Comptes Rendus
no. 4
A result on the 𝐀 ^ and elliptic genera on non-spin manifolds with circle actions
[Un résultat sur les variétés non-spinorielles de genres A ^ et elliptique munies d'actions de S1]
Haydeé Herrera1 ; Rafael Herrera2
1 Department of Mathematics, Tufts University, Medford, MA 02155, USA
2 Department of Mathematics, University of California, Riverside, CA 92521, USA
Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 371-374.

On montre que le A ^-genre d'une variété lisse, compacte munie d'un second groupe d'homotopie fini et dotée d'une action de S1 est égal à zéro. Ces variétés ne sont pas nécessairement spinorielles de sorte que ce théorème d'annulation étend le résultat d'Atiyah–Hirzebruch établi pour des variétés spinorielles avec actions de S1. La démonstration est faite à partir d'un théorème de rigidité sous des actions de S1 de genre elliptique sur ces variétés.

We prove the vanishing of the A ^-genus of compact smooth manifolds with finite second homotopy group and endowed with smooth S1 actions. These manifolds are not necessarily spin, hence, this vanishing extends that of Atiyah and Hirzebruch on spin manifolds with S1 actions. The proof is accomplished by proving a rigidity theorem under circle actions of the elliptic genus on these manifolds.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/S1631-073X(02)02480-9
Haydeé Herrera 1 ; Rafael Herrera 2

1 Department of Mathematics, Tufts University, Medford, MA 02155, USA
2 Department of Mathematics, University of California, Riverside, CA 92521, USA 
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Haydeé Herrera; Rafael Herrera. A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions. Comptes Rendus. Mathématique, Volume 335 (2002) no. 4, pp. 371-374. doi : 10.1016/S1631-073X(02)02480-9. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02480-9/

[1] M.F. Atiyah; F. Hirzebruch Spin manifolds and group actions, Essays in Topology and Related Subjects, Springer-Verlag, Berlin, 1970, pp. 18-28

[2] R. Bott; T. Taubes On the rigidity theorems of Witten, J. Amer. Math. Soc., Volume 2 (1989) no. 1, pp. 137-186

[3] G.E. Bredon Representations at fixed points of smooth actions of compact groups, Ann. of Math., Volume 89 (1969) no. 2, pp. 515-532

[4] H. Herrera, R. Herrera, A ^-genus on non-spin manifolds with S1 actions and the classification of positive quaternion-Kähler 12-manifolds, IHÉS Preprint, 2001

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

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

[7] C.R. LeBrun; S.M. Salamon Strong rigidity of positive quaternion-Kähler manifolds, Invent. Math., Volume 118 (1994), pp. 109-132

[8] S. Ochanine Sur les genres multiplicatifs définis par des intégrales elliptiques, Topology, Volume 26 (1987), pp. 143-151

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

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