In this project, the Principal Investigator will employ methods from stable homotopy theory to provide insight into the surprising relationship between p-local group theory, modular representation theory and stable homotopy theory. Specifically, the PI will first give a much simplified description of the p-local finite group model for the homotopy theory of fusion systems. This shows that a p-local finite group on a finite p-group is equivalent to map from the classifying space into a p-complete, nilpotent space with a stable retract satisfying Frobenius reciprocity up to homotopy. In recent work, joint with Radu Stancu, the PI has shown that saturated fusion systems on a finite p-group are in bijective correspondence with stable idempotents of the classifying space that satisfy Frobenius reciprocity. The PI will extend this result to a bijection between p-local finite groups and idempotents that satisfy Frobenius reciprocity up to structured homotopies. The crucial question in the field is on the existence and uniqueness of classifying spaces for saturated fusion systems (in the form of p-local finite groups). Using results from this project, the question can be approached by refining the current construction of classifying spectra from being a construction up to homotopy to producing structured classifying spectra, and steps will be taken in this direction.
Groups are fundamental objects in mathematics used to keep track of the symmetries of an object (e.g. a set or a space). A group is said to act on an object if that object exhibits the symmetries encoded by the group. Among such objects the classifying space of a group is of special importance, as its homotopical properties contain information about all possible actions of the group on topological spaces. A finite group can be characterized as the fundamental group of its classifying space --the set of paths in the space beginning and ending in a fixed point where two paths are identified if one can be continuously deformed into the other. For a given prime p, the p-local structure of a group can be thought of as the system of symmetries of the group that can be detected by actions on sets whose cardinality is a power of p. This notion is made precise by fusion systems as introduced by Puig and developed by Broto-Levi-Oliver. As conjectured by Martino-Priddy, and proved by Oliver, p-localizing the classifying space mirrors this construction in topology. More generally one can consider abstract fusion systems that do not necessarily come from groups and p-local finite groups are a model for the corresponding classifying space suggested by Broto-Levi-Oliver. In this project the PI will elucidate this correspondence by providing a simpler model for classifying spaces of abstract fusion systems and take important steps toward proving the existence of classifying spaces for arbitrary fusion systems which is a fundamental question in the field.