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 formula_1-algebras, it classifies the Fredholm modules over an algebra. 
An operator homotopy between two Fredholm modules formula_2 and formula_3 is a norm continuous path of Fredholm modules, formula_4, formula_5 Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The formula_6 group is the abelian group of equivalence classes of even Fredholm modules over A. The formula_7 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 formula_8 is formula_9