Secondary representation stability and the ordered configuration space of the once-punctured torus

TL;DR

This paper investigates the complex stability patterns in the homology of the ordered configuration space of the once-punctured torus, revealing non-trivial secondary stability phenomena using advanced algebraic and topological methods.

Contribution

It proves that the sequence of new homology generators forms a non-free, non-stably zero FIM$^{+}$-module, and shows this sequence is generated by classes on at most 4 points, demonstrating non-trivial secondary stability.

Findings

The sequence of homology generators is neither free nor stably zero.

The sequence is generated by classes on at most 4 points.

Secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.

Abstract

In this paper we study stability patterns in the homology of the ordered configuration space of the once-punctured torus. In the last decade Church and Church-Ellenberg-Farb proved that the homology groups of the ordered configuration space of a connected noncompact orientable manifold stabilize in a representation theoretic sense as the number of points in the configuration grows, with respect to a map that adds each new point "at infinity." Miller and Wilson proved that there is a secondary representation stability pattern among the unstable homology classes, with respect to adding a pair of orbiting points "near infinity." This pattern is formalized by considering sequences of homology classes as FIM$^{+}$-modules. We prove that, as FIM$^{+}$-modules, the sequence of "new" homology generators in the n-th homology of the ordered configuration space of 2n-2 points on the once-punctured torus is neither "free" nor "stably zero." We also show that this sequence is generated by homology classes on at most 4 points. Our proof uses Pagaria's work on the Betti numbers of the ordered configuration space of the torus to calculate the growth rate of the Betti numbers of the ordered configuration space of the once-punctured torus. Our computations are the first to demonstrate that secondary representation stability is a non-trivial phenomenon in positive-genus surfaces.