In the 1970s, Harder and Narasimhan calculated the Betti numbers of the moduli space of stable vector bundles on a curve of rank n and degree d, and proved that the topology of these varieties is highly dependent on d. The parallel question of topological degree-dependence was asked by Hausel in 2005 for the moduli of stable Higgs bundles. It is more subtle and interesting in the latter case, as the underlying topological spaces for different degrees can be realized by Galois conjugate character varieties for which all previously known topological invariants are degree-independent. I will discuss the background of this question; then I will discuss a topological invariant that is computable and distinguishes Galois conjugate character varieties, providing a Higgs analog of the Harder–Narasimhan result and answering Hausel’s question negatively. This is joint work with Siqing Zhang.