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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Comultiplications on free groups and wedges of circles

Authors: Martin Arkowitz and Mauricio Gutierrez
Journal: Trans. Amer. Math. Soc. 350 (1998), 1663-1680
MSC (1991): Primary 20E05, 55P45; Secondary 55P40, 18A30
MathSciNet review: 1422887
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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 $F$. We consider individual comultiplications on $F$ and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of $F$. For a comultiplication $m$ of $F$ we define a subset $\Delta _{m} \subseteq F$ of quasi-diagonal elements which is basic to our investigation of associativity. The subset $\Delta _{m}$ can be determined algorithmically and contains the set of diagonal 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 comultiplication $m$ on $F$ to be a coloop in terms of the Fox derivatives of $m$ with respect to a basis of $F$. In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group $\operatorname {Aut} F$ on the set of comultiplications of $F$. 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.

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Additional Information

Martin Arkowitz
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

Mauricio Gutierrez
Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155

Keywords: Comultiplication, co-H-space, co-H-group, cogroup, free group, equalizers, category with coproducts, wedge of circles
Received by editor(s): July 19, 1996
Additional Notes: Part of this work was done while the first-named author was a visitor at the University of Milan. In addition, the second-named author also visited Milan for a brief period. The authors would like to thank both the Department of Mathematics at the University of Milan in general, and Professor Renzo Piccinini in particular, for their hospitality.
Article copyright: © Copyright 1998 American Mathematical Society