COMULTIPLICATIONS ON FREE GROUPS AND WEDGES OF CIRCLES

By means of the fundamental group functor, a co-H-space structure or a cp-H-group structure on a wedge of circles is seen to be Equivalent t o a comultiplication or a cogroup structure on a free group F. We cons ider individual comultiplications on F and their properties such as as sociativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of F. For a comultiplication m of F w e define a subset Delta(m) subset of or equal to F of quasi-diagonal e lements which is basic to our investigation of associativity. The subs et am can be determined algorithmically and contains the set of diagon al elements D-m. We show that D-m is a basis for the largest subgroup A(m) of F on which m is associative and that A(m) is a free factor of F. We also give necessary and sufficient conditions for a comultiplica tion m on F to be a coloop in terms of the Fox derivatives of m with r espect to a basis off. In addition, we consider inverses of a comultip lication, the collection of cohomomorphisms between two free groups wi th comultiplication and the action of the group Aut F on the set of co multiplications of F. We give many examples to illustrate these notion s. We conclude by translating these results from comultiplications on free groups to co-H-space structures on wedges of circles.