Complexes whose boundaries cannot be pushed around

Abstract

Let Kn ⊂ ℝn be a triangulated n-ball. Examples are given to show that unlike in the two-dimensional case, the following hold for all n≥3: (1) there are nonconvex Kn with no convex simplexwise linear embeddings Kn → ℝn, even though there are strictly convex simplexwise linear embeddings ∂Kn → ℝn; (2) there are convex Kn, with no spanning simplices, such that not every simplexwise linear embedding f: ∂Kn → ℝn with convex image can be extended to a simplexwise linear embedding of Kn; (3) there are convex Kn such that the space of simplexwise linear homeomorphisms of Kn, fixed on ∂Kn, is not path connected. © 1989 Springer-Verlag New York Inc.