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Codimension two PL embeddings of spheres with nonstandard regular neighborhoods. (English) Zbl 1139.57024

Summary: For a given polyhedron \(K \subset M\), the notation \(R_M (K)\) denotes a regular neighborhood of \(K\) in \(M\). The authors study the following problem: find all pairs \((m, k)\) such that if \(K\) is a compact \(k\)-polyhedron and \(M\) a PL \(m\)-manifold, then \(R_M (f(K)) \cong R_M (g(K))\) for any two homotopic PL embeddings \(f, g : K \rightarrow M\). It is proved that \(R_{S^{k+2}}(S^k)\ncong S^k \times D ^{2}\) for each \(k \geq 2\) and some PL sphere \(S^k \subset S^{k+2}\) (even for any PL sphere \(S^k \subset S^{k+2}\) having an isolated non-locally flat point with the singularity \(S^{k-1} \subset S^{k+1}\) such that \(\pi _{1}(S^{k+1} - S^{k-1})\ncong \mathbb Z)\).

MSC:

57Q40 Regular neighborhoods in PL-topology
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M05 Fundamental group, presentations, free differential calculus
57N40 Neighborhoods of submanifolds
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References:

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