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 or . Then is defined to be 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, usually denotes complex K-theory whereas real K-theory is sometimes written as . The remaining discussion is focused 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 is 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 . 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, can be defined as the kernel of the map induced by the inclusion of the base point x0 into X.
extends to a long exact sequence
Let Sn be the n-th reduced suspension of a space and then define
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:
Here is with a disjoint basepoint labeled '+' adjoined.
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
- Kn 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
- The spectrum of K-theory is (with the discrete topology on ), 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.
- 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.
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 Frank 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.
There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety .
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- 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".
- Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology" (PDF).