On Intersecting Subgroups of Brunnian Link Groups

Algebraic & Geometric Topology 16 (2016) 1043–1061 msp

On intersecting subgroups of Brunnian link groups

FENGCHUN LEI JIE WU YU ZHANG

Let G.Ln/ be the link group of a Brunnian n–link Ln and Ri be the normal closure th of the i meridian in G.Ln/ for 1 i n. In this article, we show that the Ä Ä intersecting subgroup R1 R2 Rm coincides with the iterated symmetric Q \ \ \ commutator subgroup ŒŒR.1/; R.2/; : : : ; R.m/ for 2 m n using the †m Ä Ä techniques of homotopy theory.2 Moreover, we give a presentation for the intersecting subgroup R1 R2 Rn . \ \ \ 55Q40; 57M25

1 Introduction

An n–component link Ln l1; l2;:::; ln is called a Brunnian link, named for D f g Hermann Brunn[5], if it is a nontrivial link that becomes trivial when any component is removed. The simplest and best known example of Brunnian link are the Borromean rings. Milnor[25] classiﬁed Brunnian links up to link homotopy; he called them almost trivial links before Debrunner[9] named them Brunnian in 1961. The Cn –moves and the ﬁnite-type invariants have been studied by Habiro and Meilhan[13; 14; 15; 16] and Miyazawa and Yasuhara[26]. We refer to Mangum and Stanford[23], Meilhan and Yasuhara[24] and Ozawa[27] for more results. An important application of Brunnian links to mathematical chemistry was given by Chichak et al[6].

For an n–link Ln l1; l2;:::; ln , we deﬁne the link group G.Ln/ as the funda- D f g 3 mental group of the link complement S Ln . From the work of Mangum and n Stanford[23], Brunnian links are determined by their link complements, so the link group G.Ln/ plays an important role in studying a Brunnian link Ln . Let th diLn l1; l2;:::; li 1; li 1;:::; ln be the .n 1/–link obtained by deleting the i D f C g component li from the n–link Ln . There is a group homomorphism hi from G.Ln/ 3 3 to G.diLn/, induced by the inclusion of the link complements S Ln , S diLn . n ! n The kernel of hi is denoted by Ri . If we let ˛i be the normal closure of the meridian h i of li in G.Ln/, then ˛i Ri . h i D

Published: 26 April 2016 DOI: 10.2140/agt.2016.16.1043 1044 Fengchun Lei, Jie Wu and Yu Zhang

In this paper, we are interested in exploring the intersecting subgroup R1 R2 Rm th \ \ \ of a Brunnian link group G.Ln/ for 2 m n. Let ˇi be the i longitude of Ln . Ä Ä Since Ln is Brunnian, the homotopy class Œˇi is in R1 Ri 1 Ri 1 Rn \ \ \ C \ \ for each i . For some Brunnian links Ln , for instance the Whitehead link, the homotopy classes of all of the longitudes lie in the full intersecting subgroup R1 Rn . \ \ Let N and H be subgroups of a group G . Recall that the commutator subgroup ŒN; H is deﬁned as the subgroup of G generated by the commutators Œg; h g 1h 1gh for D g N and h H . If N and H are normal subgroups, then ŒN; H is contained in 2 2 the intersecting subgroup N H . Hence, for each permutation †m , the iterated \ 2 commutator subgroup

ŒŒR ; R ; : : : ; R R1 R2 Rm: .1/ .2/ .m/ Ä \ \ \ It follows that the symmetric commutator subgroup Y ŒR1; R2;:::; RmS ŒŒR ; R ; : : : ; R D .1/ .2/ .m/ †m 2 is a subgroup of R1 R2 Rm . Using a recent result of Ellis and Mikhailov[10] to- \ \ \ gether with the techniques of homotopy theory, we are able to determine the intersecting subgroup R1 R2 Rm for 2 m n. \ \ \ Ä Ä

Theorem 1.1 Suppose that Ln l1; l2;:::; ln is an n–component Brunnian link D f g th and G.Ln/ is the link group of Ln . Let Ri be the normal closure of the i meridian in G.Ln/. Then m \ Ri ŒR1;:::; RmS D i 1 for 2 m n. D Ä Ä

Let R1; R2;:::; Rm be subgroups of a group G . There is a Brown–Loday construction of non-commutative tensor product T .R1; R2;:::; Rm/ deﬁned in terms of generator- relation system[4]. The construction of T .R1; R2;:::; Rm/ will be recalled in the next section.

