Topological K theory

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.

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, 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, 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)

K ~ ( X / A ) → K ~ ( X ) → K ~ ( A )

extends to a long exact sequence

⋯ → K ~ ( S X ) → K ~ ( S A ) → K ~ ( X / A ) → K ~ ( X ) → K ~ ( A ) .

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

K ~ − n ( X ) := K ~ ( S n X ) , n ≥ 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 ) = K ~ − n ( X + ) .

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

Properties

Kⁿ respectively 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,

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

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.

K ~ n + 2 ( X ) = 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.