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Characteristic numbers of positively curved spin-manifolds with symmetry


Author: Anand Dessai
Journal: Proc. Amer. Math. Soc. 133 (2005), 3657-3661
MSC (2000): Primary 53C20; Secondary 58J26
DOI: https://doi.org/10.1090/S0002-9939-05-07928-1
Published electronically: June 6, 2005
MathSciNet review: 2163604
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $M$ be a $Spin$-manifold of positive sectional curvature and dimension $>8$. Suppose a compact connected Lie group $G$ acts smoothly on $M$. We show that the characteristic number $\hat A(M,TM)$ vanishes if $G$contains two commuting involutions acting isometrically on $M$.


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  • 1. M.F. Atiyah and F. Hirzebruch, Spin-Manifolds and Group Actions, in: Essays on Topology and Related Topics. Georges de Rham, Springer (1970), 18-28 MR 0278334 (43:4064)
  • 2. M.F. Atiyah and I.M. Singer, The index of elliptic operators. I, Ann. of Math. 87 (1968), 484-530 MR 0236950 (38:5243)
  • 3. M.F. Atiyah and I.M. Singer, The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604 MR 0236952 (38:5245)
  • 4. R. Bott and C.H. Taubes, On the rigidity theorems of Witten, J. of Amer. Math. Soc. 2 (1989), 137-186 MR 0954493 (89k:58270)
  • 5. G. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46. Academic Press (1972) MR 0413144 (54:1265)
  • 6. A. Dessai, Obstructions to positive curvature and symmetry, preprint, available at the arXiv: http://arxiv.org/abs/math.DG/0104256
  • 7. A. Dessai, Cyclic actions and elliptic genera, preprint, available at the arXiv: http://arxiv.org/abs/math.GT/0104255
  • 8. T. Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165-174 MR 0123272 (23:A600)
  • 9. M. Gromov, Curvature, diameter and Betti numbers, Comment. Math. Helvetici 56 (1981), 179-195 MR 0630949 (82k:53062)
  • 10. M. Gromov and H.B. Lawson, The classification of simply-connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980), 209-230 MR 0577131 (81h:53036)
  • 11. F. Hirzebruch and P. Slodowy, Elliptic Genera, Involutions and Homogeneous Spin Manifolds, Geom. Dedicata 35 (1990), 309-343 MR 1066570 (92a:57028)
  • 12. A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9 MR 0156292 (27:6218)
  • 13. B. Wilking, Torus actions on manifolds of positive sectional curvature, Acta Math. 191 (2003), 259-297 MR 2051400

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Additional Information

Anand Dessai
Affiliation: Department of Mathematics, University of Münster, D-48149 Münster, Germany
Email: dessai@math.uni-muenster.de

DOI: https://doi.org/10.1090/S0002-9939-05-07928-1
Keywords: Positive curvature, equivariant index theory, elliptic genera
Received by editor(s): October 24, 2003
Received by editor(s) in revised form: July 8, 2004
Published electronically: June 6, 2005
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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