# 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.

## 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, the real case being similar.

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, $\widetilde{K}(X)$, 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 $\epsilon_1$ and $\epsilon_2$, so that $E\oplus\epsilon_1\cong F\oplus\epsilon_2$. The fact that this equivalence relation results in a group follows from the fact that every vector bundle can be completed to a trivial bundle by summing with its orthogonal complement. Alternatively, $\widetilde{K}(X)$ can be defined as the kernel of the map $K(X)\to K(x_0)\cong\mathbb{Z}$ induced by the inclusion of the basepoint 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)

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

extends to a long exact sequence

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

Then define $\widetilde{K}^{-n}(X):=\widetilde{K}(S^nX)$ for $n\geq 0$ where $S^n$ is the nth reduced suspension of a space. Negative indices are chosen so that the coboundary maps increase dimension. One-point compactification extends this definition to locally compact spaces without basepoints: $K^{-n}(X)=\widetilde{K}^{-n}(X_+)$. Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

## Properties

• $K^n$ respectively $\widetilde{K}^n$ 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 (Z with the discrete topology), i.e. $K(X)\cong[X_+,\mathbb{Z}\times BU]$ where [,] denotes pointed homotopy classes and BU is the colimit of the classifying spaces $BU_n\cong Gr_n(\mathbb{C}^{\infty})$ of the unitary groups. Similarly, $\widetilde{K}(X)\cong[X,\mathbb{Z}\times BU]$. For real K-theory use BO.
• There is a natural ring homomorphism $K^*(X)\to H^{2*}(X,\mathbf{Q})$, the Chern character, such that $K^*(X)\otimes\mathbf{Q}\to H^{2*}(X,\mathbf{Q})$ 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 $K(X)\cong\widetilde{K}(T(E))$ where T(E) is the Thom space of the vector bundle E over X.
• 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

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 = CP1, i.e. the Riemann sphere.
• $\widetilde{K}^{n+2}(X)=\widetilde{K}^n(X).$
• Ω2BU ≅ BU × 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.