Mutations of quivers with 2-cycles

TL;DR

This paper introduces a mutation theory for quivers with 2-cycles using homotopies, extending existing models and allowing for infinite mutation sequences, with applications to triangulated surfaces with colored punctures.

Contribution

It develops a new mutation framework for quivers with 2-cycles using homotopies, generalizing previous models and connecting to surface triangulations.

Findings

Extended mutation theory for quivers with 2-cycles.

Constructed quivers from surface triangulations with colored punctures.

Proved flips correspond to mutations in the extended setting.

Abstract

We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further construct quivers with homotopies from triangulations of marked surfaces with colored punctures, and prove that flips correspond to mutations, extending the Fomin-Shapiro-Thurston model to the setting with 2-cycles.