Comultiplications on free groups and wedges of circles
Authors:
Martin Arkowitz and Mauricio Gutierrez
Journal:
Trans. Amer. Math. Soc. 350 (1998), 16631680
MSC (1991):
Primary 20E05, 55P45; Secondary 55P40, 18A30
MathSciNet review:
1422887
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: By means of the fundamental group functor, a coHspace structure or a coHgroup structure on a wedge of circles is seen to be equivalent to a comultiplication or a cogroup structure on a free group . We consider individual comultiplications on and their properties such as associativity, coloop structure, existence of inverses, etc. as well as the set of all comultiplications of . For a comultiplication of we define a subset of quasidiagonal elements which is basic to our investigation of associativity. The subset can be determined algorithmically and contains the set of diagonal elements . We show that is a basis for the largest subgroup of on which is associative and that is a free factor of . We also give necessary and sufficient conditions for a comultiplication on to be a coloop in terms of the Fox derivatives of with respect to a basis of . In addition, we consider inverses of a comultiplication, the collection of cohomomorphisms between two free groups with comultiplication and the action of the group on the set of comultiplications of . We give many examples to illustrate these notions. We conclude by translating these results from comultiplications on free groups to coHspace structures on wedges of circles.
 1.
Martin
Arkowitz, Co𝐻spaces, Handbook of algebraic topology,
NorthHolland, Amsterdam, 1995, pp. 1143–1173. MR 1361908
(96m:55012), http://dx.doi.org/10.1016/B9780444817792/500249
 2.
Martin
Arkowitz and Gregory
Lupton, Equivalence classes of homotopyassociative
comultiplications of finite complexes, J. Pure Appl. Algebra
102 (1995), no. 2, 109–136. MR 1354057
(96j:55008), http://dx.doi.org/10.1016/00224049(94)00074S
 3.
Joan
S. Birman, An inverse function theorem for free
groups, Proc. Amer. Math. Soc. 41 (1973), 634–638. MR 0330295
(48 #8632), http://dx.doi.org/10.1090/S00029939197303302958
 4.
Israel
Berstein and Emmanuel
Dror, On the homotopy type of nonsimplyconnected
co𝐻spaces, Illinois J. Math. 20 (1976),
no. 3, 528–534. MR 0407837
(53 #11607)
 5.
B.
Eckmann and P.
J. Hilton, Structure maps in group theory, Fund. Math.
50 (1961/1962), 207–221. MR 0132776
(24 #A2612)
 6.
B.
Eckmann and P.
J. Hilton, Grouplike structures in general categories. III.
Primitive categories, Math. Ann. 150 (1963),
165–187. MR 0153721
(27 #3682)
 7.
Ralph
H. Fox, Free differential calculus. I. Derivation in the free group
ring, Ann. of Math. (2) 57 (1953), 547–560. MR 0053938
(14,843d)
 8.
T. Ganea, Cogroups and suspensions, Inv. Math. 9 (1970), 185197. MR 112:2484
 9.
Peter
Hilton, Guido
Mislin, and Joseph
Roitberg, On co𝐻spaces, Comment. Math. Helv.
53 (1978), no. 1, 1–14. MR 483528
(80d:55014), http://dx.doi.org/10.1007/BF02566062
 10.
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 (22 #1800)
 11.
Wilhelm
Magnus, Abraham
Karrass, and Donald
Solitar, Combinatorial group theory: Presentations of groups in
terms of generators and relations, Interscience Publishers [John Wiley
& Sons, Inc.], New YorkLondonSydney, 1966. MR 0207802
(34 #7617)
 12.
𝐻spaces. Actes de la réunion de Neuchâtel
(Suisse), août 1970, Publiés par Francois Sigrist. Textes
rédigés en anglais. Lecture Notes in Mathematics, Vol. 196,
SpringerVerlag, Berlin, 1971 (French). MR 0287544
(44 #4748)
 1.
 M. Arkowitz, CoHSpaces, in Handbook of Algebraic Topology (I.M.James, ed.), Elsevier Science, 1995, pp. 11431173. MR 96m:55012
 2.
 M. Arkowitz and G. Lupton, Equivalence classes of homotopyassociative comultiplications of finite complexes, J. Pure Appl. Math. 102 (1995), 109136. MR 96j:55008
 3.
 J. Birman, An inverse function theorem for groups, Proc. Amer. Math. Soc. 41 (1974), 634638. MR 48:8632
 4.
 I. Berstein and E. Dror, On the homotopy type of nonsimplyconnected coHspaces, Illinois J. Math. 20 (1976), 528534. MR 53:11607
 5.
 B. Eckmann and P. Hilton, Structure maps in group theory, Fund. Math. 50 (1961), 207221. MR 24:A2612
 6.
 , Grouplike structures in general categories III, Math. Ann. 150 (1961), 165187. MR 27:3682
 7.
 R. Fox, Free differential calculus I, Ann. of Math. 57 (1953), 547560. MR 14:843d
 8.
 T. Ganea, Cogroups and suspensions, Inv. Math. 9 (1970), 185197. MR 112:2484
 9.
 P. Hilton, G. Mislin and J. Roitberg, On coHspaces, Comment. Math. Helv. 53 (1978), 114. MR 80d:55014
 10.
 D. Kan, Monoids and their dual, Bol. Soc. Mat. Mex. 3 (1958), 5261. MR 22:1800
 11.
 W. Magnus, A. Karrass and D. Solitar, Combinatorial Group Theory, Wiley, 1966. MR 34:7617
 12.
 J.D. Stasheff, Hspace problems, HSpaces, SpringerVerlag LNM 196, 1971, pp. 122136. MR 44:4748
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (1991):
20E05,
55P45,
55P40,
18A30
Retrieve articles in all journals
with MSC (1991):
20E05,
55P45,
55P40,
18A30
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
DOI:
http://dx.doi.org/10.1090/S0002994798019163
PII:
S 00029947(98)019163
Keywords:
Comultiplication,
coHspace,
coHgroup,
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 firstnamed author was a visitor at the University of Milan. In addition, the secondnamed 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
