Advertisement

Geometriae Dedicata

, Volume 35, Issue 1–3, pp 309–343 | Cite as

Elliptic genera, involutions, and homogeneous spin manifolds

  • F. Hirzebruch
  • P. Slodowy
Article

Abstract

We study the normalized elliptic genera Φ(X)=ϕ(X)/εk/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), GH compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

Keywords

General Formula Basic Tool Modular Function Elliptic Genus Spin Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Proc. Conf. on Elliptic Curves and Modular Forms in Algebraic Topology, Princeton, September 1986; Lecture Notes in Maths 1326, Springer-Verlag, Berlin, Heidelberg, New York, 1988.Google Scholar
  2. 2.
    Adams, J. F., Lectures on Lie Groups, Benjamin, New York, 1969.Google Scholar
  3. 3.
    Atiyah, M. F., and Hirzebruch, F., ‘Spin-manifolds and group actions’, in Memoires dédiés à Georges de Rham, Springer-Verlag, Berlin, Heidelberg, New York, 1970, pp. 18–28.Google Scholar
  4. 4.
    Atiyah, M. F., and Singer, I. M., ‘The index of elliptic operators III’, Ann. Math. 87 (1996), 546–604.Google Scholar
  5. 5.
    Borel, A. and Hirzebruch, F., ‘Characteristic classes and homogeneous spaces I’, Amer. J. Math. 80 (1958), 458–538.Google Scholar
  6. 6.
    Borel, A. and Hirzebruch, F., ‘Characteristic classes and homogeneous spaces II’, Amer. J. Math. 81 (1959), 315–382.Google Scholar
  7. 7.
    Bliss, J. G., Moody, R. V., and Pianzola, A., Appendix to this article, Geom. Dedicata 35 (1990), 345–351 (this issue).Google Scholar
  8. 8.
    Borel, A. and de Siebenthal, J., ‘Les sous-groupes fermés de rang maximum des groupes de Lie clos’, Comment Math. Helv. 23 (1949), 200–221.Google Scholar
  9. 9.
    Bott, R., ‘Vector fields and characteristic numbers’, Michigan Math. J. 14 (1967), 231–244.Google Scholar
  10. 10.
    Bott, R. and Taubes, C., ‘On the rigidity theorem of Witten’, J. Amer. Math. Soc. 2 (1989), 137–186.Google Scholar
  11. 11.
    Bourbaki, N., Groupes et algèbres de Lie, Chap. IV, V, VI. Hermann, Paris, 1968.Google Scholar
  12. 12.
    Golubitsky, M. and Guillemin, V., Stable Mappings and Their Singularities, Springer-Verlag, Berlin, Heidelberg, New York, 1973.Google Scholar
  13. 13.
    Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.Google Scholar
  14. 14.
    Hirzebruch, F., Topological Methods in Algebraic Geometry (3rd edn, with appendices by R. L. E. Schwarzenberger and A. Borel), Springer-Verlag, Berlin, Heidelberg, New York, 1966.Google Scholar
  15. 15.
    Hirzebruch, F., ‘Involutionen auf Mannigfaltigkeiten’ in Proc. Conf. on Transformation Groups, New Orleans 1967, Springer-Verlag, Berlin, Heidelberg, New York, 1968, pp. 148–166.Google Scholar
  16. 16.
    Hirzebruch, F., ‘Elliptic genera of level N for complex manifolds’, in Differential Geometrical Methods in Theoretical Physics, Kluwer, Dordrecht, 1988, pp. 37–63.Google Scholar
  17. 17.
    Hopf, H. and Samelson, H., ‘Ein Satz über die Wirkungsräume geschlossener Liescher Gruppen’, Comment. Math. Helv. 13 (1940/41), 240–251.Google Scholar
  18. 18.
    Jänich, K. and Ossa, E., ‘On the signature of an involution’, Topology 8 (1969), 27–30.Google Scholar
  19. 19.
    Kleiman, S. L., ‘The transversality of a general translate’, Comp. Math. 38 (1974), 287–297.Google Scholar
  20. 20.
    Landweber, P. S., ‘Elliptic genera: An introductory overview’, in reference [1], pp. 1–10.Google Scholar
  21. 21.
    Milnor, J., ‘Spin structures on manifolds’, L'Enseignement Math. 9 (1963), 198–203.Google Scholar
  22. 22.
    Ochanine, S., ‘Sur les genres multiplicatifs définis par des intégrales elliptiques’, Topology 26 (1987), 143–151.Google Scholar
  23. 23.
    Ochanine, S., ‘Genres elliptiques equivariants’, in [1], p. 107–122.Google Scholar
  24. 24.
    Selbach, M., Klassifikationstheorie halbeinfacher algebraischer Gruppen, Bonner Mathematische Schriften 83, Bonn, 1976.Google Scholar
  25. 25.
    Springer, T. A., ‘The classification of involutions of simple algebraic groups’, Journal Fac. Sci. University of Tokyo, Sec. IA, 34 (1987), 655–670.Google Scholar
  26. 26.
    Tits, J., ‘Sur la classification des groupes algébriques simples’, C.R.A.S. Paris 249 (1959), 1438–1440.Google Scholar
  27. 27.
    Tits, J., ‘Classification of algebraic semisimple groups’, Proc. Symp. Pure Math. 9 (1966), 33–62.Google Scholar
  28. 28.
    Witten, E., ‘The index of the Dirac operator in loop space’, in [1], p. 161–181.Google Scholar
  29. 29.
    Zagier, D., ‘Note on the Landweber-Stong Elliptic genus’, in [1], p. 216–224.Google Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • F. Hirzebruch
    • 1
  • P. Slodowy
    • 2
  1. 1.Max-Planck-Institut für MathematikBonn 3Germany
  2. 2.Mathematisches Institut BUniversität StuttgartStuttgart 80Germany

Personalised recommendations