K-homology

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 $(\mathscr{H},F_0,\mathrm{\Gamma })$ and $(\mathscr{H},F_1,\mathrm{\Gamma })$ is a norm continuous path of Fredholm modules, $t\mapsto (\mathscr{H},F_t,\mathrm{\Gamma })$, $t\in [0,1]$. 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 $(\mathscr{H},F,\mathrm{\Gamma })$ is $(\mathscr{H},-F,-\mathrm{\Gamma })$.

Title	K-homology
Canonical name	Khomology
Date of creation	2013-03-22 12:57:46
Last modified on	2013-03-22 12:57:46
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	6
Author	mhale (572)
Entry type	Topic
Classification	msc 19K33
Related topic	FredholmModule
Related topic	KTheory