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