Our second result gives a presentation of the intersecting subgroup R1 Rm in \ \ terms of the Brown–Loday construction.

Theorem 1.2 Given the same assumptions of Theorem 1.1, we have: (1) There is a short exact sequence of groups n 1 n M \ Z T .R1; R2;:::; Rn/ Ri: i 1 D Algebraic & Geometric Topology, Volume 16 (2016) On intersecting subgroups of Brunnian link groups 1045

(2) There is a short exact sequence of groups m M1 \ Z T .R1; R2;:::; Rm/ Ri i 1 for 2 m < n. D Ä Remark The combinatorial problem of determining the intersecting subgroups is technically difficult in general, and related to the longstanding unsolved classical Whitehead asphericity question in low-dimensional topology; see Bogley[3]. The question on the determination of Brunnian braids answered in Bardakov, Mikhailov, Vershinin and Wu[1], Gurzo[12], Johnson[17], Levinson[18; 19] and Li and Wu[20] concerns the intersecting subgroups of braid groups with connections to homotopy theory; see Berrick, Cohen, Wong and Wu[2] and Cohen and Wu[7; 8]. As a development of Brown and Loday’s program[4], recent progress made by Ellis and Mikhailov gives a possibility to determine certain intersecting subgroups in terms of commutator subgroups with obstructions given by homotopy groups. It was then discovered by Fang, Lei and Wu[11] and Li and Wu[20] that the quotient group 3 .R1 Rn/=ŒR1;:::; RnS is isomorphic to the homotopy group n.S / for a \ \ th strongly nonsplittable n–link L, where Ri is the normal closure of the i meridian and a link is called strongly nonsplittable if any of its nonempty sublink is nonsplittable. In other words, the determination of the intersecting subgroup of strongly nonsplittable 3 links L is equally difficult to the determination of the homotopy group n.S /, even in the simplest case that L is a Hopf n–link. Since the Brunnian links are not strongly nonsplittable, we cannot apply the results of[11; 20] to answering the question of the intersecting subgroup of Brunnian links. However, we observe in Claim 2.2 that the connectivity hypothesis Ellis and Mikhailov[10] is satisfied for Brunnian link groups. This gives us the chance to determine the intersecting subgroup of Brunnian link groups in this article. As an application to Brunnian link groups, by Theorem 1.1 and Theorem 1.2(1) there is an exact sequence

n 1 M n 1 Z T .R1; R2;:::; Rn/ G.Ln/ Fn 1; ! ! ! ! where Fn 1 is the free group of rank n 1. This gives a connection between Brown and Loday’s theory on non-commutative tensor products and Brunnian links.

We should point out that by Theorem 1.1, the connectivity hypothesis of Ellis and Mikhailov[10] is also satisﬁed for almost Brunnian link groups along the lines of the proof of Lemma 2.1, where a link is called almost Brunnian if it becomes Brunnian after removing any of its components. Of course, the full determination of the intersecting subgroup of almost Brunnian link groups will technically depend on the determination

Algebraic & Geometric Topology, Volume 16 (2016) 1046 Fengchun Lei, Jie Wu and Yu Zhang of the homotopy type of the homotopy colimit of a cubical diagram, as we will see in the case of Brunnian link groups in this article. The procedure can be continued for more general links until the intersecting subgroup differs from the symmetric commutator subgroup with obstruction given by the homotopy group of the homotopy colimit. This article highlights the homotopy theoretic methodology for determining the intersecting subgroup of link groups. In the next section, we will lay out the necessary preliminaries for proving our theorems. m In Section 3, we’ll study the homotopy type of the space X.Ln/ . The proofs of Theorems 1.1 and 1.2 will be given in Section 4.

Acknowledgements The ﬁrst and the second authors were supposed by a grant of NSFC number 11329101 of China. The second author was also supported by the Academic Research Fund of the National University of Singapore R-146-000-190-112. The third author was supported by a grant of NSFC number 11101103 of China.

2 Preliminaries

In this section we will review some results which we need to prove our main theorems, including symmetric commutator subgroups, colimits of classifying spaces and homotopy groups of homotopy colimits.

