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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 ∗ {\displaystyle C^{*}} -algebras, it classifies the Fredholm modules over an algebra.
An operator homotopy between two Fredholm modules {\displaystyle } and {\displaystyle } is a norm continuous path of Fredholm modules, t ↦ {\displaystyle t\mapsto } , t ∈ . {\displaystyle t\in.} Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The K 0 {\displaystyle K^{0}} group is the abelian group of equivalence classes of even Fredholm modules over A. The K 1 {\displaystyle K^{1}} 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 } is . {\displaystyle.}