Nested set complexes for posets and the Bier construction

TL;DR

This paper extends the concept of nested set complexes to posets, explores their topological relationship with order complexes, and generalizes existing proof methods for the Bier construction on posets.

Contribution

It introduces a generalized framework for nested set complexes in posets and establishes conditions under which these complexes relate to order complexes via subdivision, extending prior work on semilattices.

Findings

Nested set complexes can be related to order complexes via subdivision.

A sufficient condition for this subdivision relationship is provided.

The proof method for the Bier construction is generalized to all posets.

Abstract

We generalize the concept of combinatorial nested set complexes to posets and exhibit the topological relationship between the arising nested set complexes and the order complex of the underlying poset. In particular, a sufficient condition is given so that this relationship is actually a subdivision. We use the results to generalize the proof method of \v{C}uki\'c and Delucchi, so far restricted to semilattices, for a result of Bj\"orner, Paffenholz, Sj\"ostrand and Ziegler on the Bier construction on posets.