Codimension two PL embeddings of spheres with nonstandard regular neighborhoods

Details Codimension two PL embeddings of spheres with nonstandard regular neighborhoods Codimension two PL embeddings of spheres with nonstandard regular neighborhoods

2006-08-27

arxiv.org/abs/math/0608653v1

Chinese Ann. of Math., Ser. B, 28: 5 (2007), 603-608.

Mathematics

Geometric Topology

Scientific paper

10.1007/s11401-005-0476-2

For a given polyhedron $K\subset M$ the notation $R_M(K)$ denotes a regular neighborhood of $K$ in $M$. We 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 $R_M(fK)\cong R_M(gK)$, for each two homotopic PL embeddings $f,g:K\to M$. We prove that $R_{S^{k+2}}(S^k)\not\cong S^k\times D^2$ for each $k\ge2$ and {\it some} PL sphere $S^k\subset S^{k+2}$ (even for {\it 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 $\pi_1(S^{k+1}-S^{k-1})\not\cong\Z$).

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