## Algebraic & Geometric Topology

### Sheaf theory for stacks in manifolds and twisted cohomology for $S^1$–gerbes

#### Abstract

In this paper we give a sheaf theory interpretation of the twisted cohomology of manifolds. To this end we develop a sheaf theory on smooth stacks. The derived push-forward of the constant sheaf with value $ℝ$ along the structure map of a $U(1)$ gerbe over a smooth manifold $X$ is an object of the derived category of sheaves on $X$. Our main result shows that it is isomorphic in this derived category to a sheaf of twisted de Rham complexes.

#### Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 1007-1062.

Dates
Revised: 13 May 2007
Accepted: 15 May 2007
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796712

Digital Object Identifier
doi:10.2140/agt.2007.7.1007

Mathematical Reviews number (MathSciNet)
MR2336247

Zentralblatt MATH identifier
1149.14002

#### Citation

Bunke, Ulrich; Schick, Thomas; Spitzweck, Markus. Sheaf theory for stacks in manifolds and twisted cohomology for $S^1$–gerbes. Algebr. Geom. Topol. 7 (2007), no. 2, 1007--1062. doi:10.2140/agt.2007.7.1007. https://projecteuclid.org/euclid.agt/1513796712

#### References

• M Atiyah, G Segal, Twisted $K$–theory and cohomology
• M Atiyah, G Segal, Twisted $K$–theory, Ukr. Mat. Visn. 1 (2004) 287–330
• K Behrend, Cohomology of stacks, from: “Intersection theory and moduli”, ICTP Lect. Notes, XIX, Abdus Salam Int. Cent. Theoret. Phys., Trieste (2004) 249–294 (electronic)
• K A Behrend, On the de Rham cohomology of differential and algebraic stacks, Adv. Math. 198 (2005) 583–622
• K Behrend, P Xu, Differentiable stacks and gerbes
• P Bouwknegt, A L Carey, V Mathai, M K Murray, D Stevenson, Twisted $K$–theory and $K$–theory of bundle gerbes, Comm. Math. Phys. 228 (2002) 17–45
• P Bouwknegt, J Evslin, V Mathai, $T$–duality: topology change from $H$–flux, Comm. Math. Phys. 249 (2004) 383–415
• J-L Brylinski, Loop spaces, characteristic classes and geometric quantization, Progress in Mathematics 107, Birkhäuser, Boston (1993)
• U Bunke, T Schick, M Spitzweck, Inertia and delocalized twisted cohomology
• U Bunke, T Schick, M Spitzweck, $T$–duality and periodic twisted cohomology (in preparation)
• D S Freed, M J Hopkins, C Teleman, Twisted equivariant $K$–theory with complex coefficients
• J Heinloth, Notes on differentiable stacks, from: “Mathematisches Institut, Georg-August-Universität Göttingen: Seminars Winter Term 2004/2005”, Universitätsdrucke Göttingen, Göttingen (2005) 1–32
• N Hitchin, Lectures on special Lagrangian submanifolds, from: “Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999)”, AMS/IP Stud. Adv. Math. 23, Amer. Math. Soc., Providence, RI (2001) 151–182
• M Joachim, Higher coherences for equivariant $K$–theory, from: “Structured ring spectra”, London Math. Soc. Lecture Notes 315, Cambridge Univ. Press, Cambridge (2004) 87–114
• M Kashiwara, P Schapira, C Houzel, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften 292, Springer, Berlin (1990)
• G Laumon, L Moret-Bailly, Champs algébriques, Ergebnisse der Mathematik und ihrer Grenzgebiete 39, Springer, Berlin (2000)
• V Mathai, D Stevenson, Chern character in twisted $K$–theory: equivariant and holomorphic cases, Comm. Math. Phys. 236 (2003) 161–186
• V Mathai, D Stevenson, On a generalized Connes–Hochschild–Kostant–Rosenberg theorem, Adv. Math. 200 (2006) 303–335
• J P May, The geometry of iterated loop spaces, Lecture Notes in Mathematics 271, Springer, Berlin (1972)
• J P May, J Sigurdsson, Parametrized homotopy theory, Mathematical Surveys and Monographs 132, American Mathematical Society, Providence, RI (2006)
• D Metzler, Topological and smooth stacks
• M K Murray, Bundle gerbes, J. London Math. Soc. $(2)$ 54 (1996) 403–416
• M K Murray, D Stevenson, Bundle gerbes: stable isomorphism and local theory, J. London Math. Soc. $(2)$ 62 (2000) 925–937
• B Noohi, Foundations of topological stacks I
• M Olsson, Sheaves on Artin stacks, J. Reine Angew. Math. (to appear)
• D A Pronk, Etendues and stacks as bicategories of fractions, Compositio Math. 102 (1996) 243–303
• G Tamme, Introduction to étale cohomology, Universitext, Springer, Berlin (1994)
• J L Tu, P Xu, C Laurent-Gangoux, Twisted $K$–theory of differentiable stacks