Curves on surfaces and surgeries

We study collections of curves in generic position on a closed surface whose complement consists of one disk only, up to orientation-preserving homeomorphism of the surface. We define a surgery operation on the set of such collections and prove that any two of them can be connected by a sequence of such surgeries

Unicellular maps and filtrations of the mapping class group

This article first answers to questions about connectedness of a new family of graphs on unicellular maps. Answering these questions goes through a description of the mapping class group as surgeries on unicellular maps. We also show how unicellular maps encode subgroups of the mapping group and provide filtrations of the mapping class group. These facts add a layer on the ubiquitous character of unicellular maps

Connected components of the topological surgery graph of a unicellular collection

A unicellular collection on a surface is a collection of curves whose complement is a single disk. There is a natural surgery operation on unicellular collections, endowing the set of such with a graph structure where the edge relation is given by surgery. Here we determine the connected components of this graph, showing that they are enumerated by a certain homological "surgery invariant". Our approach is group-theoretic and proceeds by understanding the action of the mapping class group on unicellular collections. In the course of our arguments, we determine simple generating sets for the stabilizer in the mapping class group of a mod-$2$ homology class, which may be of independent interest.Comment: 13 pages, 12 figures. Comments welcome

Combinatorial kk-systoles on a punctured torus and a pair of pants

In this paper, $S$ denotes a surface homeomorphic to a punctured torus or a pair of pants. Our interest is the study of \emph{\textbf{combinatorial $k$-systoles}} that is closed curves with self-intersection numbers greater than $k$ and with least combinatorial length. We show that the maximal intersection number $I^c_k$ of combinatorial $k$-systoles of $S$ grows like $k$ and $\underset{k\rightarrow+\infty}{\limsup}(I^c_k-k)=+\infty$. This result, in case of a pair of pants and a punctured torus, is a positive response to the combinatorial version of the Erlandsson - Parlier conjecture, originally formulated for the geometric length.Comment: 12 pages, 8 figure

On dual unit balls of Thurston norms

International audienceThurston norms are invariants of 3-manifolds defined on their second homology and understanding the shape of their dual unit balls is a widely open problem. In this article, we provide a large family of polytopes in R^2g that appear like dual unit balls of Thurston norms, generalizing Thurston's construction for polygons in R^2

From topology curves on surfaces to unicellular maps

Cette thèse se place à l'interface entre la topologie et la combinatoire. On s'intéresse dans un premier temps au problème de réalisation des boules unités duales des normes d'intersections sur les surfaces orientables. On montre aussi un certain lien entre les normes d'intersections et la norme de Thurston sur les 3-variétés.On montre par ailleurs l'existence d'un graphe dit de chirurgie sur l'ensemble des cartes unicellulaires d'une surface orientable. Dans le cas des collections unicellulaires et de cartes cubiques unicellulaires, le graphe de chirurgie s'avère connexe.This thesis stay in between topology and combinatory. Our first concerned is the problem of realization of dual unit ball of intersection norms on orientable surfaces. We also show a certain relation between intersection norms and Thurston norms on 3-manifolds. On the other part, we show the existence of graph structure on unicellular maps on orientable surface coming from a surgery operation on unicellular maps: a surgery graph. Its happen that surgery graph on unicellular collections and cubic unicellular maps is connected

De la topologie des courbes sur les surfaces aux cartes unicellulaires

This thesis stay in between topology and combinatory. Our first concerned is the problem of realization of dual unit ball of intersection norms on orientable surfaces. We also show a certain relation between intersection norms and Thurston norms on 3-manifolds. On the other part, we show the existence of graph structure on unicellular maps on orientable surface coming from a surgery operation on unicellular maps: a surgery graph. Its happen that surgery graph on unicellular collections and cubic unicellular maps is connected.Cette thèse se place à l'interface entre la topologie et la combinatoire. On s'intéresse dans un premier temps au problème de réalisation des boules unités duales des normes d'intersections sur les surfaces orientables. On montre aussi un certain lien entre les normes d'intersections et la norme de Thurston sur les 3-variétés.On montre par ailleurs l'existence d'un graphe dit de chirurgie sur l'ensemble des cartes unicellulaires d'une surface orientable. Dans le cas des collections unicellulaires et de cartes cubiques unicellulaires, le graphe de chirurgie s'avère connexe