## Periodic Reporting for period 1 - StringHom (String topology and group (co)homology)

Reporting period: 2018-03-01 to 2020-02-29

"Homology and cohomology groups provide fundamental algebraic invariants of spaces in topology and geometry and of groups in algebra. The overall goal of this project, belonging to algebraic topology and especially string topology, with a strong focus on applications to algebra, was to deploy techniques from string topology to study the homology and cohomology of certain highly interesting families of groups, namely automorphism groups of free groups and finite groups of Lie type. The project had two components. The original aim of the first component was to detect nontrivial classes in the poorly-understood unstable homology of automorphism groups of free groups, while aim of the second component was the further development of the ""string topology of finite groups of Lie type"" discovered by Jesper Grodal and the researcher, with a view towards shedding light on the mysterious Tezuka conjecture stating that under certain conditions, the cohomologies of a finite group of Lie type (an algebraic object) and the free loop space of the classifying space of the corresponding compact Lie group (a seemingly unrelated space) agree."

"As planned, work on the first component begun with an attempt to detect nontrivial classes in the homology of automorphism groups of free groups. In the course of this work, the researcher unexpectedly discovered a novel perspective on a very classical topic in algebraic topology, namely the mod p homology of E-infinity spaces. Upon an evaluation of the mathematical significance of the material, the researcher decided to refocus the research effort in the first part of the project into the development of this perspective. The researcher then proceeded to successfully develop the foundations of the mod p homology of E-infinity spaces from the new point of view. The major research results obtained are presented in the preprint ""Homology operations revisited"" (arXiv:2001.02781 47 pages) posted on the arXiv preprint repository.

The research work in the second part, joint with Grodal, centered on the strengthening and extension of string topology of finite groups of Lie type, with a particular focus on understanding when the Tezuka conjecture holds. Using a new ""untwisting theorem,"" we showed how to dispense with the customary congruence condition in the conjecture at the expense of relating the twisted finite group of Lie type tG(q) not to the free loop space of the classifying space of the corresponding compact Lie group, but rather to the free loop space of a different l-compact group depending on G, the twisting t, and the congruence class of q modulo l. With this setup, we proved that a string topological variant of the Tezuka conjecture holds in the simply connected case for any q and any twisting t whose order is not divisible by l, except possibly in a finite number of cases requiring special attention. The results are incorporated into the preprint ""String topology of finite groups of Lie type"" (arXiv:2003.07852 58 pages) posted on the arXiv preprint repository. The researcher also mapped out a proof that the string product constructed in the paper is commutative and that it agrees with a product previously constructed by Chataur and Menichi, providing a key link between the string topology of finite groups of Lie type and previous work on string topology of classifying spaces using Chataur and Menichi's product. These enhancements will be incorporated into a future revision of the preprint.

In addition to the research work, during the project the researcher organized a workshop together with Grodal (""Workshop on classifying spaces of finite groups of Lie type,"" University of Copenhagen, July 8–12, 2019); supervised a master's thesis project concerning invertible two-dimensional topological quantum field theories; and organized mathematical gong shows showcasing junior researchers as part of the host department's outreach efforts during Copenhagen's annual Culture Night festival in October 2018 and 2019. In addition to the preprints mentioned above, the research results have been disseminated through talks the researcher has given on the work at various conferences, workshops, and seminars."

The research work in the second part, joint with Grodal, centered on the strengthening and extension of string topology of finite groups of Lie type, with a particular focus on understanding when the Tezuka conjecture holds. Using a new ""untwisting theorem,"" we showed how to dispense with the customary congruence condition in the conjecture at the expense of relating the twisted finite group of Lie type tG(q) not to the free loop space of the classifying space of the corresponding compact Lie group, but rather to the free loop space of a different l-compact group depending on G, the twisting t, and the congruence class of q modulo l. With this setup, we proved that a string topological variant of the Tezuka conjecture holds in the simply connected case for any q and any twisting t whose order is not divisible by l, except possibly in a finite number of cases requiring special attention. The results are incorporated into the preprint ""String topology of finite groups of Lie type"" (arXiv:2003.07852 58 pages) posted on the arXiv preprint repository. The researcher also mapped out a proof that the string product constructed in the paper is commutative and that it agrees with a product previously constructed by Chataur and Menichi, providing a key link between the string topology of finite groups of Lie type and previous work on string topology of classifying spaces using Chataur and Menichi's product. These enhancements will be incorporated into a future revision of the preprint.

In addition to the research work, during the project the researcher organized a workshop together with Grodal (""Workshop on classifying spaces of finite groups of Lie type,"" University of Copenhagen, July 8–12, 2019); supervised a master's thesis project concerning invertible two-dimensional topological quantum field theories; and organized mathematical gong shows showcasing junior researchers as part of the host department's outreach efforts during Copenhagen's annual Culture Night festival in October 2018 and 2019. In addition to the preprints mentioned above, the research results have been disseminated through talks the researcher has given on the work at various conferences, workshops, and seminars."

The research conducted belongs to pure mathematics, and is most likely to be exploited by other mathematicians. The mathematical significance of the work on E-infinity spaces derives from the fact that a variety of spaces of fundamental importance in topology, geometry and algebra – including, but not limited to, the classifying spaces of automorphism groups of free groups – assemble themselves into E-infinity spaces. The impact of the work is therefore likely to extend beyond the confines of string topology and the study of automorphism groups of free groups. Traditionally, the mod p homology of E-infinity spaces has been studied in terms of natural operations called Dyer–Lashof operations. The central advance made in the research is the demonstration that the subject can be profitably developed in terms of an alternative collection of operations the researcher calls E-operations. These operations offer a number of advantages over the Dyer–Lashof operations, especially in the context of E-infinity ring spaces, where they allow a simple and conceptual formulation of “mixed Adem relations” which are notoriously complicated to state in terms of the Dyer–Lashof operations.

The string topology of finite groups of Lie type developed in the second part of the project provides an exciting bridge between two seemingly unrelated topics, namely string topology of classifying spaces and the cohomology of finite groups of Lie type, and provides a way to attack the Tezuka conjecture. The research conducted in this part is inherently intradisciplinary in character, as it uses topological and homotopical techniques to study objects – finite groups of Lie type – that are algebraic in nature. As such, this research can be expected to have impact beyond algebraic topology, in particular in group theory and perhaps even representation theory.

The string topology of finite groups of Lie type developed in the second part of the project provides an exciting bridge between two seemingly unrelated topics, namely string topology of classifying spaces and the cohomology of finite groups of Lie type, and provides a way to attack the Tezuka conjecture. The research conducted in this part is inherently intradisciplinary in character, as it uses topological and homotopical techniques to study objects – finite groups of Lie type – that are algebraic in nature. As such, this research can be expected to have impact beyond algebraic topology, in particular in group theory and perhaps even representation theory.