#### Vol. 286, No. 2, 2017

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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 $dimT>{b}_{2}\left(M\right)$. We apply this result to study group actions on $M×G∕T$, 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 ${#}_{i=1}^{k}±ℂ{P}^{n}$ with $k.

##### Keywords
torus actions, Witten genus, quasitoric manifolds, torus manifolds, rigidity
##### Mathematical Subject Classification 2010
Primary: 57S15, 58J26