|
Organizers |
A PL embedding S2 --> \R4 with non-standard regular
by
Eugene Rafikov
Coauthors: Dušan Repovš
Example 1 There exists a (non-locally flat) PL embedding
f:S2®R4 such that the regular neighborhood of f(S2) in R4 is
not homeomorphic to S2×D2.
Example 1 is motivated by the following general problem, raised independently in [Rep88, Hor90, p.69] and by Shtanko (unpublished): find all pairs (m, k) such that for each k-polyhedron K and PL m-manifold M the following holds:
(*) for every two homotopic embeddings f, g:K® M the regular neighborhoods in M of f(K) and g(K) are homeomorphic.
The version of this problem where the homeomorphism between the regular neighborhoods is required to be an extension of f°g-1 from K is more famous and simplier [LiSi69] (particularly, the result of Example 1 for this similar problem is obvious). The partial cases when M=Rm (and then f and g are automatically homotopic) and/or when K is a PL manifold are most interesting. For m ³ 2k+1, (particularly, for k=2 and m ³ 5), (*) holds by [LiSi69] (since m-thickenings are classified by their tangent bundles). For k=1 (*) holds (the only case that is not covered by the above remark is m=2 and is clear). If K is a PL manifold and f, g are locally flat (particularly, if m ³ k+3), then (*) holds by [RoSa68, §4] (therefore any locally flat PL embedding S2®R4 has a standard regular neighborhood). If m=k+1 ³ 3 and K is a special polyhedron, then (*) holds by [Cas65, Cav92, Mat73, Rep88, BRS99]. The example of [Rep88] where m ³ 3, M=Rm and K=Sm-1ÚS1ÚS1 shows that (*) is not true for k=m-1. The Dunce Hat example [Zee65] shows that (*) is not true for M=R4 and some 2-polyhedron K. Example 1 is interesting because it shows that (*) is not true for M=R4 and even some 2-surface K.
Our construction is based on the following knot-theoretical lemma.
Lemma 2There exists a knot S1® S3, concordant to the
trivial one but (with any frame) not Kirby-equivalent to it.
For the proof of Kirby non-equivalence we apply the Witten invariants [PrSo97].
Let B4 be a standard ball in R4 with
Date received: June 28, 1999
Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadc-13.