From green mutation to $\mathrm{X}$-evolution: flows and foliations on cluster complexes

TL;DR

This paper introduces $ ext{X}$-evolution flows and foliations on cluster complexes of 2-Calabi-Yau categories, revealing their structure, continuity properties, and topological implications, including contractibility and fundamental group generation.

Contribution

It defines $ ext{X}$-evolution flows and foliations, linking them to green mutations, and analyzes their topological properties for Dynkin and Euclidean quivers.

Findings

$ ext{X}$-foliations are compact or semi-compact for Dynkin or Euclidean types.

Cluster complexes are shown to be spherical or contractible.

Fundamental group of the cluster exchange graph is generated by squares and pentagons.

Abstract

For any point $\mathrm{X}$ in the cluster complex $\mathrm{Cpx}(\mathcal{C})$ of a 2-Calabi-Yau category $\mathcal{C}$, we introduce $\mathrm{X}$-evolution flow on $\mathrm{Cpx}(\mathcal{C})$. We show that such a flow induces a piecewise linear one-dimensional $\mathrm{X}$-foliation with two singularities, the unique sink $\mathrm{X}$ and the unique source $\mathrm{X}[1]$. Moreover, we show that evolution flows on cluster complexes are continuous refinement/generalization of green mutations on cluster exchange graphs. For the cluster category of a Dynkin or Euclidean quiver $Q$, we prove that the $\mathrm{X}$-foliation is compact or semi-compact, for various choices of $\mathrm{X}$. As an application, we show that $\mathrm{Cpx}(\mathcal{C})$ is spherical (Dynkin case) or contractible (Euclidean case). As a byproduct, we show that the fundamental group of the cluster exchange graph of $Q$ is generated by squares and pentagons.