Codimension Two PL Embeddings of Spheres withNonstandard Regular Neighborhoods*

Abstract

For a given polyhedron K ⊂ M, the notation R _M(K) denotes a regular neigh-borhood 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)) ≅ R _M(g(K)) for each two homotopic PL embeddings f, g : K → M. It is proved that R _S k+2 (S ^k) ≇ S ^k × D ² for each k ≥ 2 and some PL sphere S ^k ⊂ S ^{k +2} (even for any PLsphere S ^k ⊂ S ^{k +2} having an isolated non-locally flat point with the singularity S ^{k -1} ⊂ S ^{k +1} such that π₁(S ^{k +1} – S ^{k -1}) ≇ ℤ).

Keywords

Embedding / Regular neighborhood / Dehn surgery / Fundamental group / 57M25 / 57Q40 / 57M05 / 57N40

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