# 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 Kk(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, ${\displaystyle {\widetilde {K}}(X)}$, 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 ε1 and ε2, so that Eε1Fε2. 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, ${\displaystyle {\widetilde {K}}(X)}$ can be defined as the kernel of the map K(X) → K({x0}) ≅ Z induced by the inclusion of the base point x0 into 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)

${\displaystyle {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A)}$

extends to a long exact sequence

${\displaystyle \cdots \to {\widetilde {K}}(SX)\to {\widetilde {K}}(SA)\to {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A).}$

Let Sn be the Template:Mvar-th reduced suspension of a space and then define

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

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

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

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

## Properties

${\displaystyle {\widetilde {K}}(X)\cong [X,\mathbf {Z} \times BU].}$
For real Template:Mvar-theory use BO.
${\displaystyle K(X)\cong {\widetilde {K}}(T(E)),}$
where T(E) is the Thom space of the vector bundle Template:Mvar over Template:Mvar.

## Bott periodicity

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

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.