nLab
groupoid K-theory

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher geometry

Contents

Idea

K-theory of stacks, often of geometric stacks such as differentiable stacks/Lie groupoids. Under mild assumptions equivalently the operator K-theory of their groupoid convolution algebras.

The twisted K-theory of geometric stacks XX assigns K-classes to maps XB 2U(1)X \to \mathbf{B}^2 U(1) to the moduli 2-stack of circle 2-bundles.

Definition

We discuss two different but related definitions. The first

realizes the K-theory of a suitable Lie groupoid (a local quotient groupoid) as a class in the Grothendieck group of suitable equivariant vector bundles (Hilbert space bundles). The second

realizes it as the operator K-theory of the groupoid convolution algebra.

In the Properties-section below we discuss how both definitions are compatible.

By topological K-theory of Hilbert bundles on groupoids

For local quotient groupoids one can define their K-theory as, essentially, the Grothendieck group of equivariant vector bundles on the groupoid, or rather, in the twisted case, by equivalence classes of sections of Fredholm bundles. (…)

(FHT, I, 3.2)

By Operator K-theory of convolution algebras of groupoids

One can define the (twisted) K-theory of a Lie groupoid (equipped with a circle 2-bundle) to be the operator K-theory of its groupoid convolution algebra.

(TuXuLG 03)

Properties

For XX a proper Lie groupoid, the two definition above agree. (TuXuLG 03, 4.6).

References

For topological and Lie groupoids/stacks

The twisted K-theory of local quotient differentiable stacks is defined in terms of Fredholm bundles in

The twisted K-theory of general differentiable stacks/Lie groupoids in terms of the operator K-theory of their twisted groupoid convolution algebras is discussed in

Functoriality of twisted K-theory of topological groupoids/Lie groupoids is discussed in

See also

  • Bram Mesland, Groupoid cocycles and K-theory, Müunster J. of Math. 4 (2011), 227-250 (arXiv:1005.3677)

For algebraic stacks

(…)

Last revised on August 30, 2018 at 09:12:13. See the history of this page for a list of all contributions to it.