Skip to main content
Log in

Rigidity of higher elliptic genera

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We prove some general rigidity theorems for both elliptic and higher elliptic genera under a natural condition on the first equivariant Pontrjagin classes. We also obtain the vanishing of some higher elliptic genera.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Atiyah, M.; Bott, R.: The moment map and equivariant cohomology. Topology 23 (1984), 1–28.

    Google Scholar 

  2. Atiyah, M.; Singer, I.: The index of elliptic operators III. Ann. Math. 87 (1968), 546–604.

    Google Scholar 

  3. Atiyah, M.; Hirzebruch, F.: Spin manifolds and group actions. In: Essays on Topology and Related Topics. Springer, NY, 1968, 18–28.

    Google Scholar 

  4. Bott, R.; Taubes, C.: On the rigidity theorems of Witten. J. Am. Math. Soc. 2 (1989), 137–186.

    Google Scholar 

  5. Browder, W.; Hsiang, W.C.: G-actions and the fundamental group. Invent. Math. 65 (1981), 411–424.

    Google Scholar 

  6. Brylinski, J.-L.: Representation of loop groups, Dirac operators on loop spaces and modular forms. Topology 29 (1990), 461–480.

    Google Scholar 

  7. Connes, A.; Gromov, M.; Moscovici, H.: Conjecture de Novikov et fibrés presque plats. C. R. Acad. Sci., Paris, Sér. I, 310 (1990), 273–277.

    Google Scholar 

  8. Gong, D.: L 2-analytic torsions, equivariant cyclic cohomology and the Novikov conjecture. Thesis, SUNY at Stony Brook, 1992.

  9. Gong, D.: Equivariant Novikov conjecture for groups acting on Euclidean buildings. Submitted 1995.

  10. Gromov, M.; Lawson, B.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math., Inst. Hautes Étud. Sci. 58 (1983), 295–408.

    Google Scholar 

  11. Hirzebruch, F.: Manifolds and Modular Forms. Aspects of Math., Vieweg, Bonn 1992.

    Google Scholar 

  12. Kac, V.G.: Infinite-dimensional Lie Algebras. Cambridge Univ. Press, London 1991.

    Google Scholar 

  13. Kahn, P.: Characteristic numbers and oriented homotopy type. Topology 3 (1965), 81–95.

    Google Scholar 

  14. Kreck, M.; Stolz, S.: HP 2-bundles and elliptic homology. Acta Math. 171 (1993), 231–261.

    Google Scholar 

  15. Krichever, I.: Generalized elliptic genera and Baker-Akhiezer functions. Math. Notes 47 (1990), 132–142.

    Google Scholar 

  16. Landweber, P.S.; Stong, R.: Circle actions on spin manifolds and characteristic numbers. Topology 27 (1988), 145–162.

    Google Scholar 

  17. Lawson, B.; Michelsohn, M.-L.: Spin Geometry. Princeton Univ. Press, NJ, 1989.

    Google Scholar 

  18. Liu, K.: On modular invariance and rigidity theorem. J. Differ. Geom. 41 (1995), 343–396.

    Google Scholar 

  19. Liu, K.: On SL(2,% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSijHikaaa!3764!\[\mathbb{Z}\]) and topology. Math. Res. Letter. 1 (1994), 53–64.

    Google Scholar 

  20. Liu, K.: On elliptic genera and theta-functions. To appear in: Topology.

  21. Liu, K.: On mod 2 and higher elliptic genera. Commun. Math. Phys. 149 (1992), 71–97.

    Google Scholar 

  22. Miščenko, A.S.: Hermitian K-theory, the theory of characteristic classes and method of functional analysis. Russ. Math. Surv. 31 (1976), 71–138.

    Google Scholar 

  23. Ochanine, S.: Generes elliptiques equivariants. In: P. Landweber (Ed.): Elliptic curves and modular forms in algebraic topology. Lect. Notes Math. 1326, Springer, NY, 1988, 107–122.

    Google Scholar 

  24. Pressley, A.; Segal, G.: Loop Groups. Oxford Univ. Press, London 1986.

    Google Scholar 

  25. Rosenberg, J.: C *-algebras, positive scalar curvature, and the Novikov conjecture. Publ. Math., Inst. Hautes Étud. Sci. 58 (1983), 409–424.

    Google Scholar 

  26. Taubes, C.: S 1-actions and elliptic genera. Commun. Math. Phys. 122 (1989), 455–526.

    Google Scholar 

  27. Witten, E.: The index of Dirac operator in loop space. In: P. Landweber (Ed.): Elliptic curves and modular forms in algebraic topology. Lect. Notes Math. 1326, Springer, NY, 1988, 161–181.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by A.S. Miščenko

Both authors are supported in part by NFS

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gong, D., Liu, K. Rigidity of higher elliptic genera. Ann Glob Anal Geom 14, 219–236 (1996). https://doi.org/10.1007/BF00054471

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00054471

Key words

MSC 1991

Navigation