EPFL Topology Seminar Spring 2024

 Location: Room MA B1 524 (Sometimes Zoom)

(For questions about the seminar, please contact the organizers:
 

Program

Date Time Place Title Speaker
20.02.2024 14:00 CET MA B1 524 Computing motivic homotopy types of families of hypersurfaces with polyhedral products William Hornslien, NTNU
27.02.2024 14:00 CET MA B1 524 Lie coalgebraic dual of a group, Milnor invariants, and linking of letters Dev Sinha, University of Oregon (Zoom).
05.03.2024     No seminar. No seminar. 
12.03.2024     No seminar. 

No seminar. 

19.03.2024 14:00 CET CM 517 Differentiable rigidity for embeddings: Haefliger’s examples Jean-Claude Hausmann, Université de Genève
26.03.2024 14:00 CET CM 517
Regularity for E1 rings, schemes, and general triangulated categories, with new obstruction results on bounded t-structures
Ruradip Biswas, Univrsity of Warwick
02.04.2024     No seminar.  No seminar.
09.04.2024 14:00 CET CM 517 Higher parametrized semiadditivity Sil Linskens, University of Bonn
16.04.2024 16:00 CET (Different !) CM 517 

Traces of dualizable categories and functoriality of the Becker-Gottlieb transfers

Marco Volpe, University of Toronto (Zoom).
23.04.2024 14:00 CET CM 517

Perfection and homotopy type

Benjamin Antieau, Northwestern University (Zoom). 

30.04.2024    

No seminar. 

No seminar.

7.05.2024 14:00 CET CM 517

Hermitian K-groups and motives

Paul Arne Oestvaer, University of Milan
14.05.2024 14:00 CET CM 517 A counterexample to the nonconnective theorem of the heart Vladimir Soslino, University of Regensburg
21.05.2024 14:00 CET CM 517 Posets of Modalities in Higher Topoi Eric Finster, University of Birmingham
28.05.2024 14:00 CET CM 517 Grothendieck lax construction Felix Loubaton, MPIM

4.06.2024

14:00 CET CM 517
 
The infinitesimal tangle hypothesis
Joost Nuiten, Université de Toulouse

11.06.2024

14:00 CET CM 517 The root functor Francesca Pratali, Université Paris XIII

18.06.2024

14:00 CET CM 517 TBA Thomas Willwacher, ETH

25.06.2024

14:00 CET CM517 TBA  Dan Petersen, University of Stockholm

Abstracts:

William Hornslien

Computing motivic homotopy types of families of hypersurfaces with polyhedral products

Polyhedral products are certain natural subspaces of products of CW complexes constructed from the combinatorial information of a simplicial complex. They play an important role in fields such as: toric varieties, homotopy theory, algebraic combinatorics, and robotics. Motivic homotopy theory is a homotopy theory for smooth algebraic varieties. Given a polynomial, it is in general not an easy task to figure out the homotopy type of the associated algebraic hypersurface. In this talk we will study a family of hypersurfaces by modeling them as polyhedral products. To do this, we generalize polyhedral products to an oo-categorical setting and prove some general results before applying them to our motivic problem.

Dev Sinha

Lie coalgebraic dual of a group, Milnor invariants, and linking of letters

(joint with Nir Gadish, Aydin Ozbek and Ben Walter) Consider two homomorphisms f, g from a free group to the rational numbers. One can then define a homomorphism {f}g on the commutator subgroup of the free group by {f}g ( [v,w] ) = f(v) g(w) – f(w) g(v). This generalizes to the notion of Lie coalgebraic dual of a group, a framework for all groups which encompasses Magnus expansion and Fox derivatives for free groups. The universal Lie coalgebraic dual of a group pairs with the lower central series Lie algebra, giving a complete set of functionals over the rational numbers for any group and a perfect duality in the finitely generated setting.

Jean-Claude Hausmann

Differentiable rigidity for embeddings: Haefliger’s examples

André Haefliger (1929-2023), a former UNIL undergraduate student, became famous in 1960 for constructing smooth embeddings (for example of S^3 into R^6) which are not smoothly isotopic, even though they are topologically isotopic. Such a “differentiable rigidity” (after that for diffeomorphisms discovered by Milnor) indeed astonished the mathematical community. We will present these examples of smooth rigidity and the ideas of Haefliger’s proof, after putting them into perspective.
 
