nLab
K-homology

Contents

Context

Index theory

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

K-theory as a generalized homology theory.

In terms of KK-theory, the KK-homology of a C*-algebra AA is KK(A,)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)(X,g) be a Riemannian manifold. Let

C τ(X)Γ 0( T *X) 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

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

be the space of square integrable sections of the exterior bundle. Write 𝒟=d+d *\mathcal{D} = d + d^\ast for the Kähler-Dirac operator and =𝒟(1+𝒟 2) 1/2\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 X *]KK(C τ(X),). [d_X + d^\ast_X] \in KK(C_\tau(X), \mathbb{C}) \,.

Now assume that XX carries a spin^c structure. Then there exists a vector bundle SXS \to X such that C τ(X)=End(S)C_\tau(X) = End(S) and hence a Morita equivalence C τ(X) MoritaC 0(X)C_\tau(X) \simeq_{Morita} C_0(X) with the algebra of continuous functions vanishing at infinity.

Let then

HL 2(S) H \coloneqq L^2(S)

and write DD for the Spin^c Dirac operator and FD(1+D 2) 1/2F \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 X *]=[D]KK(C 0(X),). [d_X + d^\ast_X] = [D] \in KK(C_0(X), \mathbb{C}) \,.

References

Last revised on October 20, 2019 at 01:31:04. See the history of this page for a list of all contributions to it.