Approximating polyhedra in codimension one spheres embedded in sⁿ by tame polyhedra

Robert Jay Daverman

Vol. 51 (1974), No. 2, 417–426

DOI: 10.2140/pjm.1974.51.417

Abstract

We investigate properties of an (n− 1)-sphere Σ topologically embedded in the n-sphere Sⁿ(n ≧ 6) implying that each (n− 3)-dimensionaI polyhedron in Σ can be homeomorphically approximated by polyhedra in Σ that are tame in Sⁿ. In case Σ bounds an n-cell, we relate these properties and the existence of homeomorphic approximations to Σ by locally flat spheres “mostly” outside this n-cell. This leads to a negative result eliminating a natural generalization to Bing’s Side Approximation Theorem.

Mathematical Subject Classification

Primary: 57C55

Secondary: 57A45

Milestones

Received: 5 January 1973

Published: 1 April 1974