Embedding Tn-like continua in Euclidean space

Abstract

AbstractMany authors have been concerned with embedding ∏-like continua in Rn where ∏ is some collection of polyhedra or manifolds. A similar concern has been embedding ∏-like continua in Rn up to shape. In this paper we prove two main theorems. Theorem: If n ⩾ 2 and X is Tn-like, then X embeds in R2n. This result was conjectured by McCord for the case H1(X) finitely generated and proved by McCord for the case that H1(X) = 0 using a theorem of Isbell. The second theorem is a shape embedding theorem. Theorem: If X is Tn-like, then X embeds in Rn+2 up to shape. This theorem is proved by showing that an n-dimensional compact connected abelian topological group embeds in Rn+2. Any Tn-like continuum is shape equivalent to a k-dimensional compact connected abelian topological group for some 0 ⩽ k ⩽ n