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.
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.
The remaining discussion is focused on complex K-theory.
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.
, defined for X a compact pointed space (cf.
This reduced theory is intuitively K(X) modulo trivial bundles.
It is defined as the group of stable equivalence classes of bundles.
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
[1] Finally, the Bott periodicity theorem as formulated below extends the theories to positive integers.
Topological K-theory has been applied in John Frank Adams’ proof of the “Hopf invariant one” problem via Adams operations.
[2] Adams also proved an upper bound for the number of linearly-independent vector fields on spheres.
[3] Michael Atiyah and Friedrich Hirzebruch proved a theorem relating the topological K-theory of a finite CW complex
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