 
Rudradip Biswas
 
Regularity for E1 rings, schemes, and general triangulated categories, with new obstruction results on bounded t-structures
 
For any essentially small triangulated category S satisfying a finiteness property (in our language, if its opposite category has finite “finitistic dimension”), we show that the existence of a bounded t-structure on S implies that it is invariant under Neeman-completion. This gives a big abstract generalization of Neeman’s own new work on the existence of bounded t-structures on perfect complexes of schemes [2] where he, using very different methods, proved a conjecture due to Antieau, Gepner, and Heller. Neeman had also shown that the derived bounded category of coherent sheaves, under some restrictive conditions on the scheme, only has one bounded t-structure up to “equivalence”. We improve and abstractly generalize this result by showing that under the same general starting conditions mentioned at the start of the paragraph, all bounded t-structures on any intermediate category between the starting category and its completion are equivalent. Perfect complexes can be completed to the derived bounded category of coherent sheaves – so our result, when specialized to schemes, improves the existing Neeman result by removing all the restrictive hypotheses on the scheme. 
 
Applying our treatment to triangulated categories arising in the study of E1 rings, DG algebras, artin algebras, etc, we get many applications and new results. The word “regularity” is used in the title because one can define the singularity category of a general triangulated category in the language of completion (we can then define regularity as corresponding to the vanishing of this singularity category), and our work shows that this singularity category is an obstruction to the existence of bounded t-structures. 
 
 
Sil Linskens
 
Higher parametrized semiadditivity
 
Given a semiadditive category, every object uniquely obtains the structure of a commutative monoid. This simple fact turns out to be crucial in the higher categorical context, where endowing an object with the structure of a commutative monoid can be highly nontrivial endeavour. More recently, the notion of higher semiadditivity has attracted considerable attention, primarily because many categories appearing in chromatic homotopy theory are higher semiadditive. Higher semiadditivity again automatically endows objects with considerable algebraic structure which has been crucial for many recent breakthroughs in chromatic homotopy theory, for example the recent disproof of the telescope conjecture. However to leverage this algebraic structure we require a “generator-and-relations style” description of the operations it entails. This is given by a theorem of Harpaz, which gives a formula for the universal higher semiadditive category on a point. In this talk I will discuss a parametrised analogue of higher semiadditivity, which simultaneously subsumes higher semiadditivity and the Wirthmüller isomorphisms in equivariant homotopy theory. I will then present the analog of the theorem of Harpaz. This is joint work with Bastiaan Cnossen and Tobias Lenz.
 
 
 
 
Marco Volpe 

Traces of dualizable categories and functoriality of the Becker-Gottlieb transfers

For any fiber bundle with compact smooth manifold fiber X ⟶ Y, Becker and Gottlieb have defined a “wrong way” map S[Y] ⟶ S[X] at the level of homology with coefficients in the sphere spectrum. Later on, these wrong way maps have been defined more generally for continuous functions whose homotopy fibers are finitely dominated, and have been since referred to as the Becker-Gottlieb transfers. It has been a long standing open question whether these transfers behave well under composition, i.e. if they can be used to equip homology with a contravariant functoriality.

In this talk, we will approach the transfers from the perspective of sheaf theory. We will recall the notion of a locally contractible geometric morphism, and then define a Becker-Gottlieb transfer associated to any proper, locally contractible map between locally contractible and locally compact Hausdorff spaces. We will then use techniques coming from recent work of Efimov on localizing invariants and dualizable stable infinity-categories to construct fully functorial “categorified transfers”. Functoriality of the Becker-Gottlieb transfers is then obtained by applying topological Hochschild homology to the categorified transfers.

If time permits, we will also explain how one can use similar methods to extend the Dwyer-Weiss-Williams index theorem for compact topological manifolds fiber bundles to proper locally contractible maps. In particular, this shows that the homotopy fibers of a proper locally contractible map are homotopy equivalent to finite CW-complexes. Therefore, it is still unclear whether functoriality of the transfers associated to maps with finitely dominated homotopy fibers holds.

