# Topological K-theory

(Redirected from KO-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.

## Definitions

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.

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, ${\displaystyle {\widetilde {K}}(X)}$, 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 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 X.

K-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 n-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.

It is often useful to have an unreduced version of these groups, simply by defining:

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

Here ${\displaystyle X_{+}}$ is ${\displaystyle X}$ with a disjoint basepoint labeled '+' adjoined.[1]

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

## Properties

• Kn respectively ${\displaystyle {\widetilde {K}}^{n}}$ 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 (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,
${\displaystyle {\widetilde {K}}(X)\cong [X,\mathbf {Z} \times BU].}$
For real K-theory use BO.
${\displaystyle K(X)\cong {\widetilde {K}}(T(E)),}$
where T(E) is the Thom space of the vector bundle E over X. This holds whenever E is a spin-bundle.

## Bott periodicity

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

• 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 = P1(C), i.e. the Riemann sphere.
• ${\displaystyle {\widetilde {K}}^{n+2}(X)={\widetilde {K}}^{n}(X).}$
• Ω2BUBU × Z.

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

## Applications

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.

## Chern Character

Atiyah and Hirzebruch proved a theorem relating the topological K-theory of a CW complex ${\displaystyle X}$ with its rational cohomology. In particular, they showed that there exists a homomorphism

${\displaystyle ch:K_{top}^{*}(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )}$

such that

{\displaystyle {\begin{aligned}K_{top}^{0}(X)\otimes \mathbb {Q} &\cong \oplus _{k}H^{2k}(X;\mathbb {Q} )\\K_{top}^{1}(X)\otimes \mathbb {Q} &\cong \oplus _{k}H^{2k+1}(X;\mathbb {Q} )\end{aligned}}}

There is an algebraic analogue relating the grothendieck group of coherent sheaves and the chow ring of a smooth projective variety ${\displaystyle X}$.