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.