An inductive approach to the Diaz-Park sharpness conjecture

We develop tools which use common fusion systems building techniques in order to compute higher limits over the centric orbit category. We apply these tools in order to study both the Diaz-Park sharpness conjecture as well as the weaker cohomological sharpness conjecture which predicts vanishing of higher limits only for the cohomology Mackey functors . Our approach leads to proving cohomological sharpness (but not sharpness) for all saturated fusion systems over p-groups of either maximal nihlpotency or of rank 2 and all polynomial, Henke-Shpectorov and van Beek fusion systems. This list includes all but 2 of the cases for which cohomological sharpness was previously known as well as most currently known families of exotic fusion systems. For the polynomial, Henke-Shpectorov and 6 of the van Beek fusion systems, sharpness is also approximated by proving vanishing of all but the first higher limits of any Mackey functor. The distinction our approach makes between sharpness and cohomological sharpness is somewhat surprising and interesting by itself. Our approach draws a new connection between cohomological sharpness and fusion system building techniques. We believe that this connection will lead to a better understanding of both fusion systems and Mackey functors over them.