Séminaire de topologie, géométrie et algèbre (archives)

Le feuilletage de Lawson est le premier feuilletage de codimension 1 sur la sphère de dimension 5, qui est construit par une modification du livre ouvert de Milnor associé à une singularité simple elliptique. Il est possible de munir chaque feuille de structure symplectique qui varie continuement, mais pour cela il faut changer la structure symplectique de la page de livre ouvert de Milnor qui vient canoniquememnt de C^3. La raison pourquoi le changement est possible est expliquée.

In joint work with Daniel Mathews, we examined complements of standard Seifert surfaces of special alternating links and enumerated those tight contact structures on them whose dividing sets are isotopic to the link. The number turns out to be the leading coefficient of the Alexander polynomial. The proof is rather combinatorial in nature; for example, the Euler classes of the contact structures are identified with `hypertrees' in a certain hypergraph. Using earlier results with Hitoshi Murakami and Alexander Postnikov, this yields a connection between contact topology and the Homfly polynomial. We also found that the contact invariants of our structures form a basis for the sutured Floer homology of the manifold.

Morsifications of real plane curve singularities, i.e., deformations with the maximal possible numer of real hyperbolic nodes, have been introduced in 70th by N. A'Campo and S, Gusein-Zade for computing important singularity invariants. We discuss the (still open) existence problem for morsifications, possible extensions to higher dimensions, and recently discovered relations to combinatorics of quivers that appears in the theory of cluster algebras. Based on joint works with P. Leviant and with S. Fomin, P. Pylyavskyy and D. Thurston.

On décrit un travail de Bourguiba, Lannes et Zarati (auquel l'orateur a modestement participé). À une action de 2-groupes abéliens élémentaires V sur un complexe fini X, on associe deux complexes, l'un géométrique, l'autre algébrique. Quand la cohomologie équivariante de X est libre comme H*V-module, ces deux complexes sont exacts et isomorphes. On décrit ainsi la cohomologie de X relative au lieu singulier de l'action, comme un foncteur de la cohomologie équivariante de X. On considèrera ensuite un cas particulier (Lannes, Hai, Nam, Schwartz, travail antérieur et en cours de rédaction) qui permet de donner une description "explicite" de certains spectres de Brown-Gitler.

Given two natural numbers n and k, consider the Thom space over the classifying space of a rank n elementary abelian 2-group associated to k copies of its real reduced regular representation. The Steinberg module of the general linear group gives rise to a stable summand of this Thom space, denoted by $L(n,k)$. Takayasu (1999) showed the existence of a cofibre sequences $$\Sigma^kL(n-1,2k+1) \to L(n,k) \to L(n,k+1),$$ which generalizes the stable splitting of Mitchell and Priddy. A cofiber sequence of the same form was proved by Arone and Mahowald by combining Goodwillie calculus with the James fibration.

I will describe in this talk how to derive the existence of the above cofibre sequences from the vanishing of some extension groups in the category of modules over the mod 2 Steenrod algebra.
This is joint work with Lionel Schwartz.

Étant donné une (p+q)-variété fermée munie d'un feuilletage F de dimension q, l'existence d'un feuilletage G supplémentaire, c'est-à-dire de codimension q et transverse à F, est bien sûr en général un problème insoluble; mais pour q>=2, si l'on affaiblit la condition de transversalité en demandant seulement que G soit limite de champs de p-plans transverses à F, je donnerai une version du h-principe de Gromov pour de tels feuilletages "quasi-supplémentaires". Il en résulte une preuve nouvelle du théorème de Mather-Thurston. Les outils, outre des méthodes classiques de construction de feuilletages, sont essentiellement la théorie des immersions de Smale, la théorie des rides d'Eliashberg-Mishachev, et une version fine de la cancellation des paires de singularités pour les fonctions de Morse.

We show that the mutant 2-component pretzel links P(p,q,-q,-p) and P(p,q,-p,-q) are not concordant for any distinct odd integers p and q greater than 1. As a corollary, we obtain a proof of the slice-ribbon conjecture for 4-stranded 2-component pretzel links. In order to distinguish mutant links up to concordance we consider 3-fold branched covers and use an obstruction based on Donaldson's diagonalization theorem. This is joint work with Min Hoon Kim, JungHwan Park and Arunima Ray.

We give simple conditions for a collection of rational homology spheres to be linearly independent in the rational homology cobordism group. These translate immediately to statements about knot concordance, recovering some results of Livingston and Naik. The key ingredient is correction terms (of either flavour). This is joint work with Kyle Larson.

Oeljeklaus-Toma manifolds are a higher dimension generalization of Inoue-Bombieri surfaces and were introduced by K. Oeljeklaus and M. Toma in 2005. They are quotients of H^s * C^t by discrete groups of affine transformations arising from a number field K and a particular choice of a subgroup of units U of K. They are commonly referred to as OT manifolds of type (s, t). OT manifolds have been of particular interest for locally conformally Kähler (lcK) geometry since they do not admit Kähler metrics, but those of type (s, 1) admit lcK metrics and for (s, t) in general, the existence of an lcK metric reduces to a numerical condition.

In this talk, we compute their de Rham and twisted cohomology and derive from this several characterization problems concerning their lcK geometry.