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

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

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