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(Non-)vanishing results for extensions between simple outer functors on free groups
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-4710-8575
(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 [en]
polynomial functor, free groups, Ext functors
National Category
Algebra and Logic
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-228744DOI: 10.48550/arXiv.2311.16881OAI: oai:DiVA.org:su-228744DiVA, id: diva2:1854277
Available from: 2024-04-24 Created: 2024-04-24 Last updated: 2024-04-26Bibliographically approved
In thesis
1. Configuration spaces on a bouquet of spheres and related moduli spaces
Open this publication in new window or tab >>Configuration spaces on a bouquet of spheres and related moduli spaces
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
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
Supervisors
Available from: 2024-05-22 Created: 2024-04-25 Last updated: 2024-05-06Bibliographically approved

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Publisher's full texthttps://arxiv.org/abs/2311.16881

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Hainaut, Louis

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