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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).


  • N. Higson and J. Roe, Analytic K-homology. Oxford University Press, 2000.

This article incorporates material from K-homology on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.