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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Microbundles, manifolds and metrisability
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by David Gauld and Sina Greenwood PDF
Proc. Amer. Math. Soc. 128 (2000), 2801-2807 Request permission

Abstract:

The notion of a microbundle was introduced in the 1960s but the theory came to an abrupt halt when it was shown that for a metrisable manifold, microbundles are equivalent to fibre bundles. In this paper we consider microbundles over non-metrisable manifolds. In some cases microbundles are equivalent to fibre bundles but in others they are not. In particular, we show that a manifold is metrisable if and only if its tangent microbundle is equivalent to a fibre bundle. We also illustrate that for some non-metrisable manifolds every trivial microbundle contains a trivial fibre bundle whereas other manifolds may support a trivial microbundle not containing a trivial fibre bundle.
References
  • Sina Greenwood and David Gauld Microbundles revisited, Proceedings of Prague TOPOSYM 1996, Topology Atlas, (1997), 114-119.
  • J. M. Kister, Microbundles are fibre bundles, Ann. of Math. (2) 80 (1964), 190–199. MR 180986, DOI 10.2307/1970498
  • J. Milnor, Topological manifolds and smooth manifolds, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 132–138. MR 0161345
  • Peter Nyikos, The theory of nonmetrizable manifolds, Handbook of set-theoretic topology, North-Holland, Amsterdam, 1984, pp. 633–684. MR 776633
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Additional Information
  • David Gauld
  • Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
  • Email: gauld@math.auckland.ac.nz
  • Sina Greenwood
  • Affiliation: Department of Mathematics, University of Auckland, Private Bag 92019, Auckland, New Zealand
  • Email: sina@math.auckland.ac.nz
  • Received by editor(s): July 8, 1997
  • Received by editor(s) in revised form: October 16, 1998
  • Published electronically: March 1, 2000
  • Additional Notes: The second author’s research was supported in part by a Marsden Fund Award, UOA611
  • Communicated by: Alan Dow
  • © Copyright 2000 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 128 (2000), 2801-2807
  • MSC (2000): Primary 57N55, 54E35, 55R60, 57N05, 57N15
  • DOI: https://doi.org/10.1090/S0002-9939-00-05343-0
  • MathSciNet review: 1664358