Combinatorics of double loop suspensions, evaluation maps and Cohen groups (Q2004920)

In this paper, the authors reformulate Milgram's model of double loop suspension in terms of a preoperad of posets, each stage of which is the poset of all ordered partitions of a finite set. More precisely, they prove that there exists a preoperad of posets \(\mathcal{L}\) such that for any connected CW complex \(X\) there is a homotopy equivalence \[ \Omega^2\Sigma^2X\simeq \coprod_k\vert \mathcal{L} (k)\vert \times X^k/\sim, \] where the right hand side is defined by the usual premonad construction for the geometric realization of \(\mathcal{L}\), and each piece \(\mathcal{L}(k)\) of \(\mathcal{L}\) is the set of all the ordered partitions of a set of size \(k\). Using this model, they also give a combinatorial model for the evaluation map \(ev:\Sigma\Omega^2\Sigma^2X\to \Omega\Sigma^2X\) and study the Cohen representation for the group of homotopy classes of maps between double loop suspensions by using it. Moreover, they can recover Wu's shuffle relation and provide a type of secondary relations in Cohen groups by using Toda brackets. In particular, they also prove that certain maps are null-homotopic by combining their relations and the classical James-Hopf invariants.