Fusion Systems

In Michaelmas, I gave a course on fusion systems. One of the key motivations for studying fusion systems is the Martino-Priddy conjecture, which states that the p-completions of the classifying spaces of two groups are homotopy equivalent if and only if the fusion systems are isomorphic, which gives geometric meaning to the famous result of Cartan and Eilenberg that the mod-p cohomology of a finite group is determined by its fusion system.

This course arose out of trying to understand the proof, due to Bob Oliver, of the Martino-Priddy conjecture. While the one side of the equivalence - fusion systems themselves - were the subject of my course two terms ago, the other side of the equivalence is the focus this time.

A very tentative syllabus is as follows.

Week 1	Introduction and simplical sets
Week 2	Homotopy theory of simplicial sets; classifying spaces
Week 3	Bousfield-Kan R-completions
Week 4	Bousfield-Kan R-completions
Week 5	Homotopy colimits
Week 6	Homology decompositions
Week 7	Spectral sequences
Week 8	p-local geometries for groups