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For a given polyhedron $K\subset M$ the notation $R_M(K)$ denotes a regularneighborhood 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 thesingularity $S^{k-1}\subset S^{k+1}$ such that$\pi_1(S^{k+1}-S^{k-1})\not\cong\Z$)

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods*