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.
Let X be a compact Hausdorff space and [math]\displaystyle{ k= \R }[/math] or [math]\displaystyle{ \Complex }[/math]. Then [math]\displaystyle{ K_k(X) }[/math] 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, [math]\displaystyle{ K(X) }[/math] usually denotes complex K-theory whereas real K-theory is sometimes written as [math]\displaystyle{ KO(X) }[/math]. 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, [math]\displaystyle{ \widetilde{K}(X) }[/math], 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 [math]\displaystyle{ \varepsilon_1 }[/math] and [math]\displaystyle{ \varepsilon_2 }[/math], so that [math]\displaystyle{ E \oplus \varepsilon_1 \cong F\oplus \varepsilon_2 }[/math]. 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, [math]\displaystyle{ \widetilde{K}(X) }[/math] can be defined as the kernel of the map [math]\displaystyle{ K(X)\to K(x_0) \cong \Z }[/math] 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 [math]\displaystyle{ X_+ }[/math] is [math]\displaystyle{ X }[/math] with a disjoint basepoint labeled '+' adjoined.[1]
Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
The phenomenon of periodicity named after Raoul Bott (see Bott periodicity theorem) can be formulated this way:
In real K-theory there is a similar periodicity, but modulo 8.
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.
Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ X }[/math].
Original source: https://en.wikipedia.org/wiki/Topological K-theory.
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