  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.
