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Hainaut, Louisorcid.org/0000-0002-4710-8575

Open this publication in new window or tab >>Configuration spaces on a bouquet of spheres and related moduli spaces### Hainaut, Louis

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##### Abstract [en]

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

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

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

Mathematics
##### Research subject

Mathematics
##### Identifiers

urn:nbn:se:su:diva-228745 (URN)978-91-8014-817-7 (ISBN)978-91-8014-818-4 (ISBN)
##### Public defence

2024-06-14, Lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 09:30 (English)
##### Opponent

### Vespa, Christine

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##### Supervisors

### Petersen, Dan

### Arone, Gregory

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Available from: 2024-05-22 Created: 2024-04-25 Last updated: 2024-05-06Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

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.

Deparment of Mathematics, Aix-Marseille University, France.

Stockholm University, Faculty of Science, Department of Mathematics.

Stockholm University, Faculty of Science, Department of Mathematics.

Open this publication in new window or tab >>The Euler characteristic of configuration spaces### Hainaut, Louis

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##### Abstract [en]

##### 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)
#####

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##### Funder

EU, European Research Council, ERC-2017-STG 759082
Available from: 2022-04-20 Created: 2022-04-20 Last updated: 2024-04-25Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

In this short note we present a generating function computing the compactly supported Euler characteristic χ_{c}(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.

Open this publication in new window or tab >>(Non-)vanishing results for extensions between simple outer functors on free groups### Hainaut, Louis

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##### Abstract [en]

##### 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)
#####

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Available from: 2024-04-24 Created: 2024-04-24 Last updated: 2024-04-26Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

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.

Open this publication in new window or tab >>Top weight cohomology of moduli spaces of Riemann surfaces and handlebodies### Hainaut, Louis

### Petersen, Dan

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##### Abstract [en]

##### 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)
#####

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Available from: 2024-04-24 Created: 2024-04-24 Last updated: 2024-04-25Bibliographically approved

Stockholm University, Faculty of Science, Department of Mathematics.

Stockholm University, Faculty of Science, Department of Mathematics.

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.