$ {\mathrm{Spin}}^{q}$ manifolds and $ {S}^{1}$ actions

Haydeé Herrera; Rafael Herrera

doi:10.1016/j.crma.2007.05.019

Topology

{Spin}^{q}

manifolds and

S^{1}

actions

Presented by: Étienne Ghys

Haydeé Herrera ¹; Rafael Herrera ²

¹ Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA
² Centro de Investigación en Matemáticas, A. P. 402, Guanajuato, Gto., C.P. 36000, Mexico

Comptes Rendus. Mathématique, Volume 345 (2007) no. 1, pp. 35-38.

Abstract
Résumé

We prove a vanishing theorem for ${Spin}^{q}$ manifolds admitting $S^{1}$ actions, generalizing those of Atiyah and Hirzebruch for Spin manifolds and Hattori for ${Spin}^{c}$ manifolds. We also prove a vanishing theorem for almost quaternionic manifolds with compatible circle actions.

On montre un théorème d'annulation pour les variétés ${Spin}^{q}$ qui admettent des actions de $S^{1}$ , ce qui généralise le théorème d'Atiyah et de Hirzebruch pour les variétés de Spin et celui de Hattori pour les variétés ${Spin}^{c}$ . De plus, on montre un théorème d'annulation pour les variétés presque quaternionienne qui admettent des actions de $S^{1}$ compatibles.

Received: 2007-03-23
Accepted: 2007-05-24
Published online: 2007-06-27

DOI: 10.1016/j.crma.2007.05.019

Author's affiliations:

Haydeé Herrera ¹; Rafael Herrera ²

¹ Department of Mathematical Sciences, Rutgers University, Camden, NJ 08102, USA
² Centro de Investigación en Matemáticas, A. P. 402, Guanajuato, Gto., C.P. 36000, Mexico

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     title = {$ {\mathrm{Spin}}^{q}$ manifolds and $ {S}^{1}$ actions},
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     pages = {35--38},
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     language = {en},
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Haydeé Herrera; Rafael Herrera. $ {\mathrm{Spin}}^{q}$ manifolds and $ {S}^{1}$ actions. Comptes Rendus. Mathématique, Volume 345 (2007) no. 1, pp. 35-38. doi : 10.1016/j.crma.2007.05.019. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.05.019/

References
Cited by

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

[2] M.F. Atiyah; I.M. Singer The index of elliptic operators. III, Ann. of Math. (2), Volume 87 (1968), pp. 546-604

[3] W. Fulton; S. Lang Riemann–Roch Algebra, Fundamental Principles of Mathematical Sciences, vol. 277, Springer-Verlag, New York, 1985 (x+203 pp)

[4] A. Hattori ${Spin}^{c}$ -structures and $S^{1}$ -actions, Invent. Math., Volume 48 (1978) no. 1, pp. 7-31

[5] H.B. Lawson; M. Michelsohn Spin Geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989 (xii+427 pp)

[6] M. Nagase ${Spin}^{q}$ structures, J. Math. Soc. Japan, Volume 47 (1995) no. 1, pp. 93-119

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