On the homeomorphism and homotopy type of complexes of multichains

TL;DR

This paper introduces a class of simplicial complexes based on multichains in a poset, analyzing their topological properties and subdivisions, revealing their homotopy equivalences and classifications.

Contribution

It defines new complexes of multichains depending on monotone functions and characterizes when they are homotopy equivalent to the original order complex.

Findings

Exactly 2^r such functions produce subdivisions of the order complex.

Half of these subdivisions are pairwise different.

Many complexes are homotopy equivalent to the original order complex.

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 from P. The simplicial complexes depend on a strictly monotone function from [r] to [2r]. We show that there 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 interval and 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.