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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Torus manifolds with non-abelian symmetries


Author: Michael Wiemeler
Journal: Trans. Amer. Math. Soc. 364 (2012), 1427-1487
MSC (2010): Primary 57S15, 57S25
Published electronically: October 20, 2011
MathSciNet review: 2869182
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Abstract: Let $ G$ be a connected compact non-abelian Lie group and $ T$ be a maximal torus of $ G$. A torus manifold with $ G$-action is defined to be a smooth connected closed oriented manifold of dimension $ 2\dim T$ with an almost effective action of $ G$ such that $ M^T\neq \emptyset $. We show that if there is a torus manifold $ M$ with $ G$-action, then the action of a finite covering group of $ G$ factors through $ \tilde {G}=\prod SU(l_i+1)\times \prod SO(2l_i+1)\times \prod SO(2l_i)\times T^{l_0}$. The action of $ \tilde {G}$ on $ M$ restricts to an action of $ \tilde {G}'=\prod SU(l_i+1)\times \prod SO(2l_i+1)\times \prod U(l_i)\times T^{l_0}$ which has the same orbits as the $ \tilde {G}$-action.

We define invariants of torus manifolds with $ G$-action which determine their $ \tilde {G}'$-equivariant diffeomorphism type. We call these invariants admissible 5-tuples. A simply connected torus manifold with $ G$-action is determined by its admissible 5-tuple up to a $ \tilde {G}$-equivariant diffeomorphism. Furthermore, we prove that all admissible 5-tuples may be realised by torus manifolds with $ \tilde {G}''$-action, where $ \tilde {G}''$ is a finite covering group of $ \tilde {G}'$.


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Additional Information

Michael Wiemeler
Affiliation: Department of Mathematics, University of Fribourg, Chemin du Musée 23, CH-1700 Fribourg, Switzerland
Address at time of publication: MPI for Mathematics, Vivatsgasse 7, D-53111 Bonn, Germany
Email: michwiem@web.de, wiemeler@mpim-bonn.mpg.de

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05463-2
PII: S 0002-9947(2011)05463-2
Keywords: Quasitoric manifolds, blow ups, compact non-abelian Lie groups
Received by editor(s): December 11, 2009
Received by editor(s) in revised form: July 16, 2010, and September 10, 2010
Published electronically: October 20, 2011
Additional Notes: Part of the research for this paper was supported by SNF Grant No. 200021-117701
Article copyright: © Copyright 2011 American Mathematical Society