Small flag complexes with torsion

TL;DR

This paper classifies small flag complexes with torsion in their first homology group, confirming a conjecture about the minimal size of posets with such properties, using a combination of theoretical and computational methods.

Contribution

It provides a classification of flag complexes up to 12 vertices with torsion, and confirms the minimal size of posets with torsion in their homology as 13 elements.

Findings

Classified all flag complexes with ≤12 vertices with torsion

Confirmed the minimal size of posets with torsion is 13 elements

Used computer-aided methods for classification

Abstract

We classify flag complexes on at most 12 vertices with torsion in the first homology group. The result is moderately computer-aided. As a consequence we confirm a folklore conjecture that the smallest poset whose order complex is homotopy equivalent to the real projective plane (and also the smallest poset with torsion in the first homology group) has exactly 13 elements.