# EPFL Topology Seminar 2017/18

Tuesdays at 10:15

CM 1 113

### Program

Date Title Speaker
15.09.2017
PH H3 33
Separable and Galois extensions in tensor triangulated categories Bregje Pauwels
Australian National University
19.09.2017
CM 012
Configuration spaces of products Kathryn Hess
EPFL
03.10.2017
CM 012
Homotopically rigid Sullivan algebras and their applications David Méndez
EPFL & Universidad de Málaga
17.10.2017
CM 012
Towards the dual motivic Steenrod algebra in positive characteristic Martin Frankland
Universität Osnabrück
24.10.2017
CM 012
Understanding rational equivariant commutative ring spectra Magdalena Kedziorek
MPIM Bonn
31.10.2017
CM 012
Motivic homotopical Galois extensions (postponed) Kathryn Hess
EPFL
7.11.2017
CM 012
The spectrum of equivariant stable homotopy theory Beren Sanders
EPFL
14.11.2017
CM 012
Motivic infinite loop spaces Adeel Khan
Universität Regensburg
14.11.2017
ELE 111
Dessins d’enfants and the generalized Hurwitz numbers George Shabat
Moscow State University
21.11.2017
CM 012
Injective and projective model structures on enriched diagram categories Lyne Moser
EPFL
28.11.2017
CM 012
Genuine equivariant conjugation Jérôme Scherer
EPFL
05.12.2017
CM 012
A model for Motivic stable homotopy in classical homotopy Nicolas Ricka
Université de Strasbourg
12.12.2017
CM 012
Real topological Hochschild homology Irakli Patchkoria
Universität Bonn
06.02.2018
CM 1 113
Equivariant Monoidal structures on stable categories Denis Nardin
Université Paris 13
13.02.2018
CM 1 113
Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum Benjamin Böhme
University of Copenhagen
13.02.2018
CM 012 @ 13:00
The Dennis trace map Aras Ergus
EPFL
20.02.2018
CM 1 113
Classical and noncommutative Voevodsky’s conjecture for cubic fourfolds and Gushel-Mukai fourfolds Laura Pertusi
University of Milan
17.04.2018
CM 1 113
Infinity-operads via symmetric sequences Rune Haugseng
University of Copenhagen
24.04.2018
CM 1 113
Six operations formalism for generalized operads Benjamin Ward
Stockholm University
01.05.2018
CM 1 113
C*-superrigidiity of nilpotent groups Sven Raum
EPFL
08.05.2018
CM 1 113
Stratified homotopy theory Sylvain Douteau
Université de Picardie
15.05.2018
CM1 113 @ 09:15
Combinatorial models for stable homotopy theory Matija Bašić
University of Zagreb
15.05.2018
CM 1 113 @ 10:15
Galois actions, purity and formality with torsion coefficients Joana Cirici
University of Barcelona
22.05.2018
CM 1 113
On Mackey 2-functors Ivo Dell’Ambrogio
Université de Lille
05.06.2018
CM 1 104
Volume variation for representations of 3-manifold groups in SL_n(C) Wolfgang Pitsch
Universitat Autònoma de Barcelona
12.06.2018
MA A1 12 @ 11:15
From elementary abelian p-group actions to p-DG modules Marc Stephan
University of British Columbia
12.06.2018
MA A1 12 @ 15:15
Rational Parametrised Stable Homotopy Theory Vincent Braunack-Mayer
University of Zurich
19.06.2018
CM 1 113
Topology of robot motion planning Michael Farber
Queen Mary University of London
20.06.2018
CM 1 113
Intersection cohomology and spectra David Chataur
Université de Picardie Jules Verne
03.07.2018
CM 1 113
Lifting G-stable endotrivial modules Joshua Hunt
University of Copenhagen

(See also the program of the topology seminar in 2011/12,  2010/11,  2009/102008/09,  2007/08, 2006/07, and 2005/06.)

