be a compact Hausdorff space. There is a natural isomorphism of abelian groups \[ [X,\Z \times BU] \xrightarrow {\cong } K^0(X). \]
Proof. We first construct the map. Let \(f\colon X\to \Z \times BU\) be continuous. Since \(X\) is compact and \(\Z \) is discrete, the composite \(X\to \Z \) has finite image. Write \(X_m\subseteq X\) for the clopen subset on which this composite has value \(m\). By Lemma 9.2.18, the restriction \(f_m\colon X_m\to BU\) factors through some stage \(\Gr _n(\C ^\infty )\). Let \(E_m\to X_m\) be the pullback of the universal rank \(n\) bundle along this factorization. We assign to \(f_m\) the class \[ [E_m]-n[\ul {\C }]+m[\ul {\C }] \in K^0(X_m). \] These classes glue over the finite clopen decomposition \(X=\bigsqcup _m X_m\) to a class in \(K^0(X)\).
This construction is independent of all choices. Passing from \(\Gr _n(\C ^\infty )\) to \(\Gr _{n+1}(\C ^\infty )\) replaces \(E_m\) by \(E_m\oplus \ul {\C }\), and hence leaves \([E_m]-n[\ul {\C }]+m[\ul {\C }]\) unchanged. If two maps are homotopic, the homotopy has constant \(\Z \)-coordinate on each \(X_m\times [0,1]\) and, again by compactness, factors through a common Grassmannian stage after stabilization. Pulling back the universal bundle along the two ends gives isomorphic bundles by Theorem 9.1.4. Thus we get a well-defined natural map \[ \Phi _X\colon [X,\Z \times BU] \to K^0(X). \]
We prove that \(\Phi _X\) is surjective. Let \(\alpha \in K^0(X)\), and let \(X=\bigsqcup _m X_m\) be the finite clopen decomposition according to the virtual rank of \(\alpha \). It suffices to work on one \(X_m\), so assume \(\alpha \) has constant rank \(m\). Choose a presentation \(\alpha =[E]-[F]\). By Theorem 9.3.6, choose \(F'\) and \(N\) such that \(F\oplus F'\cong \ul {\C }^N\). Then \[ \alpha =[E\oplus F']-N[\ul {\C }]. \] The bundle \(E\oplus F'\) has rank \(N+m\), so Theorem 9.2.4 classifies it by a map \(g\colon X_m\to \Gr _{N+m}(\C ^\infty )\subseteq BU\). Viewed as a map \(X_m\to \{m\}\times BU\), this maps under \(\Phi _{X_m}\) to \[ [E\oplus F']-(N+m)[\ul {\C }]+m[\ul {\C }]=[E]-[F]=\alpha . \] Gluing over the finitely many \(m\) proves surjectivity.
For injectivity, suppose \(f_0,f_1\colon X\to \Z \times BU\) have the same image under \(\Phi _X\). Their images in \(K^0(X)\) have the same rank function, so the \(\Z \)-coordinates of \(f_0\) and \(f_1\) agree. We again restrict to a clopen piece \(X_m\) and choose factorizations \(f_i\colon X_m\to \Gr _{n_i}(\C ^\infty )\) classifying bundles \(E_i\). The equality \(\Phi _X(f_0)=\Phi _X(f_1)\) gives \[ [E_0]-n_0[\ul {\C }] = [E_1]-n_1[\ul {\C }] \] in \(K^0(X_m)\). By the definition of the Grothendieck group, there exists a vector bundle \(F\) and an isomorphism \[ E_0\oplus \ul {\C }^{n_1}\oplus F \cong E_1\oplus \ul {\C }^{n_0}\oplus F. \] Choose \(F'\) with \(F\oplus F'\cong \ul {\C }^q\) by Theorem 9.3.6. After stabilizing \(f_0\) by \(n_1+q\) steps and \(f_1\) by \(n_0+q\) steps, the two maps land in the same Grassmannian \(\Gr _{n_0+n_1+q}(\C ^\infty )\) and classify isomorphic vector bundles. By Theorem 9.2.4, they are homotopic. These homotopies glue over the finite clopen decomposition of \(X\), proving injectivity.
The construction is compatible with addition. If maps through \(\{m\}\times \Gr _n(\C ^\infty )\) and \(\{m'\}\times \Gr _{n'}(\C ^\infty )\) classify bundles \(E\) and \(E'\), then their sum classifies \(E\oplus E'\) at stage \(n+n'\) and has \(\Z \)-coordinate \(m+m'\). Therefore \[ [E\oplus E']-(n+n')[\ul {\C }]+(m+m')[\ul {\C }] \] equals the sum of the two associated \(K^0\)-classes. Thus \(\Phi _X\) is an isomorphism of abelian groups. □
9.3.2 The Bott class and Bott periodicity
Definition 9.3.9 (Bott class). Let \(H \to \CP ^1\) be the tautological line bundle. The Bott class is \[ \beta := [H]-[\ul {\C }] \in \widetilde K^0(\CP ^1). \] By Example 9.3.5, this class generates \(\widetilde K^0(\CP ^1)\cong \Z \).
The reduced external tensor product of vector bundles induces a pairing \[ \widetilde K^0(X)\otimes \widetilde K^0(Y) \longrightarrow \widetilde K^0(X\wedge Y) \] for pointed compact Hausdorff spaces; see Hatcher (2003), p. 54. The following classical theorem is the geometric input behind the periodicity of complex K-theory.
Theorem 9.3.10 (Bott periodicity for \(\widetilde {K}^0\), Hatcher (2003), Theorem 2.11). For every pointed compact Hausdorff space \(X\), external multiplication by the Bott class induces an isomorphism \[ \widetilde K^0(X) \xrightarrow {\cong } \widetilde K^0(X\wedge \CP ^1) \cong \widetilde K^0(\Sigma ^2X). \]
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