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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 $(\hilbert,F_0,\Gamma)$ and $(\hilbert,F_1,\Gamma)$ is a norm continuous path of Fredholm modules, $t \mapsto (\hilbert,F_t,\Gamma)$ , $t \in [0,1]$ . 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 $(\hilbert, F, \Gamma)$ is $(\hilbert, -F, -\Gamma)$ .
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- N. Higson and J. Roe, Analytic K-homology.
Oxford University Press, 2000.
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"K-homology" is owned by mhale.
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Cross-references: inverse, summation, addition, odd, even, equivalence classes, abelian group, group, or operator, unitary transformations, equivalent, path, continuous, norm, homotopy, algebra, Fredholm modules, terms, vector bundles, operators, Hausdorff spaces, compact, category, theory, homology
This is version 3 of K-homology, born on 2002-08-22, modified 2004-04-16.
Object id is 3330, canonical name is KHomology.
Accessed 3544 times total.
Classification:
| AMS MSC: | 19K33 ($K$-theory :: $K$-theory and operator algebras :: EXT and $K$-homology) |
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Pending Errata and Addenda
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