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Configuration spaces on a bouquet of spheres and related moduli spaces
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-4710-8575
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

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 M2,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 Mg,ntrop and the other one computing the weight zero part of the compactly supported cohomology of Mg,n; we conjecture that this map provides an isomorphism between the two spectral sequences.

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)
Opponent
Supervisors
Available from: 2024-05-22 Created: 2024-04-25 Last updated: 2024-05-06Bibliographically approved
List of papers
1. The Euler characteristic of configuration spaces
Open this publication in new window or tab >>The Euler characteristic of configuration spaces
2022 (English)In: Bulletin of the Belgian Mathematical Society Simon Stevin, ISSN 1370-1444, E-ISSN 2034-1970, Vol. 29, no 1, p. 87-96Article in journal (Refereed) Accepted
Abstract [en]

In this short note we present a generating function computing the compactly supported Euler characteristic  χc(F(x,n),K⊠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.

Keywords
Configuration space, stratified space, Euler characteristic, constructible sheaf
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-204111 (URN)10.36045/j.bbms.211008 (DOI)000988441800005 ()2-s2.0-85148301517 (Scopus ID)
Funder
EU, European Research Council, ERC-2017-STG 759082
Available from: 2022-04-20 Created: 2022-04-20 Last updated: 2024-04-25Bibliographically approved
2. Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
Open this publication in new window or tab >>Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
(English)Manuscript (preprint) (Other academic)
Abstract [en]

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 automorphism of the free group. These representations are closely related to Hochschild-Pirashvili homology with coefficients in square-zero algebras, and they show up in seemingly unrelated parts of mathematics, from cohomology of moduli spaces of curves to polynomial functors on free groups.       

We show that these cohomology representations form a polynomial functor, and use various geometric models to compute a substantial part 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}).

Keywords
Configuration space, Hochschild-Pirashvili homology, polynomial functor
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-204141 (URN)
Funder
EU, European Research Council, ERC-2017-STG 759082
Available from: 2022-04-21 Created: 2022-04-21 Last updated: 2024-04-25Bibliographically approved
3. (Non-)vanishing results for extensions between simple outer functors on free groups
Open this publication in new window or tab >>(Non-)vanishing results for extensions between simple outer functors on free groups
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We study $\operatorname{Ext}$ groups between certain polynomial outer functors on free groups, inspired by an earlier result of Vespa in a related context. We prove certain vanishing results for these groups, and show that a Koszul-type property implied by Vespa's result no longer holds when we pass to the category of polynomial outer functors.

Keywords
polynomial functor, free groups, Ext functors
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-228744 (URN)10.48550/arXiv.2311.16881 (DOI)
Available from: 2024-04-24 Created: 2024-04-24 Last updated: 2024-04-26Bibliographically approved
4. Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies
Open this publication in new window or tab >>Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We show that a certain locus inside the moduli space $\mathcal{M}_g$ 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 $\mathcal{M}_g$, studied by Chan--Galatius--Payne, maps injectively into the cohomology of the handlebody mapping class group. 

Keywords
moduli space, mapping class group, Riemann surface, handlebody, hyperbolic geometry
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-228742 (URN)10.48550/arXiv.2305.03046 (DOI)
Available from: 2024-04-24 Created: 2024-04-24 Last updated: 2024-04-25Bibliographically approved

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Hainaut, Louis

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