Edoardo Salati

An obvious consequence of Whitehead’s theorem yields that the Eilenberg-Mac Lane spaces $K(G,1)$ correspond, up to homotopy, precisely to groups, in the sense that $\pi_1(K(G,1))\simeq G$. The situation becomes, however, much more complex by looking at the class of $p$-complete spaces in the sense of Bousfield and Kan. With regard to this, the striking Martino-Priddy conjecture, now a theorem by Oliver, tells us that $p$-fusion systems parametrize, in a similar fashion, some $p$-complete spaces and overall justifies, at the same, our interest.

Embarking on an imaginary trip through the key steps of the most modern proof of the conjecture, we will introduce partial groups, the key tool developed by Chermak for providing the closing argument to such a proof. Partial groups additionally provide an alternative to fusion systems as models of $p$-complete spaces and also very naturally suggest that we might be seeing only a shadow of a bigger picture. I will talk about what is known with regard to the category of partial groups, with a focus on its connections with groups and simplicial sets as well as on how the gathering of such information may be of use.