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 $C^{*}$ -algebras, it classifies the Fredholm modules over an algebra.

An operator homotopy between two Fredholm modules $({\mathcal {H}},F_{0},\Gamma )$ and $({\mathcal {H}},F_{1},\Gamma )$ is a norm continuous path of Fredholm modules, $t\mapsto ({\mathcal {H}},F_{t},\Gamma )$ , $t\in [0,1].$ Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The $K^{0}(A)$ group is the abelian group of equivalence classes of even Fredholm modules over A. The $K^{1}(A)$ 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 inverse of $({\mathcal {H}},F,\Gamma )$ is $({\mathcal {H}},-F,-\Gamma ).$