Subcentric linking systems

TL;DR

This paper introduces a more flexible definition of linking systems for fusion systems by defining subcentric subgroups, leading to a unique subcentric linking system that is homotopy equivalent to the classical centric one, potentially aiding classification.

Contribution

It proposes a generalized concept of linking systems using subcentric subgroups, expanding the objects beyond quasicentric subgroups, and proves the existence and uniqueness of such systems.

Findings

Existence of a unique subcentric linking system for each saturated fusion system.

Homotopy equivalence between the nerve of subcentric and centric linking systems.

Potential for new classification methods for fusion systems of characteristic p-type.

Abstract

Linking systems are crucial for studying the homotopy theory of fusion systems, but are also of interest from an algebraic point of view. We propose a definition of a linking system associated to a saturated fusion system which is more general than the one currently in the literature and thus allows a more flexible choice of objects of linking systems. More precisely, we define subcentric subgroups of fusion systems in a way that every quasicentric subgroup of a saturated fusion system is subcentric. Whereas the objects of linking systems in the current definition are always quasicentric, the objects of our linking systems only need to be subcentric. We prove that, associated to each saturated fusion system $\mathcal{F}$, there is a unique linking system whose objects are the subcentric subgroups of $\mathcal{F}$. Furthermore, the nerve of such a subcentric linking system is homotopy equivalent to the nerve of the centric linking system associated to $\mathcal{F}$. We believe that the existence of subcentric linking systems opens a new way for a classification of fusion systems of characteristic $p$-type. The various results we prove about subcentric subgroups give furthermore some evidence that the concept is of interest for studying extensions of linking systems and fusion systems.