# Homomorphism Complexes and Maximal Chains in Graded Posets

## TL;DR

This paper introduces a new topological construction for graded posets using homomorphism complexes based on maximal chains, with applications to distributive lattices and products of chains, revealing torsion-free properties.

## Contribution

It generalizes the homomorphism complex construction to graded posets and analyzes its topological properties, especially for products of chains and Boolean algebras.

## Key findings

- Homomorphism complex for Boolean algebra is isomorphic to a subcomplex of a permutahedron.
- Optimal discrete Morse matching is found for products of chains.
- The complex is proven to be torsion-free.

## Abstract

We apply the homomorphism complex construction to partially ordered sets, introducing a new topological construction based on the set of maximal chains in a graded poset. Our primary objects of study are distributive lattices, with special emphasis on finite products of chains. For the special case of a Boolean algebra, we observe that the corresponding homomorphism complex is isomorphic to the subcomplex of cubical cells in a permutahedron. Thus, this work can be interpreted as a generalization of the study of these complexes. We provide a detailed investigation when our poset is a product of chains, in which case we find an optimal discrete Morse matching and prove that the corresponding complex is torsion-free.