arXiv subjects

Marco Praderio Bova

Publications and source records attributed to Marco Praderio Bova.

5 recordsLinked to original sources

arXiv2026

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.

math.GR↗

arXiv2026

Computing higher limits over the fusion orbit category via amalgams

We study higher limits over the centric orbit category of a fusion system realized by an amalgamated product. In so doing we provide a novel technique for studying the Diaz-Park sharpness conjecture and prove it (in the case of the cohomology Mackey functors) for all the Clelland-Parker and Parker-Stroth fusion systems. This complements previous work from Henke, Libmand and Lynd. We further use the developed technique to study the Benson-Solomon fusion systems thus relating higher limits over the centric fusion orbit category of these systems with the signalizer functors described by Aschbacher and Chermak. We believe that the proposed technique can, in future work, be used as a first step in an induction argument that can bring us closer to providing an answer to this conjecture.

math.AT↗

arXiv2024

Higher limits over the fusion orbit category via centralizers of amalgams

We study the Díaz-Park sharpness conjecture for fusion systems and prove that, under certain circumstances, there exists a 4 terms exact sequence relating the first two higher limits of the contravariant part of a Mackey functor over certain fusion systems. We show how this result can be applied to the family of Benson-Solomon fusion systems thus providing another approach to studying the sharpness for this family of fusion systems.

math.AT↗

arXiv2024

Sharpness for the Benson-Solomon fusion systems

We develop tools to prove Díaz and Park's sharpness conjecture (see [8]) for fusion systems admitting tame families of fusion subsystems (see Theorem A). We use such tools to prove the conjecture for all Benson-Solomon fusion systems (see Theorem B) thus completing the work started by Henke, Libman and Lynd in [14, Theorems 1.1 and 1.4].

math.AT↗

arXiv2023

Green correspondence on centric Mackey functor over fusion systems

In this paper we give a definition of (centric) Mackey functor over a fusion system which generalizes the notion of Mackey functor over a group. In this context we prove that, given some conditions on a related ring, the centric Burnside ring over a fusion system (as defined by Diaz and Libman) acts on any centric Mackey functor. We also prove that the Green correspondence holds for centric Mackey functors over fusion systems. As a means to prove this we introduce a notion of relative projectivity for centric Mackey functors over fusion systems and provide a decomposition of a particular product in $\mathcal{O}\left(\mathcal{F}^c\right)_{\sqcup}$ in terms of the product in $\mathcal{O}\left(N_{\mathcal{F}}\left(H\right)^c\right)_{\sqcup}$.

math.RT↗