On minimal complexes

An K-complex K is called p.w.l. minimal in E^d if each proper subcomplex of K is p.W.L. is embeddable in E^d. The main purpose of this paper is to prove that for each n ≧ 2, and each d9 n + 1 ≧ d ≧ 2n, there are countably many nonhomeomorphic N-complexes, each one of which is p.W.L. minimal in Ed and is not p.W.L. embeddable there. From general position arguments it follows that if an N-complex K is P.W.L. minimal in E²ⁿ, then for each x∈ |K|, |K| − {x} is embeddable topologicallyin E²ⁿ; if an N-complex K is p.W.L. minimal in E^n+d and is not embeddable there, then the dimension of each maximal simplex of K is at least d.