Partially ordered sets and semi-simplicial complexes II Regular complexes
Sigeru OKAMOTO · Bulletin of the Faculty of Science Ibaraki University Series A Mathematics · 1971
Introduction.This is a continuation of the paper [6].Let K be an s. s. complex.Then it seems to be difficult to construct a normal s.s. complex K' whose realization I K' I is homeomorphic to |K|.However, Hu [2] has shown that the second barycentric subdivision of a CWcomplex is a simplicial polytope.Accordingly, we need to study the subdivision of an s. s. complex.On the other hand, a regular CWcomplex has a natural simplicial subdivision.Therefore, in sections 2, 3, we prepare the subdivision of an s. s. complex which has defined by Kan[3] and study a regular s. s. complex.A main theorem in section 3 can be stated as follows:If K is a regular s. s. complex, then there is a p. o. set X such that M(X) is a simplicial subdivision of |K|.In section 4, we state a similar result for a product of s. s. complexes.The terminology used here will follow that of my paper [6] hereafter referred to as I. Preliminaries.We have defined in I a chain Zn= {0,1,..., n} for a non-negative integer n.The well-known monotonic maps and are defined respectively by