2A Symmetric commutator subgroups

Let G be a group with subgroups Rj for 1 j n. The concept of the symmetric com- Ä Ä mutator subgroup of R1; R2;:::; Rn , denoted by ŒR1; R2;:::; RnS , was introduced by Li and Wu[20] with the equation Y (1) ŒR1; R2;:::; RnS ŒŒR ; R ; : : : ; R ; D .1/ .2/ .n/ †n 2 th where †n is the n symmetric group and ŒŒR.1/; R.2/; : : : ; R.n/ is the subgroup generated by the left iterated commutators ŒŒŒg1; g2; g3; : : : ; gn with gi R . 2 .i/ In the same paper, the following theorem was also shown.

Theorem [20, Theorem 1.2] Let Rj be any normal subgroup of a group G with 1 j n. Then Ä Ä Y (2) ŒR1;:::; RnS ŒŒR1; R ; : : : ; R ; D .2/ .n/ †n 1 2 where †n 1 acts on 2; 3;:::; n . f g

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A bracket arrangement of weight n in a group G is given by Magnus, Karrass and Solitar[22] as a map ˇn Gn G defined inductively by W ! 1 2 (3) ˇ idG; ˇ .g1; g2/ Œg1; g2; D D where g1 and g2 are arbitrary elements in G and Œg1; g2 is the commutator of g1 k and g2 . Inductively, suppose that the bracket arrangements of weight k , ˇ , are defined for k 1; 2;:::; n 1. Then the bracket arrangements of weight n, ˇn , is D defined by the composition

ˇk ˇn k ˇ2 (4) Gn Gk Gn k G G G D ! ! k n k for any ˇ and ˇ . Closely related to the symmetric commutator subgroup is the fat commutator subgroup. For a group G with a sequence of subgroups Rj ; 1 j n. The fat commutator Ä Ä subgroup ŒŒR1;:::; Rn is deﬁned to be the subgroup of G generated by all of the commutators

t (5) ˇ .gi1 ;:::; git /; where

(a) 1 is n, Ä Ä (b) i1;:::; it 1;:::; n , ie each integer in 1; 2;:::; n appears as at least one f g D f g f g of the integers is ,

(c) gj Rj , 2 (d) ˇt runs over all of the bracket arrangements of weight t; t n. From the deﬁnition, it is clear that the symmetric commutator subgroup is a subgroup of the fat commutator subgroup. Moreover, the following result was also shown by Li and Wu[20], and plays an important role in our proof of Theorem 1.1.

Theorem [20, Theorem 1.1] Let Rj be any normal subgroup of a group G with 1 j n. Then Ä Ä (6) ŒŒR1; R2;:::; Rn ŒR1;:::; RnS : D This theorem tells us that the fat commutator subgroup is in fact the same as the symmetric commutator subgroup. We’ll also need another result of Wu.

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Theorem [28, Corollary 3.5] Let G F.S/ be a free group generated by S and D F.Ti/ be subgroups of G freely generated by Ti S for 1 i k . Let Ri be the Ä Ä kernel of the projection homomorphism i G F.Ti/. Then W ! k \ Ri ŒŒR1; R2;:::; R : D k i 1 D

Commutator subgroups have many important applications, especially to homotopy groups and geometric groups. In[20], Li and Wu used the commutator subgroups to give a connection between links and homotopy groups. In the present paper, we apply the commutator subgroups to study the homotopy groups of the homotopy colimits of classifying spaces for Brunnian link groups.

2B Homotopy colimits of classifying spaces

In this section, we review some facts about the colimits of classifying spaces. Our terminology here is coincident with that of Ellis and Mikhailov[10].

Roughly speaking, a classifying space BG of a topological group G is a topological space that is the quotient of a weakly contractible space (ie a topological space whose homotopy groups are all trivial) by a free action of G . The existence of a classifying space for discrete groups is a classical result. For a discrete group G , the classifying space BG is a path-connected topological space whose fundamental group is isomorphic to G and all higher homotopy groups are trivial, ie BG is a K.G; 1/–space. Of course, for a group G , the classifying space is not unique. It was shown that if X is a classifying space for G and Y is a classifying space for K, then X Y is a classifying space for G K. Some of the simple examples of classifying spaces are: (1) The circle S 1 is a classifying space for the inﬁnite cyclic group Z. (2) The n–torus T n is a classifying space for the free abelian group of rank n, Zn . (3) The wedge of n circles Wn S 1 is a classifying space for the free group of rank n. th Suppose that Ln is an n–Brunnian link and Ri is the normal closure of the n Q meridian in G.Ln/. For each proper subset I 1; 2;:::; n ; G.Ln/= I Ri Q f Wn gI 1 is a free group of rank n I , so B.G.Ln/= Ri/ j j S . j j I D (4) It has been shown[11, Theorem 2.1] that a link in S 3 is nonsplittable if and only if the link complement is a K.; 1/–space. So for an n–Brunnian link Ln 3 with link group G.Ln/, BG.Ln/ S Ln is a classifying space. D n

