Homological stability for spaces of commuting elements in Lie groups

TL;DR

This paper establishes rational homological stability for spaces of commuting elements in classical Lie groups, their character varieties, and related infinite-dimensional spaces, using representation stability theory.

Contribution

It proves new rational homological stability results for spaces of commuting elements in Lie groups and their variants, with explicit bounds and stability maps.

Findings

Rational homology stabilizes for spaces of commuting elements in classical Lie groups.

Stability results extend to character varieties and infinite-dimensional analogues.

Homology isomorphisms are induced by natural maps between spaces.

Abstract

In this paper we study homological stability for spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${\rm Comm}(G)$ and ${\rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability - in particular, the theory of ${\rm FI}_W$-modules developed by Church-Ellenberg-Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.