noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A variant of algebraic K-theory for operator algebras (Banach algebras and in particular C-star algebras) is operator K-theory: it generalizes topological K-theory via the Serre-Swan theorem from commutative C-star algebras to general $C^\ast$-algebras.
geometric context | universal additive bivariant (preserves split exact sequences) | universal localizing bivariant (preserves all exact sequences in the middle) | universal additive invariant | universal localizing invariant |
---|---|---|---|---|
noncommutative algebraic geometry | noncommutative motives $Mot_{add}$ | noncommutative motives $Mot_{loc}$ | algebraic K-theory | non-connective algebraic K-theory |
noncommutative topology | KK-theory | E-theory | operator K-theory | … |
A standard textbook account is in section 2 of
Other introductions include