Pure Mathematics Seminar: Marco Praderio

Event Details

Mackey functors over fusion systems and the sharpness conjecture

Mackey functors are algebraic structures possessing operations which behave like the induction, restriction and conjugation maps in group representation theory. Such operations appear in a variety of other contexts such as group cohomology. On the other hand fusion systems are categories whose objects are subgroups of a finite p- group S and attempt to capture the p-local structure of a "fictional" finite group G containing S as a Sylow p-subgroup. Due to work from Broto Levi and Oliver and later Chenmark it is possible to associate to every fusion system F a unique (up to weak equivalence) topological space BF that plays the role of classifying space of F. In particular we have that, for every n,

H^n(BF;F_p)=lim_{O(F^c)}H^n(--;F_p)

where O(F^c) is a particular category obtained from F. Due to the existence of a particular spectral sequence this result suggests that, for i>0, the higher limits

lim^i_{O(F^c)}H^n(--;F_p)

vanish. Diaz and Park conjectured in 2014 that a stronger result (known as sharpness conjecture) may actually hold. During this talk we will go through the definitions of both fusion systems and Mackey functors and state precisely the sharpness conjecture. Time permitting we will also view the different cases in which sharpness is known to hold and explore sufficient conditions for the conjecture to hold for the Benson Solomon fusion systems.