### Abstracts

Bregje Pauwels – Separable and Galois extensions in tensor triangulated categories

I will consider separable and Galois extensions of commutative monoids in tensor triangulated categories, and show how they pop up in various settings. In stable homotopy theory, separable extensions of commutative S-algebras have been studied extensively by Rognes. In modular representation theory, restriction to a subgroup can be thought of as extension along a separable monoid in the (stable or derived) module category. In algebraic geometry, separable monoids correspond to étale extensions of schemes, alowing us to define a generalized- étale site for any tensor triangulated category.

Kathryn Hess – Configuration spaces of products

I will explain the construction of a new model for the configuration space of a product of two closed manifolds in terms of the configuration spaces of each factor separately.  The key to the construction is the lifted Boardman-Vogt tensor product of modules over operads, developed earlier in joint work with Dwyer.

(Joint work with Bill Dwyer and Ben Knudsen.)

David Méndez – Homotopically rigid Sullivan algebras and their applications

The group of homotopy self-equivalences of a space is rarely trivial. Kahn was the first to obtain an example of one such space with non-trivial rational homology in the seventies. Later, Arkowitz and Lupton came across an example of a Sullivan algebra (equivalently, a rational homotopy type) with trivial homotopy self-equivalences. This algebra was used by Costoya and Viruel to solve Kahn’s group realisability problem for finite groups, thus obtaining for any finite group G a rational space X whose group of homotopy self-equivalences is isomorphic to G. This construction also provide a way to obtain an infinite amount of homotopically rigid spaces. However, they all share their level of connectivity with the example of Arkowitz and Lupton.

The objective of this talk is to illustrate that

(i) Homotopically rigid spaces are not as rare as they were though to be. We are able to obtain an infinite family of homotopically rigid spaces, showing a level of connectivity as high as desired.

(ii) Building blocks other than the example of Arkowitz and Lupton can be used to solve Kahn’s realisability problem.

We can also apply the obtained results to differential geometry by enlarging the class of inflexible manifolds existing in literature and building new examples of strongly chiral manifolds.

Reference: C. Costoya, D. Méndez, A. Viruel, Homotopically rigid Sullivan algebras and their applications, arXiv:1701.03705 [math.AT].

Martin Frankland – Towards the dual motivic Steenrod algebra in positive characteristic

Several tools from classical topology have useful analogues in motivic homotopy theory. Voevodsky computed the motivic Steenrod algebra and its dual over a base field of characteristic zero. Hoyois, Kelly, and Ostvaer generalized those results to a base field of characteristic p, as long as the coefficients are mod l with l \neq p. The case l = p remains conjectural.

In joint work with Markus Spitzweck, we show that over a base field of characteristic p, the conjectured form of the mod p dual motivic Steenrod algebra is a retract of the actual answer. I will sketch the proof and possible applications. I will also explain how this problem is closely related to the Hopkins-Morel-Hoyois isomorphism, a statement about the algebraic cobordism spectrum MGL.

Magdalena Kedziorek – Understanding rational equivariant commutative ring spectra

Recently, there has been some new understanding of various possible levels of commutative ring G-spectra. In this talk I will recall these possibilities and discuss the most naive (or trivial) commutative ring G-spectra. Then I will sketch the main ingredients coming into the proof that if G is finite and we work rationally these objects correspond to (the usual) commutative differential algebras in the algebraic model for rational G-spectra. This is joint work with David Barnes and John Greenlees.

Kathryn Hess – Motivic homotopical Galois extensions

(Joint work with Agnès Beaudry, Magdalena Kedziorek, Mona Merling, and Vesna Stojanoska)  I will sketch a formal framework for homotopical Galois extensions, motivated by the case of commutative ring spectra developed by Rognes,  within which we can prove invariance of Galois extensions under extension of coefficients and  the forward part of a Galois correspondence.  I will explain why both motivic spaces and motivic spectra fit into this framework, then provide explicit examples of motivic homotopical Galois extensions, some of which have no classical analogue.

