K-homology

K-homology is a homology theory on the category of compact Hausdorff spaces.It classifies the elliptic pseudo-differential operators acting on thevector bundles over a space.In terms of $C^{*}$-algebras, it classifies the Fredholm modules over an algebra.

An operator homotopy between two Fredholm modules $(\mathscr{H},F_0,\mathrm{\Gamma })$ and $(\mathscr{H},F_1,\mathrm{\Gamma })$is a norm continuous path of Fredholm modules, $t\mapsto (\mathscr{H},F_t,\mathrm{\Gamma })$, $t\in [0,1]$.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 classesof even Fredholm modules over A.The $K^1(A)$ group is the abelian group of equivalence classesof odd Fredholm modules over A.Addition is given by direct summation of Fredholm modules,and the inverse of $(\mathscr{H},F,\mathrm{\Gamma })$ is $(\mathscr{H},-F,-\mathrm{\Gamma })$.