Homotopy groups of moduli spaces of stable quiver representations

TL;DR

This paper introduces a method to compute the homotopy groups of moduli spaces of stable quiver representations, revealing their triviality up to certain dimensions and applying results to polygon spaces.

Contribution

It provides a novel approach for calculating homotopy groups of stable quiver representation spaces and establishes their triviality in low dimensions.

Findings

Homotopy groups are trivial up to a certain dimension depending on the quiver.

Computed low-dimensional homotopy groups of moduli spaces.

Applied theory to the space of non-degenerate polygons in 3D.

Abstract

The purpose of this paper is to describe a method for computing homotopy groups of the space of $\alpha$-stable representations of a quiver with fixed dimension vector and stability parameter $\alpha$. The main result is that the homotopy groups of this space are trivial up to a certain dimension, which depends on the quiver, the choice of dimension vector, and the choice of parameter. As a corollary we also compute low-dimensional homotopy groups of the moduli space of $\alpha$-stable representations of the quiver with fixed dimension vector, and apply the theory to the space of non-degenerate polygons in three-dimensional Euclidean space.