Noncompact codimension-1 real algebraic manifolds
type: Item
instance of: scholarly article
title: Noncompact codimension-1 real algebraic manifolds
zbMATH Open document ID: 1085.14048
published in: Michigan Mathematical Journal
publication date: 2004-11-01T00:00:00Z
review text: The central result of the article is the following statement. Let \(X^n\) be a smooth compact \(n\)-dimensional manifold with boundary. Then, the interior \(\text{Int}(X^n)\) of \(X^n\) is diffeomorphic to a nonsingular real algebraic subset of \(\mathbb{R}^{n+1}\) if and only if \(X^n\) admits a smooth proper embedding \(X^n \hookrightarrow D^{n+1}\) in an \((n+1)\)-dimensional closed disc \(D^{n+1}\). If such an embedding exists, then \(\text{Int}(X^n)\) is isotopic to a nonsingular real algebraic subset of \(\text{Int}(D^{n+1})\approx\mathbb{R}^{n+1}\). Here, by a proper map the authors mean a map sending boundary points to boundary points, being transverse at boundaries, and such that the inverse image of any compact set is compact. The article contains several applications of the result formulated above. The first application is related to a classical theorem of Seifert which asserts that every smooth closed codimension 1 submanifold of \(\mathbb{R}^{n+1}\) is smoothly isotopic to a nonsingular real algebraic set [see \textit{H. Seifert}, Math. Z. 41, 1--17 (1936; JFM 62.0807.02)]. The authors of the article under review consider a noncompact analogue of Seifert's theorem and prove that a smooth proper \(n\)-dimensional submanifold \(M^n\subset\mathbb{R}^{n+1}\) is isotopic to a nonsingular real algebraic subset of \(\mathbb{R}^{n+1}\) if and only if the pair \((\mathbb{R}^{n+1}, M^n)\) admits a completion \((\overline{\mathbb{R}^{n+1}},\overline{M^n})\) such that \(\overline{\mathbb{R}^{n_1}}\) is diffeomorphic to \(D^{n+1}\). Here, a completion of a pair \((A,B)\), where \(A\) is a smooth manifold without boundary, \(B\subset A\) is a smooth proper submanifold (without boundary), is a smooth compact manifold pair \((\overline A,\overline B)\) and a smooth embedding \(i:(A,B)\hookrightarrow (\overline A,\overline B)\) such that \(i(A,B)=(\text{Int}(\overline A),\text{Int} (\overline B))\), the manifold \(\overline B\) intersects the boundary \(\partial A\) of …
Mathematics Subject Classification ID: 14P05
Mathematics Subject Classification ID: 14P25
Mathematics Subject Classification ID: 57R40
Mathematics Subject Classification ID: 57R52
Mathematics Subject Classification ID: 57R55
zbMATH DE Number: 2112906
author: Jack S. Calcut
author: Henry C. King
reviewed by: Ilia Itenberg
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1091112080
OpenAlex ID: W2021388195
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DOI: 10.1307/MMJ/1091112080
description: scientific article; zbMATH DE number 2112906