Positively curved $ {\pi }_{2}$-finite manifolds

Haydeé Herrera

doi:10.1016/j.crma.2007.10.021

Differential Geometry

Positively curved

π_{2}

-finite manifolds
[Variétés avec

π_{2}

fini et courbure positive]

Présenté par : Étienne Ghys

Haydeé Herrera ¹

¹ Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA

Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 499-502.

Résumé
Abstract

Soit M une variété lisse avec un deuxième groupe d'homotopie fini, de courbure sectionnelle positive et de dimension plus grande que 8. Soit G un groupe de Lie compact et connexe qui agit de façon $C^{\infty}$ sur M. On démontre que le nombre caractéristique $\hat{A} (M, T M)$ s'annule si G contient deux involutions qui commutent entre elles.

Let M be a smooth manifold with finite second homotopy group, positive sectional curvature, dimension greater than 8, and assume that a compact connected Lie group G acts smoothly on M. We prove the vanishing of the characteristic number $\hat{A} (M, T M)$ if G contains two commuting involutions.

Reçu le : 2006-02-08
Accepté le : 2007-10-03
Publié le : 2007-11-01

DOI : 10.1016/j.crma.2007.10.021

Affiliations des auteurs :

Haydeé Herrera ¹

¹ Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA

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     title = {Positively curved $ {\pi }_{2}$-finite manifolds},
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     pages = {499--502},
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     number = {9},
     year = {2007},
     doi = {10.1016/j.crma.2007.10.021},
     language = {en},
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Haydeé Herrera. Positively curved $ {\pi }_{2}$-finite manifolds. Comptes Rendus. Mathématique, Volume 345 (2007) no. 9, pp. 499-502. doi : 10.1016/j.crma.2007.10.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.10.021/

Bibliographie
Cité par

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[3] T. Frankel Manifolds with positive curvature, Pacific J. Math., Volume 11 (1961), pp. 165-174

[4] H. Herrera; R. Herrera $\hat{A}$ -genus on non-spin manifolds with $S^{1}$ actions and the classification of positive quaternion-Kähler 12-manifolds, J. Differential Geom., Volume 61 (2002) no. 3, pp. 341-364

[5] H. Herrera; R. Herrera The signature and the elliptic genus of $π_{2}$ -finite manifolds with circle actions, Topology Appl., Volume 136 (2004) no. 1–3, pp. 251-259

[6] F. Hirzebruch; T. Berger; R. Jung Manifolds and Modular Forms, Aspects of Mathematics, Vieweg, 1992

[7] A. Petrunin; W. Tuschmann Diffeomorphism finiteness, positive pinching, and second homotopy, Geom. Funct. Anal., Volume 9 (1999) no. 4, pp. 736-774

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[9] E. Witten Elliptic genera and quantum field theory, Comm. Math. Phys., Volume 109 (1987), p. 525

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