# Workshop on Homology of Configuration Spaces and related topics

**May 15th--17th 2023**

Configuration spaces are classical objects in algebraic topology, but the study of their homotopy type remains a difficult question. The purpose of this conference is to bring together experts on that study, with a focus on the phenomenon known as homological stability. They will be invited to give talk about recent developments on the subject and we hope that the workshop will open up discussions about new avenues of research.

This three-day workshop will consist in talks from invited speakers and a mini-course on secondary stability. It will provide ample times for informal discussions.

## Practical information

### Schedule

Dates: May 15th to May 17th 2023.

Download the above calendar in iCal format here. Each talk will be 50 minutes long, with 10 additional minutes for questions.

### Registration

~~To register, please fill the following form: https://forms.office.com/r/kwm6CJXsvy before April 3rd.~~ Registration is now closed.

*Registration is free and open to all, but please register if you wish to attend.*

### Location

Street address: Campus Pierre et Marie Curie, 4 place Jussieu, 75006 Paris.

To get there, stop at the metro stop "Jussieu," which is located on lines 7 and 10. Maps and timetables can be found here. There are also several bus stops nearby.

To takes the metro or the bus inside Paris itself, either buy individual t+ tickets (2.20€), or buy an "easy pass" (2€) and load it with tickets (2.10€/ticket or 16.90€/10 tickets). If you are taking public transportation from/to outside Paris proper, e.g., the airport, you need another (more expensive) kind of ticket that is valid only for a single route. Of course, you can also walk, use the city's bike sharing offer Vélib (3€ for one trip, 20€ for a 3-day pass)... There are many options!

Room: 15-16-**101** (⚠️ different room than announced previously).

Once you have entered the campus, find either tower 15 or 16, climb the stairs up to the first floor, and enter the corridor 15-16 that connects the two towers to find room 101. Refer to the map below for the location.

Coffee breaks will be held in room 417, three floors above. Lunch will be at the brasserie L'Ardoise, also located in the Jussieu campus.

## Speakers

### Andrea Bianchi (Copenhagen University): Polynomial stability of the homology of Hurwitz spaces

This is joint work with Jeremy Miller. For a conjugation-invariant subset $Q$ of a finite group $G$ we consider the Hurwitz spaces $\mathrm{Hur}_n(Q)$, parametrising $G$-branched covers of the plane with exactly $n$ branch points and local monodromies in $Q$. We are interested in stability phenomena of the homology groups $H_i(\mathrm{Hur}*n(Q))$ for fixed $i$ and increasing $n$. There are several stabilisation maps $\mathrm{Hur} n(Q) \to \mathrm{Hur}{n+1}(Q)$, one for each element of $Q$; our main result is that for $n$ large enough (compared to $i$), each class in $H_i(\mathrm{Hur}*{n+1}(Q))$ is a combination of images of homology classes in $H_i(\mathrm{Hur}_n(Q))$ along (possibly different) stabilisation maps. Taking coefficients in a field $F$, we show that the dimension of $H_i(\mathrm{Hur}_n(Q);F)$ agrees, for $n$ large enough, with a quasi-polynomial function of $n$ of controlled period and degree. Our work extends previous results of Ellenberg--Venkatesh--Westerland and rely on techniques introduced by them and by Hatcher--Wahl.

### Luciana Bonatto (MPIM Bonn): Diff-equivariant homology of configuration spaces

We will discuss the equivariant homology of the space of configurations on a manifold with respect to the action of its diffeomorphism group. We show these exhibit homological stability and completely describe their stable homology. We also show how these techniques can be generalized to study the Diff-equivariant homology of configurations spaces with summable labels, giving equivariant homological stability for factorization homology.

### Rachael Boyd (University of Glasgow): Homology of diagram algebras

I will survey recent work on the homology of algebras, including joint work with Hepworth and Patzt. In this work, we abstract the notion of homological stability to sequences of algebras, and show that sequences of Temperley--Lieb, Brauer and Partition diagram algebras satisfy homological stability.

