Topological K-Theory

Topological K-theory is the cohomology theory built from stable classes of vector bundles. For a compact Hausdorff space , complex is the Grothendieck group of the monoid of complex vector bundles over under direct sum. Thus elements are formal differences , with stable bundle equivalence built into the group. Its higher groups are periodic with period 2 (Hatcher 2017). For a point, is , determined by vector space dimension.

For a unital C^*-algebra , similarly uses stable classes of projections in matrices over , and uses stable homotopy classes of unitary matrices over (Higson and Roe 2000). In particular, . The analytic targets of assembly maps use this topological K-theory of operator algebras, which is distinct from algebraic K-theory of an arbitrary ring.