K-Theory

-theory is a branch of mathematics which brings together ideas from algebraic geometry, linear algebra, and number theory. Its two main types are topological K-theory and algebraic K-theory.

For a compact T₂-space , complex topological K-theory defines as the Grothendieck group of complex vector bundles over under Whitney sum. Its elements are formal differences of complex vector bundles. The Whitney sum gives addition, and the fiberwise vector space tensor product gives multiplication, making a ring. Using real vector bundles instead gives the real group .

For a chosen basepoint , the reduced group is the kernel of restriction ; the real version is defined similarly. For connected , this amounts to stable equivalence of vector bundles after trivial summands are ignored (Hatcher 2017, pp. 39-40). Topological K-theory extends to a generalized cohomology theory. It applies to the problem of vector fields on spheres and to the J-homomorphism in homotopy theory.

For a compact T₂-space , let be the C^*-algebra of continuous functions from to . Swan's correspondence identifies complex vector bundles over with finitely generated projective modules over , by taking their sections. Real vector bundles give the analogous correspondence using real-valued continuous functions (Swan 1962, Theorems 1 and 2). This relates topological K-theory to algebraic K-theory, whose group is formed from finitely generated projective modules over a ring .

The Quillen-Lichtenbaum conjecture connects algebraic K-theory to étale cohomology.