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#### Open Access in DiVA

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Hainaut, Louis
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Hainaut, Louis
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Department of Mathematics
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Configuration spaces on a bouquet of spheres and related moduli spacesPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
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2024 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Place, publisher, year, edition, pages

Stockholm: Department of Mathematics, Stockholm University , 2024. , p. 31
##### Keywords [en]

Configuration space, moduli space, polynomial functors
##### National Category

Mathematics
##### Research subject

Mathematics
##### Identifiers

URN: urn:nbn:se:su:diva-228745ISBN: 978-91-8014-817-7 (print)ISBN: 978-91-8014-818-4 (electronic)OAI: oai:DiVA.org:su-228745DiVA, id: diva2:1854510
##### Public defence

2024-06-14, Lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 09:30 (English)
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PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt557",{id:"formSmash:j_idt557",widgetVar:"widget_formSmash_j_idt557",multiple:true}); Available from: 2024-05-22 Created: 2024-04-25 Last updated: 2024-05-06Bibliographically approved
##### List of papers

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 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 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 M_{g,n}^{trop} and the other one computing the weight zero part of the compactly supported cohomology of M_{g,n}; we conjecture that this map provides an isomorphism between the two spectral sequences.

1. The Euler characteristic of configuration spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay1652837",{id:"formSmash:j_idt608:0:j_idt612",widgetVar:"overlay1652837",target:"formSmash:j_idt608:0:partsLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

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