Configuration spaces of commuting elements

TL;DR

This paper introduces the configuration space of commuting elements in topological groups, demonstrating rational homological and cohomological stability for classical groups, and provides cohomology computations in the unstable range.

Contribution

It establishes stability properties of these configuration spaces and extends understanding of their cohomology, including explicit computations.

Findings

Rational homological stability for unitary, special unitary, and symplectic groups.

Cohomological rational representation stability for finite products of these groups.

Explicit cohomology computations in the unstable range.

Abstract

In this article we introduce the space of configurations of commuting elements in a topological group and show that it satisfies rational homological stability for the sequences of unitary, special unitary and symplectic groups. We also prove that it satisfies cohomological rational representation stability with respect to the number of elements in the tuple for finite products of such groups, in particular cohomological rational stability for the space of unordered configurations of commuting elements. Finally we present some computations of cohomology in the unstable range.