We reformulate Milgram's model of a 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. Using this model, we give a combinatorial model for the evaluation map and use it to study the Cohen representation for the group of homotopy classes of maps between double loop suspensions. Demonstrating the general theory, we recover Wu's shuffle relations and further provide a type of secondary relations in Cohen groups by using Toda brackets. In particular, we prove certain maps are null-homotopic by combining our relations and the classical James–Hopf invariants.
The first named author is supported by Postdoctoral International Exchange Program for Incoming Postdoctoral Students under Chinese Postdoctoral Council and Chinese Postdoctoral Science Foundation. He is also supported in part by Chinese Postdoctoral Science Foundation (Grant Nos. 2018M631605 and 2019T120145), and National Natural Science Foundation of China (Grant No. 11801544). The second named author is partially supported by National Natural Science Foundation of China (Grant No. 11971144) and High-level Scientific Research Foundation of Hebei Province (Grant No. 13113093).
"Combinatorics of double loop suspensions, evaluation maps and Cohen groups." J. Math. Soc. Japan 72 (3) 847 - 889, July, 2020. https://doi.org/10.2969/jmsj/81678167