Polyhedral joins and graph complexes

TL;DR

This paper explores the topological properties of polyhedral joins and graph complexes, providing decomposition formulas, connectivity bounds, and applications to graph homotopy types, advancing understanding in combinatorial topology.

Contribution

It introduces a decomposition of the suspension of polyhedral joins in terms of polyhedral smash products and analyzes their homotopy types and connectivity bounds.

Findings

Decomposition formula for the suspension of polyhedral joins

Lower bounds for connectivity of polyhedral joins

Homotopy type analysis of forest filtrations in graph products

Abstract

We give a decomposition of the suspension of a polyhedral join in terms of the polyhedral smash product of the suspension of the family of pairs, and study some cases in which the formula can be desuspended, particularly for polyhedral joins over independence complexes of graphs. We also give some lower bounds for the connectivity of polyhedral joins. We use these results to study the homotopy type of the forest filtration for some lexicographic products of graphs.