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We are interested in the homotopy colimits of classifying spaces. In order to have an intuitive image of homotopy colimits, we describe the homotopy push-out (colimit) of a diagram of spaces g f Z X Y: ! We want the homotopy push-out to preserve the homotopy type of X . So, we will ﬁrst thicken X to get X Œ0; 1 and then identify X X 0 with f .X / Y and ' f g X X 1 with g.X / Z. Therefore, the homotopy push-out is just the quotient ' f g space Q Y .X Œ0; 1/ Z= ; D q q where .x; 0/ f .x/ and .x; 1/ g.x/. In a more general sense than homotopy push-out, the homotopy colimit will be a functor from the category of diagrams of spaces into the category of topological spaces ᐀. Suppose that Ᏸ is a small category, ie a category where both the class of objects and class of morphisms are actually just sets. Denote by ᐀Ᏸ the category of functors F Ᏸ ᐀ and natural transformations. That is, ᐀Ᏸ is the category of diagrams of W ! spaces with the shape of Ᏸ. Our homotopy colimit is a functor

hocolim ᐀Ᏸ ᐀ ᏰW ! which satisﬁes the following properties:

(1) (Homotopy invariance property) Suppose that F; F Ᏸ ᐀ are two functors 0W ! and there is a natural transformation F F such that F.d/ F .d/ W ! 0 d W ! 0 is a homotopy equivalence for all d Ᏸ, then 2 hocolim.F/ hocolim.F /: ' 0 (2) If all the maps between spaces are coﬁbrations, then the homotopy colimit is coincident with the usual colimit, ie if F.˛/ F.d/ F.d / is a coﬁbration for W ! 0 every ˛ d d in Ᏸ, then W ! 0 hocolim.F/ colim.F/: '

What inspired us was the work on homotopy colimits given by cubical diagrams by Ellis and Mikhailov[10]. In the next section, we will review some of their results and settle down some important lemmas we need later for the proofs of our theorems.

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2C Homotopy groups of homotopy colimits

Let G be a group. We recall the deﬁnition of connectedness for m–tuples of normal subgroups .R1; R2;:::; Rm/ from[10].

Deﬁnition The m–tuple .R1; R2;:::; Rm/ is connected if either m 2 or m 3 Ä and for all subsets I; J 1; 2;:::; m with I 2; J 1 we have f g j j j j Â \ Ã Y \Â Y Ã (7) Ri Rj Ri Rj : D i I j J i I j J 2 2 2 2 Q In this paper, we study the homotopy colimit X of the spaces B.G= i I Ri/, where I ranges over all strict subsets I ¨ 1; 2;:::; m . The following theorem2 of Ellis and f g Mikhailov is important for our proofs.

Theorem [10, Theorem 1] If the .m 1/–tuple .R1;:::; Ri;:::; Rm/ is connected O for each 1 i m, then the homotopy colimit X satisﬁes Ä Ä R1 ::: Rm (8) m.X / \ \ ; Q ŒT R ; T R Š I J 1;2;:::;m ;I J ∅ i I i j J j [ Df g \ D 2 2 (9) m 1.X / ker.@ T .R1;:::; Rm/ G/: C Š W ! The connectivity condition in this theorem is crucial. The following lemma ensures the connectivity property of the subgroups in our results.

Lemma 2.1 For a Brunnian link Ln l1; l2;:::; ln as described above, the k –tuple D f g .Ri ;:::; Ri / is connected for each 1 k n 1 and ij 1;:::; n . 1 k Ä Ä 2 f g Before we prove the lemma, we make the following claim for simplicity.

