Homotopy type of free group character varieties

TL;DR

This paper demonstrates that the space of closed orbits in the character variety of a free group into a real reductive group has the same homotopy type as the space into its maximal compact subgroup, via a natural deformation retraction.

Contribution

It constructs a natural strong deformation retraction showing the homotopy equivalence between these character variety spaces.

Findings

Hom(F,G)/G and Hom(F,K)/K have the same homotopy type.

A natural strong deformation retraction exists between the spaces.

The result simplifies understanding the topology of character varieties.

Abstract

Let G be a real reductive algebraic group with maximal compact subgroup K, and let F be a rank r free group. Here, we summarize the construction of a natural strong deformation retraction from the space of closed orbits in Hom(F,G)/G to the orbit space Hom(F,K)/K. In particular, these spaces have the same homotopy type.