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See detailBad representations and homotopy groups of Character Varieties
Guerin, Clément UL; Lawton, Sean; Ramras, Daniel

E-print/Working paper (2019)

Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr ... [more ▼]

Let G be a connected, reductive, complex affine algebraic group, and let Xr denote the moduli space of G-valued representations of a rank r free group. We first characterize the singularities in Xr resolving conjectures of Florentino-Lawton. In particular, we compute the codimension of the orbifold singular locus using facts about Borel-de Siebenthal groups. We then use this codimension to calculate some higher homotopy groups of the smooth locus of Xr, proving conjectures of Florentino-Lawton-Ramras. Lastly, using the earlier analysis of Borel-de Siebenthal groups, we prove a conjecture of Sikora about CI Lie groups. [less ▲]

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