Habilitation thesis: Moduli spaces and their fundamental groups: homology, representations and lower central series

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Abstract

The overarching goal of my research so far has been to understand the topology of moduli spaces through algebraic invariants, primarily homology and fundamental groups. This thesis presents six different results, three concerned with the homology of moduli spaces and three concerned with studying the fundamental groups of moduli spaces (motion groups and mapping class groups) via their representations and their lower central series.

In the preliminary Chapter O, I first give a brief overview of my main research results since my PhD thesis. Chapters 1–6 then form the core of the thesis and develop six of these results in full detail.

Chapter 1 proves homological stability for two different flavours of asymptotic monopole moduli spaces, namely moduli spaces of framed Dirac monopoles and moduli spaces of ideal monopoles. The former are Gibbons-Manton torus bundles over configuration spaces whereas the latter are obtained from them by replacing each circle factor of the fibre with a monopole moduli space by the Borel construction. They form boundary hypersurfaces in a partial compactification of the classical monopole moduli spaces. These results follow from a general homological stability result for configuration spaces equipped with non-local data (non-local configuration spaces). This chapter corresponds to a joint paper with U. Tillmann [PalmerTillmann2023] published in the Proceedings of the Royal Society A.

Moduli spaces of manifolds with marked points were proven in [Tillmann2016] to be homologically stable as the number of marked points goes to infinity. Chapter 2 generalises this result to moduli spaces of manifolds with conical singularities. (Marked points may be thought of as inessential conical singularities, since a disc neighbourhood of a marked point is the cone on its boundary sphere.) This is deduced as a special case of a more general homological stability result for classifying spaces of symmetric diffeomorphism groups of manifolds, with respect to parametric connected sum, an operation generalising ordinary connected sum and surgery (including Dehn surgery).

The key input for the proof of this result is homological stability for moduli spaces of submanifolds as the number of components of the submanifold goes to infinity, which was proven in my PhD thesis and published in [Palmer2021]. The relation to conical singularities is given by collapsing tubular neighbourhoods of submanifolds to isolated points. The results of Chapter 2 correspond to the preprint [Palmer2018], which is submitted for publication.

Chapter 3 is concerned with "big mapping class groups", i.e. mapping class groups of infinite-type surfaces, and corresponds to a joint paper with X. Wu [PalmerWu2024] accepted for publication in Documenta Mathematica. In this chapter we prove that, for any infinite-type surface S, the integral homology of the closure of the compactly-supported mapping class group PMap_c(S) and of the Torelli group T(S) is uncountable in every positive degree. By our earlier results in [PalmerWu2022], and other known computations, such a statement cannot be true for the full mapping class group Map(S) for all infinite-type surfaces S. However, we are still able to prove that the integral homology of Map(S) is uncountable in all positive degrees for a large class of infinite-type surfaces S. The key property of this class of surfaces is, roughly, that the space of ends of the surface S contains a limit point of topologically distinguished points. This result includes in particular all finite-genus surfaces having countable end spaces with a unique point of maximal Cantor-Bendixson rank α, where α is a successor ordinal.

Understanding the lower central series of a group is, in general, a difficult task. However, successfully computing the lower central series and the associated Lie algebras of a group or of some of its subgroups can lead to a deep understanding of the underlying structure of that group. The goal of Chapter 4 is to showcase several techniques aimed at carrying out part of this task. In particular, we seek to answer the following question: when does the lower central series stop? We introduce a number of tools to answer this question that we then apply to partitioned surface braid groups on any surface and with respect to any partition. The path from our general techniques to their application is far from straightforward, and a certain amount of tenacity is required to deal with all of the cases encountered along the way. We finally arrive at an answer to our question for every one of these groups, with the sole exception of one family of partitioned braid groups on the projective plane. In a number of cases, we even compute completely the lower central series. This chapter corresponds to a part of the monograph [DarnePalmerSoulie2022], joint with J. Darné and A. Soulié, which is accepted for publication in the Memoirs of the American Mathematical Society.

Next, turning from lower central series to representations of motion groups, in Chapter 5 we give a simple topological construction of the Burau representations of the loop braid groups. There are four versions: defined either on the non-extended or extended loop braid groups, and in each case there is an unreduced and a reduced version. Three are not surprising, and one could easily guess the correct matrices to assign to generators. The fourth is more subtle, and does not seem combinatorially obvious, although it is topologically very natural. This chapter corresponds to a joint paper with A. Soulié [PalmerSoulie2022] published in the Comptes Rendus Mathématique.

Chapter 6 is concerned with constructing homological representations of mapping class groups of surfaces, and corresponds to a joint paper with C. Blanchet and A. Shaukat [BlanchetPalmerShaukat2023] accepted for publication in Contemporary Mathematics. In previous work with the same co-authors [BlanchetPalmerShaukat2021], we constructed twisted representations of mapping class groups of surfaces, depending on a choice of representation V of the Heisenberg group H. For certain V we were able to untwist these mapping class group representations. In this chapter, we study the restrictions of our twisted representations to different subgroups of the mapping class group. Notably, we prove that these representations may be untwisted on the Torelli group for any given representation V of H. In the case when V is the Schrödinger representation, we also construct untwisted representations of subgroups defined as kernels of crossed homomorphisms studied by Earle and Morita.

In the final Chapter F, I describe various open problems and questions related to the topics of the thesis, some of which are immediately approachable and others of which are very difficult.