For some reasons, people are interested in studying vector bundles over spaces. At first consideration, it might seem as if there cannot be anything particularly interesting about vector bundles, since they are a special instance of fiber bundles. But the set of vector bundles have an additive structure, given by the direct sum, and also a multiplicative structure, given by the tensor product. Indeed, to each compact Hausdorff space \( X \) can be associated a ring \( K(X) \) coming from the vector bundles of \( X \). It turns out that \( K(-) \) is a contravariant functor and enjoys various common properties with cohomology theories.
From a more historical viewpoint, another motivation for developing a theory of vector bundles is to understand (differentiable) manifolds. Every differentiable manifold has a tangent bundle, and it is a natural question to ask about its topological properties. For instance, when does a manifold have trivial tangent bundle? The following theorem was proved by Adams [Ada62] in 1962.
Theorem 1. The maximum number of linearly independent vector fields on \( S^{n-1} \) is \( 2^c + 8d - 1 \), where \( n = (2a+1) 2^{c + 4d} \) and \( 0 \le c \lt 4 \).
As a direct corollary, \( S^{n-1} \) has trivial tangent bundle if and only if \( n = 1, 2, 4, 8 \). Although we shall not develop the theory far enough to prove, or even give a sketch of the proof of this result, it is motivating to know that \( K \)-theory has such an interesting consequence.
The purpose of this short paper is to give an introductory overview of topological \( K \)-theory and develop the basic notions. In the course, some details will be omitted and we refer the reader to other sources.
1. Vector bundles#
A vector bundle is a fiber bundle having a vector space as a fiber, with the additional condition that transition maps are linear. For our purposes, we allow the vector bundle to have fibers of different dimension on each connected component.
Definition 2. A (real or complex) vector bundle over a base space \( X \) is space \( E \) with a map \( \pi : E \rightarrow X \), such that
- for any point \( x \in X \) the fiber \( \pi^{-1}(x) \) has a structure of a finite-dimensional \( \mathbb{K} \)-vector space,
- for any point \( x \in X \) there exists a neighborhood \( U \) such that there is a nonnegative integer \( n \) and an homeomorphism \( f \) for which commutes and is linear on each fiber,
where \( \mathbb{K} = \mathbb{R} \) if the vector bundle is real and \( \mathbb{K} = \mathbb{C} \) if it is complex.
Example 1. For any space \( X \), the cross product with projection \( X \times \mathbb{K}^n \rightarrow X \) is a vector bundle. This is called the trivial bundle.
Example 2. For \( 0 \le k \le n \), consider the Grassmannian \[ \displaystyle \begin{aligned} G_{\mathbb{K}}(k, n) & \displaystyle= { \Lambda \subseteq \mathbb{K}^n : \dim \Lambda = k }, \end{aligned} \] which has a natural smooth (or complex) structure. There is a tautological bundle over \( G(k, n) \) given by \[ \displaystyle \begin{aligned} \tau_{k,n} : { (p, \Lambda) : p \in \Lambda \subseteq \mathbb{K}^n, \dim \Lambda = k } \rightarrow G(k, n); \quad (p, \lambda) \mapsto \Lambda. \end{aligned} \] This can be checked to be a vector bundle with fiber \( \mathbb{K}^k \).
Given two vector bundles \( E, F \rightarrow X \), we can construct more vector bundles over \( X \) by the following operations:
- \( E \oplus F \rightarrow X \) fiberwise being the direct sum,
- \( E \otimes F \rightarrow X \) fiberwise being the tensor product,
- \( E^\vee \rightarrow X \) fiberwise being the dual.
Also given a continuous map \( f : Y \rightarrow X \), we can construct the pullback bundle \( f^\ast \pi : f^\ast E \rightarrow Y \) that naturally satisfies \[ \displaystyle \begin{aligned} \pi^{-1}(f(y)) & \displaystyle\cong (f^\ast \pi)^{-1}(y) \end{aligned} \] for all \( y \in Y \).
Just like vector spaces, the operations satisfies the following properties.
Proposition 3. For any continuous \( f : Y \rightarrow X \), \( g : Z \rightarrow Y \) and vector bundles \( E, F, G \rightarrow X \), there are natural isomorphisms
- \( (E \oplus F) \otimes G \cong (E \otimes G) \oplus (F \otimes G) \),
- \( f^\ast(E \oplus F) \cong f^\ast E \oplus f^\ast F \),
- \( f^\ast(E \otimes F) \cong f^\ast E \otimes f^\ast F \),
- \( (fg)^\ast(E) \cong g^\ast(f^\ast E) \).
