Fusion Systems, Group Cohomology and Classifying Spaces

Sammanfattning

We first motivate fusion systems and provide basic definitions. Then we discuss saturated fusion systems and the equivalence of various definitions found in the literature. We then give the definitions of group (co)homology and introduce (co)induced modules, restriction, conjugation and transfer homomorphisms, Shapiro’s lemma, and prove the Cartan-Eilenberg stable elements theorem. Thereafter, we construct the classifying space of a discrete group via simplicial methods. The (co)homology of simplicial sets with coefficients in an abelian group is defined. It is shown that the (co)homology of the classifying space of a discrete group with coefficients in an abelian group coincides with the usual definitions of group (co)homology. Bockstein and transfer homomorphisms in topology are introduced, and examples of cohomology rings are given. Thereafter, it is proved how the Cartan-Eilenberg stable elements theorem can be reinterpreted in terms of fusion systems. Lastly, the Martino-Priddy conjecture (now a theorem) is stated.