Séminaire de topologie, géométrie et algèbre (archives)

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.

We discuss a link between the geometry of hyperbolic three-manifolds and their Floer theoretic invariants, provided by spectral geometry. In particular, we provide sufficient conditions for a hyperbolic three-manifold to be an L-space (i.e. the Floer homology group has the least possible rank) in terms of its volume and the geodesic spectrum (i.e. the set of lengths of closed geodesics). We discuss several explicit (numerical) examples in which this criterion can be applied. This is joint work in progress with Michael Lipnowski.

Avec D. Pecker, nous avons montré que tout nœud de R^3 est un nœud de Chebyshev, ie admet une représentation polynomiale de la forme ( Ta(t),Tb(t), Tc(t+\phi) ) où a,b,c sont des entiers, \phi est un rationnel et Tn(x)= cos( n arcos x) est un polynôme de Chebyshev.

Avec F. Rouillier et C. Tran, nous avons proposé un algorithme pour identifier les nœuds de Chebyshev à 2 ponts (cas a=3, (a,b)=1, c fixé et \phi variant).

L'exposé abordera les différents aspects de ce problème : géométrique, calculatoire - et évoquera la question plus générale qui est de déterminer, pour les nœuds rationnels, une représentation polynomiale de degré minimal, sujet étudié avec E. Brugallé.