By S. Hutter

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1 provides a calculation of hol∗ τ K([G/G]). 9. We consider a regular and admissible twist τ of M . Let I : [G/G] → [G/G] be the map which is given by g → g −1 on the level of spaces, and by the identity on the level of groups. We call the twist τ odd, if I ∗ hol∗ τ ∼ = −hol∗ τ . 2. Orientations. 1. Let E be a real euclidian vector space. By Cliff(E) we denote the associated complex Clifford algebra. It comes with an embedding of E → Cliff(E) and a ∗-operation. Let Cliff(E)∗ denote the group of unitary elements.

Then we have a cartesian diagram [G/G] → [G/G × G] p↓ q↓ . d [∗/G] → [∗/G × G] The −τ (G)-K-orientation of q induces a −σ(G)-K-orientation of p : [G/G] → [∗/G], where σ(G) := d∗ τ (G). We consider the sequence i p [∗/G] → [G/G] → [∗/G] . The canonical K-orientation of the composition p ◦ i = id and the −σ(G)-Korientation of p induce a p∗ σ(G)-K-orientation of i. 13. The normal bundle of the map ∗ → [∗/G] is Lie(G) → ∗ placed in degree one. 10. The unique Spinc structure on the vector bundle Lie(G) → ∗ induces the K-orientation of ∗ → [∗/G] .

Thus H˜1 defines a projective bundle on M , which gives the gerbe. References [AS] [Bre94] [Cra] [Del74] [DK00] [FHT] [Fri82] M. Atiyah & G. KT/0407054. L. Breen – On the classification of 2-gerbes and 2-stacks, Astérisque (1994), no. 225, p. 160. M. DG/0008064. P. Deligne – Théorie de Hodge. III, Inst. Hautes Études Sci. Publ. Math. (1974), no. 44, p. 5–77. J. J. Duistermaat & J. A. C. Kolk – Lie groups, Universitext, SpringerVerlag, Berlin, 2000. D. Freed, M. Hopkins & C. AT/0312155. E. M. Friedlander – Étale homotopy of simplicial schemes, Annals of Mathematics Studies, vol.

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