### Sophie d'Espalungue (Université Paris Cité): A tensor product for the interchange of categorical operads

Boardman and Vogt introduces the tensor product of (set theorical or topological) operads in order to represent the category of $P$-algebras *in* the category of $Q$-algebras as a category of algebras over an operad $P\otimes Q$, providing a better understanding of objects on which $P$ and $Q$ act in a compatible way. However, this construction failed to provide effective information about the structure of $n$-fold loop spaces in terms of its $n$ distinct compatible structures of $1$-fold loop spaces : while $Set$-theorical operads do not carry enought homotopical information to describe the structure of iterated loop spaces ($As \otimes As \cong Com$), topological operads lack of explicit description by generators and relations, so that as mentionned by Boardman and Vogt, the structure of $P\otimes Q$ is far from clear and few examples are known.

We suggest to fix those issues by considering operads defined in the $2$-category of categories, which we equip with a convenient system of generators and relations, so that in particular the categorical $E^n$-operads admit such a presentation. We construct a tensor product both at the level of operads and at the level of presentations in a compatible way, so that we obtain an explicit presentation of the tensor product of operads in terms of the presentation of each factors. We obtain isomorphisms $E^p \otimes E^q \cong E^{p+q}$ without cofibrancy hypothesis and deduce an explicit presentation of a cofibrant resolution $\hat E^n$ of each $E^n$. We finally provide an explicit $\hat E^n$-algebra structure on $n$-fold loop spaces by constructing a morphism of operads $\hat E^n \rightarrow (C^n)^\ast$, where $(C^n)^\ast$ is a categorical (hence truncated) version of the little $n$-cubes operad.

### Aurélien Djament (Université Sorbonne Paris Nord): Finitude homologique des foncteurs sur une catégorie additive

Les foncteurs (non nécessairement additifs) d'une petite catégorie additive vers une catégorie de modules interviennent dans plusieurs parties de l'algèbre et de la topologie ; ils forment une catégorie abélienne aux bonnes propriétés, notamment homologiques. Les calculs connus de groupes d'extensions entre tels foncteurs donnent ainsi accès à certains groupes de cohomologie stable de groupes linéaires à coefficients tordus. Dans cet exposé, nous nous intéresserons à des propriétés homologiques qualitatives des foncteurs, notamment l'existence de résolutions projectives de type fini. Nous présenterons des résultats démontrés récemment avec A. Touzé, qui fournissent de vastes généralisations de théorèmes obtenus par L. Schwartz, S. Betley et T. Pirashvili au début des années 1990. Ils s'appuient notamment sur des techniques simpliciales à la Dold--Kan--Puppe et sur un théorème de structure pour les foncteurs à valeurs dans des espaces vectoriels de dimension finie établi avec A. Touzé et C. Vespa.

### Nir Gadish (University of Michigan): A Serre spectral sequence for moduli spaces of tropical curves

The moduli space of genus g tropical curves with n marked points is a fascinating topological space, with a combinatorial flavor and deep algebro-geometric meaning. In the algebraic world, forgetting the n marked points gives a fibration whose fibers are configuration spaces of a surface, and Serre's spectral sequence lets one compute the cohomology "in principle". In joint work with Bibby, Chan and Yun, we construct a surprising tropical analog of this spectral sequence, manifesting as a graph complex and featuring the cohomology of compactified configuration spaces on graphs.

### Louis Hainaut (Stockholm University): The compactly supported cohomology of configuration spaces on wedges of spheres

For every positive integer $n$, the functor sending $X$ a finite wedge sum of spheres (of arbitrary dimensions) to the compactly supported cohomology of the configuration space of n distinct points on $X$ is equipped with a natural filtration, and its associated graded admits a factorization as the composition of the functor sending $X$ to its reduced cohomology, followed by a polynomial functor in this reduced cohomology.

I will explain how to prove this polynomiality result, and how it can be used to obtain new results in related topics, such as Hochschild--Pirashvili (co)homology, representation theory of $\mathrm{Out}(F_n)$, as well as top weight cohomology of the moduli space $\mathcal{M}_{2,n}$. (Joint with N. Gadish).

