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 $K k (X)$ 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 $ε 1$ and $ε 2$ , so that $E \oplus ε 1 ≅ F \oplus ε 2$ . 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, $\widetilde{K}(X)$ can be defined as the kernel of the map $K (X) \to K ({x 0}) ≅ Z$ 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 $(X, A)$

\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).

Let $S n$ be the $n$ -th reduced suspension of a space and then define

\widetilde{K}^{-n}(X):=\widetilde{K}(S^nX), \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:

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 $BU \times Z$ (with the discrete topology on $Z$ ), i.e. $K (X) ≅ [X +, Z \times BU]$ , where $[ , ]$ denotes pointed homotopy classes and $BU$ is the colimit of the classifying spaces of the unitary groups: $BU (n) ≅ Gr(n, C \infty)$ . Similarly,

\widetilde{K}(X) \cong [X, \mathbf{Z} \times BU].

For real

K

-theory use

BO

There is a natural ring homomorphism $K 0 (X) \to H 2* (X, Q)$ , the Chern character, such that $K 0 (X) \otimes Q \to H 2* (X, Q)$ 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

. 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

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

$K (X \times S 2) = K (X) \otimes K (S 2)$ , 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.
$\widetilde{K}^{n+2}(X)=\widetilde{K}^n(X).$
$Ω 2 BU ≅ BU \times Z$ .

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.

References

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, ISBN 978-3-540-30436-4, MR 2182598
Max Karoubi (1978), K-theory, an introduction Springer-Verlag
- Max Karoubi (2006), "K-theory. An elementary introduction", arXiv:math/0602082
Allen Hatcher, Vector Bundles & K-Theory, (2003)
Maxim Stykow, Connections of K-Theory to Geometry and Topology, (2013)