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.

DefinitionsEdit

Let X be a compact Hausdorff space and   or  . Then   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,   usually denotes complex K-theory whereas real K-theory is sometimes written as  . 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,  , 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   and  , so that  . 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,   can be defined as the kernel of the map   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)

 

extends to a long exact sequence

 

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

 

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:

 

Here   is   with a disjoint basepoint labeled '+' adjoined.[1]

Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.

PropertiesEdit

  •   (respectively,  ) 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  
  • The spectrum of K-theory is   (with the discrete topology on  ), i.e.   where [ , ] denotes pointed homotopy classes and BU is the colimit of the classifying spaces of the unitary groups:   Similarly,
     
    For real K-theory use BO.
  • There is a natural ring homomorphism   the Chern character, such that   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
     
    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 periodicityEdit

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

  •   and   where H is the class of the tautological bundle on   i.e. the Riemann sphere.
  •  
  •  

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

ApplicationsEdit

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 characterEdit

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

 

such that

 

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

See alsoEdit

ReferencesEdit

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