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

 

Microbundles, manifolds and metrisability


Authors: David Gauld and Sina Greenwood
Journal: Proc. Amer. Math. Soc. 128 (2000), 2801-2807
MSC (2000): Primary 57N55, 54E35, 55R60, 57N05, 57N15
Published electronically: March 1, 2000
MathSciNet review: 1664358
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05343-0
PII: S 0002-9939(00)05343-0
Keywords: Metrisability, microbundle, non-metrisable manifold, tangent microbundle
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
Article copyright: © Copyright 2000 American Mathematical Society