Carles Broto

## Algebraic structure of finite loop spaces

video: comming soon.
abstract:
We show that every finite loop space admits a natural $$p$$-local compact group structure at each prime $$p$$. In other words, given a finite loop space $$X$$ and a prime $$p$$, we attach to $$X$$ a Sylow $$p$$-subgroup $$S$$ which is a $$p$$-toral discrete group and a system of conjugacy relations among the subgroups of $$S$$ satisfying the usual properties of fusion in finite groups or compact Lie groups. This algebraic structure determines and it is determined by the $$p$$-completed classifying space $$BX^{\wedge}_{p}$$. (Joint work with Ran levi and Bob Oliver.)