nLab groupoid K-theory

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 $X$ assigns K-classes to maps $X \to \mathbf{B}^2 U(1)$ to the moduli 2-stack of circle 2-bundles.

Definition

We discuss two different but related definitions. The first

• By topological K-theory of Hilbert bundles

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

• By operator K-theory of convolution algebras

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 $X$ 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

• Daniel Freed, Michael Hopkins, Constantin Teleman, part I of Loop Groups and Twisted K-Theory

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

• Jean-Louis Tu, Ping Xu, Camille Laurent-Gengoux, Twisted K-theory of differentiable stacks (arXiv:math/0306138)

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

• Alexander Alldridge, Jeffrey Giansiracusa, Contravariant functoriality for twisted K-theory of stacks, Oberwolfach Reports no. 46 (2006) 2796 – 2800 (pdf)

• Joost Nuiten, Cohomological quantization of local prequantum boundary field theory, master thesis, August 2013

See also

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

For algebraic stacks

(…)