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| In [[mathematics]], '''K-homology''' is a [[homology (mathematics)|homology]] theory on the [[Category (mathematics)|category]] of locally [[compact space|compact]] [[Hausdorff space]]s. It classifies the elliptic [[pseudo-differential operator]]s acting on the [[vector bundle]]s over a space. In terms of [[C*-algebra|<math>C^*</math>-algebras]], it classifies the [[Fredholm module]]s over an [[algebra]].
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| An '''operator [[homotopy]]''' between two Fredholm modules <math>(\mathcal{H},F_0,\Gamma)</math> and <math>(\mathcal{H},F_1,\Gamma)</math> is a [[norm (mathematics)|norm]] [[Continuous function|continuous]] [[Path (topology)|path]] of Fredholm modules, <math>t \mapsto (\mathcal{H},F_t,\Gamma)</math>, <math>t \in [0,1].</math> Two Fredholm modules are then equivalent if they are related by [[unitary transformation]]s or operator homotopies. The <math>K^0(A)</math> [[group (mathematics)|group]] is the [[abelian group]] of [[equivalence relation|equivalence classes]] of even Fredholm modules over A. The <math>K^1(A)</math> group is the abelian group of equivalence classes of odd Fredholm modules over A. Addition is given by [[Direct sum of modules|direct summation]] of Fredholm modules, and the [[Inverse (mathematics)|inverse]] of <math>(\mathcal{H}, F, \Gamma)</math> is <math>(\mathcal{H}, -F, -\Gamma).</math>
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| == References ==
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| * N. Higson and J. Roe, ''Analytic K-homology''. Oxford University Press, 2000.
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| {{PlanetMath attribution|id=3330|title=K-homology}}
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| [[Category:K-theory]]
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Latest revision as of 19:12, 11 January 2015
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