The crosscut poset

TL;DR

This paper introduces the crosscut poset, a new combinatorial invariant that refines the crosscut complex and has applications in generalizing the crosscut theorem and analyzing fixed point properties.

Contribution

The paper presents the crosscut poset, a novel combinatorial invariant with broad applications in poset theory and topological combinatorics.

Findings

Generalizes Björner's crosscut theorem

Provides new results on fixed point and fixed simplex properties

Demonstrates the utility of the crosscut poset in combinatorial topology

Abstract

We introduce a new combinatorial invariant, which we call crosscut poset, that is finer than the crosscut complex. We exhibit many applications of the crosscut poset which include a generalization of Bj\"orner's crosscut theorem and two results concerning the fixed point property and the fixed simplex property.