Torsion in the space of commuting elements in a Lie group

Kishimoto, Daisuke  ; Takeda, Masahiro

Canadian journal of mathematics, Tome 76 (2024) no. 3, pp. 1033-1061

Résumé

Let G be a compact connected Lie group, and let $\operatorname {Hom}({\mathbb {Z}}^m,G)$ be the space of pairwise commuting m-tuples in G. We study the problem of which primes $p \operatorname {Hom}({\mathbb {Z}}^m,G)_1$, the connected component of $\operatorname {Hom}({\mathbb {Z}}^m,G)$ containing the element $(1,\ldots ,1)$, has p-torsion in homology. We will prove that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ for $m\ge 2$ 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 $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ and show that $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$ always has 2-torsion in homology whenever G is simply-connected and simple. Our computation is based on a new homotopy decomposition of $\operatorname {Hom}({\mathbb {Z}}^m,G)_1$, which is of independent interest and enables us to connect torsion in homology to the combinatorics of the Weyl group.

Détail

DOI : 10.4153/S0008414X23000317

Mots-clés : Space of commuting elements, Lie group, Weyl group, homotopy colimit, Bousfield–Kan spectral sequence, extended Dynkin diagram