Q Claim 2.2 Let G G= Rj ; x x be the quotient map. Equation (7) in W ! j J 7! the deﬁnition of connectivity is equivalent2 to the equality \ \ (10) Ri Ri: D i I i I 2 2 Proof First, suppose that (10) is true, ie \ Â\ Ã Y \Â Y Ã \ Ri Ri = Rj Ri= Rj Ri: D D D i I i I j J i I j J i I 2 2 2 2 2 2

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T Q Q T Q Notice that . i I Ri/. j J Rj / Ri. j J Rj /. Therefore . i I Ri/. j J Rj / T Q 2 2 Â 2 2 2 Â i I .Ri. j J Rj //. 2 2 T Q Q On the other hand, we pick any x i I .Ri. j J Rj //, ie x Ri. j J Rj / 2 2 2 2 T 2 for any i . The image of x, .x/, belongs to Ri for all i . So .x/ i I Ri T T 2 2 D i I Ri , where the last equality follows from (10). There exists y i I Ri such that 2 1 1 Q 2 2 1 .y/ .x/. So .y x/ e, ie y x ker j J Rj . Then x y y x T D Q D 2 D 2 D 2 . i I Ri/. j J Rj /. Equation (7) holds. 2 2 T T For the other direction, we suppose (7) is true. i I Ri Ri for any i , so i I Ri T T 2 Â 21 Â i I Ri . Now for any x i I Ri , x Ri for any i . Then x .x/ T 2 Q T2 2 Q 2 2 i I .Ri. j J Rj // . i I Ri/. j J Rj /, where the last equality follows from 2 2 T D 2 2 (7). So .x/ i I Ri . Equation (10) holds. 2 2

Proof of Lemma 2.1 For the cases that n 3, the proofs are trivial. So without loss Ä of generality, we assume n > 3; I 1;:::; n 2 and J n 1 . For other cases, D f g D f g the proofs are similar. Q Suppose that G G.Ln/= j J Rj . We notice that G is a free group gener- D 2 ated by ˛i ˛i i J since J is nonempty. So G F.˛1;:::; ˛n 2; ˛n/ f D j 62 g D D F.˛1; : : : ; ˛n 2; ˛n/. There is a group homomorphism

F.˛1;:::; ˛n 2/ F.˛1;:::; ˛i 1; ˛i 1;::: ˛n 2/; ! C so that Ri ˛i is the kernel. By[1, Lemma 5.1], we have R1 Rn 2 D h i Tn 2 \ \ D ŒR1;:::; Rn 2S . We also notice that ŒR1;:::; Rn 2S k 1 Rk . Now, if we think Ä of the quotient map D q G.Ln/ G; ! we have \ Â \ Ã (11) Rk q Rk q.ŒR1; R2;:::; Rn 2S / D k I k I 2 2 \ ŒR1; R2;:::; Rn 2S Rk : D D k I 2 The other direction is obvious.

The group T .R1; R2;:::; Rm/ in[10, Theorem 1] is generated by symbols

a A;B b; ˝

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where A B 1; 2;:::; m ; a RA; b RB (here for any I 1; 2;:::; m ; RI T t D f g 2 2 f g D i I Ri ), and it satisfies the relations 2 1 a A;B b .b B;A a/ ; ˝ D ˝ a a aa A;B b . a A;B b/.a A;B b/; 0 ˝ D 0 ˝ ˝ u 1 u w 1 w v 1 v . Œu ; v U V;W w/. Œw ; u W U;V v/. Œv ; w V W;U u/ 1; ˝ [ ˝ [ ˝ [ D 1 Œa;b Œa;b .a A;B b/.a0 A ;B b0/.a A;B b/ a0 A ;B b0 ˝ ˝ 0 0 ˝ D ˝ 0 0 for A B A B 1; 2;:::; m ; a N ; a N ; b N ; b N ; U V W 0 0 A 0 A0 B 0 B0 t D t D f g 2 2 2 x 2 1 t t D 1; 2;:::; m ; u NU ; v NV ; w NW . Here we define y xyx . f g 2 2 2 D m For an m–tuple 0; 1 , there is an m–cube of spaces B B.G; R1;:::; Rm/ 2 f g D W 0; 1 m (spaces) for which f g ! Â G Ã B B Q D i Ri 2 Q is the classifying space of the quotient group G= Ri . Here i means the i 2 i th coordinate of equals 1. This idea can be easily2 extended to an m–tuple 2 1; 0; 1 m . It was shown[4; 21] that each m–cube of spaces corresponds to an f g m–cube of fibrations F 1; 0; 1 m (spaces). The homomorphism W f g ! T .R1;:::; Rm/ 1.F 1; 1;:::; 1;0/ G; x y Œx; y ! Ä ˝ 7! has a crossed module structure @ T .R1;:::; Rm/ G; x y Œx; y such that for W ! ˝ 7! any g G; g.x y/ .gx gy/. 2 ˝ D ˝ m 3 Homotopy type of X.Ln/

m In this section we will study the homotopy type of the classifying space X.Ln/ . 2 3 3 When n 2, it’s immediate to see that X.L2/ S , because S is the push-out of D 3 3 3 ' K.; 1/–spaces S d2L2 S L2 S d1L2 . We choose the case n 4 as n n ! n D an experimental example; the case n 3 is similar. The more general cases will be D studied inductively.

