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 $\mathbb {R} ,\mathbb {C}$ . 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.

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^{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:

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

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

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 Frank 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

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

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

such that

{\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 $X$ .

References

↑ Hatcher. Vector Bundles and K-theory (PDF). p. 57. Retrieved 27 July 2017.

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. doi:10.1007/978-3-540-27855-9. ISBN 978-3-540-30436-4. MR 2182598.
Karoubi, Max (1978). K-theory: an introduction. Springer-Verlag. doi:10.1007/978-3-540-79890-3. ISBN 0-387-08090-2.
Karoubi, Max (2006). "K-theory. An elementary introduction". arXiv:math/0602082.
Hatcher, Allen (2003). "Vector Bundles & K-Theory".

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[1] Hatcher. Vector Bundles and K-theory (PDF). p. 57. Retrieved 27 July 2017.