process

346 U. Bunke, T. Schick, M. Spitzweck, and A. ThomThis group is exactly the group ker. /=C in the notation of [BRS, Theorem 2.24(3)].It follows from (65) that the action of H 3 .BI Z/ on Q E induces an action of E onQ E which preserves the fibres of f . In [BRS, Theorem 2.24(3)] we have shown that italso acts on Triple E .B/ and preserves the subsets Ext.E; h/.Let us fix a Picard stack P 2 Q E , and let Oc WD Oc.P/ and h 2 H 3 .EI Z/ be suchthat f.P/ 2 Ext.E; h/. From (65) we see that the group E acts simply transitivelyon the setA E; Oc WD ¹Q 2 f1 .h/ jOc.Q/ DOcº:By [BRS, Theorem 2.24(3)] it also acts simply transitively on the setB E; Oc WD ¹t 2 Ext.E; h/ jOc.t/ DOcº:By Lemma 6.18 we have ‰.A E; Oc / B E; Oc . By Lemma 6.20 the map ‰ is E -equivariant. Hence it must induce a bijection between A E; Oc and B E; Oc . Ifwelet Oc runover all possible choices (solutions of Oc [c.E/ D 0) we see that ‰ W Q E ! Triple E .B/is a bijection. In view of (74) we now also get ˆ ı ‰ D id.Acknowledgment. We thank Tony Pantev for the crucial suggestion which initiatedthe work on this paper.References[Brd97] Glen E. Bredon, Sheaf theory, Grad. Texts in Math. 170, Springer-Verlag, New York1997.[Bre69] Lawrence Breen, Extensions of abelian sheaves and Eilenberg-MacLane algebras, Invent.Math. 9 (1969/1970), 15–44.[Bre76] Lawrence Breen, Extensions du groupe additif sur le site parfait, in Algebraic surfaces(Orsay, 1976–78), Lecture Notes in Math. 868, Springer-Verlag, Berlin 1981, 238–262.[Bre78] Lawrence Breen, Extensions du groupe additif, Inst. Hautes Études Sci. Publ. Math.48 (1978), 39–125.[Bro82] Kenneth S. Brown, Cohomology of groups, Grad. Texts in Math. 87, Springer-Verlag,New York 1982.[BRS] Ulrich Bunke, Philipp Rumpf, and Thomas Schick, The topology of T -duality forT n -bundles, Rev. Math. Phys. 18 (2006), 1103–1154.[BS05] Ulrich Bunke and Thomas Schick, On the topology of T -duality, Rev. Math. Phys. 17(2005), 77–112.[BSS] Ulrich Bunke, Thomas Schick, and Markus Spitzweck, Sheaf theory for stacks inmanifolds and twisted cohomology for S 1 -gerbes, Algebr. Geom. Topol. 7 (2007),1007–1062.[Del] P. Deligne, La formule de dualité globale, in Théorie des topos et cohomologie étaledes schémas Tome 3, Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964(SGA 4), Lecture Notes in Math. 305, Springer-Verlag, Berlin 1973, 481–587