A note on torus actions and the Witten genus

Michael Wiemeler

Vol. 286 (2017), No. 2, 499–510

DOI: 10.2140/pjm.2017.286.499

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 $dimT>b_2\left(M\right)$. We apply this result to study group actions on $M\times G∕T$, 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 ${\#}_{i=1}^k\pm ℂP^n$ 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