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# 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 $(\mathord{\mathcal{H}},F_{0},\Gamma)$ and $(\mathord{\mathcal{H}},F_{1},\Gamma)$ is a norm continuous path of Fredholm modules, $t\mapsto(\mathord{\mathcal{H}},F_{t},\Gamma)$, $t\in[0,1]$. 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 $(\mathord{\mathcal{H}},F,\Gamma)$ is $(\mathord{\mathcal{H}},-F,-\Gamma)$.

# References

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

## Mathematics Subject Classification

19K33*no label found*

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