Sean Lawton

The topology of moduli spaces of free group representations

For any complex affine reductive group G and a fixed choice of maximal compact subgroup K, we show that the G-character variety of a free group strongly deformation retracts to the corresponding K-character space, which is a real semi-algebraic set. Combining this with constructive invariant theory and classical topological methods, we show that the

Topology of character varieties of Abelian groups

{{\rm SLm}(3, mathbb {C})}

Topology of moduli spaces of free group representations in real reductive groups

-character variety of a rank 2 free group is homotopic to an 8 sphere and the

Algebraic independence in SL(3,C) character varieties of free groups

{{\rm SLm}(2, mathbb {C})}

Spin networks and SL(2,ℂ)-character varieties

-character variety of a rank 3 free group is homotopic to a 6 sphere.

Fundamental group of moduli spaces of representations

Let Xr be the moduli space of SLn, SUn, GLn, or Un-valued representations of a rank r free group. We classify the algebraic singular stratification of Xr . This comes down to showing that the singular locus corresponds exactly to reducible representations if there exist singularities at all. Then by relating algebraic singularities to topological singularities, we show the moduli spacesXr generally are not topological manifolds, except for a few examples we explicitly describe.