## On symmetric commutator subgroups, braids, links and homotopy groups

HTML articles powered by AMS MathViewer

- by J. Y. Li and J. Wu PDF
- Trans. Amer. Math. Soc.
**363**(2011), 3829-3852 Request permission

## 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

## Additional Information

**J. Y. Li**- Affiliation: Institute of Mathematics and Physics, Shijiazhuang Railway Institute, Shijiazhuang 050043, People’s Republic of China
- Email: yanjinglee@163.com
**J. Wu**- Affiliation: Department of Mathematics, National University of Singapore, Block S17 (SOC1), 10, Lower Kent Ridge Road, Republic of Singapore
- Email: matwuj@nus.edu.sg
- Received by editor(s): August 4, 2009
- Received by editor(s) in revised form: January 25, 2010, and February 28, 2010
- Published electronically: February 25, 2011
- Additional Notes: The first author was partially supported by the National Natural Science Foundation of China 10971050

The second author was partially supported by the AcRF Tier 1 (WBS No. R-146-000-101-112 and R-146-000-137-112) of MOE of Singapore and a grant (No. 11028104) of NSFC - © Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**363**(2011), 3829-3852 - MSC (2010): Primary 55Q40, 20F12; Secondary 20F36, 57M25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05339-0
- MathSciNet review: 2775829