3A The experimental example n 4 D 2 To get X.L4/ we study the cubical diagram

h 3 1- W3 1 BG.L4/ S L4 B.G.L4/=R1/ S D n D h2 W3 1 - 2 B.G.L4/=R2/ S X.L4/ D

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where h1 and h2 are induced from the group homomorphisms h1 and h2 . 2 In order to get the homotopy type of the push-out X.L4/ we need to look into the structure of B.G.L4/=R1/ and B.G.L4/=R2/. Let d1L4 l2; l3; l4 be the 3–link D f g 3 obtained by deleting the ﬁrst component of L4 . The link complement S d1L4 has W3 1 W2 2 n same the homotopy type as . S / . S / since d1L4 is splittable; we refer _ to[11, Proposition 2.2] for more details. The 2–sphere S 2; i 2; 3, is a separating i D sphere which separates li from the other components. Then the classifying space W3 1 W2 2 B.G.L4/=R1/ is obtained by ﬁlling in the 2–spheres from . S / . S / with 3– 3 S2 3 W3 1 W2_ 2 S2 3 balls, ie B.G.L4/=R1/ .S d1L4/ . D / .. S / . S / . D / 3 D n [ D _ [ D W S 1 . The left side of Figure 1 will show us the idea more clearly. S 2 D3 3 2 S3 l3 D3 l3 l 1 1 4 S l4 S S 1 S 1 D3 S 1 S 1 S 1 S 1 l l1 l22 l1 S 2 S 2 2 1 l2 D3

Figure 1: The classifying space B.G.L4/=R1/ is obtained by ﬁlling in the W3 1 W2 2 2–spheres from . S / . S / with 3–balls (left); B.G.L4/=R2/ is _ obtained similarly (right).

3 D S 2 D3 3 l3 l4 S 1 1 3 3 S D D D3 1 S S 1 l1 2 l2 S1 2 S2 S 1

2 W2 1 W3 3 Figure 2: X.L4/ is homotopic to . S / . S /. _ By the same argument, we can determine the homotopy type and structure of the space B.G.L4/=R2/, which is illustrated on the right in Figure 1.

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2 W2 1 W3 3 By deﬁnition, the homotopy push-out X.L4/ is homotopic to . S / . S / _ as indicated in Figure 2.

2 Actually, the space X.L4/ is obtained from the disjoint union of B.G.L4/=R1/ W3 1 W2 2 S2 3 W3 1 W2 2 S2 3D .. S / . S // . D /, B.G.L4/=R2/ .. S / . S // . D / 3 _ [ 3 D _ 3 [ and .S L4/ Œ0; 1 by identifying .S L4/ 0 (respectively .S L4/ 1 ) with 3 n n f g3 n f g h .S L4/ B.G.L4/=R1/ (respectively h .S L4/ B.G.L4/=R2/). Notice 1 n 2 n that now the component l1 (respectively l2 ) is ﬁlled in, so one side of the 2–sphere 2 2 3 3 3 3 S (respectively S ) is .D l1/ l1 D (respectively .D l2/ l2 D ) while 1 2 n [ D n [ D the other side is also a 3–ball, ie there is a 3–sphere.

