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.
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References
Allday C, Puppe V. Cohomological Methods in Transformation Groups. Cambridge Stud Adv Math, Vol 32. Cambridge: Cambridge Univ Press, 1993
Amann M, Kennard L. Topological properties of positively curved manifolds with symmetry. Geom Funct Anal, 2014, 24: 1377–1405
Atiyah M F. K-Theory. 2nd ed. Advanced Book Classics. Upper Saddle River: Addison-Wesley, 1989
Atiyah M F, Bott R. A Lefschetz fixed point formula for elliptic complexes. II. Ann of Math, 1967, 88: 451–491
Atiyah M F, Bott R. The moment map and equivariant cohomology. Topology, 1984, 23: 1–28
Atiyah M F, Bott R, Shapiro A. Clifford modules. Topology, 1964, 3(suppl 1): 3–38
Atiyah M F, Hirzebruch F. Riemann-Roch theorems for differentiable manifolds. Bull Amer Math Soc, 1959, 65: 276–281
Atiyah M F, Hirzebruch F. Vector bundles and homogeneous spaces. In: Proc Sympos Pure Math, Vol III. Providence: Amer Math Soc, 1961, 7–38
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
Atiyah M F, Segal G B. The index of elliptic operators: II. Ann of Math, 1968, 87: 531–545
Atiyah M F, Segal G B. Equivariant K-theory and completion. J Differential Geom, 1969, 3: 1–18
Atiyah M F, Singer I M. The index of elliptic operators: I. Ann of Math, 1968, 87: 484–530
Atiyah M F, Singer I M. The index of elliptic operators: III. Ann of Math, 1968, 87: 546–604
Berline N, Vergne M. Classes caractéristiques équivariantes. Formules de localisation en cohomologie équivariante. C R Acad Sci Paris, 1982, 295: 539–541
Boardman J M. On manifolds with involution. Bull Amer Math Soc, 1967, 73: 136–138
Bochner S. Vector fields and Ricci curvature. Bull Amer Math Soc, 1946, 52: 776–797
Borel A. Seminar on Transformation Groups. Ann of Math Stud, No 46. Princeton: Princeton Univ Press, 1960
Bott R. Vector fields and characteristic numbers. Michigan Math J, 1967, 14: 231–244
Bott R. A residue formula for holomorphic vector-fields. J Differential Geom, 1967, 1: 311–330
Bott R, Taubes C H. On the rigidity theorems of Witten. J Amer Math Soc, 1989, 2: 137–186
Bredon G. Introduction to Compact Transformation Groups. New York: Academic Press, 1972
Conner P E, Floyd E E. Differentiable Periodic Maps. Ergebnisse Series, 33. Berlin: Springer, 1964
Conner P E, Floyd E E. Maps of odd period. Ann of Math, 1966, 84: 132–156
Dessai A. The Witten genus and S 3-actions on manifolds. Preprint 94/6, Univ of Mainz, 1994
Dessai A. Rigidity Theorems for Spin c-Manifolds. Topology, 2000, 39: 239–258
Dessai A. Cyclic actions and elliptic genera. Preprint, arXiv: math/0104255
Dessai A. Obstructions to positive curvature and symmetry. Preprint, arXiv: math/0104256
Dessai A. Elliptic genera, positive curvature and symmetry. Habilitationsschrift, 2002
Dessai A. Obstructions to positive curvature and symmetry. Adv Math, 2007, 210: 560–577
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
Dessai A. Preprint (in preparation)
Dessai A, Wiemeler M. Complete intersections with S 1-action. Transform Groups (to appear)
tom Dieck T. Bordism of G-manifolds and integrality theorems. Topology, 1970, 9: 345–358
tom Dieck T. Transformation Groups. de Gruyter Stud Math, 8. Berlin: de Gruyter, 1987
Frankel T. Manifolds with positive curvature. Pacific J Math, 1961, 11: 165–174
Gromov M. Curvature, diameter and Betti numbers. Comment Math Helv, 1981, 56: 179–195
Guillemin VW, Sternberg S. Supersymmetry and Equivariant de Rham Theory. Berlin: Springer, 1999
Hattori A. Spinc-structures and S 1-actions. Invent Math, 1978, 48: 7–31
Hattori A, Yoshida T. Lifting compact group actions in fiber bundles. Jpn J Math, 1976, 2: 13–25
Hirzebruch F. Involutionen auf Mannigfaltigkeiten. In: Proceedings of the Conference on Transformation Groups, New Orleans. 1968, 148–166
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
Hirzebruch F, Berger Th, Jung R. Manifolds and Modular Forms. Aspects Math, Vol E20. Wiesbaden: Friedr Vieweg, 1992
Hirzebruch F, Slodowy P. Elliptic genera, involutions and homogeneous spin-manifolds. Geom Dedicata, 1990, 35: 309–343
