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Quillen's Conjecture

Project description

Study seeks to prove Quillen’s conjecture

Funded by the Marie Skłodowska-Curie Actions programme, the QUILCON project aims to prove the Quillen subgroup-poset conjecture that appeared in 1978. The conjecture states that for a finite group G and a prime number p, if the poset of the non-trivial p-subgroups of G is contractible, then G has a non-trivial normal p-subgroup. To prove this conjecture, the project will use tools from higher category theory and combinatorial homotopy. The conjecture is important for linking finite group theory, algebraic topology and combinatorics.

Objective

Quillen's Conjecture

The ultimate goal of this project it to prove the Quillen subgroup-poset conjecture, open since 1978. The conjecture is easy to state: for G a finite group and p a prime number, if the poset of non-trivial p-subgroups of G is contractible, then G has a non-trivial normal p-subgroup. The novel approach of this proposal is to prove Quillen's conjecture by dealing with it in a more abstract categorical setting of certain cartesian fibrations of 2-categories over the 2-category of finite groups, where tools from higher category theory and 'homotopy combinatorics' can be used. Carrying out this plan thus requires generalising relevant parts of the classical theory from posets to certain fibred categories. For one of the main tools, Möbius inversion, this theory has already been developed by the applicant. The conjecture is important for the interface between finite group theory, algebraic topology, and combinatorics, and the tools developed promise to be useful to further these interactions. The research will be carried out at the University of Copenhagen, at one of the strongest topology groups in the world, supervised by Jesper Møller, one of the leading experts on Quillen's conjecture. The applicant, Joachim Kock, has ample expertise in category theory applied to algebraic topology and combinatorics. By applying his skills to solve an important open problem, he will fill a gap in his research profile and take the step from high-quality to top-quality research.

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Coordinator

KOBENHAVNS UNIVERSITET
Net EU contribution
€ 219 312,00
Address
Norregade 10
1165 Kobenhavn
Denmark

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Region
Danmark Hovedstaden Byen København
Activity type
Higher or Secondary Education Establishments
Links
Other funding
€ 0,00