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 $(\mathscr{H},-F,-\mathrm{\Gamma})$.
References
- 1 N. Higson and J. Roe, Analytic^{} K-homology. Oxford University Press, 2000.
Title | K-homology |
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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 |