#### Volume 16, issue 2 (2016)

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On intersecting subgroups of Brunnian link groups

### Fengchun Lei, Jie Wu and Yu Zhang

Algebraic & Geometric Topology 16 (2016) 1043–1061
##### Abstract

Let $G\left({L}_{n}\right)$ be the link group of a Brunnian $n$–link ${L}_{n}$ and ${R}_{i}$ be the normal closure of the ${i}^{th}$ meridian in $G\left({L}_{n}\right)$ for $1\le i\le n$. In this article, we show that the intersecting subgroup ${R}_{1}\cap {R}_{2}\cap \cdots \cap {R}_{m}$ coincides with the iterated symmetric commutator subgroup ${\prod }_{\sigma \in {\Sigma }_{m}}\left[\left[{R}_{\sigma \left(1\right)},{R}_{\sigma \left(2\right)}\right],\dots ,{R}_{\sigma \left(m\right)}\right]$ for $2\le m\le n$ using the techniques of homotopy theory. Moreover, we give a presentation for the intersecting subgroup ${R}_{1}\cap {R}_{2}\cap \cdots \cap {R}_{n}$.

##### Keywords
Brunnian link, homotopy colimit, link group, symmetric commutator subgroup
Primary: 55Q40
Secondary: 57M25
##### Publication
Received: 8 April 2015
Accepted: 27 July 2015
Published: 26 April 2016
##### Authors
 Fengchun Lei School of Mathematical Sciences Dalian University of Technology Dalian, 116024 China Jie Wu Department of Mathematics National University of Singapore 10 Lower Kent Ridge Road Singapore 119076 Singapore Yu Zhang Department of Mathematics Harbin Institute of Technology Harbin, 150001 China