Iterated suspensions are coalgebras over the little disks operad

We study the Eckmann-Hilton dual of the little disks algebra structure on iterated loop spaces: With the right definitions, every $n$-fold suspension is a coalgebra over the little $n$-disks operad. This structure induces non-trivial cooperations on the rational homotopy groups of an $n$-fold suspension. We describe the Eckmann-Hilton dual of the Browder bracket, which is a cooperation that forms an obstruction for an $n$-fold suspension to be an $(n+1)$-fold suspension, i.e. if this cooperation is non-zero then the space is not an $(n+1)$-fold suspension. We prove several results in equivariant rational homotopy theory that play an essential role in our results. Namely, we prove a version of the Sullivan conjecture for the Maurer-Cartan simplicial set of certain $L_\infty$-algebras equipped with a finite group action, and we provide rational models for fixed and homotopy fixed points of (mapping) spaces under some connectivity assumptions in the context of finite groups. We further show that by using the Eckmann-Hilton dual of the Browder operation we can use the rational homotopy groups to detect the difference between certain spaces that are rationally homotopy equivalent, but not homotopy equivalent.