### Zachary Himes (University of Cambridge): Secondary homological stability and periodic homological stability for unordered configuration spaces

Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary stability for unordered configuration spaces of manifolds. The main difficulty is the compact case (the non-compact case was previously known by some experts). In the compact case, there are no obvious stabilization maps and the homology does not stabilize but is periodic. We resolve this issue by constructing a chain-level stabilization map for configuration spaces of compact manifolds.

### Geoffroy Horel (Université Sorbonne Paris Nord): An algebro-geometric model for configuration categories

The configuration category is a fine invariant of smooth manifolds that records the homotopy types of the configuration spaces of this manifold together with the data of collisions between points. In good cases, knowing the configuration categories of two manifolds is good enough to compute the homotopy type of the space of embeddings between them thanks to a technology called manifold calculus due to Goodwillie and Weiss. When the manifold is an algebraic variety over the complex numbers, I will give an algebro-geometric model for the configuration category using the theory of log-schemes. This implies the collapse of the Goodwillie--Weiss spectral sequence in some cases. One consequence of this collapse is an explicit conmputation of the universal finite type invariant of long knots in a 3-manifold of the form surface times $\mathbb{R}$. This is joint work with Pedro Boavida de Brito and Danica Kosanovic.

### Manuel Krannich (Karlsruhe Institute of Technology): Automorphisms of the little d-discs operad

The operad $E_d$ of little $d$-discs, whose space of $k$-ary operations is equivalent to the ordered configuration space of $k$ points in Euclidean $d$-space, is ubiquitous in homotopy theory and has become increasingly relevant in geometric topology. Nevertheless, our understanding of the space of its (derived) automorphisms is still fairly poor. I will give an overview of what we know and what we wish to know about this space of automorphisms, in parts based on joint works with Kupers as well as with Horel and Kupers.

### Dan Petersen (Stockholm University): Hyperelliptic curves, the scanning map, and moments of quadratic L-functions

I will explain a calculation of the stable homology of the moduli space of hyperelliptic curves, with coefficients in an arbitrary symplectic local system. The result is closely related to, and provides a geometric interpretation of, a series of conjectures on asymptotics of moments of families of quadratic L-functions. (Joint with J. Bergström, A. Diaconu and C. Westerland).

### Geoffrey Powell (Université d’Angers): Lie theory for functors on free groups

This talk develops the interplay between functors on free groups and modules over the Lie operad, motivated by examples coming from geometry. The base field is taken to be of characteristic zero and the above relationship can be seen as a generalization of the Schur--Weyl correspondence between representations of the general linear groups and of the symmetric groups; a significant difference is that the categories considered are not semisimple and have a rich structure.

I will start by explaining the relationship between polynomial functors on free groups and representations of the PROP associated to the Lie operad, in both the covariant and contravariant cases, and how to extend this to treat non-polynomial functors. This then applies to consider bifunctors on free groups, which allows various categories to be modelled.

The theory will be illustrated by examples, showing how the Lie theory provides insight and allows for generalization.

### Oscar Randal-Williams (University of Cambridge): Homological stability via higher algebra (mini-course)

There are many proofs of homological stability for unordered configuration spaces of a manifold with boundary. My main goal is to explain yet another one. It proceeds by considering the collection $C(M)$ of all configuration spaces of a $d$-manifold $M$ as a kind of module over the collection $C(D^d)$ of all configuration spaces of the $d$-disc, and analysing its "module indecomposables". These control how $C(M)$ is constructed as a $C(D^d)$-module, and I will explain how to deduce that "many" kinds of stability properties of $C(D^d)$ are also enjoyed by $C(M)$: certainly homological stability, but also "secondary homological stability" and so on. To finish, I will explain how the analogous method can be applied to many other situations which do not have anything to do with configurations.

The talks will, by some standards, have a somewhat abstract flavour: the abstraction is to clarify the general nature of the method. But they will be essentially self-contained.

