Torus actions, fixed-point formulas, elliptic genera and positive curvature

Abstract

We study fixed points of smooth torus actions on closed manifolds using fixed point formulas and equivariant elliptic genera. We also give applications to positively curved Riemannian manifolds with symmetry.

Key words： Torus actions; fixed point formulas; equivariant indices; elliptic genera; rigidity and vanishing

References

1 Allday C, Puppe V. Cohomological Methods in Transformation Groups. Cambridge Stud Adv Math, Vol 32. Cambridge: Cambridge Univ Press, 1993

2 Amann M, Kennard L. Topological properties of positively curved manifolds with symmetry. Geom Funct Anal, 2014, 24: 1377–1405

3 Atiyah M F. K-Theory. 2nd ed. Advanced Book Classics. Upper Saddle River: Addison-Wesley, 1989

4 Atiyah M F, Bott R. A Lefschetz fixed point formula for elliptic complexes. II. Ann of Math, 1967, 88: 451–491

5 Atiyah M F, Bott R. The moment map and equivariant cohomology. Topology, 1984, 23: 1–28

6 Atiyah M F, Bott R, Shapiro A. Clifford modules. Topology, 1964, 3(suppl 1): 3–38

7 Atiyah M F, Hirzebruch F. Riemann-Roch theorems for differentiable manifolds. Bull Amer Math Soc, 1959, 65: 276–281

8 Atiyah M F, Hirzebruch F. Vector bundles and homogeneous spaces. In: Proc Sympos Pure Math, Vol III. Providence: Amer Math Soc, 1961, 7–38

9 Atiyah M F, Hirzebruch F. Spin-Manifolds and group actions. In: Essays on Topology and Related Topics. Memoires dédiés à Georges de Rham. Berlin: Springer, 1970, 18–28

10 Atiyah M F, Segal G B. The index of elliptic operators: II. Ann of Math, 1968, 87: 531–545

11 Atiyah M F, Segal G B. Equivariant K-theory and completion. J Differential Geom, 1969, 3: 1–18

12 Atiyah M F, Singer I M. The index of elliptic operators: I. Ann of Math, 1968, 87: 484–530

13 Atiyah M F, Singer I M. The index of elliptic operators: III. Ann of Math, 1968, 87: 546–604

14 Berline N, Vergne M. Classes caractéristiques équivariantes. Formules de localization en cohomologie équivariante. C R Acad Sci Paris, 1982, 295: 539–541

15 Boardman J M. On manifolds with involution. Bull Amer Math Soc, 1967, 73: 136–138

16 Bochner S. Vector fields and Ricci curvature. Bull Amer Math Soc, 1946, 52: 776–797

17 Borel A. Seminar on Transformation Groups. Ann of Math Stud, No 46. Princeton: Princeton Univ Press, 1960

18 Bott R. Vector fields and characteristic numbers. Michigan Math J, 1967, 14: 231–244

19 Bott R. A residue formula for holomorphic vector-fields. J Differential Geom, 1967, 1: 311–330

20 Bott R, Taubes C H. On the rigidity theorems of Witten. J Amer Math Soc, 1989, 2: 137–186

21 Bredon G. Introduction to Compact Transformation Groups. New York: Academic Press, 1972

22 Conner P E, Floyd E E. Differentiable Periodic Maps. Ergebnisse Series, 33. Berlin: Springer, 1964

23 Conner P E, Floyd E E. Maps of odd period. Ann of Math, 1966, 84: 132–156

24 Dessai A. The Witten genus and S3-actions on manifolds. Preprint 94/6, Univ of Mainz, 1994

25 Dessai A. Rigidity Theorems for Spinc-Manifolds. Topology, 2000, 39: 239–258

26 Dessai A. Cyclic actions and elliptic genera. Preprint, arXiv: math/0104255

27 Dessai A. Obstructions to positive curvature and symmetry. Preprint, arXiv: math/0104256

28 Dessai A. Elliptic genera, positive curvature and symmetry. Habilitationsschrift, 2002

29 Dessai A. Obstructions to positive curvature and symmetry. Adv Math, 2007, 210: 560–577

30 Dessai A. Some geometric properties of the Witten genus. In: Proceedings of the Third Arolla Conference on Algebraic Topology August 18-24, 2008. Contemp Math, Vol 504. Providence: Amer Math Soc, 2009, 99–115

31 Dessai A. Preprint (in preparation)

32 Dessai A, Wiemeler M. Complete intersections with S1-action. Transform Groups (to appear)

33 tom Dieck T. Bordism of G-manifolds and integrality theorems. Topology, 1970, 9: 345–358

34 tom Dieck T. Transformation Groups. de Gruyter Stud Math, 8. Berlin: de Gruyter, 1987

35 Frankel T. Manifolds with positive curvature. Pacific J Math, 1961, 11: 165–174

36 Gromov M. Curvature, diameter and Betti numbers. Comment Math Helv, 1981, 56: 179–195

37 Guillemin VW, Sternberg S. Supersymmetry and Equivariant de Rham Theory. Berlin: Springer, 1999

38 Hattori A. Spinc-structures and S1-actions. Invent Math, 1978, 48: 7–31

39 Hattori A, Yoshida T. Lifting compact group actions in fiber bundles. Jpn J Math, 1976, 2: 13–25

40 Hirzebruch F. Involutionen auf Mannigfaltigkeiten. In: Proceedings of the Conference on Transformation Groups, New Orleans. 1968, 148–166

