On the Homeomorphism and Homotopy Type of Complexes of Multichains

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 [2 r ]. 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 rth 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 .