Approximating maps of 2-manifolds with zero-dimensional nondegeneracy sets
type: Item
instance of: scholarly article
title: Approximating maps of 2-manifolds with zero-dimensional nondegeneracy sets
zbMATH Open document ID: 0754.57004
DOI: 10.1016/0166-8641(92)90059-9
author: Matthew V. Brahm
published in: Topology and its Applications
publication date: 1992-09-27T00:00:00Z
review text: Let \(f\) be a map of a 2-manifold \(M^ 2\) to a 3-manifold \(M^ 3\) such that \(N_ f=\{x\in M^ 2: f^{-1}f(x)\neq x\}\) is 0-dimensional. This paper attacks the conjecture that \(f\) can be approximated by an embedding, confirming it when \(N_ f\) is closed and otherwise establishing its equivalence with two related conjectures. The key contribution is a complicated but useful and original technical device which shows, among other things, that such a map \(f\) can be approximated by a map having point preimages of arbitrarily small diameter.
reviewed by: Robert J. Daverman
Mathematics Subject Classification ID: 57M35
Mathematics Subject Classification ID: 57N35
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57N12
Mathematics Subject Classification ID: 54B15
Mathematics Subject Classification ID: 57M30
zbMATH DE Number: 64611
zbMATH Keywords: Dehn disk
zbMATH Keywords: recognition problem
zbMATH Keywords: map of a 2-manifold to a 3-manifold
zbMATH Keywords: approximation by an embedding
zbMATH Keywords: point preimages
MaRDI profile type: Publication
full work available at URL: 0166-8641(92)90059-9
OpenAlex ID: W2031353638
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description: scientific article; zbMATH DE number 64611