On the fundamental group of $${{\rm Hom}({\mathbb Z}^k,G)}$$

Abstract

Let G be a compact Lie group. Consider the variety Hom(Z k , G) of representations of Z k into G. We can see this as a based space by taking as base point the trivial representation 1. The goal of this paper is to prove that π 1 (Hom(Z k , G)) is naturally isomorphic to π 1 (G) k .

On the structure of spaces of commuting elements in compact Lie groups

“…In particular, the restriction to the first ℓ − 1 homogeneous coordinates is equal to the standard inclusion. The conclusion follows again from [28] and a commutative diagram similar to (23). For the case ℓ = 3 see Remark 6.4.…”

“…be the tuple consisting of the first ℓ − 2 entries of n ∨ ℓ−1 . We are going to describe f ℓ−1 explicitly, and show that it fits into a commutative diagram (23) CP…”

“…AA acknowledges support from NSERC prior to October 2019. from geometry and homotopy theory. See for example [2,8,12,23,26,38,42,46], among others.…”

Fundamental groups of character varieties: surfaces and tori

“…In the case when is free Abelian, the main results in [16] and [29] imply that the k-th factor inclusions induce an isomorphism π 1 (DG) r ∼ = −→ π 1 (Hom 0 ( , DG)). …”

“…By the main results in [16] and [29], Hom 0 (Z k , DG) is simply connected. Note here that DG has the form F k × H 1 × · · · × H n , where F is either R or C and the H i are simply connected, simple Lie groups (see [25,Section 2], for instance).…”