Beren Sanders – The spectrum of equivariant stable homotopy theory

In this talk, I will discuss the spectrum of the G-equivariant stable homotopy category, for G a finite group.  In joint work with P. Balmer, we were able to describe this space, as a set, for all finite groups and gain a lot of information about its topology, obtaining a complete answer for groups of square-free order. We also reduced the problem of understanding the topology for arbitrary finite groups to understanding a specific question about the topology for p-groups.  Understanding this unresolved question for p-groups boils down to understanding an interesting phenomenon — that the Tate construction performs a chromatic “blue shift”.  Recently, T. Barthel, M.  Hausmann, N. Naumann, T.  Nikolaus, J. Noel and N. Stapleton have clarified such behavior, showing that an earlier “blue shift” conjecture of ours was too naive, and have thereby succeeded in computing the spectrum for all finite abelian groups. The task that remains is to find and prove a correct “blue shift” conjecture for nonabelian p-groups.

Adeel Khan – Motivic infinite loop spaces

Given a topological space X, May’s recognition principle says that a structure of infinite loop space on X is equivalent to a group-like $E_\infty$-monoid structure.  We will discuss an analogous result in motivic homotopy theory which says that a structure of infinite P^1-loop space on a motivic space X is equivalent to a homotopy coherent system of certain transfer maps.  This is based on an analogue of the Pontrjagin-Thom construction which identifies stable homotopy groups of spheres with groups of framed bordisms.  This is joint work with E. Elmanto, M. Hoyois, V. Sosnilo and M. Yakerson.

George Shabat – Dessins d’enfants and the generalized Hurwitz numbers

A multi-graph, embedded into a closed connected oriented surface, is called  a ‘dessin d’enfant’ if its complement is homeomorphic to a disjoint union of cells. The (appropriately defined) category of dessins d’enfants turned out to be equivalent to the category of ‘Belyi pairs’ that belongs to arithmetic geometry; a first part of the talk will be devoted to the foundations of this theory.

Then the certain problems of enumeration of dessins (and their generalizations) will be introduced. The particular cases of these problems, e.g., recursions for the so-called ‘Hurwitz numbers’, were studied intensively during the last decades; the corresponding recent results will be mentioned. However, the general case is currently out of reach, and during the second part of the talk a certain project, based on the above category equivalence, will be presented. Hopefully, realization of this project will promote the understanding of general case.

Lyne Moser – Injective and projective model structures on enriched diagram categories

K. Hess, M. Kedziorek, E. Riehl, and B. Shipley have developed methods to induce model structures from an adjunction which they apply to prove the existence of injective and projective model structures on categories of diagrams in accessible model categories. In this talk, I will explain how to adapt their proof to an enriched setting, in order to prove the existence of injective and projective model structures on some enriched diagram categories. I will talk in particular about the case of enriched diagrams from a small simplicial category to the category of pointed simplicial sets, and then generalize it by replacing the category of pointed simplicial sets by other symmetric monoidal categories, which are locally presentable bases and accessible model categories.

Jérôme Scherer – Genuine equivariant conjugation

This is joint work with Wolfgang Pitsch and Nicolas Ricka. I will first introduce conjugation spaces as they have originally been defined by Haussmann, Holm, and Puppe. Roughly speaking they are spaces equiped with a nice action by a cyclic group of order 2, so that the fixed points look like “half the space’’ through the eyes of mod 2 cohomology (think of real projective spaces as fixed points of complex ones under conjugation). I will then explain how tools from (genuine) equivariant stable homotopy theory allow us to give a more conceptual characterization of conjugation spaces in terms of purity. A priori one only looks at the mod 2 cohomology of a conjugation space as a graded vector space. Our approach highlights the compatibility with the cup product and the action of the Steenrod algebra.

Nicolas Ricka – A model for Motivic stable homotopy in classical homotopy

Morel-Voevodsky’s motivic stable category of schemes plays a crutial role in today’s stable homotopy. For instance, a recent work of Isaksen-Wang push our knowledge of stable stems up to dimension 93. In this joint work with Achim Krause and Bogdan Gheorghe, we construct a new model category, equivalent to the cellular motivic stable category, but whose construction lies entierly in the realm of classical stable homotopy theory. Moreover, this construction emphasizes the close relationship between cellular motivic homotopy theory and the theory of BP-resolutions. If time permits, I will talk about a new computation of the motivic dual Steenrod algebra using this new model, as well as the perspectives opened by similar constructions.

