On the topology of a boolean representable simplicial complex

TL;DR

This paper investigates the topological properties of boolean representable simplicial complexes, showing their fundamental groups are free, characterizing their homotopy type in dimension 2, and providing conditions for shellability and Cohen-Macaulayness.

Contribution

It establishes new topological and combinatorial characterizations of boolean representable simplicial complexes, including their fundamental groups, homotopy types, and shellability conditions.

Findings

Fundamental groups are free and their rank depends on the graph of flats.

In dimension 2, complexes have the homotopy type of a wedge of spheres.

Conditions for shellability and Cohen-Macaulayness are characterized.

Abstract

It is proved that fundamental groups of boolean representable simplicial complexes are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\geq 2$. In the case of dimension 2, it is shown that boolean representable simplicial complexes have the homotopy type of a wedge of spheres of dimensions 1 and 2. Also in the case of dimension 2, necessary and sufficient conditions for shellability and being sequentially Cohen-Macaulay are determined. Complexity bounds are provided for all the algorithms involved.