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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Comultiplications on free groups and wedges of circles
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by Martin Arkowitz and Mauricio Gutierrez PDF
Trans. Amer. Math. Soc. 350 (1998), 1663-1680 Request permission


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
  • Email:
  • Mauricio Gutierrez
  • Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
  • Email:
  • 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.
  • © Copyright 1998 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 350 (1998), 1663-1680
  • MSC (1991): Primary 20E05, 55P45; Secondary 55P40, 18A30
  • DOI:
  • MathSciNet review: 1422887