A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions

Herrera, Haydeé; Herrera, Rafael

doi:10.1016/S1631-073X(02)02480-9

A result on the

\hat{𝐀}

and elliptic genera on non-spin manifolds with circle actions
[Un résultat sur les variétés non-spinorielles de genres

\hat{A}

et elliptique munies d'actions de S¹]

Herrera, Haydeé ¹ ; Herrera, Rafael ²

¹ Department of Mathematics, Tufts University, Medford, MA 02155, USA
² Department of Mathematics, University of California, Riverside, CA 92521, USA

Comptes Rendus. Mathématique, Tome 335 (2002) no. 4, pp. 371-374.

Résumé
Abstract

On montre que le $\hat{A}$ -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 $\hat{A}$ -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 : 2002-04-12
Accepté le : 2002-05-27
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02480-9

Affiliations des auteurs :

Herrera, Haydeé ¹ ; Herrera, Rafael ²

¹ Department of Mathematics, Tufts University, Medford, MA 02155, USA
² Department of Mathematics, University of California, Riverside, CA 92521, USA

@article{CRMATH_2002__335_4_371_0,
     author = {Herrera, Hayde\'e and Herrera, Rafael},
     title = {A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {371--374},
     publisher = {Elsevier},
     volume = {335},
     number = {4},
     year = {2002},
     doi = {10.1016/S1631-073X(02)02480-9},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/S1631-073X(02)02480-9/}
}

TY  - JOUR
AU  - Herrera, Haydeé
AU  - Herrera, Rafael
TI  - A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions
JO  - Comptes Rendus. Mathématique
PY  - 2002
SP  - 371
EP  - 374
VL  - 335
IS  - 4
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/S1631-073X(02)02480-9/
DO  - 10.1016/S1631-073X(02)02480-9
LA  - en
ID  - CRMATH_2002__335_4_371_0
ER  -

%0 Journal Article
%A Herrera, Haydeé
%A Herrera, Rafael
%T A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions
%J Comptes Rendus. Mathématique
%D 2002
%P 371-374
%V 335
%N 4
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/S1631-073X(02)02480-9/
%R 10.1016/S1631-073X(02)02480-9
%G en
%F CRMATH_2002__335_4_371_0

Herrera, Haydeé; Herrera, Rafael. A result on the $ \hat{\mathbf{A}}$ and elliptic genera on non-spin manifolds with circle actions. Comptes Rendus. Mathématique, Tome 335 (2002) no. 4, pp. 371-374. doi : 10.1016/S1631-073X(02)02480-9. http://archive.numdam.org/articles/10.1016/S1631-073X(02)02480-9/

Bibliographie
Cité par

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

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

[3] Bredon, G.E. 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, $\hat{A}$ -genus on non-spin manifolds with S¹ actions and the classification of positive quaternion-Kähler 12-manifolds, IHÉS Preprint, 2001

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

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

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

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

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

Cité par Sources :