On minimal complexes

Abstract

An ^-complex K is called p.w.l. minimal in Ed if each proper subcomplex of K is p.w.l. is embeddable in Ed. 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 ^-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 ^-complex K is p.w.l. minimal in E2n, then for each xe K |, | K — {x} is embeddable topologically in E2n; if an ^-complex K is p.w.l. minimal in En+d and is not embeddable there, then the dimension of each maximal simplex of K is at least d. Here Ed denotes the Euclidean d-space, an ^-complex is a finite ^-dimensional simplicial complex. \K denotes the underlying point set of the complex K in some Ed, and in case where there is no confusion, |JSΓ| will be replaced by K. Cnm denotes the complete ncomplex with m vertices. A subset X of Ed is called cellular if there exists a sequence {Qi}T=i of closed d-cells, such that Qi+ί(zIntQiy for each i, and X = ΠΓ-i Qi] where Int means interior. A cellular decomposition G of En is an upper semicontinuous (u.s.c.) decomposition of En, such that each element of G is cellular; an u.s.c. decomposition is finite if it has only finitely many nondegenerate elements, see [1],