[Paper Review] On the fundamental group of Hom(Z^k,G)

José Manuel Gómez, Alexandra Pettet+1 more|arXiv (Cornell University)|Jun 15, 2010

Advanced Algebra and Geometry6 references13 citations

TL;DR

This paper establishes that the fundamental group of the space of commuting k-tuples in a compact Lie group G, based at the trivial representation, is isomorphic to the k-fold product of the fundamental group of G. Using the Weyl group action and a key surjective map σₖ from a twisted product G ×_{N(T)} Tᵏ to Hom(ℤᵏ, G)₁, the authors prove this via reduction to the simply connected case and covering space techniques, resolving a conjecture for specific groups and extending it generally.

ABSTRACT

Let G be a compact Lie group, and consider the variety Hom(Z^k,G) of representations of Z^k into G. We view this as a based space by designating the trivial representation to be its base point. We prove that the fundamental group of this space is naturally isomorphic to π_1(G)^k.

Motivation & Objective

• To determine the fundamental group of the space of commuting k-tuples in a compact Lie group G, with base point at the trivial representation.
• To generalize a result by Torres-Giese and Sjerve on SO(3), SU(2), and U(2) to all compact Lie groups.
• To show that π₁(Hom(ℤᵏ,G)₁) ≅ π₁(G)ᵏ holds naturally for all k ≥ 1 and compact Lie groups G.
• To clarify the topological structure of Hom(ℤᵏ,G) by analyzing its connected components and fundamental group behavior.

Proposed method

• Use the natural inclusion i: G₀ → G to reduce the problem to the case where G is connected, since π₁(Hom(ℤᵏ,G₀)) → π₁(Hom(ℤᵏ,G)) is an isomorphism.
• Define the map σₖ: G ×_{N(T)} Tᵏ → Hom(ℤᵏ,G)₁, which sends (g,t₁,…,tk) to the conjugated k-tuple (gt₁g⁻¹,…,gtk g⁻¹), and show it is continuous and surjective.
• Prove σₖ is π₁-surjective using a general position argument, showing every loop in Hom(ℤᵏ,G)₁ lifts under σₖ.
• For simply connected G, show that generators of π₁(G ×_{N(T)} Tᵏ) map to trivial loops under π₁(σₖ), implying π₁(Hom(ℤᵏ,G)₁) ≅ π₁(G)ᵏ.
• Extend the result to non-simply connected G by passing to the universal cover G̃ and analyzing the induced map on Hom(ℤᵏ,G̃)₁ and Hom(ℤᵏ,G)₁.
• Use Weyl’s covering theorem and the structure of regular elements to ensure the map σₖ is a diffeomorphism onto the regular part, aiding in homotopy analysis.

Experimental results

Research questions

• RQ1 What is the fundamental group of the space of commuting k-tuples in a compact Lie group G, when the base point is the trivial representation?
• RQ2 Does the isomorphism π₁(Hom(ℤᵏ,G)₁) ≅ π₁(G)ᵏ hold for all compact Lie groups G and all k ≥ 1?
• RQ3 How does the topology of Hom(ℤᵏ,G) depend on the choice of base point, especially when not in the component of the trivial representation?
• RQ4 Can the result be extended from simply connected groups to general compact Lie groups using covering space theory?
• RQ5 What topological obstructions prevent Hom(ℤᵏ,G) from being path-connected, and how do they affect the fundamental group?

Key findings

• The fundamental group π₁(Hom(ℤᵏ,G)₁) is naturally isomorphic to π₁(G)ᵏ for any compact Lie group G and k ≥ 1.
• The result holds even when G is simply connected but Hom(ℤᵏ,G) is not path-connected, as long as the base point lies in the component of the trivial representation.
• For SU(n) with n ≥ 2, Hom(ℤᵏ,SL(n,ℂ)) is connected and simply connected for all k ≥ 1, as a corollary.
• The map σₖ: G ×_{N(T)} Tᵏ → Hom(ℤᵏ,G)₁ is π₁-surjective, a key technical step in the proof.
• When G is not simply connected, the fundamental group of Hom(ℤᵏ,G)₁ is isomorphic to π₁(G)ᵏ via lifting to the universal cover and analyzing the covering map.
• The result fails if the base point is not in the component of the trivial representation; for example, in Hom(ℤ³,Spin(7)), the exotic component B₃ has fundamental group (ℤ/2)⁴.