Higher Order Representation Stability and Disk Configuration Spaces

Abstract

Given a manifold X, the ordered configuration space of n points in X, denoted F_{n}(X), is the space of ways of putting n labeled non-overlapping points in X. Church--Ellenberg--Farb showed that if X is a connected non-compact orientable finite type manifold of dimension d at least 2, then (H_{k}(F_{n}(X)))_{n} stabilizes as a sequence of symmetric group representations. Miller--Wilson extended this first-order representation stability to non-orientable manifolds, and proved that there is a stability pattern among the unstable terms of the sequence. In this thesis we prove that there exists a manifold such that this secondary stability sequence is neither free nor stably-zero, providing the first example of such a phenomenon.

In the second half of this thesis, we turn to disk configuration spaces, specifically the ordered configuration space of open unit-diameter disks on the infinite strip of width w. In the spirit of Arnol'd and Cohen, we provide a finite presentation for the rational homology groups of this ordered configuration space as a twisted algebra. We use this presentation to prove that for all w at least 2 the ordered configuration space of open unit-diameter disks in the infinite strip of width w exhibits a notion of first-order representation stability similar to Church--Ellenberg--Farb and Miller--Wilson's first-order representation stability for the ordered configuration space of points in a manifold. This extends a result of Alpert in the case w=2. Additionally, we prove that for large w this disk configuration space exhibits notions of second- (and higher) order representation stability.