K-homology explained

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

(l{H},F_0,\Gamma)

and

(l{H},F_1,\Gamma)

is a norm continuous path of Fredholm modules,

t\mapsto(l{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

(l{H},F,\Gamma)

(l{H},-F,-\Gamma).

References

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