# 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.

## Contents

## Definitions[edit]

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 *x*_{0} 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

Let *S ^{n}* 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.^{[1]}

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

## Properties[edit]

*K*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^{n}

- 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*.

- 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. 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[edit]

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

- and where
*H*is the class of the tautological bundle on i.e. the Riemann sphere.

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

## Applications[edit]

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.

## Chern character[edit]

Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a CW complex with its rational cohomology. In particular, they showed that there exists a homomorphism

such that

There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety .

## See also[edit]

## References[edit]

**^**Hatcher.*Vector Bundles and K-theory*(PDF). p. 57. Retrieved 27 July 2017.

- 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".
- Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology" (PDF).