Combinatorics of double loop suspensions, evaluation maps and Cohen groups

TL;DR

This paper introduces a combinatorial model for double loop suspensions using posets, enabling new insights into evaluation maps and Cohen groups, including relations and null-homotopy results.

Contribution

It reformulates Milgram's model with posets, providing a combinatorial approach to evaluation maps and Cohen groups, and derives new relations and null-homotopy results.

Findings

Recovered Wu's shuffle relations

Established secondary relations in Cohen groups

Proved certain maps are null-homotopic

Abstract

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.