K-homology
In mathematics, K-homology is a homology theory on the category of locally compact Hausdorff spaces. It classifies the elliptic pseudo-differential operators acting on the vector bundles over a space. In terms of -algebras, it classifies the Fredholm modules over an algebra. An operator homotopy between two Fredholm modules   and   is a norm continuous path of Fredholm modules,  ,   Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The  group is the abelian group of equivalence classes of even Fredholm modules over A. The  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 inverse of   is

References 
 N. Higson and J. Roe, Analytic K-homology. Oxford University Press, 2000. K-theory
Homology theory