# Some remarks on the subrack lattice of finite racks

## TL;DR

This paper investigates the topological structure of the lattice of subracks in finite racks, establishing conditions under which their order complexes resemble spheres, with specific results for group and conjugacy class racks.

## Contribution

It introduces a condition linking the decomposability of racks to the homotopy type of their subrack lattice complexes, extending known results to new classes of racks.

## Key findings

- Homotopy type of subrack lattice is a sphere under certain conditions.
- Decomposability of racks is necessary for the main topological result.
- Determined homotopy types for subrack lattices of symmetric and alternating groups.

## Abstract

The set of all subracks $\mathcal{R}(X)$ of a finite rack $X$ form a lattice under inclusion. We prove that if a rack $X$ satisfies a certain condition then the homotopy type of the order complex of $\mathcal{R}(X)$ is a $(m-2)$-sphere, where $m$ is the number of maximal subracks of $X$. The rack $X$ satisfying the condition of this general result is necessarily decomposable. Two particular instances occur when \begin{itemize} \item $X=G$ is a group rack, and when \item $X=C$ is a conjugacy class rack of a nilpotent group. \end{itemize} We also studied the subrack lattices of indecomposable racks by focusing on the conjugacy class racks of symmetric or alternating groups and determined the homotopy types of the corresponding order complexes in some cases.