Tags: #todo #higher-algebra/K-theory Refs: K-theory

Why study K theory

Tags: #higher-algebra/K-theory

Why study K-theory?

See K-theory. Examples of results gleaned from the Adams operations.

It played a major role in the second proof of the Aatiyah-Singer Index Theorem (circa 1962). In 1955, Jean-Pierre Serre had used the analogy of vector bundles with Serre’s conjecture on vector bundles,

Geometrically

• Finitely generated projective (modules) modules \(\rightleftharpoons\) vector bundles over \({\mathbf{A}}^N\),
• Free modules \(\rightleftharpoons\) trivial vector bundles.

See also Serre-Swan.

Affine space is topologically contractible, so It admits no non-trivial topological vector bundles. It also admits no non-trivial holomorphic vector bundles. Jean-Pierre Serre remarked that the corresponding question was not known for algebraic vector bundle:

"It is not known whether there exist projective (modules) \(A{\hbox{-}}\)modules of finite type, where \(A = k[x_1, ..., x_n]\) is a polynomial ring over a field.