Irakli Patchkoria – Real topological Hochschild homology

This talk will define the real topological Hochschild homology (THR), introduced by Hesselholt and Madsen. THR is an invariant for rings with anti-involution and is a genuine Z/2-equivariant refinement of the classical topological Hochschild homology. THR approximates the real algebraic K-theory KR and hence on fixed points one gets an approximation for the Hermitian K-theory. In this this talk we will concentrate on foundations of THR. We will compare different models and discuss tools for computations. Along the way we will introduce necessary equivariant homotopy theory background. At the end we will compute the group of components of THR, THR of finite fields and the geometric fixed points of THR of integers. This work is joint with Dotto, Moi and Reeh.

Denis Nardin – Equivariant Monoidal structures on stable categories

Many important objects in homotopy theory can be endowed with genuine equivariant structures: algebraic K-theory, Thom spectra etc. In this talk we will explore how this interacts with the monoidal structures and what is one possible definition of an “equivariant monoidal structure”. As examples we will present how to put a Galois-commutative ring structure on algebraic K-theory and how to recover the S^1-equivariant structure on THH.

Benjamin Böhme – Multiplicativity of the idempotent splittings of the Burnside ring and the G-sphere spectrum

I provide a complete characterization of the equivariant commutative ring structures of all the idempotent summands of the G-equivariant sphere spectrum, including their Hill-Hopkins-Ravenel norms, where G is any finite group. My results describe explicitly how these structures depend on the subgroup lattice and conjugation in G. Algebraically, my analysis characterizes the multiplicative transfers on the localization of the Burnside ring of G at any idempotent element, which is of independent interest to group theorists. As an application, I obtain an explicit description of the incomplete sets of norm functors which are present in the idempotent splitting of the equivariant stable homotopy category.

Aras Ergus – The Dennis trace map

Algebraic K-theory is an invariant of rings (or in general, ring spectra) which appears in many areas of mathematics. An important approach to understanding algebraic K-theory is relating it to certain other invariants which are easier to compute. One such invariant is topological Hochschild homology, the analogue of the usual Hochschild homology in higher algebra. The aim of this talk is to give an exposition of the Dennis trace map from algebraic K-theory to topological Hochschild homology which is the starting point of the above mentioned “trace methods” in the study of algebraic K-theory.

Laura Pertusi – Classical and noncommutative Voevodsky’s conjecture for cubic fourfolds and Gushel-Mukai fourfolds

In a seminal paper, Voevodsky introduced the smash-nilpotence equivalence relation on the group of algebraic cycles on a smooth projective variety. He also conjectured that the nilpotence equivalence corresponds to the classical numerical equivalence on cycles. More recently, Bernardara, Marcolli and Tabuada defined a noncommutative version of this conjecture for smooth and proper dg categories. They proved the equivalence between the classical conjecture and their noncommutative version for the unique enhancement of the derived category of perfect complexes on a smooth projective k-scheme.

The aim of this talk is to prove Voevodsky’s conjecture for cubic fourfolds and generic Gushel-Mukai fourfolds. Then, we deduce the noncommutative version of this conjecture for the K3 subcategory appearing in the semiorthogonal decomposition of the derived category of perfect complexes on a cubic fourfold and on a generic GM fourfold, introduced by Kuznetsov and Kuznetsov-Perry. Finally, we apply this result to deduce Voevodsky’s conjecture for special classes of GM fourfolds. This is a joint work with Mattia Ornaghi.

Rune Haugseng – Infinity-operads via symmetric sequences

A useful description of operads is that they are associative monoids in symmetric sequences. I’ll discuss an analogous description of (enriched) infinity-operads; this gives rise to a bar-cobar adjunction for infinity-operads, with potential applications to Koszul duality.

Benjamin Ward – Six operations formalism for generalized operads

In this talk I will fill in an analogy between Verdier duality for sheaves and Koszul duality for algebras over operads.  To make this analogy precise we will consider the underlying categorical structure present in both situations.  Time permitting I will explain how understanding Koszul duality for modular operads from this perspective can be used to do computations in graph homology.

