Homological stability for spaces of commuting elements in Lie groups

Daniel A. Ramras, Mentor Stafa

Published 2018-05-03Version 1

In this paper we study homological stability for spaces of pairwise commuting n-tuples in a Lie group G. We prove that for each n>0, 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, Comm(G) and Bcom(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, J. Wilson's theory of FI$_W$-modules. 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.