Definition 4. Given two vector bundles \( E, F \rightarrow X \), a morphism between vector bundles is a continuous map \( f : E \rightarrow F \) such that \( \pi _ F \circ f = \pi _ E \) and \( f \) is linear on each fiber.
It can be readily checked that the category of vector bundles over \( X \), \( \mathsf{Vect} _ {\mathbb{K}}(X) \), is an additive category, with the direct sum being the biproduct. However, it is not true in general that kernels and cokernels exist, because the linear maps on fibers may have different rank at nearby points.
Despite not being an abelian category, under suitable hypotheses for \( X \) the notion of monomorphisms and epimorphisms exists, and they can be checked fiberwise. Kernels of epimorphisms and cokernels of monomorphisms are also well-defined. In particular, it makes sense to talk about short exact sequences.
Proposition 5. Let \( X \) be a paracompact space. (We take this mean that \( X \) is Hausdorff and there exists a partition of unity subordinate to an arbitrary open cover. All CW-complexes are paracompact. All compact Hausdorff spaces are paracompact. Paracompact spaces are normal and closed subsets of paracompact spaces are paracompact.) Then any short exact sequence \( 0 \rightarrow E \rightarrow F \rightarrow G \rightarrow 0 \) of vector bundles over \( X \) splits.
Proof. Suppose \( { U _ \alpha } \) is an open cover of \( X \) such that \( F \) has local trivializations on each \( U _ \alpha \). Take a positive-definite (hermitian if \( \mathbb{K} = \mathbb{C} \)) metric on each \( U _ \alpha \times \mathbb{K}^n \) and glue them together with a partition of unity. This gives a metric on \( F \). The orthogonal complement \( E^\perp \subseteq F \) is then isomorphic to \( G \). ▨
Proposition 6. Let \( X \) be compact Hausdorff. For any vector bundle \( E \rightarrow X \) there exists a vector bundle \( F \rightarrow X \) such that \( E \oplus F \) is trivial, i.e., isomorphic to a trivial bundle.
Proof. Take a finite open cover \( U _ \alpha \) such that \( E \) is trivial over each \( U _ \alpha \). Let \( \varphi _ \alpha \) be a partition of unity subordinate to \( U _ \alpha \). For each \( \alpha \), take sections \( s _ {\alpha\beta} \) of \( E \vert _ {U _ \alpha} \) that generate it. By construction, \( s _ {\alpha\beta} \varphi _ \alpha \) can be regarded as global sections of \( E \), and they span \( E \). This gives a epimorphism \( X \times \mathbb{R}^N \rightarrow E \rightarrow 0 \). By Proposition 5, we may take \( F \) to be its kernel. ▨
Theorem 7. Let \( f, g : Y \rightarrow X \) be a continuous map with \( Y \) paracompact and let \( E \rightarrow X \) be a vector bundle. If \( f \) and \( g \) are homotopic, then \( f^\ast E \cong g^\ast E \).
Proof. See [Ati67] Lemma 1.4.3 or [Hat09] Theorem 1.6. ▨
As a consequence, if \( X \) and \( Y \) are homotopy equivalent, then there is a bijection between the isomorphism classes of vector bundles over \( X \) and \( Y \). In particular, if \( X \) is contractible, any vector bundle over \( X \) is trivial and thus determined up to isomorphism by the dimension of its fiber.
2. The \( K \)-group#
Given two vector bundles \( E, F \rightarrow X \), we can take the direct sum. This gives the set of equivalence classes of vector bundles over \( X \) a structure of an abelian monoid under addition.
Definition 8. For a compact Hausdorff space \( X \), we define its \( K \)-group \( K(X) \) as the groupification of the abelian monoid of equivalence classes of complex vector bundles. In other words, \[ \displaystyle \begin{aligned} K(X) & \displaystyle= { \text{formal differences } [E] - [F] } / \sim_s, \end{aligned} \] where \( [E _ 1] - [F _ 1] \sim _ s [E _ 2] - [F _ 2] \) if \( E _ 1 \oplus F _ 2 \oplus G \cong E _ 2 \oplus F _ 1 \oplus G \) for some \( G \rightarrow X \).
Clearly \( K(X) \) is an abelian group. It further has a multiplicative structure coming from the tensor product: \[ \displaystyle \begin{aligned} ([E_1] - [F_1]) \cdot ([E_2] - [F_2]) & \displaystyle= [(E_1 \otimes E_2) \oplus (F_1 \otimes F_2)] - [(E_1 \otimes F_2) \oplus (E_2 \otimes F_1)]. \end{aligned} \] By Proposition 3 this makes \( K(X) \) into a commutative ring, with the trivial bundle \( X \times \mathbb{C} \rightarrow X \) as the identity.
