Vol. 286, No. 2, 2017

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ISSN: 0030-8730
A note on torus actions and the Witten genus

Michael Wiemeler

Vol. 286 (2017), No. 2, 499–510
Abstract

We show that the Witten genus of a string manifold M vanishes if there is an effective action of a torus T on M such that dimT > b2(M). We apply this result to study group actions on M × GT, where G is a compact connected Lie group and T a maximal torus of G.

Moreover, we use the methods which are needed to prove these results to the study of torus manifolds. We show that up to diffeomorphism there are only finitely many quasitoric manifolds M with the same cohomology ring as #i=1k ± Pn with k < n.

Keywords
torus actions, Witten genus, quasitoric manifolds, torus manifolds, rigidity
Mathematical Subject Classification 2010
Primary: 57S15, 58J26
Milestones
Received: 17 July 2015
Revised: 23 May 2016
Accepted: 17 August 2016
Published: 15 January 2017
Authors
Michael Wiemeler
Institut für Mathematik
Universität Augsburg
Universitätsstrasse 14
D-86159 Augsburg
Germany