Sven Raum – C*-superrigidiity of nilpotent groups

It is a classical problem to recover a discrete group from various rings or algebras associated with it, such as the integral group ring (cf. the Whitehead group and the Whitehead torsion). By analogy, in an operator algebraic framework we want to recover torsion-free groups from certain topological completions of the complex group ring, such as the reduced group C*-algebra. Groups for which this is possible are called C*-superrigid. In recent joint work with Caleb Eckhardt, we could prove C*-superrigidity for arbitrary finitely generated, torsion-free, 2-step nilpotent groups by combining K-theoretic methods with certain bundle decompositions of C*-algebras.

In this talk, I will introduce the relevant notion of reduced group C*-algebras and put it in the context of the more familiar complex group algebra.  Further, I will provide a first motivation to study C*-superrigidity. Then I will discuss our result with Caleb Eckhardt, focusing on methods that have analogies in topology such as bundle decompositions and topological K-theory. The talk will finish with a persepective on relevance of C*-superrigidity in other areas of mathematics.

Sylvain Douteau – Stratified homotopy theory

Stratified spaces appear as natural objects in singularity theory. Goresky and MacPherson introduced intersection cohomology to extend cohomological properties of closed manifolds to stratified spaces, and it proved to be a powerful tool to study those objects. However, intersection cohomology is not homotopy invariant, rather it is invariant with respect to homotopies that “preserve” the stratification. This begs the question : does there exist a model category of stratified spaces which reflects this stratified notion of homotopy, and if so, is intersection cohomology representable in it?

We answer the first part of this question using a simplicial model category of filtered simplicial sets. As a category, it is only the category of simplicial sets over the classifying space of some fixed poset, but as a presheaf category, it inherits a model structure constructed using a natural cylinder object. We show that this category is simplicial, then we get stratified versions of Kan complexes and of homotopy groups that characterise fibrations and weak equivalences.

Matija Bašić – Combinatorial models for stable homotopy theory

We will recall the definition of dendroidal sets as a generalization of simplicial sets, and present the connection (Quillen equivalence) to connective spectra which gives a factorization of the so-called K-theory spectrum functor from symmetric monoidal categories to spectra. We will present a common generalization of  two results of Thomason: 1) posets model all homotopy types; 2) symmetric monoidal categories model all connective spectra. We will introduce a notion of multiposets (special type of coloured operads) and of the subdivision of dendroidal sets which can be used to show that multiposets model all connective spectra. If time permits we will mention homology of dendroidal sets as it provides a means to define equivalences of multiposets in an internal combinatorial way.

Joana Cirici – Galois actions, purity and formality with torsion coefficients

I will explain how to use the theory of weights on étale cohomology to study formality with torsion coefficients of objects arising from algebraic geometry. As an application, I will discuss formality with torsion coefficients of the dg-algebra of singular cochains of configuration spaces of points in the complex space. This is joint work in progress with Geoffroy Horel.

Ivo Dell’Ambrogio – On Mackey 2-functors

(Joint work with Paul Balmer.) We propose a categorification of the notion of Mackey functor for finite groups, intended to capture “higher” equivariant phenomena occurring throughout mathematics which are invisible to ordinary Mackey functors. Examples of such structures include familiar equivariant objects such as: (abelian and derived) categories of linear group representations, stable module categories, equivariant topological spectra, equivariant KK-theory of C*-algebras, etc. Among the phenomena covered we find: ambidexterity (a.k.a. Wirthmüller isomophisms), the separable monadicity of restriction functors, block decompositions, etc.  In this talk I will motivate our axioms with examples and explain the first results of the theory.

Wolfgang Pitsch – Volume variation for representations of 3-manifold groups in SL_n(C)

Let M be a compact oriented three-manifold whose interior is hyperbolic of finite volume. In recent years the variety of representations of the fundamental group \pi_1(M) into semisimple Lie groups, most notably SL_2(C) and SL_n(C) has attracted a lot of attention. It turns out that many invariants of the variety M such as the volume, the Reidemeister torsion etc. can be extended to arbitrary representations. In this talk I will concentrate on the Volume of a representation and show, in case M has toric boundary, how we can derive a formula for the variation of the volume along a differentiable path in the variety of representations. This is joint work with J. Porti.

