Branch of algebraic topology
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
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)
![{\displaystyle {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04358198a37dc687afea2e0a2230293c8eafe093)
extends to a
long exact sequence
![{\displaystyle \cdots \to {\widetilde {K}}(SX)\to {\widetilde {K}}(SA)\to {\widetilde {K}}(X/A)\to {\widetilde {K}}(X)\to {\widetilde {K}}(A).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/95e40c379bce1dfcf6062973491cc1f054e7241f)
Let Sn be the n-th
reduced suspension of a space and then define
![{\displaystyle {\widetilde {K}}^{-n}(X):={\widetilde {K}}(S^{n}X),\qquad n\geq 0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dbecf2352dd6f2019140d62a39175eb08c081c9f)
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:
![{\displaystyle K^{-n}(X)={\widetilde {K}}^{-n}(X_{+}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8e8522599bbb91cf14cffae63170ab41ef6536)
Here
is
with a disjoint basepoint labeled '+' adjoined.
[1]
Finally, the
Bott periodicity theorem as formulated below extends the theories to positive integers.
Properties
(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 ![{\displaystyle \mathbb {Z} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89f4f38f32c2068bca9dc701d13b03dd4a5d52ab)
- 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, ![{\displaystyle {\widetilde {K}}(X)\cong [X,\mathbb {Z} \times BU].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/035a00294eb3c68d1132027458e751be51038d31)
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
![{\displaystyle K(X)\cong {\widetilde {K}}(T(E)),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c71330bb25d0b71f9f63167124ae88f2cbfa8a18)
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:
and
where H is the class of the
tautological bundle on
i.e. the
Riemann sphere.
![{\displaystyle {\widetilde {K}}^{n+2}(X)={\widetilde {K}}^{n}(X).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b7926971a38dd920f664c3b3243443847c2a87e)
![{\displaystyle \Omega ^{2}BU\cong BU\times \mathbb {Z} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c43779512cb59e90ca3abf9a2273e2f235930f38)
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
with its rational cohomology. In particular, they showed that there exists a homomorphism
![{\displaystyle ch:K_{\text{top}}^{*}(X)\otimes \mathbb {Q} \to H^{*}(X;\mathbb {Q} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20bc8c438337b22a9acec6deaa9bee3989d95aba)
such that
![{\displaystyle {\begin{aligned}K_{\text{top}}^{0}(X)\otimes \mathbb {Q} &\cong \bigoplus _{k}H^{2k}(X;\mathbb {Q} )\\K_{\text{top}}^{1}(X)\otimes \mathbb {Q} &\cong \bigoplus _{k}H^{2k+1}(X;\mathbb {Q} )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7c873adb0cf6a7dbaa022779e33b61f907d4de)
There is an algebraic analogue relating the Grothendieck group of coherent sheaves and the Chow ring of a smooth projective variety
.
See also
References