Open Access
Translator Disclaimer
July, 2002 Â-Genus on Non-Spin Manifolds with S1 Actions and the Classification of Positive Quaternion-Kähler 12-Manifolds
Haydeé Herrera, Rafael Herrera
J. Differential Geom. 61(3): 341-364 (July, 2002). DOI: 10.4310/jdg/1090351527

Abstract

We prove that the Â-genus vanishes on certain non-spin manifolds. Namely, Â(M) vanishes on any oriented, compact, connected, smooth manifold M with finite second homotopy group and endowed with non-trivial (isometric) smooth S1 actions. This result extends that of Atiyah and Hirzebruch on spin manifolds endowed with smooth S1 actions [1] to manifolds which are not necessarily spin.

We prove such vanishing by means of the elliptic genus defined by Ochanine [23, 24], showing that it also has the special property of being "rigid under S1 actions" on these (not necessarily spin) manifolds.

We conclude with a non-trivial application of this new vanishing theorem by classifying the positive quaternion-Kähler 12-manifolds. Namely, we prove that every quaternion-Kähler 12-manifold with a complete metric of positive scalar curvature must be a symmetric space.

Citation

Download Citation

Haydeé Herrera. Rafael Herrera. "Â-Genus on Non-Spin Manifolds with S1 Actions and the Classification of Positive Quaternion-Kähler 12-Manifolds." J. Differential Geom. 61 (3) 341 - 364, July, 2002. https://doi.org/10.4310/jdg/1090351527

Information

Published: July, 2002
First available in Project Euclid: 20 July 2004

zbMATH: 1071.53027
MathSciNet: MR1979364
Digital Object Identifier: 10.4310/jdg/1090351527

Rights: Copyright © 2002 Lehigh University

JOURNAL ARTICLE
24 PAGES


SHARE
Vol.61 • No. 3 • July, 2002
Back to Top