- Topological K-theory
-
In mathematics, topological K-theory is a branch of algebraic topology. It was founded to study vector bundles on general 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.
Contents
Definitions
Let X be a compact Hausdorff space and
or
. Then Kk(X) is the Grothendieck group of the commutative monoid whose elements are the isomorphism classes of finite dimensional k-vector bundles on X with the operation
for vector bundles E, F. Usually, Kk(X) is denoted KO(X) in real case and KU(X) in the complex case.
More explicitly, stable equivalence, the equivalence relation on bundles E and F on X of defining the same element in K(X), occurs when there is a trivial bundle G, so that
.
Under the tensor product of vector bundles K(X) then becomes a commutative ring.
The rank of a vector bundle carries over to the K-group. Define the homomorphism
where
is the 0-group of Čech cohomology which is equal to the group of locally constant functions with values in
.
If X has a distinguished basepoint x0, then the reduced K-group (cf. reduced homology) satisfies
and is defined as either the kernel of
(where
is basepoint inclusion) or the cokernel of
(where
is the constant map).
When X is a connected space,
.
The definition of the functor K extends to the category of pairs of compact spaces (in this category, an object is a pair (X,Y), where X is compact and
is closed, a morphism between (X,Y) and (X',Y') is a continuous map
such that
)
The reduced K-group is given by x0 = {Y}.
The definition
gives the sequence of K-groups for
, where S denotes the reduced suspension.
Properties
- Kn is a contravariant functor.
- The classifying space of
is BOk(BO, in real case; BU in complex case), i.e.
- The classifying space of K is
(
with discrete topology), i.e.
- There is a natural ring homomorphism
, the Chern character, such that
is an isomorphism.
- 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 for Raoul Bott (see Bott periodicity theorem) can be formulated this way:
and
where H is the class of the tautological bundle on the
, i.e. the Riemann sphere as complex projective line.
In real K-theory there is a similar periodicity, but modulo 8.
See also
- KR-theory
References
- M. Karoubi, K-theory, an introduction, 1978 - Berlin; New York: Springer-Verlag
- M.F. Atiyah, D.W. Anderson K-Theory 1967 - New York, WA Benjamin
- A. Hatcher Vector Bundles & K-Theory
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