Archives Séminaire de topologie algébrique

Grégoire Naisse (MPIM Bonn), Nov 24

Title: A construction of odd Khovanov homology from classical algebraic topology

Slides

Teams

Khovanov homology is a well-studied link homology theory categorifying the Jones polynomial. There exists many constructions of it (higher rep. theory, algebraic geometry, ...), but it was originally combinatorially defined using a TQFT. It is well-known that cohomology of nice even-dimensional manifolds give Frobenius algebras, and thus TQFTs. In particular Khovanov homology TQFT comes from the complex projective line CP^1.
By constrast, odd Khovanov homology, a mysterious cousin of (even) Khovanov homology defined by Osvath, Rasmussen and Szabo also categorifying the Jones polynomial, is by far less understood. In particular, ORS construction is ad hoc. In a attempt to understand better its origin, we gave a construction of odd Khovanov homology based on the geometry of the *real* projective line RP^1.
My goal for this talk will be to explain to you how one gets a TQFT from the cohomology of a nice even-dimensional manifold, and how this can be generalized to an odd dimensional one, giving a chronological TQFT. For this, we will need to explore some classical results of algebraic topology (in the first part of the talk) and how graded monoidal categories work (in the second part).

No knowledge of link homology theory is needed for this talk.

This talk based on a joint work with Jens N. Eberhardt and Arik Wilbert.

Léo Schelstraete (UCLouvain), Oct 13

Title: Supercategorification and Odd Khovanov Homology

Slides

In 2000, Khovanov discovered a new invariant of knots, now called Khovanov homology. As the name suggests, it associates to any knot a certain graded homology. In some sense, this construction gives a categorification of the Jones polynomial: indeed, the Jones polynomial turns out to be the Euler characteristic of Khovanov homology. On the other hand, the Jones polynomial is a quantum invariant, as it can be derived through the representation theory of the quantum algebra Uq(sl2). Later works by Webster and Lauda, Queffelec and Rose showed that the two approaches can be unified: one can use a categorification of the representation theory of Uq(sl2) to derive Khovanov homology.

In 2013, Oszváth, Rasmussen and Szabó constructed a different version of Khovanov homology using exterior algebras, called odd Khovanov homology. This construction also categorifies the Jones polynomial, and over Z/2Z the resulting invariant coincides with (even) Khovanov homology. However, the two homologies are distinct. In March 2020, Naisse and Putyra extended odd Khovanov homology to tangles. At the time being though, no representation theory construction similar to the even case is known. Exploring this question is the goal of my current research.

The end goal of this talk is to introduce the invariant of knots developed in my master's thesis, defined thanks to a categorification of a quantum algebra and conjectured to coincide with odd Khovanov homology. This construction uses the categorical notion of 2-supercategories, hence the name supercategorification. The first part of the talk (~35 min) aims to introduce Khovanov homology and the idea of categorification and categorified quantum algebras, only giving hints toward my work. Prerequisites are at the master's level (notably a sense of what is the homology of a topological space and what is a category). The second part of the talk (~25 min) delves deeper into the construction of odd Khovanov homology through supercategorification.

Note that no acquaintance with quantum algebras is needed to follow any part of the talk.

Jacques Darné (UCLouvain), Jun 15

Title: Group filtrations and invariants: braids and generalizations

Slides

Milnor invariants of degree at most d distinguish braids up to elements of the (d+1)-th term of the lower central series of the pure braid group. The goal of this talk will be to present a proof of this classical result. This will involve talking about group filtrations and how they encode invariants such as finite-type invariants of braids and their generalizations. We will also discuss results and conjectures about similar statements in various contexts.

Najib Idrissi, Université Paris Diderot, Jul 04

Title: Homologie de factorisation et espaces de configuration

L'homologie de factorisation est une théorie homologique pour les variétés structurées (orientées, parallélisées...) qui trouve ses origines dans les théories topologique et conformes des champs (Beilinson–Drinfeld, Salvatore, Lurie, Ayala–Francis, Costello–Gwilliam...). Après l'avoir définie et donné une idée de ses propriétés, j'expliquerai comment on peut la calculer sur ℝ grâce au modèle de Lambrechts--Stanley des espaces de configuration et/ou grâce à des complexes de graphes dans le cas des variétés fermées parallélisées, des variétés fermées orientées, et des variétés à bord parallélisées. [En partie en collaboration avec R. Campos, J. Ducoulombier, P. Lambrechts, T. Willwacher]