For m 3, we refer to the cubical diagram in Figure 3. D

3 W3 1 BG.L4/ S L4 B.G.L4/=R1/ S D n D

2 X.L4/

W3 1 W2 1 B.G.L4/=R2/ S B.G.L4/=R1R2/ S D D

W3 1 W2 1 B.G.L4/=R3/ S B.G.L4/=R1R3/ S D D

P4;3

W2 1 3 B.G.L4/=R2R3/ S X.L4/ D Figure 3

2 W2 1 W3 3 There is a natural inclusion from X.L4/ . S / . S / to B.G.L4/=R1R2/ W2 1 D _ D S . Actually the space B.G.L4/=R1R2/ can be thought of as obtained from 2 3 X.L4/ by ﬁlling in the three S components by 4–balls, ie B.G.L4/=R1R2/ 2 S3 4 W2 1 W3 3 S3 4 W2 1 D X.L4/ . D / .. S / . S // . D / S . [ D _ [ D Now, let’s study the bottom level of the cubical diagram. We have B.G.L4/=R3/ 3 S2 3 W3 1 D .S d3L4/ . D / S . B.G.L4/=R1R3/ (respectively B.G.L4/=R2R3/) n [ D can be thought of as obtained from the space B.G.L4/=R3/ by ﬁlling in the l1 3 (respectively l2 ) component and then capping off the resulting S boundary by a 4 3 S2 3 4 W2 1 D , ie B.G.L4/=R1R3/ .S d3L4/ . D / l1 D S (respectively 3 D n S2 [3 4[ W[2 1D B.G.L4/=R2R3/ .S d3L4/ . D / l2 D S ), as shown in Figure 4. D n [ [ [ D

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By the deﬁnition of homotopy push-out, the space P4;3 is determined by the diagram

W3 1 - W2 1 B.G.L4/=R3/ S B.G.L4/=R1R3/ S D D W2 1 - B.G.L4/=R2R3/ S P4;3; D 1 and P4;3 is isomorphic to S ; see Figure 5. On the other hand, P4;3 can also be 2 thought of as obtained from X.L4/ by removing the l3 component and capping off all three S 3 components by 4–balls; see Figure 6.

3 Then the space X.L4/ is determined by the diagram 2 - W2 1 X.L4/ B.G.L4/=R1R2/ S D 1 - 3 P4;3 S X.L4/ ; D 2 where the space B.G.L4/=R1R2/ is obtained from X.L4/ by ﬁlling in the three 3 3 S components by 4–balls, as shown in Figure 7. We conclude that X.L4/ D S 1 .W3 S 4/. _

l3 l3

l l4 1 4 S 1 S S 1 S 1 D3 D3 D3 1 1 S S 1 S S 1 l1 l1 l 2 l2 2 2 S 2 S1 2 1 S2 S2 D3 D3

l4 S 1 S 1 D3 D3 S 1 S 1 l1 2 l2 S1 2 S2 D3

Figure 4: The classifying spaces B.G.L4/=R3 , B.G.L4/=R1R3 , and B.G.L4/=R2R3 , respectively

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4 The last case left is X.L4/ , which is determined by the following diagram:

3 1 W3 4 - 3 S3 5 1 X.L4/ S . S / B.G.L4/=R1R2R3/ X.L4/ . D / S D _ D [ D ? ? - 4 P4;4 X.L4/ ;

3 4 where the space B.G.L4/=R1R2R3/ is obtained from X.L4/ by capping off the S components by 5–balls, and the space P4;4 is obtained by removing the l4 component

3 3 D 2 S S3 3 D l3

l3 l4 l4 1 3 S 3 S S 1 S1 3 S 3 S1 S 3 3 D S D 3 3 3 D 3 D D 1 1 D 3 S S D3 D 1 1 l1 S S l2 S 2 2 l1 1 S2 l2 S 2 2 1 S2 Figure 6: On the other hand, P4;3 can be thought of as 2 Figure 5: The pushout P4;3 obtained from X.L4/ by 1 is isomorphic to S . removing the l3 component and capping off all the three S 3 components by D4 ’s.

3 3 D 2 S S3 3 D l3

l4 1 3 S 3 S S S1 3 3 D D 3 3 D D S1 S1 l1 2 l2 S1 2 S2

2 Figure 7: The space B.G.L4/=R1R2/ is obtained from X.L4/ by ﬁlling in the three S 3 components by D4 ’s.

Algebraic & Geometric Topology, Volume 16 (2016) On intersecting subgroups of Brunnian link groups 1057

3 4 from X.L4/ and capping off the S components by 5–balls, ie P4;4 is isomorphic 4 W3 5 to a point. So X.L4/ S . D

3B The general cases

Based on the experimental example we studied in detail in the previous section, we’ll sketch the procedure of ﬁnding the homotopy types of the homotopy colimits very roughly in this section.

