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.
Let X be a compact Hausdorff space and k=R, C. Then Kk(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, , 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 and , so that . The fact that this equivalence relation results in a group follows from the fact that every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, can be defined as the kernel of the map induced by the inclusion of the basepoint 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)
extends to a long exact sequence
Then define for where is the nth reduced suspension of a space. Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without basepoints: . Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
- respectively 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 (Z with the discrete topology), i.e. where [,] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups. Similarly, . For real K-theory use BO.
- There is a natural ring homomorphism , the Chern character, such that 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.
- 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.
- 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 = CP1, i.e. the Riemann sphere.
- Ω2BU ≅ BU × Z.
In real K-theory there is a similar periodicity, but modulo 8.
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.
- 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, ISBN 978-3-540-30436-4, MR 2182598
- Max Karoubi (1978), K-theory, an introduction Springer-Verlag
- Allen Hatcher, Vector Bundles & K-Theory, (2003)
- Maxim Stykow, Connections of K-Theory to Geometry and Topology, (2013)