The homotopy type of the moment-angle complex associated to the complex of injective words

TL;DR

This paper computes the homotopy type of the moment-angle complex related to injective words, revealing a deep link between topology and combinatorics.

Contribution

It determines the homotopy type of the moment-angle complex for injective words and introduces a generalized homotopy fibration for ordered simplicial complexes.

Findings

Homotopy type is determined by the h-vector of complexes of injective words.

Constructs a homotopy fibration generalizing known fibrations for polyhedral products.

Provides insights into the topological structure of complexes derived from directed graphs.

Abstract

Topological methods have emerged as valuable tools for analyzing the structural properties of directed graphs, particularly connectome data in computational neuroscience. This paper investigates the construction of topological spaces from combinatorial data of directed graphs using the polyhedral product functor, with particular emphasis on understanding their homotopy type, which is also of independent interest in topology and combinatorics. Specifically, we compute the homotopy type of the moment-angle complex over the face poset of the complex of injective words. This reveals a tight connection between homotopy and combinatorics: its homotopy type is determined by the $h$-vector of complexes of injective words. We also construct an associated homotopy fibration of polyhedral products associated to ordered simplicial complexes, which in a way generalizes the analogous homotopy fibration for polyhedral products over abstract simplicial complexes.