# Stabilization of the Homotopy Groups of the Moduli Spaces of $k$-Higgs Bundles

## TL;DR

This paper proves the stabilization of homotopy groups for moduli spaces of rank 2 and 3 k-Higgs bundles, extending previous results and providing new proofs that do not rely on stratification coincidences.

## Contribution

It offers a new proof of homotopy group stabilization for moduli spaces of k-Higgs bundles and extends the results from rank two to rank three.

## Key findings

- Homotopy groups of $\
- $
ightarrow$ stabilization established for ranks 2 and 3.
- New proof methods avoiding stratification coincidence assumptions.

## Abstract

The work of Hausel proves that the Bia\l{}ynicki-Birula stratification of the moduli space of rank two Higgs bundles coincides with its Shatz stratification. He uses that to estimate some homotopy groups of the moduli space of $k$-Higgs bundles of rank two. Unfortunately, those two stratifications do not coincide in general. Here, the objective is to present a different proof of the stabilization of the homotopy groups of $\mathcal{M}^k(2,d)$, and generalize it to $\mathcal{M}^k(3,d)$, the moduli spaces of $k$-Higgs bundles of degree $d$, and ranks two and three respectively, using the results from the works of Hausel and Thaddeus, among other tools.