# On the topology of bi-cyclopermutohedra

## TL;DR

This paper investigates the topological structure of bi-cyclopermutohedra, a CW complex related to partitions of a set, and computes its homology using discrete Morse theory.

## Contribution

It introduces the bi-cyclopermutohedron as a new topological object and provides an optimal discrete Morse function to analyze its homology.

## Key findings

- Homology computed with integer coefficients
- Homology computed with mod 2 coefficients
- Contains subcomplexes homeomorphic to moduli spaces of planar polygons

## Abstract

Motivated by the work of Panina and her coauthors on cyclopermutohedron we study a poset whose elements correspond to equivalence classes of partitions of the set $\{1,\cdots, n+1\}$ up to cyclic permutations and orientation reversion. This poset is the face poset of a regular CW complex which we call bi-cyclopermutohedron and denote it by $\mathrm{QP}_{n+1}$. The complex $\mathrm{QP}_{n+1}$ contains subcomplexes homeomorphic to moduli space of certain planar polygons with $n+1$ sides up to isometries. In this article we find an optimal discrete Morse function on $\mathrm{QP}_{n+1}$ and use it to compute its homology with $\mathbb{Z}$ as well as $\mathbb{Z}_2$ coefficients.