Comultiplications on free groups and wedges of circles

Abstract

By means of the fundamental group functor, a co-H-space structure or a co-H-group structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group F F . We consider individual comultiplications on F F and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of F F . For a comultiplication m m of F F we define a subset Δ m ⊆ F \Delta _{m} \subseteq F of quasi-diagonal elements which is basic to our investigation of associativity. The subset Δ m \Delta _{m} can be determined algorithmically and contains the set of diagonal elements D m D_{m} . We show that D m D_{m} is a basis for the largest subgroup A m A_{m} of F F on which m m is associative and that A m A_{m} is a free factor of F F . We also give necessary and sufficient conditions for a comultiplication m m on F F to be a coloop in terms of the Fox derivatives of m m with respect to a basis of F F . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group Aut ⁡ F \operatorname {Aut} F on the set of comultiplications of F F . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.