2020/03/17 by Grodal, Jesper, Lahtinen, Anssi
#20J06 (Primary) 20D06 #55P50 (Secondary) #55R35 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2003.07852
We show that the mod ℓ cohomology of any finite group of Lie type in characteristic p different from ℓ admits the structure of a module over the mod ℓ cohomology of the free loop space of the classifying space BG of the corresponding compact Lie group G, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain fundamental class in the homology of the finite group of Lie type is non-trivial, then this module structure becomes free of rank one, and provides a structured isomorphism between the two cohomology rings equipped with the cup product, up to a filtration. We verify the nontriviality of the fundamental class in a range of cases, including all simply connected untwisted classical groups over the field of q elements, with q congruent to 1 mod ℓ. We also show how to deal with twistings and get rid of the congruence condition by replacing BG by a certain ℓ-compact fixed point group depending on the order of q mod ℓ, without changing the finite group. With this modification, we know of no examples where the fundamental class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.