A map \( f : X \rightarrow Y \) induces a map \[ \displaystyle \begin{aligned} f^\ast & \displaystyle: K(Y) \rightarrow K(X) \end{aligned} \] by simply pulling back the vector bundles along \( f \).
Proposition 9. The map \( X \mapsto K(X) \) defines a functor \[ \displaystyle \begin{aligned} K(-) & \displaystyle: \mathsf{hCptHaus}^\mathrm{op} \rightarrow \mathsf{CRing}. \end{aligned} \]
Proof. The pullback maps behave well because of Proposition 3. The pullback maps depend only on the homotopy class because of Theorem 7. ▨
We can further define a reduced version of \( K \).
Definition 10. Let \( (X, x _ 0) \) be a pointed compact Hausdorff space. We define the reduced \( \widetilde{K}(X) \) as the kernel \[ \displaystyle \begin{aligned} 0 & \displaystyle\rightarrow \widetilde{K}(X) \rightarrow K(X) \rightarrow K(x_0) \cong \mathbb{Z} \rightarrow 0, \end{aligned} \] where the map \( K(X) \rightarrow K(x _ 0) \) is induced from the inclusion \( {x _ 0} \hookrightarrow X \).
Then \( \tilde{K}(X) \) is an ideal of \( K(X) \) and hence has a structure of a (non-unital) ring. Also observe that the short exact sequence splits though the map \[ \displaystyle \begin{aligned} \mathbb{Z} \cong K(x_0) \rightarrow K(X); \quad n \mapsto [X \times \mathbb{C}^n]. \end{aligned} \] Thus there is an isomorphism \( K(X) \cong \widetilde{K}(X) \oplus \mathbb{Z} \) as abelian groups. To recover \( K \) from \( \widetilde{K} \), we can simply take \( K(X) = \widetilde{K}(X _ +) \), where \( X _ + = X \amalg \ast \).
Proposition 11. The map \( X \mapsto \widetilde{K}(X) \) defines a functor \[ \displaystyle \begin{aligned} \widetilde{K}(-) & \displaystyle: \mathsf{hCptHaus}_{\ast}^{\mathrm{op}} \rightarrow \mathsf{CRng}. \end{aligned} \]
For \( X \) compact Hausdorff, Proposition 6 implies that \( \tilde{K}(X) \) can be identified with the set of vector bundles under the equivalence relation \( E \sim F \) if and only if \( E \oplus (X \times \mathbb{C}^n) \cong F \oplus (X \times \mathbb{C}^m) \) for some \( n, m \ge 0 \).
We now turn to some properties of \( K(-) \) that resemble cohomology.
Theorem 12. Let \( {X _ \alpha} \) be a finite collection of pointed compact Hausdorff spaces. Then there is a natural isomorphism \[ \displaystyle \begin{aligned} \widetilde{K}({\textstyle \bigvee_\alpha} X_\alpha) & \displaystyle\cong \prod_{\alpha}^{} \widetilde{K}(X_\alpha). \end{aligned} \]
Proof. To get the map \( \widetilde{K}(\bigvee _ \alpha X _ \alpha) \rightarrow \prod _ {\alpha}^{} \widetilde{K}(X _ \alpha) \), simply restrict the bundles. To get the inverse map, represent each element in \( \tilde{K}(X _ \alpha) \) as \( [E _ \alpha] - [X _ \alpha \times \mathbb{C}^N] \) for some large \( N \) and glue the bundles \( [E _ \alpha] \). It is clear that the ring structures are preserved. ▨
Theorem 13. Let \( X \) be a pointed compact Hausdorff space. Suppose \( X _ 1, X _ 2 \) are two closed subsets of \( X \) containing the basepoint, with \( X _ 1 \cup X _ 2 = X \). If \( \alpha _ 1 \in \widetilde{K}(X _ 1) \) and \( \alpha _ 2 \in \widetilde{K}(X _ 2) \) restrict to the same element in \( \widetilde{K}(X _ 1 \cap X _ 2) \), then there is an element \( \alpha \in \widetilde{K}(X) \) such that \( \alpha \vert _ {X _ 1} = \alpha _ 1 \) and \( \alpha \vert _ {X _ 2} = \alpha _ 2 \).
