K-homology is a homology theory on the category of compactPlanetmathPlanetmath Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of C*-algebras, it classifies the Fredholm modules over an algebra.

An operatorMathworldPlanetmath homotopy between two Fredholm modules (,F0,Γ) and (,F1,Γ) is a norm continuousMathworldPlanetmathPlanetmath path of Fredholm modules, t(,Ft,Γ), t[0,1]. Two Fredholm modules are then equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath if they are related by unitary transformations or operator homotopies. The K0(A) group is the abelian group of equivalence classesMathworldPlanetmath of even Fredholm modules over A. The K1(A) group is the abelian group of equivalence classes of odd Fredholm modules over A. Addition is given by direct summation of Fredholm modules, and the inversePlanetmathPlanetmathPlanetmath of (,F,Γ) is (,-F,-Γ).


  • 1 N. Higson and J. Roe, AnalyticPlanetmathPlanetmath K-homology. Oxford University Press, 2000.
Title K-homology
Canonical name Khomology
Date of creation 2013-03-22 12:57:46
Last modified on 2013-03-22 12:57:46
Owner mhale (572)
Last modified by mhale (572)
Numerical id 6
Author mhale (572)
Entry type Topic
Classification msc 19K33
Related topic FredholmModule
Related topic KTheory