Cristina Ana-Maria Palmer-Anghel, Univesrity of Oxford, Mar 20

title: Topological models for Uq(sl2)-quantum invariants

The theory of quantum invariants for knots started with the Jones polynomial. After that, Reshetikhin and Turaev introduced an algebraic method which having as input a quantum group, leads to a link invariant. The representation theory of Uq(sl(2)) at generic q leads through this method to a family of link invariants {JN(L, q)}_{N∈\mathbb{N}} called coloured Jones polynomials. The first term of this sequence is the original Jones polynomial. On the other hand, the quantum group Uq(sl(2)) at roots of unity leads to a sequence of invariants, called coloured Alexander polynomials, having the original Alexander invariant as the first term. On the topological side, R. Lawrence introduced a sequence of braid group representations based on the homology of coverings of configurations spaces and using these, Bigelow and Lawrence gave a homological model for the original Jones polynomial. We present a topological model for all coloured Jones polynomials, showing that they can be described as graded intersection pairings between two homology classesin a covering of the configuration space in the punctured disc. If time permits, we will show some directions towards a topological description for Uq(sl(2)) quantum invariants at roots of unity.

Krzysztof Putyra (UZH), march 13

title: An equivalence between gl(2)-foams and cobordisms

The original construction of the Khovanov homology of a link can be seen as a formal complex in the category of flat tangles and surfaces between them. There is a way how to associate a chain map with a link cobordism, but only up to a sign. Blanchet has fixed this by introducing the category of gl(2)-foams, certain singular cobordisms between planar trivalent graphs. Originating in the representation theory of quantum groups, foams are usually thought as algebraic objects. In my talk I will bring topology back by interpreting foams as two surfaces transverse to each other. This description leads to a quick proof that gl(2)-foams can be evaluated, a construction of a natural basis of foams, and an explicit equivalence between the category of gl(2)-foams and cobordisms between flat tangles. This is a joint work with Anna Beliakova, Matthew Hogancamp and Stephan Wehrli.

Federico Cantero Morán (BGSMath), Nov 22

Title: Low-dimensional spatial refinements of Khovanov functors

Khovanov spectra of links were introduced in 2014 by Lipshitz and Sarkar. They are link invariants up to stable homotopy, and their cohomology is isomorphic to Khovanov homology. Their construction was later simplified in a joint work with Lawson. In this talk we will explain how to give a further simplification in the first non-trivial quantum gradings. This is joint work with Marithania Silvero.

Yves Félix (UCL), Oct 19

Title : The rational homotopy of mapping spaces - an introduction to the subject

We will begin with a description of the approaches of Haefliger and Brown-Szczarba, and ends with a presentation of Lie models in the spirit of Lazarev-Markl, Getzler and Berglund)

Miradain Atontsa Nguemo (UCL) , Oct 11

Title: Algebraic characterization of homogeneous functors

Goodwillie developped, in a serie of papers in 1990, 1992 and 2003, an approach to give polynomial approximations of functors F: Top ----> Top. In the Taylor tower that he has constructed:

F ... ----->P_nF -----> P_n-1 F ------> .... ------> P_1F

he was able to characterize explicitely the homotopy fibres D_nF of the maps in the tower. In this talk, I will give an overall idea of the Goodwillie approximation for functors F: C ----> D, when C and D are any of the model categories: "Chain complexes", "DGL=differential graded Lie algebras", "Alg_O=Algebras over the operad O". I will end by giving the characterization of homogeneous functors from C to D.

David Méndez (University of Malaga), Oct 04

Title: Realising groups in arrow categories of graphs and spaces

Whenever an algebraic structure arises in any mathematical context, it is natural to ask how to characterise the algebraic objects that may appear in that context. For instance, given any category, we may ask ourselves which groups can appear as automorphism groups of objects in that category. The Kahn realisation problem goes in that direction: it asks which groups may appear as the group of self-homotopy equivalences (that is, the automorphism group in HoTop) of a space X. A partial solution was attained by Costoya-Viruel in 2014, showing that every finite group arises as self-homotopy equivalences of a space (which is in fact rational). In this talk we consider the more general problem of realising groups in arrow categories of a given category C, Arr(C). In Arr(C), objects are arrows in C and morphisms between two arrows are commutative squares, thus given by two arrows in C. In this context, we ask: Given G1, G2 groups and H a subgroup of G1xG2, is there an arrow f:X1--->X2 in C such that Aut(Xi)=Gi and Aut(f)=H? We build a solution in the category of Graphs and show how to translate it to spaces, by means of algebraic models of rational homotopy types of spaces.