Noncompact codimension-1 real algebraic manifolds
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 \(\overline A\) in the boundary \(\partial\overline B\) of \(\overline B\), and this intersection is transverse. Using this statement, the authors obtain a complete answer to the noncompact Seifert problem in ambient dimension \(<4\) and in the higher dimensional simply connected case. Another application is the following result: there exists an exotic \(\mathbb{R}^4\) diffeomorphic to a nonsingular real algebraic subset of \(\mathbb{R}^5\) if and only if there exists an exotic \(\mathbb{R}^4\) diffeomorphic to the interior of a compact manifold.
scientific article; zbMATH DE number 2112906
scientific article