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  $$(\mathcal{H}, -F, -\Gamma).$$