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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[edit]

Let $X$ be a compact Hausdorff space and $k=\mathbb {R}$ or $\mathbb {C}$ . Then $K_{k}(X)$ is defined to be 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 focused on complex $K$ -theory.

As a first example, note that the $K$ -theory of a point is 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 is 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 $\varepsilon _{1}$ and $\varepsilon _{2}$ , so that $E\oplus \varepsilon _{1}\cong F\oplus \varepsilon _{2}$ . 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 $K(X)\to K(x_{0})\cong \mathbb {Z}$ 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[edit]

$K^{n}$ (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 $\mathbb {Z} .$
The spectrum of $K$ -theory is $BU\times \mathbb {Z}$ (with the discrete topology on $\mathbb {Z}$ ), i.e. $K(X)\cong \left[X_{+},\mathbb {Z} \times BU\right],$ where $[ , ]$ denotes pointed homotopy classes and $BU$ is the colimit of the classifying spaces of the unitary groups: $BU(n)\cong \operatorname {Gr} \left(n,\mathbb {C} ^{\infty }\right).$ Similarly, ${\widetilde {K}}(X)\cong [X,\mathbb {Z} \times BU].$ For real $K$ -theory use $BO$ .
There is a natural ring homomorphism $K^{0}(X)\to H^{2*}(X,\mathbb {Q} ),$ the Chern character, such that $K^{0}(X)\otimes \mathbb {Q} \to H^{2*}(X,\mathbb {Q} )$ 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[edit]

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

$K(X\times \mathbb {S} ^{2})=K(X)\otimes K(\mathbb {S} ^{2}),$ and $K(\mathbb {S} ^{2})=\mathbb {Z} [H]/(H-1)^{2}$ where H is the class of the tautological bundle on $\mathbb {S} ^{2}=\mathbb {P} ^{1}(\mathbb {C} ),$ i.e. the Riemann sphere.
${\widetilde {K}}^{n+2}(X)={\widetilde {K}}^{n}(X).$
$\Omega ^{2}BU\cong BU\times \mathbb {Z} .$

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

Applications[edit]

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[edit]

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

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

such that

{\begin{aligned}K_{\text{top}}^{0}(X)\otimes \mathbb {Q} &\cong \bigoplus _{k}H^{2k}(X;\mathbb {Q} )\\K_{\text{top}}^{1}(X)\otimes \mathbb {Q} &\cong \bigoplus _{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[edit]

^ 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. Classics in Mathematics. 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".
Stykow, Maxim (2013). "Connections of K-Theory to Geometry and Topology".

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

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