Codimension two PL embeddings of spheres with nonstandard regular neighborhoods
by
Arkady Skopenkov
Kolmogorov College
Coauthors: Dusan Repovs

For a subpolyhedron K of M the notation R_M(K) denotes a regular neighborhood of K in M. The following problem was posed in [Mas59] (in the smooth category) and by Horvatich in 1985, Repovs in 1988 and Shtanko (in the PL category): find all pairs (m, k) such that if K is a compact k-polyhedron and M a PL m-manifold, then R_M(fK) is homeomorphic to R_M(gK), for each two homotopic PL embeddings f, g of K in M. Note that R_S^k+2(S^k) is S^k×D² for any DIFF or PL locally flat embedding of S^k in S^k+2 by the Wall 1967 result.

Theorem. [ReSk] R_S^k+2(S^k) is not homeomorphic to S^k×D², if   a) k > 1 and S^k is the suspension over a locally flat PL knot S^k-1 --> S^k+1 whose group is distinct from Z, or   b) k > 2 and S^k is a PL k-sphere in S^k+2 having an isolated non-locally flat point with the singularity S^k-1 --> S^k+1 whose group is distinct from Z.

The proof is based on the following Lemma. Let k > 1 and S^k be a PL sphere in S^k+2, non-locally flat at a point x. If either k > 2 and x is an isolated non-locally flat point, or S^k is the suspension (one of whose vertices is x) over a knot S^k-1 --> S^k+1, then the fundamental group of the boundary of R_S^k+2(S^k) maps onto the group of the singularity at x.

References

[Mas59] W. S. Massey, On the normal bundle of a sphere imbedded in Euclidean space, Proc. of the Amer. Math. Soc. 10 (1959), 959-964.

[ReSk] D. Repovs and A. B. Skopenkov, Codimension two PL embeddings of spheres with nonstandard regular neighborhoods, submitted.

Date received: June 19, 2000

Copyright © 2000 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 # caeh-28.