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 [math]\displaystyle{ C^* }[/math]-algebras, it classifies the Fredholm modules over an algebra. An operator homotopy between two Fredholm modules [math]\displaystyle{ (\mathcal{H},F_0,\Gamma) }[/math] and [math]\displaystyle{ (\mathcal{H},F_1,\Gamma) }[/math] is a norm continuous path of Fredholm modules, [math]\displaystyle{ t \mapsto (\mathcal{H},F_t,\Gamma) }[/math], [math]\displaystyle{ t \in [0,1]. }[/math] Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The [math]\displaystyle{ K^0(A) }[/math] group is the abelian group of equivalence classes of even Fredholm modules over A. The [math]\displaystyle{ K^1(A) }[/math] 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 [math]\displaystyle{ (\mathcal{H}, F, \Gamma) }[/math] is [math]\displaystyle{ (\mathcal{H}, -F, -\Gamma). }[/math]

References

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