Carles Broto

Algebraic structure of finite loop spaces

video: comming soon.
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.)