Contents

Idea

In terms of KK-theory, the $K$-homology of a C*-algebra $A$ is $KK(A,\mathbb{C})$.

Presentations

There are various useful ways to present K-homology classes.

By geometric cycles

See at Baum-Douglas geometric cycle.

By Fredholm modules and Dirac operators

Let $(X,g)$ be a Riemannian manifold. Let

$C_\tau(X) \coloneqq \Gamma_0(\wedge^\bullet T^\ast X)$

be the algebra of continuous sections of the exterior bundle vanishing at infinity, and let

$\mathcal{H} \coloneqq L^2(\wedge^\bullet T^\ast X)$

be the space of square integrable? sections of the exterior bundle. Write $\mathcal{D} = d + d^\ast$ for the Kähler-Dirac operator and $\mathcal{F} = \mathcal{D} (1 + \mathcal{D}^2)^{-1/2}$. Then $(\mathcal{H}, \mathcal{F})$ is a Fredholm-Hilbert module which hence represents an element

$[d_X + d^\ast_X] \in KK(C_\tau(X), \mathbb{C}) \,.$

Now assume that $X$ carries a spin^c structure. Then there exists a vector bundle $S \to X$ such that $C_\tau(X) = End(S)$ and hence a Morita equivalence $C_\tau(X) \simeq_{Morita} C_0(X)$ with the algebra of continuous functions vanishing at infinity.

Let then

$H \coloneqq L^2(S)$

and write $D$ for the Spin^c Dirac operator and $F \coloneqq D (1+ D^2)^{-1/2}$.

Then under the above Morita equivalence these two Fredholm-Hilbert modules represent the same element in K-homology

$[d_X + d^\ast_X] = [D] \in KK(C_0(X), \mathbb{C}) \,.$

References

Last revised on December 23, 2016 at 05:06:17. See the history of this page for a list of all contributions to it.