On symmetric commutator subgroups, braids, links and homotopy groups
Authors:
J. Y. Li and J. Wu
Journal:
Trans. Amer. Math. Soc. 363 (2011), 38293852
MSC (2010):
Primary 55Q40, 20F12; Secondary 20F36, 57M25
Published electronically:
February 25, 2011
MathSciNet review:
2775829
Fulltext PDF
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Additional Information
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.
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 J. A. Berrick, F. R. Cohen, Y. L. Wong and J. Wu, Braids, configurations and homotopy groups, J. Amer. Math. Soc. 19 (2006), 265326. MR 2188127 (2007e:20073)
 2.
 V. G. Bardakov, R. Mikhailov, V. V. Vershinin and J. Wu, Brunnian braids on surfaces, preprint.
 3.
 R. Brown and J.L. Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987), 311335. MR 899052 (88m:55008)
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 F. R. Cohen and J. Wu, On braids, free groups and the loop space of the sphere, Progress in Mathematic Techniques 215 (2004), Algebraic Topology: Categorical Decompositions, 93105. MR 2039761 (2005b:55011)
 5.
 F. R. Cohen and J. Wu, On braid groups and homotopy groups, Geometry & Topology Monographs 13 (2008), 169193. MR 2508205
 6.
 G. Ellis and R. Mikhailov, A colimit of classifying spaces, Adv. Math. 223 (2010), no. 6, 20972113. MR 2601009
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 E. Fadell, Homotopy groups of configuration spaces and the string problem of Dirac, Duke Math. J. 29 (1962), 231242. MR 0141127 (25:4538)
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 E. Fadell and J. Van Buskirk, The braid groups of and , Duke Math. J. 29 (1962), 243257. MR 0141128 (25:4539)
 9.
 E. Fadell and L. Neuwirth, Configuration spaces, Math. Scand 10 (1962), 111118. MR 0141126 (25:4537)
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 B. Gray, A note on the HiltonMilnor theorem, Topology 10 (1971), 199201. MR 0281202 (43:6921)
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 I. M. James, Reduced product spaces, Ann. Math. 62 (1953), 170197. MR 0073181 (17:396b)
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 D. L. Johnson, Towards a characterization of smooth braids, Math. Proc. Cambridge Philos. Soc. 92 (1982), 425427. MR 677467 (84c:20048)
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 H. W. Levinson, Decomposable braids and linkages, Trans. AMS 178 (1973), 111126. MR 0324684 (48:3034)
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 H. W. Levinson, Decomposable braids as subgroups of braid groups, Trans. AMS 202 (1975), 5155. MR 0362287 (50:14729)
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 T. Stanford, Brunnian braids and some of their generalizations, arXiv: math/9907072v1
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 20.
 J. Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002), 645662. MR 1888426 (2003e:20041)
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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
DOI:
http://dx.doi.org/10.1090/S000299472011053390
Keywords:
Symmetric commutator subgroup,
homotopy group,
link group,
Brunnian braid,
free group,
surface group
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. R146000101112 and R146000137112) of MOE of Singapore and a grant (No. 11028104) of NSFC
Article copyright:
© Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
