# Topological K-theory

In mathematics, **topological Template:Mvar-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 Template:Mvar-theory is due to Michael Atiyah and Friedrich Hirzebruch.

## Definitions

Let Template:Mvar 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 Template:Mvar-vector bundles over Template:Mvar under Whitney sum. Tensor product of bundles gives Template:Mvar-theory a commutative ring structure. Without subscripts,

*K*(

*X*) usually denotes complex Template:Mvar-theory whereas real Template:Mvar-theory is sometimes written as

*KO*(

*X*). The remaining discussion is focussed on complex Template:Mvar-theory, the real case being similar.

As a first example, note that the Template:Mvar-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 Template:Mvar-theory, , defined for Template:Mvar 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 Template:Mvar and Template:Mvar are said to be **stably isomorphic** if there are trivial bundles *ε*_{1} and *ε*_{2}, so that *E* ⊕ *ε*_{1} ≅ *F* ⊕ *ε*_{2}. 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 *K*(*X*) → *K*({*x*_{0}}) ≅ **Z** induced by the inclusion of the base point *x*_{0} into Template:Mvar.

Template:Mvar-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 Template:Mvar-th reduced suspension of a space and then define

Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without base points:

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

## Properties

*K*respectively is a contravariant functor from the homotopy category of (pointed) spaces to the category of commutative rings. Thus, for instance, the Template:Mvar-theory over contractible spaces is always^{n}**Z**.- The spectrum of Template:Mvar-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 Template:Mvar-theory use
*BO*.

- There is a natural ring homomorphism
*K*^{ ∗}(*X*) →*H*^{ 2∗}(*X*,**Q**), the Chern character, such that*K*^{ ∗}(*X*) ⊗**Q**→*H*^{ 2∗}(*X*,**Q**) is an isomorphism. - The equivalent of the Steenrod operations in Template:Mvar-theory are the Adams operations. They can be used to define characteristic classes in topological Template:Mvar-theory.
- The Splitting principle of topological Template:Mvar-theory allows one to reduce statements about arbitrary vector bundles to statements about sums of line bundles.
- The Thom isomorphism theorem in topological Template:Mvar-theory is

- where
*T*(*E*) is the Thom space of the vector bundle Template:Mvar over Template:Mvar.

- The Atiyah-Hirzebruch spectral sequence allows computation of Template:Mvar-groups from ordinary cohomology groups.
- Topological Template:Mvar-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*×**S**^{2}) =*K*(*X*) ⊗*K*(**S**^{2}), and*K*(**S**^{2}) =**Z**[*H*]/(*H*− 1)^{2}where*H*is the class of the tautological bundle on**S**^{2}=**P**^{1}(**C**), i.e. the Riemann sphere.- Ω
^{2}*BU*≅*BU*×**Z**.

In real Template:Mvar-theory there is a similar periodicity, but modulo 8.

## Applications

The two most famous applications of topological Template:Mvar-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

## References

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- Max Karoubi (1978), K-theory, an introduction Springer-Verlag
- Max Karoubi (2006), "K-theory. An elementary introduction", Template:Arxiv

- Allen Hatcher,
*Vector Bundles & K-Theory*, (2003) - Maxim Stykow,
*Connections of K-Theory to Geometry and Topology*, (2013)