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| Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods |
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Citation： |
Matija CENCELJ,Dušan REPOVŠ,Arkadiy B. SKOPENKOV.Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods[J].Chinese Annals of Mathematics B,2007,28(5):603~608 |
| Page view： 1163
Net amount： 1004 |
Authors： |
Matija CENCELJ; Du?an REPOV?;Arkadiy B. SKOPENKOV |
Foundation： |
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). |
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| Abstract： |
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 each two homotopic PL embeddings $f,g:K\to M$. It is proved that $R_{S^{k+2}}(S^k)\not\cong
S^k\times D^2$ for each $k\ge2$ 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 $\pi_1(S^{k+1}-S^{k-1})\not\cong\Z$). |
Keywords： |
Embedding, Regular neighborhood, Dehn surgery, Fundamental group |
Classification： |
57M25, 57Q40, 57M05, 57N40 |
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