Abstract

  Fusion systems are structures that encode the properties of conjugation between p-subgroups of a group, for p any prime number, and represent the modern approach to the study of the p-local structure of finite groups. To every saturated fusion system F defined on the p-group S, one can associate a topological space BF that plays the role that classifying spaces play for finite groups. In analogy with finite groups, it is possible to reconstruct such a classifying space BF by gluing together classifying spaces BP, where P runs over a suitably chosen collection of subgroups of S. In 1998, Dwyer showed that if G is a finite group and the collection considered is the one of p-centric subgroups of G, then the corresponding homology decomposition is sharp, making it easier to describe the classifying space BG. In 2015 Diaz and Park established a conjecture that extends Dwyer’s sharpness result in two ways: they consider fusion systems instead of groups and Mackey functors in place of cohomology functors.