EPFL Topology Seminar 2017/18

Tuesdays at 10:15

CM 1 113

Program

(See also the program of the topology seminar in 2011/12,  2010/11,  2009/10, 2008/09,  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