Marc Stephan – From elementary abelian p-group actions to p-DG modules

A basic open problem in equivariant topology is to determine lower bounds for the sum of mod-p Betti numbers of finite CW-complexes with a free (Z/p)^n-action. For p=2, Carlsson connected chain complexes with a free (Z/p)^n-action to DG modules over a polynomial ring in order to establish lower bounds using commutative algebra.

In joint work with Jeremiah Heller, we extend this connection to all primes using p-DG modules as developed in the Hopfological algebra framework by Khovanov and Qi. In this talk, I will provide an introduction to Hopfological algebra and explain its relevance for our work.

Vincent Braunack-Mayer – Rational Parametrised Stable Homotopy Theory

Rational homotopy theory is a simplification of homotopy theory in which all torsion phenomena are systematically ignored. Under some mild hypotheses, celebrated results of Quillen and Sullivan provide complete descriptions of the rational homotopy category in terms of algebraic data. Quillen’s approach identifies the rational homotopy type of a 1-connected space with a dg coalgebra or, equivalently, with a dg Lie algebra, whereas Sullivan’s approach identifies the rational homotopy theory of nilpotent spaces of finite type with finite type cochain algebras. Stably, the situation is much simpler: the stable rational homotopy category is identified with graded rational vector spaces.
In this talk, I present recent results on the rational homotopy theory of parametrised spectra which unify these established models for stable and unstable rational homotopy theory. A parametrised spectrum is a family of spectra parametrised by a fixed parameter space, representing a cohomology theory twisted by an unstable homotopy type. After discarding torsion, I demonstrate that both Quillen’s and Sullivan’s approaches to rational homotopy theory can be lifted to provide algebraic characterisations of the rational homotopy category of spectra parametrised by a 1-connected space. The underlying idea is that whereas a parametrised X-spectrum P is a family of spectra twisted by or acted upon by X, after disregarding torsion this becomes the information of a graded rational vector space acted upon by an algebraic avatar of X.
I conclude by discussing an application of these results to M-theory, where we obtain a rational lift to M-theory of the twisted K-theory classification of D-brane charges in 10-dimensional superstring theory.

Michael Farber – Topology of robot motion planning

I will survey topological results relevant to the task of designing stable motion planning algorithms and will emphasise some outstanding challenges.

David Chataur – Intersection cohomology and spectra

Intersection cohomology has become an important tool in algebraic geometry and geometric representation theory. This theory has its roots in algebraic topology as a powerful gadget to restore Poincaré duality for spaces with singularities and extend the theory of characteritic classes from manifolds to singular spaces. A series of foundational questions remain open since its introduction in the beginning of the 1980’s by M. Goresky and R. MacPherson. In particular, they asked for a homotopical treatment of this theory. In this talk I plan to present a homotopical treatment based on joint works with M. Saralegui (Lens) and D. Tanré (Lille) and of recent results of S. Douteau (Amiens). As a byproduct, I will explain that intersection cohomology is representable by a spectrum in a stable homotopy category of stratified simplicial sets, a kind of stratified Eilenberg-MacLane spectrum. This answers positively to a problem of Goreky and MacPherson on the stratified homotopic representability of intersection cohomology, and could lead in the future to the construction of generalized intersection cohomology theories.

Joshua Hunt – Lifting G-stable endotrivial modules

Endotrivial modules are indecomposable kG-modules M for which End(M) splits as a trivial module direct sum a projective. Such modules play an important role in modular representation theory, and form an abelian group T(G) under tensor product. The structure of T(S) for S a finite p-group is known through many years of effort, culminating in the work of Carlson-Thévenaz. For an arbitrary finite group G, Balmer and Grodal have described the kernel of the restriction from T(G) to its Sylow T(S). In this talk we study the image of this restriction map. We provide a complete description when p = 2 and get partial information at odd primes. This verifies conjectures of Carlson-Mazza-Thévenaz when p = 2 but provides counterexamples at infinitely many odd primes. This is joint work with Tobias Barthel and Jesper Grodal.