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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Vector bundles with infinitely many souls


Author: Igor Belegradek
Journal: Proc. Amer. Math. Soc. 131 (2003), 2217-2221
MSC (2000): Primary 53C20
Published electronically: February 20, 2003
MathSciNet review: 1963770
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Abstract: We construct the first examples of manifolds, the simplest one being $S^3\times S^4\times\mathbb{R} ^5$, which admit infinitely many complete nonnegatively curved metrics with pairwise nonhomeomorphic souls.


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  • [CE00] D. Crowley and C. M. Escher, A classification of $S^3$-bundles over $S^4$, to appear in Differential Geom. Appl.
  • [CG72] J. Cheeger and D. Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. MR 46:8121
  • [Coh73] M. M. Cohen, A course in simple-homotopy theory, Springer-Verlag, 1973. MR 50:14762
  • [GM74] D. Gromoll and W. Meyer, An exotic sphere with nonnegative sectional curvature, Ann. of Math. 100 (1974), 401-406. MR 51:11347
  • [GT01] D. Gromoll and K. Tapp, Nonnegatively curved metrics on $S^2\times\mathbb R^2$, to appear in Geom. Dedicata.
  • [GZ00] K. Grove and W. Ziller, Curvature and symmetry of Milnor spheres, Ann. Math. 151 (2000), 1-36. MR 2000i:53047
  • [Hae61] A. Haefliger, Plongements différentiables de variétés dans variétés, Comment. Math. Helv. 36 (1961), 47-82.
  • [Kam77] B. N. P. Kamerich, Transitive transformation groups of products of two spheres, Ph.D. thesis, Catholic University of Nijmegen, 1977.
  • [Mil56] J. Milnor, On manifolds homeomorphic to the $7$-sphere, Ann. of Math. 64 (1956), 399-405. MR 18:498d
  • [Mim95] M. Mimura, Homotopy theory of Lie groups, Handbook of algebraic topology, North-Holland, 1995, pp. 951-991. MR 97c:57038
  • [MM79] I. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, 1979. MR 81b:57014
  • [Oni94] A. L. Onishchik, Topology of transitive transformation groups, Johann Ambrosius Barth Verlag GmbH, 1994. MR 95e:57058
  • [Sha79] V. A. Sharafutdinov, Convex sets in a manifold of nonnegative curvature, Math. Notes 26 (1979), no. no. 1-2, 556-560. MR 81d:53039
  • [Sie69] L. Siebenmann, On detecting open collars, Trans. Amer. Math. Soc. 142 (1969), 201-227. MR 39:7605
  • [Tam58] I. Tamura, Homeomorphy classification of total spaces of sphere bundles over spheres, J. Math. Soc. Japan 10 (1958), 29-43. MR 20:2717

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Additional Information

Igor Belegradek
Affiliation: Department of Mathematics, 253-37, California Institute of Technology, Pasadena, California 91125
Email: ibeleg@its.caltech.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-03-06863-1
PII: S 0002-9939(03)06863-1
Keywords: Nonnegative curvature, soul
Received by editor(s): June 17, 2001
Received by editor(s) in revised form: September 25, 2001
Published electronically: February 20, 2003
Communicated by: Wolfgang Ziller
Article copyright: © Copyright 2003 American Mathematical Society