Proof. We may assume that \( \alpha _ k \in \widetilde{K}(X _ 1) \) is represented by \( [E _ k] - [X _ k \times \mathbb{C}^{n}] \) for \( k = 1, 2 \) with the \( n \). Then adding more trivial bundles, we may as well assume that \( E _ 1 \vert _ {X _ 1 \cap X _ 2} \cong E _ 2 \vert _ {X _ 1 \cap X _ 2} \). After this, we can simply glue \( E _ 1 \) and \( E _ 2 \). ▨
Corollary 14. Let \( f : A \hookrightarrow X \) be a pointed inclusion of compact Hausdorff spaces. There exists a coexact sequence \( A \rightarrow X \rightarrow C _ f \), and this induces an exact sequence \[ \displaystyle \begin{aligned} \widetilde{K}(A) & \displaystyle\leftarrow \widetilde{K}(X) \leftarrow \widetilde{K}(C_f) \end{aligned} \] of abelian groups.
Proof. To show that the composite is zero, simply note that \( A \rightarrow X \rightarrow C _ f \) is homotopic to \( A \rightarrow \ast \rightarrow C _ f \). Now consider an arbitrary element \( \alpha \in \widetilde{K}(X) \) such that \( f^\ast \alpha = 0 \). The restriction of \( 0 \in \widetilde{K}(CA) \) to \( A \) is clearly also zero. Then by Theorem 13 there exists a \( \beta \in \widetilde{K}(C _ f) \) that restricts to \( \alpha \in \widetilde{K}(X) \). ▨
Corollary 15. For any pointed inclusion \( f : A \hookrightarrow X \) of compact Hausdorff spaces, there is an exact sequence \[ \displaystyle \begin{aligned} \widetilde{K}(A) & \displaystyle\leftarrow \widetilde{K}(X) \leftarrow \widetilde{K}(C_f) \leftarrow \widetilde{K}(\Sigma A) \leftarrow \widetilde{K}(\Sigma X) \leftarrow \widetilde{K}(\Sigma C_f) \leftarrow \cdots \end{aligned} \] of abelian groups.
Proposition 16. If \( A \hookrightarrow X \) is a pointed inclusion of compact Hausdorff spaces and \( A \) is contractible, then \( X \rightarrow X / A \) induces and isomorphism \( \widetilde{K}(X/A) \cong \widetilde{K}(X) \).
Proof. In fact, we shall prove that \( X \rightarrow X/A \) gives a bijection at the level of isomorphism classes of vector bundles. Every vector bundle over \( A \) is trivial. Thus every vector bundle over \( X \) has a trivialization at least over \( A \). Contracting this to a single vector space gives a vector bundle over \( X/A \). It can be checked that this construction is independent of the choice of the trivialization, and that it is an inverse to the pullback along \( X \rightarrow X/A \). ▨
Corollary 17. For any pointed inclusion \( A \hookrightarrow X \) of compact Hausdorff spaces, let us write \( \widetilde{K}(X, A) = \widetilde{K}(X/A) \). There is an exact sequence \[ \displaystyle \begin{aligned} \widetilde{K}(A) & \displaystyle\leftarrow \widetilde{K}(X) \leftarrow \widetilde{K}(X, A) \leftarrow \widetilde{K}(\Sigma A) \leftarrow \widetilde{K}(\Sigma X) \leftarrow \widetilde{K}(\Sigma X, \Sigma A) \leftarrow \cdots \end{aligned} \] of abelian groups.
There is no reason for us to only look at complex vector bundles. We likewise construct \( KO(X) \) in the same way, using real vector bundles instead of complex vector bundles. It again has a natural ring structure coming from the tensor product. We can also define the reduced version \( \widetilde{KO}(X) \). All the theorems and corollary we have stated so far holds equally for \( KO \) and \( \widetilde{KO} \). However, we restrict our attention to \( K \) and \( \widetilde{K} \) simply because complex vector bundles are easier to work with. The difference becomes eminent in Section 4.
3. Representation of \( \widetilde{K} \)#
Theorems 12 and 13 seem to beg us to use Brown representability. But we cannot immediately apply the theorem, because our functor \( \widetilde{K}(-) \) is defined only on \( \mathsf{hCptHaus} _ \ast \), not on \( \mathsf{hTop} _ \ast \). However, there is a version of Brown representability that operates on functors defined only on finite CW-complexes. This shows that \( \widetilde{K}(-) \) must be represented by some space.
