Louis Hainaut: Configuration spaces on a bouquet of spheres and related moduli spaces

Abstract.

This thesis is a compilation of four papers, revolving primarily around the cohomology of certain configuration spaces and moduli spaces.

Paper I studies the Euler characteristic of configuration spaces over a large family of base spaces \(X\), with any constructible complex of sheaves as coefficients. This paper generalizes a previous formula of Gal, which applies to the restricted case when \(X\) is a finite simplicial complex.

Paper II, written jointly with Nir Gadish, studies configuration spaces on a bouquet of spheres \(X\) via their compactly supported cohomology. We prove that, as a vector space, this compactly supported cohomology can be expressed as a certain polynomial functor applied to the reduced cohomology of \(X\), and we relate the coefficients of this polynomial functor to so-called bead representations introduced by Turchin–Willwacher. Moreover we perform partial computations of these coefficients, and these computations lead us to detect a large number of homology classes for the moduli space \(\mathcal{M}_{2,n}\); these classes live in the virtual cohomological dimension as well as one degree below.

Paper III studies cohomological properties of a certain category of polynomial outer functors, and more precisely the \(\operatorname{Ext}\)-groups between the simple objects of this category. In this paper I prove vanishing results in a certain range, and also detect that certain terms do not vanish outside that range. This contrasts with results of Vespa about the whole category of (non-necessarily outer) polynomial functors.

Paper IV, written jointly with Dan Petersen, studies the handlebody mapping class group. In this paper we give a novel geometric model for a classifying space for these groups, using hyperbolic geometry, and use this description to detect a vast number of classes in their homology. At the end of the paper we use the classifying space constructed to provide a map between two spectral sequences, one computing the compactly supported cohomology of the tropical moduli space \(\mathcal{M}_{g,n}^{\text{trop}}\) and the other one computing the weight zero part of the compactly supported cohomology of \(\mathcal{M}_{g,n}\); we conjecture that this map provides an isomorphism between the two spectral sequences.