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
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.References
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Additional Information
- Martin Arkowitz
- Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
- Email: martin.arkowitz@dartmouth.edu
- Mauricio Gutierrez
- Affiliation: Department of Mathematics, Tufts University, Medford, Massachusetts 02155
- Email: mgutierr@tufts.edu
- 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: https://doi.org/10.1090/S0002-9947-98-01916-3
- MathSciNet review: 1422887