But we are not satisfied with just knowing that this space exists; we want a concrete description, and fortunately this is possible. For convenience, suppose for a moment that \( X \) is connected. Recall (Proposition 6) that for any vector bundle \( \pi : E \rightarrow X \) of a compact Hausdorff space \( X \), there is a embedding into a trivial bundle \( 0 \rightarrow E \hookrightarrow X \times \mathbb{C}^n \). If \( \dim \pi^{-1}(x _ 0) = k \), then every fiber is a \( k \)-dimensional subspace of \( \mathbb{C}^n \) by connectivity of \( X \). Consider the map \[ \displaystyle \begin{aligned} f & \displaystyle: X \rightarrow G_{\mathbb{C}}(k, n); \quad x \mapsto \pi^{-1}(x) \in G(k, n). \end{aligned} \] It can be easily seen that \( E \cong f^\ast(\tau _ {k,n}) \), where \( \tau _ {k,n} \) is the tautological bundle over \( G _ {\mathbb{C}}(k,n) \). That is, every vector bundle is the pullback of a tautological bundle.
In the construction of \( f : X \rightarrow G _ {\mathbb{C}}(k, n) \) we have made a choice of a sufficiently large \( n \) dependent on \( E \rightarrow X \). To avoid this choice, we consider the limit \[ \displaystyle \begin{aligned} G_{\mathbb{C}}(k, \infty) & \displaystyle= \varinjlim ( \cdots \hookrightarrow G_{\mathbb{C}}(k, n) \hookrightarrow G_{\mathbb{C}}(k, n+1) \hookrightarrow \cdots ). \end{aligned} \] We may glue the tautological vector bundles \( \tau _ {k,n} \) all together to form a rank \( k \) vector bundle \( \tau _ {k,\infty} \) over \( G _ {\mathbb{C}}(k, \infty) \). Set-theoretically \( G _ {\mathbb{C}}(k, \infty) \) and \( \tau _ {k,\infty} \) may be thought of as \[ \displaystyle \begin{aligned} G_{\mathbb{C}}(k,\infty) & \displaystyle= { \Lambda \subseteq \mathbb{C}^\infty : \dim \Lambda = k } \end{aligned} \] and its tautological bundle, where \( \mathbb{C}^\infty \) is the direct limit \( \mathbb{C}^\infty = \varinjlim \mathbb{C}^n \). Using the infinite Grassmannian, we can state that every rank \( k \) vector bundle \( E \rightarrow X \) is some pullback of the tautological bundle \( \tau _ {k,\infty} \) over \( G _ {\mathbb{C}}(k, \infty) \).
Theorem 18. Let \( X \) be a connected compact Hausdorff space. Denote by \( \mathrm{Vect} _ {\mathbb{C}}^k(X) \) the isomorphism classes of rank \( k \) vector bundles over \( X \). Then the map \[ \displaystyle \begin{aligned} {X, G_\mathbb{C}(k, \infty)} & \displaystyle\rightarrow \mathrm{Vect} _ {\mathbb{C}}^k(X); \quad [f] \mapsto f^\ast \tau_{k,\infty} \end{aligned} \] is a bijection.
Proof. We have already proved that the map is surjective. Suppose that \( f _ 1^\ast \tau _ {k,\infty} \cong f _ 2^\ast \tau _ {k,\infty} \) for some \( f _ 1, f _ 2 : X \rightarrow G _ {\mathbb{C}}(k,\infty) \). Because \( X \) is compact, the images of \( f _ 1 \) and \( f _ 2 \) both lie in some \( G _ {\mathbb{C}}(k, n) \). The isomorphisms \( E \cong f _ 1^\ast \tau _ {k,\infty} \cong f _ 2^\ast \tau _ {k,\infty} \) give two maps \( g _ 1, g _ 2 : E \rightarrow \mathbb{C}^\infty \) that are linear embeddings on each fiber. Then the image of \( g _ 1 \) and \( g _ 2 \) both lie in \( \mathbb{C}^n \subseteq \mathbb{C}^\infty \). There is a homotopy \( h \) between \( g _ 1 \) and \( g _ 2 \) given by \[ \displaystyle \begin{aligned} h_t(v) & \displaystyle= \begin{cases} ([(1-2t)g_1(v)], [2t g_1(v)], 0, 0, \ldots) & \displaystyle 0 \le t \le 1/2, \\ ([(2t-1) g_2(v)], [(2-2t) g_1(v)], 0, 0, \ldots) & \displaystyle 1/2 \le t \le 1, \end{cases} \end{aligned} \] where \( g _ i(v) \) is considered to be a \( n \)-tuple of complex numbers. It is clear that \( h _ t(v) \) is a linear embedding on each fiber for each fixed \( t \). Thus the homotopy \( h \) induces a homotopy between \( f _ 1 \) and \( f _ 2 \). ▨
We also write \( G _ {\mathbb{C}}(k, \infty) = BU(k) \), because it is the classifying space for the unitary group \( U(k) \).