## Registered participants

## Click here for the list of registered participants.

- Hossein Abbaspour (Université de Nantes)
- Alexis Aumonier (University of Copenhagen)
- Djament Aurélien (CNRS)
- Christian Ausoni (Université Sorbonne Paris Nord)
- Etienne Batelier (WWU Muenster, NRW, Germany)
- Andrea Bianchi (Copenhagen University)
- Ouriel Bloede (Université d'Angers)
- Luciana Bonatto (MPIM Bonn)
- Rachael Boyd (University of Glasgow)
- Adrien Brochier (Université Paris Cité)
- Luigi Caputi (University of Torino)
- Clovis Chabertier (Université Paris Cité)
- Sophie D'Espalungue (Université Paris Cité)
- Marie-Camille Delarue (Université Paris-Cité)
- Pierre Elis (University of Copenhagen)
- Coline Emprin (LAGA - ENS)
- Nathan Fauvette (Université Paris Cité)
- Antoine Feltz (University of Strasbourg)
- Vincent Franjou (Nantes Université)
- Nir Gadish (The University of Michigan)
- Thomas Gaujal (Université de Lille )
- Lorenzo Guerra (Università di Roma Tor Vergata)
- Yves Guiraud (INRIA, Université Paris Cité)
- Nicolas Guès (?)
- Louis Hainaut (Stockholm University)
- Rodrigue Haya Enriquez (UCLouvain)
- Lander Hermans (University of Antwerp)
- Kaif Hilman (MPIM Bonn)
- Zachary Himes (University of Cambridge)
- Eric Hoffbeck (Université Sorbonne Paris Nord)
- Geoffroy Horel (Université Sorbonne Paris Nord)
- Najib Idrissi (Université Paris Cité)
- Hyeonhee Jin (MPIM Bonn)
- Florian Kranhold (Karlsruhe Institute of Technology)
- Manuel Krannich (Karlsruhe Institute of Technology)
- Christian Kremer (Max Planck Institute for Mathematics Bonn)
- Guillaume Laplante-Anfossi (University of Melbourne)
- Erik Lindell (Stockholm University)
- Muriel Livernet (université paris cité)
- João Lobo Fernandes (Karlsruhe Institute for Technology)
- Jean-Pierre Magnot (université d'Angers)
- Zhouhang Mao (Institut Mathématique d'Orsay)
- Verstraete Marvin (Université de Lille)
- Jan McGarry Furriol (Stockholm University)
- Fadi Mezher (University of Copenhagen)
- Samuel Muñoz Echániz (University of Cambridge)
- François Métayer (IRIF)
- Dan Petersen (Stockholm University)
- Azélie Picot (ENS )
- Andrea Pizzi (University of Rome 'Tor Vergata')
- Hugo Pourcelot (Université Sorbonne Paris Nord)
- Geoffrey Powell (Université d’Angers)
- Francesca Pratali (Université Sorbonne Paris Nord)
- Oscar Randal-Williams (University of Cambridge)
- Tommaso Rossi (Università di Roma Tor Vergata)
- Ismael Sierra (University of Cambridge)
- Christopher-Lloyd Simon (The Pennsylvania State University)
- Arthur Soulié (IBS-CGP)
- Andreas Stavrou (University of Cambridge)
- Robin Stoll (Stockholm University)
- Daniel Tanré (Université de Lille)
- Bruno Vallette (Université Sorbonne Paris Nord)
- Christine Vespa (Aix-Marseille Université)
- Nicholas Wawrykow (University of Michigan)
- Leo Yoshioka (Graduate School of Mathematical Sciences, The University of Tokyo)
- Claudia Yun (MPI MiS Leipzig)

This list is updated manually. If you registered recently, it may not be reflected here yet.

## Organizers

- Adrien Brochier (Université Paris Cité & IMJ-PRG)
- Yves Guiraud (INRIA & IMJ-PRG)
- Najib Idrissi (Université Paris Cité & IMJ-PRG)
- Christine Vespa (Aix-Marseille Université & I2M)

## Thanks

This workshop is possible thanks to the support of the ANR project "Structure and Homotopy of Configuration Spaces" (SHoCoS), the Institut de Mathématiques de Jussieu, Université Paris Cité, and Sorbonne Université.

We thank very much Claire Lavollay (IMJ-PRG) for handling the administrative side of this conference.

The background of this webpage was generated by DALL-E 2 after the prompt "impressionist oil painting of a group of researchers watching a mathematician give a talk at a blackboard."