arXiv subjects
Publications and source records attributed to Jules Chenal.
In this article, we investigate the relationship between the fundamental group of a space and its continuous transformations. To be more precise, we show that if a continuous action of a Lie group on a space admits a simply connected cross-section, then we can build the universal covering of the space using an extension of the Lie group by a discrete group.
In the present article, we investigate the topology of real toric varieties, especially those whose torus is not split over the field of real numbers. We describe some canonical fibrations associated to their real loci. Then, we establish various properties of their cohomology provided that their real loci are compact and smooth. For instance, we compute their Betti numbers, show that their cohomology is totally algebraic, and extend a criterion of orientability. In addition, we provide the topological classification of equivariant embeddings of non-split tridimensional tori.
A T-hypersurface is a combinatorial hypersurface of the real locus of a projective toric variety $Y$. It is constructed from a primitive triangulation $K$ of a moment polytope $P$ of $Y$ and a $0$-cochain $\varepsilon$ on $K$ with coefficients in the field with two elements $\mathbb{F}_2$, called a sign distribution. O. Viro showed that when $K$ is convex the T-hypersurface is ambiantly isotopic to a real algebraic hypersurface of $Y$. A. Renaudineau and K. Shaw gave upper bounds on the Betti numbers of T-hypersurfaces in terms of the Hodge numbers of a generic section of the ample line bundle $L$ associated with the moment polytope. In particular, the number of connected components of a T-hypersurface cannot exceed the geometric genus of a generic section of $L$ plus one. In this article we investigate whether this upper bound is attainable. We are able to characterise the couples $(K;\varepsilon)$ leading to T-hypersurfaces realising the Renaudineau-Shaw upper bound on the number of connected components. This theorem generalises B. Haas' theorem for T-curves. In contrast with this results we find that the upper bound is not always attainable on every primitive triangulations. For some of those on which it is not attainable we provide a sharper upper bound. Finally we use our characterisation to show that there always exists a triangulation and a sign distribution on the standard simplex that reach the Renaudineau-Shaw upper bound. We also study the growth of the expected number of connected components of a T-hypersurface as we dilate the moment polytope by $d$ (i.e. we tensorise the line bundle $d$-times with itself) and show that it is always of the order of $d^n$ where $n$ is the dimension of $P$.
In this article, we present two structural results about the Renaudineau-Shaw spectral sequence that computes the cohomology of T-hypersurfaces. The first is a Poincar\'e duality satisfied by all its pages of positive index. The second is a vanishing criterion. It reformulates the vanishing of the boundary operators of the spectral sequence as the injectivity of some morphisms induced in cohomology by the inclusion of the T-hypersurface in its surrounding toric variety. It implies that the Renaudineau-Shaw spectral sequence of a T-hypersurface degenerates at the second page if and only if the T-hypersurface satisfies a real version of the Lefschetz Hyperplane Section Theorem.
In a first time we present a version of the Poincar{é}-Lefschetz theorem for certain cellular cosheaves on a particular subdivision of a CW-complex K. To that end we construct a cellular sheaf on K whose cohomology with compact support is isomorphic to the homology of the initial cosheaf. In a second time we use the first result to generalise the tropical version of the Lefschetz hyperplane section theorem to singular tropical toric varieties and singular tropical hypersurfaces.