This is a joint work with Maxime Ramzi and Sebastian Wolf.

 

Benjamin Antieau

Perfection and homotopy type

I will discuss the role of perfection in classifying homotopy types by their integral cohomology, building on work of Sullivan, Kriz, Mandell, and Yuan. The main idea involves binomial rings, which in the p-complete case are delta-rings for which the associated Frobenius endomorphism is the identity. This idea has been discovered independently by Horel and by Kubrak—Shuklin—Zakharov.

 
 
 
Paul Arne Øestvær
 
Hermitian K-groups and motives
 
Hermitian K-theory is closely related to the classical theory of quadratic forms. We will give an overview of recent calculations of higher Hermitian K-groups of fields and rings of integers in number fields through motivic homotopy theory. The answers involve arithmetic data, while the calculational methods are rooted in homotopy theory. Joint work with Haakon Kolderup, Jonas Kylling, and Oliver Roendigs.
 
 
 
Vladimir Soslino 
 
A counterexample to the nonconnective theorem of the heart
 

Let C be a stable infinity-category equipped with a bounded t-structure with the heart denoted by A. Antieau, Gepner, and Heller conjectured that the map of nonconnective K-theory spectra K(A) —-> K(C) is always an equivalence. Barwick’s theorem of the heart implies that this map is an equivalence on connective covers, and both sides are known to be connective if A is a noetherian abelian category.

To any spectrum M we functorially assign a stable infinity-category C_M such that the spectrum K(C_M) is equivalent to M.

Using this result and some basic chromatic homotopy theory, we construct a counterexample to the conjecture above.

 

Eric Finster

Posets of Modalities in Higher Topoi
 
A *modality* in a higher topos can be described as a (unique) factorization system in which the factorization of a map is stable under base change.  Modalities can be ordered by the inclusion of the left or right classes and I will describe some properties of (and structures on) the resulting posets.  The sub-poset of *left-exact* modalities, in which the factorization is stable under all finite limits, correspond bijectively with sub-topoi, and are therefore of particular interest.
 
 
 
Félix Loubaton
 
Grothendieck lax construction
 
The Grothendieck construction is a fundamental of category theory. In this talk, I will give a generalization of it in the framework of $(\infty,\omega)$-categories. To this end, I’ll introduce some important concepts of $(\infty,\omega)$-category theory, such as lax transformations, lax (co)limits, and lCartesian fibrations. Finally, I’ll explain how this result can be used to give explicit calculations of Kan lax extensions.
 
 
 
Joost Nuiten 
 
The infinitesimal tangle hypothesis
 
The tangle hypothesis is a variant of the cobordism hypothesis that considers cobordisms embedded in some finite-dimensional Euclidean space (together with framings). Such tangles of codimension d can be organized into an E_d-monoidal n-category, where n is the maximal dimension of the tangles. The tangle hypothesis then asserts that this category of tangles is the free E_d-monoidal n-category with duals generated by a single object. In this talk, based on joint work in progress with Yonatan Harpaz, I will describe an infinitesimal version of the tanglehypothesis: Instead of showing that the E_d-monoidal category of tangles is freely generated by an object, we show that its cotangent complex is free of rank 1. This provides support for the tangle hypothesis (of which it is a direct consequence), but can also be used to reduce the tangle hypothesis to a statement at the level of E_d-monoidal (n+1, n)-categories by means of obstruction theory.
 
 
Francesca Pratali
 
The root functor
 
By a well known result of Joyal, any oo-category is equivalent to the localization of the nerve of a strict category. In today’s talk, I will explain how one can generalize this result to the operadic case. More specifically, by working with the dendroidal formalism we show how, given any dendroidal set X, we can functorially construct an operad \Omega/X whose dendroidal nerve admits a map into X. We call this the ‘root functor’. After explaining how this extends an analogous map built for simplicial sets, we prove that it induces a  weak operadic equivalence between X and the localization of the nerve of \Omega/X  at a certain class of 1-morphisms. In a nutshell, we prove that any oo-operad is the localization of the nerve of a strict operad! If time permits, we will conclude the exposition by presenting possiblefuture applications.