Assuming that \( X \) is pointed connected compact Hausdorff, the abelian group \( \widetilde{K}(X) \) can be identified with the set of vector bundles under the equivalence relation \( E _ 1 \sim _ s E _ 2 \) if and only if \( E _ 1 \oplus (X \times \mathbb{C}^{n _ 1}) \cong E _ 2 \oplus (X \times \mathbb{C}^{n _ 2}) \) for some \( n _ 1, n _ 2 \). That is, \( \widetilde{K}(X) \) can be considered as the direct limit \[ \displaystyle \begin{aligned} \widetilde{K}(X) & \displaystyle= \varinjlim (\cdots \rightarrow \mathrm{Vect} _ {\mathbb{C}}^k(X) \rightarrow \mathrm{Vect} _ {\mathbb{C}}^{k+1}(X) \rightarrow \cdots), \end{aligned} \] where the connecting maps are taking the direct sum with \( X \times \mathbb{C} \).
Under the identification \( [X, BU(k)] = \mathrm{Vect} _ {\mathbb{C}}^k(X) \), the connecting map \( \mathrm{Vect} _ {\mathbb{C}}^k(X) \rightarrow \mathrm{Vect} _ {\mathbb{C}}^{k+1}(X) \) is represented by the map \[ \displaystyle \begin{aligned} BU(k) & \displaystyle\hookrightarrow BU(k+1); \quad (\Lambda \subseteq \mathbb{C}^\infty) \mapsto (\mathbb{C} \oplus \Lambda \subseteq \mathbb{C} \oplus \mathbb{C}^\infty \cong \mathbb{C}^\infty). \end{aligned} \] Since \( X \) is compact, \[ \displaystyle \begin{aligned} \widetilde{K}(X) & \displaystyle= \varinjlim (\cdots \rightarrow \mathrm{Vect} _ {\mathbb{C}}^k(X) \rightarrow \mathrm{Vect} _ {\mathbb{C}}^{k+1}(X) \rightarrow \cdots) \\ & \displaystyle= \varinjlim ( \cdots \rightarrow [X, BU(k)] \rightarrow [X, BU(k+1)] \rightarrow \cdots) \\ & \displaystyle= [X, \varinjlim (\cdots \hookrightarrow BU(k) \hookrightarrow BU(k+1) \hookrightarrow \cdots)]. \end{aligned} \] Therefore, if we write the direct limit as \[ \displaystyle \begin{aligned} BU & \displaystyle= \varinjlim_k BU(k) = \varinjlim_{k,n} G_{\mathbb{C}}(k,n+k), \end{aligned} \] we get the following.
Proposition 19. For any pointed compact Hausdorff \( X \), there is a natural bijection \( \widetilde{K}(X) = [X, BU \times \mathbb{Z}] \).
Proof. For the connected component \( X _ 0 \subseteq X \) containing the basepoint, \( \widetilde{K}(X _ 0) = [X _ 0, BU] = [X _ 0, BU \times \mathbb{Z}] \). For any other component \( X _ 1 \subseteq X \), we do not have control over the dimension, and hence \( \widetilde{K}(X _ {1+}) = [X _ {1+}, BU \times \mathbb{Z}] \). ▨
It is now natural to define general \( K \)-theory in terms of the space \( BU \times \mathbb{Z} \).
Definition 20. For a general (pointed) CW-complex, we define \[ \displaystyle \begin{aligned} \widetilde{K}(X) & \displaystyle= [X, BU \times \mathbb{Z}], \quad K(X) = [X_+, BU \times \mathbb{Z}]. \end{aligned} \]
There is a slight caveat in this definition. It is possible to define a "\( K \)-group" for general spaces by simply groupifying the commutative monoid of equivalence classes of vector bundles. However, this might not agree with our definition of \( K \) if the base space is not compact. In particular, the infinitary wedge axiom for "\( \widetilde{K} \)" fails, whereas it must be true for \( \widetilde{K} \).
The sets \( K(X) \) and \( \widetilde{K}(X) \) have addition and multiplication. Indeed, the space \( BU \times \mathbb{Z} \) is a ring space with satisfying the ring axioms up to homotopy. We will not describe it here in detail. A brief description is given in [May99], Section 24.1.
