Topological K-theory

Properties

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

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

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 ${\displaystyle X}$ with its rational cohomology. In particular, they showed that there exists a homomorphism

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

such that

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

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