## Algebraic & Geometric Topology

### On intersecting subgroups of Brunnian link groups

#### Abstract

Let $G(Ln)$ be the link group of a Brunnian $n$–link $Ln$ and $Ri$ be the normal closure of the $ith$ meridian in $G(Ln)$ for $1 ≤ i ≤ n$. In this article, we show that the intersecting subgroup $R1 ∩ R2 ∩⋯ ∩ Rm$ coincides with the iterated symmetric commutator subgroup $∏ σ∈Σm[[Rσ(1),Rσ(2)],…,Rσ(m)]$ for $2 ≤ m ≤ n$ using the techniques of homotopy theory. Moreover, we give a presentation for the intersecting subgroup $R1 ∩ R2 ∩⋯ ∩ Rn$.

#### Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 1043-1061.

Dates
Received: 8 April 2015
Accepted: 27 July 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841160

Digital Object Identifier
doi:10.2140/agt.2016.16.1043

Mathematical Reviews number (MathSciNet)
MR3493415

Zentralblatt MATH identifier
1337.55013

Subjects
Primary: 55Q40: Homotopy groups of spheres
Secondary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}

#### Citation

Lei, Fengchun; Wu, Jie; Zhang, Yu. On intersecting subgroups of Brunnian link groups. Algebr. Geom. Topol. 16 (2016), no. 2, 1043--1061. doi:10.2140/agt.2016.16.1043. https://projecteuclid.org/euclid.agt/1510841160

#### References

• V,G Bardakov, R Mikhailov, V,V Vershinin, J Wu, Brunnian braids on surfaces, Algebr. Geom. Topol. 12 (2012) 1607–1648
• A,J Berrick, F,R Cohen, Y,L Wong, J Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006) 265–326
• W,A Bogley, J,H,C Whitehead's asphericity question, from “Two dimensional homotopy and combinatorial group theory” (C Hog-Angeloni, W Metzler, editors), London Math. Soc. Lecture Note Ser. 197, Cambridge Univ. Press, Cambridge (1993) 309–334
• R Brown, J-L Loday, Van Kampen theorems for diagrams of spaces, Topology 26 (1987) 311–335
• H Brunn, Üeber Verkettung, Sitzungberichte der Bayerischer Akad. Wiss. Math.-Phys. 22 (1892) 77–99
• K,S Chichak, S,J Cantrill, A,R Pease, S-H Chiu, G,W,V Cave, J,L Atwood, J,F Stoddart, Molecular Borromean Rings, Science 304 (2004) 1308–1312
• F,R Cohen, J Wu, On braid groups and homotopy groups, from “Groups, homotopy and configuration spaces” (N Iwase, T Kohno, R Levi, D Tamaki, J Wu, editors), Geom. Topol. Monogr. 13 (2008) 169–193
• F,R Cohen, J Wu, Artin's braid groups, free groups, and the loop space of the 2-sphere, Q. J. Math. 62 (2011) 891–921
• H Debrunner, Links of Brunnian type, Duke Math. J. 28 (1961) 17–23
• G Ellis, R Mikhailov, A colimit of classifying spaces, Adv. Math. 223 (2010) 2097–2113
• F Fang, F Lei, J Wu, The symmetric commutator homology of link towers and homotopy groups of $3$–manifolds, Commun. Math. Stat. 3 (2015) 497–526
• G,G Gurzo, The group of smooth braids, from “16th All-Union Algebra Conference”, Abstract II, Leningrad (1981) 39–40
• K Habiro, Claspers and finite type invariants of links, Geom. Topol. 4 (2000) 1–83
• K Habiro, Brunnian links, claspers and Goussarov–Vassiliev finite type invariants, Math. Proc. Cambridge Philos. Soc. 142 (2007) 459–468
• K Habiro, J-B Meilhan, On the Kontsevich integral of Brunnian links, Algebr. Geom. Topol. 6 (2006) 1399–1412
• K Habiro, J-B Meilhan, Finite type invariants and Milnor invariants for Brunnian links, Internat. J. Math. 19 (2008) 747–766
• D,L Johnson, Towards a characterization of smooth braids, Math. Proc. Cambridge Philos. Soc. 92 (1982) 425–427
• H Levinson, Decomposable braids and linkages, Trans. Amer. Math. Soc. 178 (1973) 111–126
• H Levinson, Decomposable braids as subgroups of braid groups, Trans. Amer. Math. Soc. 202 (1975) 51–55
• J,Y Li, J Wu, On symmetric commutator subgroups, braids, links and homotopy groups, Trans. Amer. Math. Soc. 363 (2011) 3829–3852
• J-L Loday, Spaces with finitely many nontrivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179–202
• W Magnus, A Karrass, D Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience, New York (1966)
• B Mangum, T Stanford, Brunnian links are determined by their complements, Algebr. Geom. Topol. 1 (2001) 143–152
• J-B Meilhan, A Yasuhara, Whitehead double and Milnor invariants, Osaka J. Math. 48 (2011) 371–381
• J Milnor, Link groups, Ann. of Math. 59 (1954) 177–195
• H,A Miyazawa, A Yasuhara, Classification of $n$–component Brunnian links up to $C\sb n$–move, Topology Appl. 153 (2006) 1643–1650
• M Ozawa, On a genus of a closed surface containing a Brunnian link, Tokyo J. Math. 31 (2008) 347–349
• J Wu, Combinatorial descriptions of homotopy groups of certain spaces, Math. Proc. Cambridge Philos. Soc. 130 (2001) 489–513