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Proceedings of the American Mathematical Society

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

 

 

Topological triviality of a family of zero-sets


Authors: Michael A. Buchner and Wojciech Kucharz
Journal: Proc. Amer. Math. Soc. 102 (1988), 699-705
MSC: Primary 58C25
DOI: https://doi.org/10.1090/S0002-9939-1988-0929006-5
MathSciNet review: 929006
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Abstract: This paper gives conditions on a mapping $ F:U \times {{\mathbf{R}}^k} \to {{\mathbf{K}}^p}\left( {U \subset {{\mathbf{K}}^n}{\text{open, }}{\mathbf{K}} = {\mathbf{R}}\;{\text{or }}{\mathbf{C}}} \right)$ such that the family $ F_t^{ - 1}\left( 0 \right)$ is a topologically trivial family (i.e., does not change topologically as $ t \in {{\mathbf{R}}^k}$ varies). As an application an easy proof is given of a counterexample to a conjecture of Thom concerning the number of topologically different realizations of a given jet.


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DOI: https://doi.org/10.1090/S0002-9939-1988-0929006-5
Article copyright: © Copyright 1988 American Mathematical Society