We will host a series of lectures in geometric topology from 30 March to 2 April, taking place in the afternoons.
Speakers
Alexis Aumonier
Marie-Camille Delarue
Christian Kremer
Fadi Mezher
Each speaker will give 2–3 lectures. The series will explore applications of homological stability, embedding calculus, and Goodwillie calculus to geometric topology.
Rooms
30 March — DDW 1.30
31 March — KBG Atlas
1 April — DDW 1.30
2 April — KBG Atlas
Abstracts
Speaker: Alexis Aumonier
Title: Calculus of mapping spaces and applications to some moduli spaces
Abstract: The calculus of functors gives us natural towers that can be used to perform computations, compare functors one bit at a time, or even simply help us ask questions. The goal of my lectures will be to present examples where these ideas are applied to moduli spaces. I will particularly focus on the Goodwillie towers of mapping spaces, showing old and new results.
Roughly, I will cover the following topics:
1- Some basics of Goodwillie calculus in one and two variables, and the tower of the suspension spectrum of a section space.
2- Moduli spaces of maps and how to scan them using Goodwillie calculus.
3- Interactions between Goodwillie and Weiss calculus.
Speaker: Marie-Camille Delarue
Title: Scanning in stable homology
Abstract: The homology groups of mapping class groups of surfaces are quite difficult to compute individually, and it came as a surprise when Tillmann showed the homology stabilized to that of an infinite loop space. The infinite loop space in question was conjectured by Madsen and Tillman and computed by Madsen and Weiss using scanning methods. The idea of constructing a scanning map coalesced from the Pontryagin—Thom construction and McDuff and Segal’s work on configuration spaces and homotopy types of spaces of holomorphic functions of certain closed surfaces. This map compares a space of embeddings of certain objects (such as surfaces) into high-dimensional space to the « local pictures » of said embedded objects. We will recall this construction and the main steps needed to prove the Madsen—Weiss theorem. We will also explain how to adapt this idea to various other settings. For instance, Galatius computes the stable homology of automorphism groups of free groups by applying these techniques to a case where the embedded objects are not manifolds. We may also in this way compute the homology of generalized Thompson groups, also known as Stein’s groups.
Speaker: Christian Kremer
Title: h-Principles, Smoothing Theory and Dimension 4
Kirby-Siebenmann's smoothing theory provides a homotopy theoretic description of the "difference" of the category of topological and smooth manifolds. It is an indespensible tool in the modern study of moduli spaces of manifolds, and an important instance of the h-principle. It is well known that smoothing theory in dimension four is much more subtle, and Kirby-Siebenmann's machinery does not extend to this case.
In my series of talks, I will give a general overview over h-principles, aimed at homotopy theorists, with smoothing theory being the main example. I will give an overview over its failure in dimension 4, and present some joint ongoing work with Sander Kupers about a "stabilised" version of smoothing theory in that case.
Speaker: Fadi Mezher
Title: Arithmeticity and residual finiteness of mapping class groups of high dimensional manifolds.
Abstract: Given a smooth manifold M, the smooth resp. topological mapping class groups of M, i.e. the groups of diffeomorphisms resp. homeomorphisms considered up to isotopy, are central objects in geometric topology. In the current lectures, we plan to investigate the properties of these discrete groups, with a particular emphasis on studying the difference between the smooth and topological category. Of particular interest is the question of arithmeticity; the above two groups come close to being arithmetic groups. Upon further investigation, there arises an interesting interaction between the Kervaire-Milnor exotic spheres and the failure of the smooth mapping class group to be an arithmetic group. In contrast, we show that the topological mapping class groups are arithmetic for a large class of manifolds. The talks will combine techniques from embedding calculus, surgery theory, rational homotopy theory and profinite homotopy theory.
Talk 1: Group theoretic preliminaries. In this talk, I would like to discuss the main theme for the following three talks. Given a smooth manifold M, we consider the two incarnations of the mapping class group of M, namely \pi_0 Homeo(M) and \pi_0 Diff(M); our general outlook will be to compare these two discrete groups, starting from classical results. I will discuss arithmetic groups, and discuss how close these two groups come to being arithmetic, using rational homotopy theory and surgery. We then introduce the notion of residual finiteness and discuss why this notion obstructs the smooth mapping class group from being arithmetic; the obstructions take the form of exotic spheres, an object of central importance in geometric topology.
Talk 2: Embedding calculus. We will start by motivating this talk by discussing the theorem of Sullivan and Serre regarding the group of homotopy automorphisms of a simply connected finite space. We then construct the embedding calculus tower as a tower of functors from the category of manifolds to certain presheaves on discs, filtered by arity. We discuss smoothing theory in the context of embedding calculus. We identify the layers of this tower, and discuss convergence for the embedding calculus tower, in various incarnations (namely, for manifolds with boundary, and for the category of topological manifolds which admit smooth structures).
Talk 3: Main results. We discuss the Weiss fibre sequence, and discuss how it allows us to employ the embedding calculus tower in order to tackle the question of residual finiteness of mapping class groups of certain manifolds. We show the main result, stating that \pi_0 Homeo(M) is a residually finite group (under certain mild assumptions on M). Time permitting, we show how the techniques that arise in the above proof allows us to answer specific questions about the detectability of exotic spheres in embeding calculus.