The $ \stackrel{ˆ}{A}$-genus of $ {S}^{1}$-manifolds with finite second homotopy group

Manuel Amann; Anand Dessai

doi:10.1016/j.crma.2010.01.011

Geometry/Topology

The

\hat{A}

-genus of

S^{1}

-manifolds with finite second homotopy group
[Le

\hat{A}

-genre de

S^{1}

-variétés avec le deuxième groupe homotopie fin]

Présenté par : Jean-Pierre Demailly

Manuel Amann ¹ ; Anand Dessai ²

¹ University of Münster, Department of Mathematics, Einsteinstraße 62, 48149 Münster, Germany
² University of Fribourg, Department of Mathematics, Chemin du Musée 23, CH-1700 Fribourg, Switzerland

Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 283-285.

Résumé
Abstract

Nous construisons des variétés M simplement connexes de dimension $4 k ⩾ 8$ avec les propriétés suivantes : le deuxième groupe d'homotopie $π_{2} (M)$ est fini, M admet une action lisse du cercle $S^{1}$ et le $\hat{A}$ -genre $\hat{A} (M)$ est non nulle.

We construct simply connected smooth manifolds M of dimension $4 k ⩾ 8$ with the following properties: the second homotopy group $π_{2} (M)$ is finite, M admits a smooth action by the circle $S^{1}$ and the $\hat{A}$ -genus $\hat{A} (M)$ is non-zero.

Reçu le : 2009-09-29
Accepté le : 2010-01-09
Publié le : 2010-02-19

DOI : 10.1016/j.crma.2010.01.011

Affiliations des auteurs :

Manuel Amann ¹ ; Anand Dessai ²

@article{CRMATH_2010__348_5-6_283_0,
     author = {Manuel Amann and Anand Dessai},
     title = {The $ \stackrel{{\textasciicircum}}{A}$-genus of $ {S}^{1}$-manifolds with finite second homotopy group},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {283--285},
     publisher = {Elsevier},
     volume = {348},
     number = {5-6},
     year = {2010},
     doi = {10.1016/j.crma.2010.01.011},
     language = {en},
}

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%A Anand Dessai
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Manuel Amann; Anand Dessai. The $ \stackrel{ˆ}{A}$-genus of $ {S}^{1}$-manifolds with finite second homotopy group. Comptes Rendus. Mathématique, Volume 348 (2010) no. 5-6, pp. 283-285. doi : 10.1016/j.crma.2010.01.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.01.011/

Bibliographie
Cité par

[1] M.F. Atiyah; F. Hirzebruch Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, 1970, pp. 18-28

[2] A. Borel; F. Hirzebruch Characteristic classes and homogeneous spaces. II, Amer. J. Math., Volume 81 (1959), pp. 315-382

[3] R. Bott; C. Taubes On the rigidity theorems of Witten, J. Amer. Math. Soc., Volume 2 (1989), pp. 137-186

[4] G.E. Bredon Representations at fixed points of smooth actions of compact groups, Ann. of Math. (2), Volume 89 (1969), pp. 515-532

[5] W. Browder Surgery on Simply Connected Manifolds, Springer, 1972

[6] 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), pp. 341-364

[7] H. Herrera, R. Herrera, Erratum to [6], in press

[8] F. Hirzebruch; P. Slodowy Elliptic genera, involutions and homogeneous spin manifolds, Geom. Dedicata, Volume 35 (1990), pp. 309-343

[9] S. Salamon Quaternion-Kähler geometry, Essays on Einstein Manifolds, Surveys in Differential Geometry, vol. VI, Int. Press, 1999, pp. 83-121

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