Contents

cohomology

# 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 $X$ 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

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. (…)

### 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.

## 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

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