Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopy type of shellable $q$-complexes and their homology groups

Sudhir R. Ghorpade, Rakhi Pratihar, Tovohery H. Randrianarisoa, Hugues Verdure, Glen Wilson

Abstract

The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for $q$-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends the study of shellability to $q$-matroid complexes and determines singular homology groups for a subclass of these $q$-simplicial complexes. In this paper, we determine the homotopy type of shellable $q$-simplicial complexes. Moreover, we establish the shellability of order complexes from lexicographically shellable $q$-simplicial complexes, that include the $q$-matroid complexes. This results in a comprehensive determination of the homology groups for any lexicographically shellable $q$-complexes.

Homotopy type of shellable $q$-complexes and their homology groups

Abstract

The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for -simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends the study of shellability to -matroid complexes and determines singular homology groups for a subclass of these -simplicial complexes. In this paper, we determine the homotopy type of shellable -simplicial complexes. Moreover, we establish the shellability of order complexes from lexicographically shellable -simplicial complexes, that include the -matroid complexes. This results in a comprehensive determination of the homology groups for any lexicographically shellable -complexes.
Paper Structure (10 sections, 21 theorems, 29 equations)

This paper contains 10 sections, 21 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Shellability of order complexes of q-complexes
  3. Preliminaries
  4. The order complex of a lexicographically shellable q-complex
  5. The homotopy type of a pure shellable q-complex
  6. Background on topology
  7. Relevant constructions interpreted with model categories
  8. Proof of the result
  9. Homology of some classes of shellable q-complexes
  10. Conclusion and open question

Key Result

Proposition 2.6

The binary relation $\prec_q$ of definition def:ordering is the same as the ordering defined in GPR21. As such, it defines a total ordering on the subspaces of same dimension of $\mathbb{F}_q^n$.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Remark 2.7
  • Theorem 2.8
  • Definition 2.9
  • ...and 50 more