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K homology

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Lukasz grabowski the atiyah problem for k homology gradients


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.

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An operator homotopy between two Fredholm modules ( H , F 0 , Γ ) and ( H , F 1 , Γ ) is a norm continuous path of Fredholm modules, t ( H , F t , Γ ) , t [ 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 ( H , F , Γ ) is ( H , F , Γ ) .

Nigel higson k homology and the quantization commutes with reduction problem


References

K-homology Wikipedia