Bonn Topology Group - Abstracts

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Talk

July 2nd 2019
Jesper Grodal (Københavns Universitet, Denmark): String topology of finite groups of Lie type

Abstract

Finite groups of Lie type, such as SL_n(F_q), Sp_n(F_q)..., are ubiquitous in mathematics, and calculating their cohomology has been a central theme over the years. Without any structural reasons as to why, it has calculationally been observed that, when calculable, their mod ell cohomology agree with the mod ell cohomology of LBG(C), the free loop space on BG(C), the classifying space of the corresponding complex algebraic group G(C), as long as q is congruent to 1 mod ell. This despite that LBG(C) and BG(F_q) are vastly different spaces, also at a prime ell, ruling out some space-level equivalence. In recent joint work with Anssi Lahtinen, that combines ell–compact groups with string topology à la Chas–Sullivan, we give a general structural relationship between these two cohomologies, which, suitably formulated, even works without any congruence condition on q, as long as it is prime to ell. We use this to prove structured versions of previous calculations, and establish isomorphism in new cases. The isomorphism conjecture in general hinges on the fate of a single cohomology class in exceptional Lie groups at small primes. My talk will begin to tell this story, as we know it so far...

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