On the Homeomorphism and Homotopy Type of Complexes of Multichains

Abstract

In this paper we define and study for a finite partially ordered set P a class of simplicial complexes on the set \(P_r\) of r-element multichains of P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there are exactly \(2^r\) such functions which yield subdivisions of the order complex of P, of which \(2^{r-1}\) are pairwise different. Within this class are, for example, the order complexes of the intervals in P, the zig-zag poset of P, and the \(r{\hbox {th}}\) edgewise subdivision of the order complex of P. We also exhibit a large subclass for which our simplicial complexes are order complexes and homotopy equivalent to the order complex of P.