Topological K-theory

In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological K-theory is due to Michael Atiyah and Friedrich Hirzebruch.

Definitions

Let X be a compact Hausdorff space and k = R, C. Then K_k(X) is the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional k-vector bundles over X under Whitney sum. Tensor product of bundles gives K-theory a commutative ring structure. Without subscripts, K(X) usually denotes complex K-theory whereas real K-theory is sometimes written as KO(X). The remaining discussion is focussed on complex K-theory, the real case being similar.

As a first example, note that the K-theory of a point are the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natural numbers are the integers.

There is also a reduced version of K-theory, $\widetilde{K}(X)$, defined for X a compact pointed space (cf. reduced homology). This reduced theory is intuitively K(X) modulo trivial bundles. It is defined as the group of stable equivalence classes of bundles. Two bundles E and F are said to be stably isomorphic if there are trivial bundles ε₁ and ε₂, so that E ⊕ ε₁ ≅ F ⊕ ε₂. This equivalence relation results in a group since every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, $\widetilde{K}(X)$ can be defined as the kernel of the map K(X) → K({x₀}) ≅ Z induced by the inclusion of the base point x₀ into X.

K-theory forms a multiplicative (generalized) cohomology theory as follows. The short exact sequence of a pair of pointed spaces (X, A)

$\widetilde{K}(X/A)\to\widetilde{K}(X)\to\widetilde{K}(A)$

extends to a long exact sequence

$\cdots \to \widetilde{K}(SX) \to \widetilde{K}(SA) \to \widetilde{K}(X/A) \to \widetilde{K}(X) \to \widetilde{K}(A).$

Let Sⁿ be the n-th reduced suspension of a space and then define

$\widetilde{K}^{-n}(X):=\widetilde{K}(S^nX), \qquad n\geq 0.$

Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without base points:

$K^{-n}(X)=\widetilde{K}^{-n}(X_+).$

Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

Properties

• Kⁿ respectively $\widetilde{K}^n$ is a contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the K-theory over contractible spaces is always Z.
• The spectrum of K-theory is BU × Z (with the discrete topology on Z), i.e. K(X) ≅ [X₊, Z × BU], where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: BU(n) ≅ Gr(n, C^∞). Similarly,

$\widetilde{K}(X) \cong [X, \mathbf{Z} \times BU].$

For real K-theory use BO.

• There is a natural ring homomorphism K⁰(X) → H^2∗(X, Q), the Chern character, such that K⁰(X) ⊗ Q → H^2∗(X, Q) is an isomorphism.
• The equivalent of the Steenrod operations in K-theory are the Adams operations. They can be used to define characteristic classes in topological K-theory.
• The Splitting principle of topological K-theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
• The Thom isomorphism theorem in topological K-theory is

$K(X)\cong\widetilde{K}(T(E)),$

where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle.

• The Atiyah-Hirzebruch spectral sequence allows computation of K-groups from ordinary cohomology groups.
• Topological K-theory can be generalized vastly to a functor on C*-algebras, see operator K-theory and KK-theory.

Bott periodicity

The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:

• K(X × S²) = K(X) ⊗ K(S²), and K(S²) = Z[H]/(H − 1)² where H is the class of the tautological bundle on S² = P¹(C), i.e. the Riemann sphere.
• $\widetilde{K}^{n+2}(X)=\widetilde{K}^n(X).$
• Ω²BU ≅ BU × Z.

In real K-theory there is a similar periodicity, but modulo 8.

Applications

The two most famous applications of topological K-theory are both due to J. F. Adams. First he solved the Hopf invariant one problem by doing a computation with his Adams operations. Then he proved an upper bound for the number of linearly independent vector fields on spheres.

See also

• KR-theory
• Atiyah–Singer index theorem
• Snaith's theorem
• Algebraic K-theory

References

• Max Karoubi (1978), K-theory, an introduction Springer-Verlag
  • Max Karoubi (2006), "K-theory. An elementary introduction", arXiv:math/0602082
• Allen Hatcher, Vector Bundles & K-Theory, (2003)