Â-Genus on Non-Spin Manifolds with S1 Actions and the Classification of Positive Quaternion-Kähler 12-Manifolds
Herrera, Haydeé ; Herrera, Rafael
J. Differential Geom., Tome 60 (2002) no. 1, p. 341-364 / Harvested from Project Euclid
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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.
Publié le : 2002-07-14
Classification:  
@article{1090351527,
     author = {Herrera, Hayde\'e and Herrera, Rafael},
     title = {\^A-Genus on Non-Spin Manifolds with S<sup>1</sup> Actions and the Classification of Positive Quaternion-K\"ahler 12-Manifolds},
     journal = {J. Differential Geom.},
     volume = {60},
     number = {1},
     year = {2002},
     pages = { 341-364},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1090351527}
}

Herrera, Haydeé; Herrera, Rafael. Â-Genus on Non-Spin Manifolds with S1 Actions and the Classification of Positive Quaternion-Kähler 12-Manifolds. J. Differential Geom., Tome 60 (2002) no. 1, pp.  341-364. http://gdmltest.u-ga.fr/item/1090351527/