4. Bott periodicity and the spectrum \( KU \)#
For two compact Hausdorff spaces \( X \) and \( Y \), there is a natural map \[ \displaystyle \begin{aligned} \mu_{X,Y} : K(X) \otimes K(Y) & \displaystyle\rightarrow K(X \times Y); \quad [E] \otimes [F] \mapsto [p_X^\ast E \otimes p_Y^\ast F]. \end{aligned} \] It is clear that this is ring morphism.
Theorem 21 (Bott perioditicity). For compact Hausdorff \( X \), the map \[ \displaystyle \begin{aligned} \mu & \displaystyle: K(X) \otimes K(S^2) \rightarrow K(X \times S^2) \end{aligned} \] is an isomorphism.
Proof. There is a long list of proofs, complied in [MO8800]. Bott’s original proof using Morse theory can be found in [Bot59]. An elementary proof is given in both [Ati67] and [Hat09]. ▨
Corollary 22. For pointed compact Hausdorff \( X \), there is a natural isomorphism \[ \displaystyle \begin{aligned} \widetilde{K}(X) \otimes \widetilde{K}(S^2) \rightarrow \widetilde{K}(\Sigma^2 X) \end{aligned} \] of abelian groups.
Proof. We first note that \( \mu \) be decomposed into \[ \displaystyle \begin{aligned} \mu : (\widetilde{K}(X) \otimes \widetilde{K}(S^2)) \oplus \widetilde{K}(X) \oplus \widetilde{K}(S^2) \oplus \mathbb{Z} \rightarrow \widetilde{K}(X \times S^2) \oplus \mathbb{Z}. \end{aligned} \] Also there is an short exact sequence \[ \displaystyle \begin{aligned} 0 \leftarrow \widetilde{K}(X \vee Y) \leftarrow \widetilde{K}(X \times Y) \leftarrow \widetilde{K}(X \wedge Y) \leftarrow 0 \end{aligned} \] since the map \( \widetilde{K}(\Sigma X \vee \Sigma Y) \rightarrow \widetilde{K}(X \wedge Y) \) is zero. This shows that \( \mu \) being an isomorphism implies that \[ \displaystyle \begin{aligned} \tilde{\mu} & \displaystyle: \widetilde{K}(X) \otimes \widetilde{K}(S^2) \rightarrow \widetilde{K}(\Sigma^2 X) \end{aligned} \] is an isomorphism. ▨
Here, we can compute \( \widetilde{K}(S^2) \) fairly easily.
Proposition 23. Let \( H = \tau _ {1,2} \) be the tautological line bundle over \( \mathbb{C}P^1 \). Then \( K(S^2) \cong \mathbb{Z}[H] / ((H-1)^2) \) as rings and thus \( \widetilde{K}(S^2) \cong \mathbb{Z}\langle H-1 \rangle \).
Proof. Any rank \( n \) vector bundle \( E \rightarrow S^2 \) has a local trivialization on the two hemispheres \( D _ + \) and \( D _ - \). The bundle \( E \) only depends on the gluing datum, which lives in \( [S^1 \rightarrow \mathrm{GL}(n, \mathbb{C})] \). Thus there are \( \mathbb{Z} \)-many rank \( n \) vector bundles, and it can be checked that the ring structure is \( \mathbb{Z}[H] / ((H-1)^2) \). ▨
Corollary 24. For pointed compact Hausdorff \( X \), there is a natural isomorphism \( \widetilde{K}(X) \cong \widetilde{K}(\Sigma^2 X) \) as abelian groups.
The multiplicative structure might not agree, as multiplication on \( \widetilde{K}(S^0) \) is multiplication in \( \mathbb{Z} \) whereas multiplication in \( \widetilde{K}(S^2) \) is zero.
Using Bott periodicity, we can define the cohomology theory \( \widetilde{K}^\ast \).
Definition 25. For a nonnegative integer \( n \), we define \[ \displaystyle \begin{aligned} \widetilde{K}^{-n}(X) & \displaystyle= \widetilde{K}(\Sigma^n X). \end{aligned} \]
Bott periodicity then states that \( \widetilde{K}^{-n-2} = \widetilde{K}^{-n} \). Thus it is natural to use this identity to extend \( \widetilde{K}^n \) for positive \( n \) as well.
