On symmetric commutator subgroups, braids, links and homotopy groups

Li, J.; Wu, J.

doi:10.1090/S0002-9947-2011-05339-0

On symmetric commutator subgroups, braids, links and homotopy groups
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by J. Y. Li and J. Wu

Trans. Amer. Math. Soc. 363 (2011), 3829-3852

DOI: https://doi.org/10.1090/S0002-9947-2011-05339-0

Published electronically: February 25, 2011

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Abstract:

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.

References

A. J. Berrick, F. R. Cohen, Y. L. Wong, and J. Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006), no. 2, 265–326. MR 2188127, DOI 10.1090/S0894-0347-05-00507-2
V. G. Bardakov, R. Mikhailov, V. V. Vershinin and J. Wu, Brunnian braids on surfaces, preprint.
Ronald Brown and Jean-Louis Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), no. 3, 311–335. With an appendix by M. Zisman. MR 899052, DOI 10.1016/0040-9383(87)90004-8
F. R. Cohen and J. Wu, On braid groups, free groups, and the loop space of the 2-sphere, Categorical decomposition techniques in algebraic topology (Isle of Skye, 2001) Progr. Math., vol. 215, Birkhäuser, Basel, 2004, pp. 93–105. MR 2039761
F. R. Cohen and J. Wu, On braid groups and homotopy groups, Groups, homotopy and configuration spaces, Geom. Topol. Monogr., vol. 13, Geom. Topol. Publ., Coventry, 2008, pp. 169–193. MR 2508205, DOI 10.2140/gtm.2008.13.169
Graham Ellis and Roman Mikhailov, A colimit of classifying spaces, Adv. Math. 223 (2010), no. 6, 2097–2113. MR 2601009, DOI 10.1016/j.aim.2009.11.003
Edward Fadell, Homotopy groups of configuration spaces and the string problem of Dirac, Duke Math. J. 29 (1962), 231–242. MR 141127
Edward Fadell and James Van Buskirk, The braid groups of $E^{2}$ and $S^{2}$, Duke Math. J. 29 (1962), 243–257. MR 141128
Edward Fadell and Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962), 111–118. MR 141126, DOI 10.7146/math.scand.a-10517
Brayton Gray, A note on the Hilton-Milnor theorem, Topology 10 (1971), 199–201. MR 281202, DOI 10.1016/0040-9383(71)90004-8
I. M. James, Reduced product spaces, Ann. of Math. (2) 62 (1955), 170–197. MR 73181, DOI 10.2307/2007107
D. L. Johnson, Towards a characterization of smooth braids, Math. Proc. Cambridge Philos. Soc. 92 (1982), no. 3, 425–427. MR 677467, DOI 10.1017/S0305004100060138
H. Levinson, Decomposable braids and linkages, Trans. Amer. Math. Soc. 178 (1973), 111–126. MR 324684, DOI 10.1090/S0002-9947-1973-0324684-X
H. Levinson, Decomposable braids as subgroups of braid groups, Trans. Amer. Math. Soc. 202 (1975), 51–55. MR 362287, DOI 10.1090/S0002-9947-1975-0362287-3
Jingyan Li and Jie Wu, Artin braid groups and homotopy groups, Proc. Lond. Math. Soc. (3) 99 (2009), no. 3, 521–556. MR 2551462, DOI 10.1112/plms/pdp005
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
G. S. Makanin, Kourovka Notebook (Unsolved Questions in Group Theory) Seventh edition. Novosibirsk. 1980, \text{Question 6.23.} p. 78.
T. Stanford, Brunnian braids and some of their generalizations, arXiv: math/9907072v1
J. Wu, Combinatorial descriptions of homotopy groups of certain spaces, Math. Proc. Cambridge Philos. Soc. 130 (2001), no. 3, 489–513. MR 1816806, DOI 10.1017/S030500410100487X
Jie Wu, A braided simplicial group, Proc. London Math. Soc. (3) 84 (2002), no. 3, 645–662. MR 1888426, DOI 10.1112/S0024611502013370