K-homology

In mathematics, K-homology is a homology theory on the category of 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 $(mathcal\{H\},\; -F,\; -Gamma).$

References

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

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