Mackey functors over fusion systems

Praderio, Marco and Mazza, Nadia (2024) Mackey functors over fusion systems. PhD thesis, Lancaster University.

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Abstract

In this thesis we study the properties of Mackey functors over fusion systems as opposed to Mackey functors over groups. Given a fusion system F we start by defning the Mackey algebra of F. We then use it in order to provide defnitions for Mackey functors over F and F-centric Mackey functors (also known as Fc -restricted Mackey functors) which coincide with those in the literature. We go on to proving that several results such as Higman's criterion and the Green correspondence can be translated from Mackey functors over groups to F-centric Mackey functors. We also show that the methods used to perform this translation cannot be used to prove similar results for Mackey functors over fusion systems in general. In the second part of this thesis we focus our efforts on the sharpness conjecture for fusion systems. We do so by using spectral sequences in order to provide suffcient conditions (in terms of fusion subsystems of F) for the conjecture to be satisfied for F. We then use the developed tools in order to prove that the sharpness conjecture is satisfed for all Benson-Solomon fusion systems thus completing previous work of Henke, Libman and Lynd.