Hirzebruch F, Zagier D. The Atiyah-Singer Theorem and Elementary Number Theory. Math Lecture Series 1. Boston: Publish or Perish, 1974
Hopf H. Vektorfelder in n-dimensionalen Mannigfaltigkeiten. Math Ann, 1926, 96: 225–250
Hsiang W Y. Cohomology Theory of Topological Transformation Groups. Ergeb Math Grenzgeb. Berlin: Springer, 1975
Jänich K, Ossa E. On the signature of an involution. Topology, 1969, 8: 27–30
Kawakubo K. The Theory of Transformation Groups. Oxford: Oxford Univ Press, 1992
Kennard L. On the Hopf conjecture with symmetry. Geom Topol, 2013, 17: 563–593
Kobayashi S. Transformation Groups in Differential Geometry. Ergeb Math Grenzgeb. Berlin: Springer, 1972
Kosniowski C. Applications of the holomorphic Lefschetz formula. Bull Lond Math Soc, 1970, 2: 43–48
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
Kosniowski C, Stong R E. Involutions and characteristic numbers. Topology, 1978, 17: 309–330
Kričever I M. Formal groups and the Atiyah-Hirzebruch formula. Math USSR Investija, 1974, 6: 1271–1285
Kričever I M. Obstructions to the existence of S 1-actions. Bordism of ramified coverings. Math USSR Investija, 1977, 10: 783–797
Landweber P S, ed. Elliptic Curves and Modular Forms in Algebraic Topology. Proceedings Princeton 1986. Lecture Notes in Math, Vol 1326. Berlin: Springer, 1988
Lashof R. Poincaré duality and cobordism. Trans Amer Math Soc, 1963, 109: 257–277
Lawson H B, Michelsohn M -L. Spin Geometry. Princeton Math Ser 38. Princeton: Princeton Univ Press, 1989
Lichnerowicz A. Spineurs harmoniques. C R Acad Sci, 1963, 257: 7–9
Liu K. On modular invariance and rigidity theorems. J Differential Geom, 1995, 41: 343–396
Lusztig G. Remarks on the holomorphic Lefschetz numbers. In: Analyse globale. Sém Math Supérieures, No 42. Montréal: Presses Univ Montréal, 1971, 193–204
Milnor J. On the cobordism ring Ω* and a complex analogue. Part I. Amer J Math, 1960, 82: 505–521
Milnor J, Stasheff J. Characteristic classes. Ann of Math Stud, Vol 76. Princeton: Princeton Univ Press, 1974
Musin O R. On rigid Hirzebruch genera. Mosc Math J, 2011, 11: 139–147
Novikov S P. Some problems in the topology of manifolds connected with the theory of Thom spaces. Soviet Math Dokl, 1960, 1: 717–720
Petrie T. Smooth S 1-actions on homotopy complex projective spaces and related topics. Bull Amer Math Soc, 1972, 78: 105–153
Quillen D. The spectrum of an equivariant cohomology ring. I. Ann of Math, 1971, 98: 549–571
Quillen D. The spectrum of an equivariant cohomology ring. II. Ann of Math, 1971, 98: 573–602
Segal G B. Equivariant K-theory. Publ Math Inst Hautes Études Sci, 1968, 34: 129–151
Stewart T E. Lifting group actions in fibre bundles. Ann of Math, 1961, 74: 192–198
Stolz St. A conjecture concerning positive Ricci curvature and the Witten genus. Math Ann, 1996, 304: 785–800
Su J C. Transformation groups on cohomology projective spaces. Trans Amer Math Soc, 1963, 106: 305–318
Switzer R M. Algebraic Topology—Homotopy and Homology. Grundlehren MathWiss, 212. Berlin: Springer, 1975
Taubes C H. S 1 Actions and elliptic genera. Comm Math Phys, 1989, 122: 455–526
Thom R. Espaces fibrés en sphères et carrés de Steenrod. Ann Sci Éc Norm Supér, III Sér, 1952, 69: 109–182
Thom R. Quelques propriétés globales des variétés différentiables. Comment Math Helv, 1954, 28: 17–86
Tu L W. What is... equivariant cohomology? Notices Amer Math Soc, 2011, 58: 423–426
Weisskopf N. Positive curvature and the elliptic genus. Preprint, arXiv: 1305.5175
Wilking B. Torus actions on manifolds of positive sectional curvature. Acta Math, 2003, 191: 259–297
Wilking B. Nonnegatively and positively curved manifolds. In: Metric and Comparison Geometry. Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 25–62
Witten E. The index of the Dirac operator in loop space. In: Lecture Notes in Math, Vol 1326. Berlin: Springer, 1988, 161–181
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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Dessai, A. Torus actions, fixed-point formulas, elliptic genera and positive curvature. Front. Math. China 11, 1151–1187 (2016). https://doi.org/10.1007/s11464-016-0583-2
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DOI: https://doi.org/10.1007/s11464-016-0583-2