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Relative self-equivalences and graph complexes
Stockholm University, Faculty of Science, Department of Mathematics.ORCID iD: 0000-0002-2068-6228
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of three papers.

In Paper I, we identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of Sk × Sl, where 3 ≤ k < l ≤ 2k - 2. We express the result in terms of Lie graph complex homology.

In Paper II, we construct a rational model for the classifying space BautA(X) of homotopy automorphisms of a simply connected finite CW-complex X relative to a simply connected subcomplex A. Using this model, we provide a purely algebraic description of the cohomology of this classifying space. This constitutes an important input for the results of Paper I.

In Paper III, we show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. As an application, we provide the foundations of a Koszul duality theory for modular operads.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University , 2024. , p. xxix
National Category
Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-228529ISBN: 978-91-8014-793-4 (print)ISBN: 978-91-8014-794-1 (electronic)OAI: oai:DiVA.org:su-228529DiVA, id: diva2:1853120
Public defence
2024-06-13, Lärosal 4, hus 1, Albano, Albanovägen 28, Stockholm, 14:00 (English)
Opponent
Supervisors
Available from: 2024-05-21 Created: 2024-04-21 Last updated: 2024-04-29Bibliographically approved
List of papers
1. The stable cohomology of self-equivalences of connected sums of products of spheres
Open this publication in new window or tab >>The stable cohomology of self-equivalences of connected sums of products of spheres
2024 (English)In: Forum of mathematics, sigma, ISSN 2050-5094, Vol. 12, article id e1Article in journal (Refereed) Published
Abstract [en]

We identify the cohomology of the stable classifying space of homotopy automorphisms (relative to an embedded disk) of connected sums of Sk×Sl, where 3≤k<l2k−2. The result is expressed in terms of Lie graph complex homology.

National Category
Geometry
Identifiers
urn:nbn:se:su:diva-226075 (URN)10.1017/fms.2023.113 (DOI)001136559700001 ()
Available from: 2024-02-01 Created: 2024-02-01 Last updated: 2024-04-21Bibliographically approved
2. Equivariant algebraic models for relative self-equivalences
Open this publication in new window or tab >>Equivariant algebraic models for relative self-equivalences
(English)Manuscript (preprint) (Other academic)
National Category
Geometry
Research subject
Mathematics
Identifiers
urn:nbn:se:su:diva-228528 (URN)
Available from: 2024-04-21 Created: 2024-04-21 Last updated: 2024-04-21
3. MODULAR OPERADS AS MODULES OVER THE BRAUER PROPERAD
Open this publication in new window or tab >>MODULAR OPERADS AS MODULES OVER THE BRAUER PROPERAD
2022 (English)In: Theory and Applications of Categories, ISSN 1201-561X, Vol. 38, no 40, p. 1538-1607Article in journal (Refereed) Published
Abstract [en]

We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore, we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

Keywords
Modular operads, properads, Koszul duality
National Category
Mathematics
Identifiers
urn:nbn:se:su:diva-214565 (URN)000904058600001 ()
Available from: 2023-02-06 Created: 2023-02-06 Last updated: 2024-04-21Bibliographically approved

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Stoll, Robin

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