An n-complex K is called
p.w.1. minimal in Ed if each proper subcomplex of K is p.w.1. is embeddable in Ed.
The main purpose of this paper is to prove that for each n ≧ 2, and each d,
n + 1 ≦ d ≦ 2n, there are countably many nonhomeomorphic n-complexes,
each one of which is p.w.1. minimal in Ed and is not p.w.1. embeddable
there. From general position arguments it follows that if an n-complex K is
p.w.1. minimal in E2n, then for each x ∈|K|, |K|−{x} is embeddable
topologically in E2n; if an n-complex K is p.w.1. minimal in En+d and is not
embeddable there, then the dimension of each maximal simplex of K is at least
d.
