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K-homology

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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 ∗ {\displaystyle C^{*}} -algebras, it classifies the Fredholm modules over an algebra.

An operator homotopy between two Fredholm modules ( H , F 0 , Γ ) {\displaystyle ({\mathcal {H}},F_{0},\Gamma )} and ( H , F 1 , Γ ) {\displaystyle ({\mathcal {H}},F_{1},\Gamma )} is a norm continuous path of Fredholm modules, t ↦ ( H , F t , Γ ) {\displaystyle t\mapsto ({\mathcal {H}},F_{t},\Gamma )} , t ∈ [ 0 , 1 ] . {\displaystyle t\in .} Two Fredholm modules are then equivalent if they are related by unitary transformations or operator homotopies. The K 0 ( A ) {\displaystyle K^{0}(A)} group is the abelian group of equivalence classes of even Fredholm modules over A. The K 1 ( A ) {\displaystyle 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 ) {\displaystyle ({\mathcal {H}},F,\Gamma )} is ( H , − F , − Γ ) . {\displaystyle ({\mathcal {H}},-F,-\Gamma ).}

References

  • N. Higson and J. Roe, Analytic K-homology. Oxford University Press, 2000.

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