Example 3. Let us compute the cohomology of \( S^0 \). We already know that \( \widetilde{K}^{2n}(S^0) = \mathbb{Z} \). To compute \( \widetilde{K}^{2n+1}(S^0) = \widetilde{K}(S^1) \), we use the fact stated in the proof of Proposition 23, namely that rank \( n \)-vector bundles over \( S^1 \) are classified by \( [S^0, \mathrm{GL}(n, \mathbb{C})] \). Since \( \mathrm{GL}(n, \mathbb{C}) \) is connected, we immediately see that \( \widetilde{K}(S^1) = 0 \). Therefore \[ \displaystyle \begin{aligned} \widetilde{K}^\ast(S^0) & \displaystyle= \begin{cases} \mathbb{Z} & \displaystyle \text{if } \ast = 2k, \\ 0 & \displaystyle \text{if } \ast = 2k+1. \end{cases} \end{aligned} \]
This is different from the cohomology theories we are used to. For most cohomology theories, \( H^\ast(S^0) = 0 \) for \( \ast \neq 0 \). If a cohomology theory satisfies \( H^{\ast \neq 0}(S^0) = 0 \), then it is called ordinary. Because this is not true for \( \widetilde{K} \), it is an example of an "extraordinary cohomology theory".
Let us think what Bott periodicity means in terms of the spectrum associated to this cohomology theory. The Bott isomorphism is a natural map \[ \displaystyle \begin{aligned} {X, BU \times \mathbb{Z}} = \widetilde{K}(X) \xrightarrow{\cong} \widetilde{K}(\Sigma^2 X) = [\Sigma^2 X, BU \times \mathbb{Z}] = [X, \Omega^2(BU \times \mathbb{Z})] \end{aligned} \] for pointed compact Hausdorff spaces \( X \). This must be induced by a homotopy equivalence \[ \displaystyle \begin{aligned} \beta & \displaystyle: BU \times \mathbb{Z} \rightarrow \Omega^2(BU \times \mathbb{Z}) = \Omega^2 BU \end{aligned} \] that respects the structure of H-spaces.
Let us consider the space \[ \displaystyle \begin{aligned} U & \displaystyle= \varinjlim U(n), \end{aligned} \] which has the obvious H-space structure. The space \( BU \) is the classifying space of \( U \) and thus we have a homotopy equivalence \( \Omega BU \simeq U \).
Proposition 26. There is a homotopy equivalence \( BU \times \mathbb{Z} \simeq \Omega U \) that respects the structure of H-spaces.
This allows us to define the \( \Omega \)-spectrum \( KU \) as \[ \displaystyle \begin{aligned} KU_{2n} & \displaystyle= BU \times \mathbb{Z}, \quad KU_{2n+1} = U. \end{aligned} \] The periodicity of this spectrum is a restatement of Bott periodicity. Indeed, the original proof of Bott [Bot59] was by exhibiting the homotopy equivalence in Proposition 26.
Let us briefly outline the theory of \( \widetilde{KO}(-) \), which is more complicated than the theory of \( \widetilde{K}(-) \).
Theorem 27 (Bott periodicity). For pointed compact Hausdorff spaces \( X \), there is a natural isomorphism \[ \displaystyle \begin{aligned} \widetilde{KO}(X) \cong \widetilde{KO}(\Sigma^8 X). \end{aligned} \]
The space representing \( \widetilde{KO}(-) \) is similarly given as \( BO \times \mathbb{Z} \). The chain of homotopy equivalences in this case is \[ \displaystyle \begin{aligned} \Omega (BO \times \mathbb{Z}) & \displaystyle\simeq O, & \displaystyle \Omega(BSp \times \mathbb{Z}) & \displaystyle\simeq Sp, \\ \Omega O & \displaystyle\simeq O/U, & \displaystyle \Omega Sp & \displaystyle\simeq Sp / U, \\ \Omega (O / U) & \displaystyle\simeq U/Sp, & \displaystyle \Omega (Sp / U) & \displaystyle\simeq U/O, \\ \Omega(U/Sp) & \displaystyle\simeq BSp \times \mathbb{Z}, & \displaystyle \Omega(U/O) & \displaystyle\simeq BO \times \mathbb{Z}. \end{aligned} \]
References#
[Ada62] J. F. Adams, Vector fields on spheres, Ann. of Math. (2) 75 (1962), 603–632.
[Ati67] M. F. Atiyah, K-theory, Lecture notes by D. W. Anderson, W. A. Benjamin, Inc., New York-Amsterdam, 1967.
[Bot59] Raoul Bott, The stable homotopy of the classical groups, Ann. of Math. (2) 70 (1959), 313–337.
[Hat09] Allen Hatcher, Vector bundles and K-theory, 2009.
[May99] J. P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1999.
[MO8800] Eric Peterson (http://mathoverflow.net/users/1094/eric-peterson), Proofs of bott periodicity, MathOverflow, URL:http://mathoverflow.net/q/8800 (version: 2010-03-05).