Torsion in the space of commuting elements in a Lie group

2021/03/22 by Daisuke Kishimoto, Masahiro Takeda, Kishimoto, Daisuke +1
Mathematics · #55P65 #57S05 #Algebraic Topology (math.AT) #Algebraic structures and combinatorial models #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2103.11662

openalex publication_date 2021/03/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a compact connected Lie group, and let Hom(ℤ^m,G) be the space of pairwise commuting m-tuples in G. We study the problem of which primes p Hom(ℤ^m,G)₁, the connected component of Hom(ℤ^m,G) containing the element (1,…,1), has p-torsion in homology. We will prove that Hom(ℤ^m,G)₁ for m≥ 2 has p-torsion in homology if and only if p divides the order of the Weyl group of G for G=SU(n) and some exceptional groups. We will also compute the top homology of Hom(ℤ^m,G)₁ and show that Hom(ℤ^m,G)₁ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of Hom(ℤ^m,G)₁, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

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