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 $$k= \R$$ or $$\Complex$$. 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 $$K(X)\to K(x_0) \cong \Z$$ 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.

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.

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. $$K(X) \cong \left [ X_+, \Z \times BU \right ],$$ where $[, ]$ denotes pointed homotopy classes and $BU$ is the colimit of the classifying spaces of the unitary groups: $$BU(n) \cong \operatorname{Gr} \left (n, \Complex^{\infty} \right ).$$ Similarly, $$\widetilde{K}(X) \cong [X, \Z \times BU].$$ 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:


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

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 finite CW complex $$X$$ with its rational cohomology. In particular, they showed that there exists a homomorphism


 * $$ch : K^*_{\text{top}}(X)\otimes\Q \to H^*(X;\Q)$$

such that


 * $$\begin{align}

K^0_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k}(X;\Q) \\ K^1_{\text{top}}(X)\otimes \Q & \cong \bigoplus_k H^{2k+1}(X;\Q) \end{align}$$

There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety $$X$$.