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.

Keywords

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

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