## Comultiplications on free groups and wedges of circles

HTML articles powered by AMS MathViewer

- 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

- Martin Arkowitz,
*Co-$H$-spaces*, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 1143–1173. MR**1361908**, DOI 10.1016/B978-044481779-2/50024-9 - Martin Arkowitz and Gregory Lupton,
*Equivalence classes of homotopy-associative comultiplications of finite complexes*, J. Pure Appl. Algebra**102**(1995), no. 2, 109–136. MR**1354057**, DOI 10.1016/0022-4049(94)00074-S - Joan S. Birman,
*An inverse function theorem for free groups*, Proc. Amer. Math. Soc.**41**(1973), 634–638. MR**330295**, DOI 10.1090/S0002-9939-1973-0330295-8 - Israel Berstein and Emmanuel Dror,
*On the homotopy type of non-simply-connected co-$H$-spaces*, Illinois J. Math.**20**(1976), no. 3, 528–534. MR**407837** - B. Eckmann and P. J. Hilton,
*Structure maps in group theory*, Fund. Math.**50**(1961/62), 207–221. MR**132776**, DOI 10.4064/fm-50-2-207-221 - B. Eckmann and P. J. Hilton,
*Group-like structures in general categories. III. Primitive categories*, Math. Ann.**150**(1963), 165–187. MR**153721**, DOI 10.1007/BF01470843 - P. Hebroni,
*Sur les inverses des éléments dérivables dans un anneau abstrait*, C. R. Acad. Sci. Paris**209**(1939), 285–287 (French). MR**14** - Wilhelm Müller,
*Zum Problem der Längsbewegung eines Flugszeugs*, Z. Angew. Math. Mech.**19**(1939), 193–202 (German). MR**112**, DOI 10.1002/zamm.19390190402 - Peter Hilton, Guido Mislin, and Joseph Roitberg,
*On co-$H$-spaces*, Comment. Math. Helv.**53**(1978), no. 1, 1–14. MR**483528**, DOI 10.1007/BF02566062 - T. Genčev,
*Über die unzerlegbaren Vektoren gewisser Kegel*, Bŭlgar. Akad. Nauk Izv. Mat. Inst.**3**(1958), no. 1, 69–88 (1958) (Bulgarian, with Russian and German summaries). MR**0110932** - Wilhelm Magnus, Abraham Karrass, and Donald Solitar,
*Combinatorial group theory: Presentations of groups in terms of generators and relations*, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR**0207802** *$H$-spaces. Actes de la réunion de Neuchâtel (Suisse), août 1970*, Lecture Notes in Mathematics, Vol. 196, Springer-Verlag, Berlin-New York, 1971 (French). Publiés par Francois Sigrist; Textes rédigés en anglais. MR**0287544**

## 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