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Maps with Locally Flat Singular Sets

Published online by Cambridge University Press:  20 November 2018

J. G. Timourian
J. G. Timourian*
Affiliation:
The University of Tennessee, Knoxville, Tennessee
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A map f : M → N is topologically equivalent tog: X → Y if there exist homeomorphisms α: M → X and β: N → Y such that βfα–1 = g. At x ∊ M, f is locally topologically equivalent to g if, for every neighbourhood W ⊂ M of x, there exist neighbourhoods U ⊂ W of x and V of f(x) such that f|U: U → V is topologically equivalent to g.

1.1. Definition. Given a map f: M → N and x ∊ M, let F be the component of f–1(f(x)) containing x. The singular set Af is defined as follows: x ∊ M – Af if and only if there are neighbourhoods U of F and V of f(x) such that f| U: U → F is topologically equivalent to the product projection map of V × F onto V.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 916 - 921
Copyright
Copyright © Canadian Mathematical Society 1969

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