Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods

For a given polyhedron K(?)M,the notation RM(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 PL sphere S~k(?)S~(K 2)having an isolated non-locally flat point with the singularity S~(k-1)(?) S~(k 1)such thatπ_1(S~(k 1)-S~(k-1))(?)Z).

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods

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 PL sphere 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). * Project supported by the Pierre Deligne Fund based on 2004 Balzan Prize in Mathematics, INTAS Grants (No. YSF-2002-393), the Russian Foundation for Basic Research (Nos. 05-01-00993, 04-01-00682, 06-01-72551-NCNILa), President of the Russian Federation Grants (Nos. MD-3938.2005.1, NSH-1988.2003.1, MD-4729.2007.1) and the Slovenian Research Agency (Nos. BI-RU/05-07-04, BI-RU/05-07-13).