A nice acyclic matching on the nerve of the partition lattice

TL;DR

This paper constructs an equivariant acyclic matching on the nerve of the partition lattice, providing a more elementary method to analyze the homotopy type of the space quotient by a subgroup, and develops new tools in Equivariant Discrete Morse Theory.

Contribution

It introduces an $S_1\times S_{n-1}$-equivariant acyclic matching on the nerve of the partition lattice and develops new methods in Equivariant Discrete Morse Theory.

Findings

Constructed an equivariant acyclic matching on the nerve of the partition lattice.

Provided a description of the critical simplices in the matching.

Developed new methods for Equivariant Discrete Morse Theory.

Abstract

The author has already proven that the space $\Delta(\Pi_n)/G$ is homotopy equivalent to a wedge of spheres of dimension $n-3$ for all natural numbers $n\geq 3$ and all subgroups $G\subset S_1\times S_{n-1}$. We construct an $S_1\times S_{n-1}$-equivariant acyclic matching on $\Delta(\Pi_n)$ together with a description of its critical simplices. This is also a more elementary approach to determining the number of spheres. We also develop new methods for Equivariant Discrete Morse Theory by adapting the Patchwork Theorem and poset maps with small fibers from Discrete Morse Theory.