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 $ℂ$. 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, $\tilde{K}(X)$, 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 $E\oplus {\epsilon }_1\cong F\oplus {\epsilon }_2$. 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, $\tilde{K}(X)$ 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 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)

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

extends to a long exact sequence

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

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

${\tilde{K}}^{-n}(X):=\tilde{K}(S^{n}X),n\ge 0.$

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:

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

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, ${\tilde{K}}^n$) 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. $K(X)\cong [X_{+},\mathbb{Z}\times BU],$ where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups: $BU(n)\cong \mathrm{Gr}(n,{ℂ}^{\mathrm{\infty }}).$ Similarly, $$\tilde{K}(X)\cong [X,\mathbb{Z}\times BU].$$ 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 $$K(X)\cong \tilde{K}(T(E)),$$ 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 $K({𝕊}^2)=\mathbb{Z}[H]/(H-1{)}^2$ where H is the class of the tautological bundle on ${𝕊}^2={\mathbb{P}}^1(ℂ),$ i.e. the Riemann sphere.
• ${\tilde{K}}^{n+2}(X)={\tilde{K}}^n(X).$
• ${\Omega }^{2}BU\cong BU\times \mathbb{Z}.$

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

Applications

Topological K-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations. Adams also 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

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

such that

$\begin{array}{cc}{K}_{\text{top}}^0(X)\otimes \mathbb{Q}& \cong \underset{k}{⨁}{H}^{2k}(X;\mathbb{Q})\\ K_{\text{top}}^1(X)\otimes \mathbb{Q}& \cong \underset{k}{⨁}{H}^{2k+1}(X;\mathbb{Q})\end{array}$

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

See also

• Atiyah–Hirzebruch spectral sequence (computational tool for finding K-theory groups)
• KR-theory
• Atiyah–Singer index theorem
• Snaith's theorem
• Algebraic K-theory