The Discrete Fundamental Group of the Order Complex of $B_n$

TL;DR

This paper computes the discrete fundamental group of the order complex of the Boolean lattice, introducing combinatorial techniques that simplify the analysis and provide new insights into the topology of related polytopes.

Contribution

It introduces combinatorial methods using words over transpositions to compute discrete fundamental groups of polytopes, specifically applied to the Boolean lattice and permutahedron.

Findings

Discrete fundamental group of the Boolean lattice's order complex computed.

New combinatorial techniques for analyzing discrete homotopy groups introduced.

A combinatorial proof of the Betti number for the 3-equal arrangement complement provided.

Abstract

A few years ago Kramer and Laubenbacher introduced a discrete notion of homotopy for simplicial complexes. In this paper, we compute the discrete fundamental group of the order complex of the Boolean lattice. As it turns out, it is equivalent to computing the discrete homotopy group of the 1-skeleton of the permutahedron. To compute this group we introduce combinatorial techniques that we believe will be helpful in computing discrete fundamental groups of other polytopes. More precisely, we use the language of words, over the alphabet of simple transpositions, to obtain conditions that are necessary and sufficient to characterize the equivalence classes of cycles. The proof requires only simple combinatorial arguments. As a corollary, we also obtain a combinatorial proof of the fact that the first Betti number of the complement of the 3-equal arrangement is equal to $2^{n-3}(n^2-5n+8)-1.$ This formula was originally obtained by Bj\"orner and Welker in 1995.