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Configuration spaces on a wedge of spheres and Hochschild-Pirashvili homology
Stockholm University, Faculty of Science, Department of Mathematics. (Algebra, Geometry, Topology and Combinatorics)ORCID iD: 0000-0002-4710-8575
(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 [en]
Configuration space, Hochschild-Pirashvili homology, polynomial functor
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-204141OAI: oai:DiVA.org:su-204141DiVA, id: diva2:1653462
Funder
EU, European Research Council, ERC-2017-STG 759082Available from: 2022-04-21 Created: 2022-04-21 Last updated: 2024-04-25Bibliographically approved
In thesis
1. Some computations of compact support cohomology of configuration spaces
Open this publication in new window or tab >>Some computations of compact support cohomology of configuration spaces
2022 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This licentiate thesis consists of two papers related to configuration spaces of points.

In paper I a general formula for the Euler characteristic of configuration spaces on any topologically stratified space X is obtained in terms of geometric and combinatorial data about the strata. More generally this paper provides a formula for the Euler characteristic of the cohomology with compact support of these configuration spaces with coefficients in a constructible complex of sheaves K on X. The formula for the classical Euler characteristic is then obtained by taking K to be the dualizing complex of X. This formula generalizes similar results about configuration spaces on a manifold or on a simplicial complex, as well as another formula for any Hausdorff space X when the complex of sheaves K is trivial.

In paper II we study the cohomology with compact support of configuration spaces on a wedge sum of spheres X, with rational coefficients. We prove that these cohomology groups are the coefficients of an analytic functor computing the Hochschild--Pirashvili homology of X with certain coefficients. Moreover, we prove that, up to a filtration, these same cohomology groups are a polynomial functor in the reduced cohomology of X, with coefficients not depending on X. Contrasting the information provided by two different models we are able to partially compute these coefficients, and in particular we obtained a complete answer for configurations of at most 10 points. The coefficients thus obtained can be used to compute the weight 0 part of the cohomology with compact support of the moduli space M_{2,n}.

Place, publisher, year, edition, pages
Stockholm: Stockholm University, 2022. p. 112
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-204142 (URN)978-91-7797-999-9 (ISBN)
Presentation
2022-05-13, 10:00 (English)
Opponent
Supervisors
Available from: 2022-04-22 Created: 2022-04-22 Last updated: 2022-04-22Bibliographically approved
2. 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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