Séminaire de topologie, géométrie et algèbre (archives)

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from Riemann surfaces with boundary to a symplectic manifold, with boundary conditions and various constraints on boundary and interior marked points. The presence of boundary leads to bubbling phenomena that pose a fundamental obstacle to invariance. In a joint work with J. Solomon, we developed a general approach to defining genus zero OGW invariants. For real symplectic manifolds in dimensions 2 and 3, these invariants are strongly related to Welschinger's invariants.

In dimension 3, the theory of codimension 2 contact submanifolds is better known as the transverse knot theory of a contact manifold, a theory which has a complete description in terms of braid theory. In higher dimensions, almost nothing is known, but there is a small (but growing) list of results. I will explain a method developed with A. Kaloti to use open books and Lefschetz fibrations to study codimension 2 contact embeddings. I will present as some initial applications and some interesting behaviors.

A higher genus version of Welschinger invariant was defined by Shustin for del Pezzo surfaces. These invariants count real curves of positive genera with signs. We study the properties of higher genus Welschinger invariants under Morse simplification. When the signs are defined by the number of solitary nodes, we prove that these higher genus Welschinger invariants depend only on the total number of real interpolated points. The result follows from a reduction of the genus and Brugallé's result on the invariance of genus zero Welschinger invariants.

On étudie la version réelle suivante d'un théorème célèbre d'Abhyankar-Moh : quelles applications rationnelles de la droite affine dans le plan affine, dont le lieu réel est un plongement fermé non singulier de R dans R^2, sont équivalentes, à difféomorphisme birationnel du plan près, au plongement trivial ? Dans ce cadre, on montre qu'il existe des plongements non équivalents. Certains d'entre eux sont détectés pas la non-négativité de la dimension de Kodaira réelle du complémentaire de leur image. Ce nouvel invariant est dérivé des propriétés topologiques de « faux plans réels » particuliers associés à ces plongements. (Travail en commun avec Adrien Dubouloz.)

Everything has been said about Stein fillings of lens spaces when they are endowed with their standard (tight) contact structure. Nevertheless, lens spaces support many more tight structures, that are all classified, but for which a complete list of their Stein fillings is still missing. I will present a series of methods and techniques which can be used to investigate the topology of these fillings, giving constraints, for example, on their Euler characteristic and fundamental group.

Oriented knots are said to be concordant if they cobound an embedded cylinder in the interval times the 3–sphere. This defines an equivalence relation under which the set of knots becomes an abelian group with the connected sum operation. The importance of this group lies in its strong connection with the study of 4-manifolds. Indeed, many questions pertaining to 4–manifolds with small topology (like the 4–sphere) can be addressed in terms of concordance. A powerful tool for studying the algebraic structure of this group comes from satellite operations or the process of tying a given knot P along another knot K to produce a third knot P(K).

To each coprime pair of natural numbers is associated a rational homology 4-ball B_{p,q}; these are of interest in algebraic geometry and in constructions of smooth 4-manifolds. Evans and Smith have completely determined which of these may be embedded symplectically in CP^2; the answer coincides with an algebraic geometric result of Hacking and Prokhorov, and is described in terms of solutions to the Markov diophantine equation.

Using double branched covers, we exhibit an infinite family of such balls which embed smoothly but not symplectically in CP^2. We also describe an obstruction using Donaldson’s diagonalisation which may be used to show that no two of our examples may be embedded disjointly.

Hamiltonian homeomorphisms are those homeomorphisms of a symplectic manifold which can be written as uniform limits of Hamiltonian diffeomorphisms. One difficulty in studying Hamiltonian homeomorphisms (particularly in dimensions greater than two) has been that we possess fewer tools for studying them. For example, (filtered) Floer homology, which has been a very effective tool for studying Hamiltonian diffeomorphisms, is not well-defined for homeomorphisms. We will show in this talk that using barcodes and persistence homology one can indirectly define (filtered) Floer homology for Hamiltonian homeomorphisms. This talk is based on joint projects with Buhovsky-Humiliére and Le Roux-Viterbo.