arXiv subjects
Publications and source records attributed to Louis Hainaut.
We give explicit formulas for the asymptotic Betti numbers, over an arbitrary field, of the ordered configuration spaces of a graph. In characteristic zero, we further give explicit formulas for the asymptotic multiplicities in homology of many irreducible representations of the symmetric group, in the spirit of representation stability.
In this article we study cohomological properties of the category of polynomial outer functors on free groups, which are the functors from the category of finitely generated free groups to the category of rational vector spaces which send all inner automorphisms to the identity morphism, and which satisfy a certain polynomiality property. More precisely, we prove vanishing and non-vanishing results for the Ext groups between simple polynomial outer functors. This work is inspired by an earlier result of Vespa for the category of all polynomial functors from finitely generated free groups to rational vector spaces; it follows in particular from her results that, in this larger category, the Ext groups between simple functors are always concentrated in a specific single degree. Our main results show that, when we pass to the full subcategory of polynomial outer functors, Ext groups between simple functors are sometimes non-trivial outside of this specific degree.
We show that a certain locus inside the moduli space $M_g$ of hyperbolic surfaces, given by surfaces with "sufficiently many" short geodesics, is a classifying space of the handlebody mapping class group. A consequence of the construction is that the top weight cohomology of $M_g$, studied by Chan-Galatius-Payne, maps injectively into the cohomology of the handlebody mapping class group.
We study the compactly supported rational cohomology of configuration spaces of points on wedges of spheres, equipped with natural actions of the symmetric group and the group $Out(F_g)$ of outer automorphisms of the free group. These representations show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups and Hochschild-Pirashvili cohomology. We show that these cohomology representations form a polynomial functor, and use various geometric models to compute many of its composition factors. We further compute the composition factors completely for all configurations of $n\leq 10$ particles. An application of this analysis is a new super-exponential lower bound on the symmetric group action on the weight $0$ component of $H^*_c(M_{2,n})$.
In this short note we present a generating function computing the compactly supported Euler characteristic $χ_c(F(X, n), K^{\boxtimes n})$ of the configuration spaces on a topologically stratified space $X$, with $K$ a constructible complex of sheaves on $X$, and we obtain as a special case a generating function for the Euler characteristic $χ(F(X, n))$. We also recall how to use existing results to turn our computation of the Euler characteristic into a computation of the equivariant Euler characteristic.