A strong collapse increasing the geometric simplicial Lusternik-Schnirelmann category

TL;DR

This paper provides an example demonstrating that the geometric simplicial Lusternik-Schnirelmann (LS) category can increase under strong collapses, showing it is not invariant under strong homotopy, contrary to prior conjecture.

Contribution

The authors construct a specific simplicial complex that exhibits an increase in geometric simplicial LS category during a strong collapse, proving it is not a strong homotopy invariant.

Findings

An explicit example where geometric simplicial LS category increases under strong collapse

Demonstration that geometric simplicial LS category is not strong homotopy invariant

Clarification of the relationship between simplicial and geometric simplicial LS categories

Abstract

In [3], after defining notions of LS category in the simplicial context, the authors show that the geometric simplicial LS category is non-decreasing under strong collapses. However, they do not give examples where it increases strictly, but they conjecture that such an example should exist, and thus that the geometric simplicial LS category is not strong homotopy invariant. The purpose of this note is to provide with such an example. We construct a simplicial complex whose simplicial and geometric simplicial LS categories are different, and using this, we provide an example of a strong collapse that increases the geometric simplicial LS category, thus settling the geometric simplicial LS category not being strong homotopy invariant.