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.

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

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

Negative indices are chosen so that the coboundary maps increase dimension.

It is often useful to have an unreduced version of these groups, simply by defining:

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

Here $X_{+}$ is $X$ with a disjoint basepoint labeled '+' adjoined.

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

• Atiyah, Michael Francis (1989). K-theory. Advanced Book Classics (2nd ed.). Addison-Wesley. ISBN 978-0-201-09394-0. MR 1043170.
• Friedlander, Eric; Grayson, Daniel, eds. (2005). Handbook of K-Theory. Berlin, New York: Springer-Verlag. doi:10.1007/978-3-540-27855-9. ISBN 978-3-540-30436-4. MR 2182598.
• Karoubi, Max (1978). K-theory: an introduction. Springer-Verlag. doi:10.1007/978-3-540-79890-3. ISBN 0-387-08090-2.
• Karoubi, Max (2006). "K-theory. An elementary introduction". arXiv:math/0602082.
• Hatcher, Allen (2003). "Vector Bundles & K-Theory".