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

Pedro Conceição

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.

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

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 -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.
Paper Structure (10 sections, 25 theorems, 41 equations)

This paper contains 10 sections, 25 theorems, 41 equations.

Table of Contents

  1. Organization.
  2. The main objects and their preliminaries
  3. Ordered simplicial complexes and simplicial posets.
  4. Face vectors of a simplicial cell complex.
  5. Properties of the complex of injective words.
  6. Polyhedral product spaces.
  7. Colimits and homotopy colimits.
  8. Cohomology of moment angle complexes.
  9. Homotopy type of $\mathcal{Z}_{\Gamma_n}$
  10. A homotopy fibration of polyhedral products over ordered simplicial complexes

Key Result

Theorem A

where $h_k$ is the $k$th element of the $h$-vector of the complex of injective words.

Theorems & Definitions (42)

  • Theorem A
  • Theorem B
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8
  • ...and 32 more