41 Hirzebruch F. Elliptic genera of level N for complex manifolds. In: Bleuler K, Werner M, eds. Differential Geometrical Methods in Theoretical Physics (Como 1987). NATO Adv Sci Inst Ser C: Math Phys Sci, 250. Armsterdam: Kluwer, 1988, 37–63

42 Hirzebruch F, Berger Th, Jung R. Manifolds and Modular Forms. Aspects Math, Vol E20. Wiesbaden: Friedr Vieweg, 1992

43 Hirzebruch F, Slodowy P. Elliptic genera, involutions and homogeneous spin-manifolds. Geom Dedicata, 1990, 35: 309–343

44 Hirzebruch F, Zagier D. The Atiyah-Singer Theorem and Elementary Number Theory. Math Lecture Series 1. Boston: Publish or Perish, 1974

45 Hopf H. Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Math Ann, 1926, 96: 225–250

46 Hsiang W Y. Cohomology Theory of Topological Transformation Groups. Ergeb Math Grenzgeb. Berlin: Springer, 1975

47 Jänich K, Ossa E. On the signature of an involution. Topology, 1969, 8: 27–30

48 Kawakubo K. The Theory of Transformation Groups. Oxford: Oxford Univ Press, 1992

49 Kennard L. On the Hopf conjecture with symmetry. Geom Topol, 2013, 17: 563–593

50 Kobayashi S. Transformation Groups in Differential Geometry. Ergeb Math Grenzgeb. Berlin: Springer, 1972

51 Kosniowski C. Applications of the holomorphic Lefschetz formula. Bull Lond Math Soc, 1970, 2: 43–48

52 Kosniowski C. Fixed points and group actions. In: Algebraic Topology, Proc Conf, Aarhus 1982. Lecture Notes in Math, Vol 1051. Berlin: Springer, 1984, 603–609

53 Kosniowski C, Stong R E. Involutions and characteristic numbers. Topology, 1978, 17: 309–330

54 Kriˇcever I M. Formal groups and the Atiyah-Hirzebruch formula. Math USSR Investija, 1974, 6: 1271–1285

55 Kriˇcever I M. Obstructions to the existence of S1-actions. Bordism of ramified coverings. Math USSR Investija, 1977, 10: 783–797

56 Landweber P S, ed. Elliptic Curves and Modular Forms in Algebraic Topology. Proceedings Princeton 1986. Lecture Notes in Math, Vol 1326. Berlin: Springer, 1988

57 Lashof R. Poincaré duality and cobordism. Trans Amer Math Soc, 1963, 109: 257–277

58 Lawson H B, MichelsohnM-L. Spin Geometry. Princeton Math Ser 38. Princeton: Princeton Univ Press, 1989

59 Lichnerowicz A. Spineurs harmoniques. C R Acad Sci, 1963, 257: 7–9

60 Liu K. On modular invariance and rigidity theorems. J Differential Geom, 1995, 41: 343–396

61 Lusztig G. Remarks on the holomorphic Lefschetz numbers. In: Analyse globale. Śem Math Sup´erieures, No 42. Montŕeal: Presses Univ Montŕeal, 1971, 193–204

62 Milnor J.On the cobordism ring Ω∗ and a complex analogue. Part I. Amer J Math, 1960, 82: 505–521

63 Milnor J, Stasheff J. Characteristic classes. Ann of Math Stud, Vol 76. Princeton: Princeton Univ Press, 1974

64 Musin O R. On rigid Hirzebruch genera. Mosc Math J, 2011, 11: 139–147

65 Novikov S P. Some problems in the topology of manifolds connected with the theory of Thom spaces. Soviet Math Dokl, 1960, 1: 717–720

66 Petrie T. Smooth S1-actions on homotopy complex projective spaces and related topics. Bull Amer Math Soc, 1972, 78: 105–153

67 Quillen D. The spectrum of an equivariant cohomology ring. I. Ann of Math, 1971, 98: 549–571

68 Quillen D. The spectrum of an equivariant cohomology ring. II. Ann of Math, 1971, 98: 573–602

69 Segal G B. Equivariant K-theory. Publ Math Inst Hautes Études Sci, 1968, 34: 129–151

70 Stewart T E. Lifting group actions in fibre bundles. Ann of Math, 1961, 74: 192–198

71 Stolz St.A conjecture concerning positive Ricci curvature and the Witten genus. Math Ann, 1996, 304: 785–800

72 Su J C. Transformation groups on cohomology projective spaces. Trans Amer Math Soc, 1963, 106: 305–318

73 Switzer R M. Algebraic Topology—Homotopy and Homology. Grundlehren MathWiss, 212. Berlin: Springer, 1975

74 Taubes C H. S1 Actions and elliptic genera. Comm Math Phys, 1989, 122: 455–526

75 Thom R. Espaces fibrés en sphères et carrés de Steenrod. Ann Sci Éc Norm Supér, III Sér, 1952, 69: 109–182

76 Thom R. Quelques propriétés globales des variétés différentiables. Comment Math Helv, 1954, 28: 17–86

77 Tu L W. What is ... equivariant cohomology? Notices Amer Math Soc, 2011, 58: 423–426

78 Weisskopf N. Positive curvature and the elliptic genus. Preprint, arXiv: 1305.5175

79 Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259–297

80 Wilking B. Nonnegatively and positively curved manifolds. In: Metric and Comparison Geometry. Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 25–62

81 Witten E. The index of the Dirac operator in loop space. In: Lecture Notes in Math, Vol 1326. Berlin: Springer, 1988, 161–181

82 Ziller W. Examples of Riemannian manifolds with non-negative sectional curvature. In: Metric and Comparison Geometry. Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 63–102