For any n > 4 and 2 m n, we want to show that Ä Ä Ân m Ã Ân 1 Ã m _ 1 _ m 1 X.Ln/ S S ; D _ C noticing that for n 2; 3; 4 this equality holds. Now, for any n > 4, we assume that kD the spaces X.Ln/ ; 2 k m 1 satisfy this equation. By induction, we notice that m Ä Ä X.Ln/ is determined by the diagram

m 1 Wn m 1 1 B.G.Ln/=R1 Rm 1/ X.Ln/ . C S / - m 1 Sn 1 m 1 D Wn 1 m X.Ln/ . D C / . S / D Wn m 1 [1 _ C S D

? ? - m Pn;m X.Ln/ ;

m 1 where B.G.Ln/=R1 Rm 1/ can be thought of as obtained from X.Ln/ by m m 1 ﬁlling in the S components, and Pn;m is obtained from X.Ln/ by removing the lm component and capping off all the sphere components with solid balls. Then Wn m 1 m Wn m 1 Wn 1 m 1 Pn;m S , and the colimit X.Ln/ . S / . S /. D D _ C

4 Proofs of theorems

In this section, we ﬁrst study the nth and .n 1/st homotopy groups of the homotopy C colimits, then the proofs of our theorems are just corollaries.

4A The homotopy groups of the homotopy colimits

m Wn m 1 Wn 1 m 1 We have already shown that X.Ln/ . S / . S /. D _ C

Algebraic & Geometric Topology, Volume 16 (2016) 1058 Fengchun Lei, Jie Wu and Yu Zhang

n For the case m n, the homotopy groups of X.Ln/ can be obtained by applying the D n Hurewicz theorem. In fact, the homology groups of X.Ln/ satisfy the equalities

Ân 1 Ã n 1 n _ n 1 M n 1 (12) Hn.X.Ln/ / Hn S Hn.S / 0; D C D C D Ân 1 Ã n 1 n 1 n _ n 1 M n 1 M (13) Hn 1.X.Ln/ / Hn 1 S C Hn 1.S C / Z: C D C D C D

n n Ln 1 So by the Hurewicz theorem, n.X.Ln/ / 0 and n 1.X.Ln/ / Z. D C D m Wn m 1 Wn 1 m 1 For the case m

4B Proof of Theorem 1.1

According to (8),

m R1 Rm 0 m.X.Ln/ / \ \ Q ŒT R ; T R D Š I J 1;2;:::;m ;I J ∅ i I i j J j [ Df g \ D 2 2 for all m, ie R R Q ŒT R ; T R . Obviously, 1 m I J 1;2;:::;m ;I J ∅ i I i j J j \ \ D [ Df g \ D 2 2 ŒR1;:::; RmS R1 Rm . The following lemma will conclude the proof. Ä \ \

Lemma 4.1 Based on the assumptions of Theorem 1.1,

Y Ä \ \ ŒR1;:::; RmS Ri; Rj R1 Rm: D \ \ I J 1;2;:::;m ;I J ∅ i I j J [ Df g \ D 2 2

T T Proof According to[28, Corollary 3.5], i I Ri ŒŒRi1 ;:::; Ri I and j J Rj 2 D j j 2 D ŒŒRj1 ;:::; Rj J , so j j Y Ä \ \ Ri; Rj ŒŒR1;:::; Rm ŒR1;:::; RmS : Ä D I J 1;2;:::;m ;I J ∅ i I j J [ Df g \ D 2 2

Algebraic & Geometric Topology, Volume 16 (2016) On intersecting subgroups of Brunnian link groups 1059

4C Proof of Theorem 1.2

According to (9), Ln 1 m Z when m n ker.@ T .R1;:::; Rm/ G.Ln// m 1.X.Ln/ / L D W ! D C D 1 Z when m < n.

For all disjoint subsets A; B 1; 2;:::; m and any x RA; y RB , we have T T f g 2 2 @.x y/ Œx; y Œ i A Ri; j B Rj . By the arbitrariness of A and B , we claim ˝ D Q 2 2 2 T T that Im.@/ Œ Ri; Rj . Further, we notice that D I J 1;2;:::;m ;I J ∅ i I j J there is an exact sequence[ Df g \ D 2 2 m ker.@/ m 1.X.Ln/ / T .R1; R2;:::; Rm/ D C Y Ä \ \ Im.@/ Ri; Rj D I J 1;2;:::;m ;I J ∅ i I j J m [ Df g \ D 2 2 \ m Ri Coker.@/ m.X.Ln/ / 0: ! D D i 1 D This concludes the proof.

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School of Mathematical Sciences, Dalian University of Technology Dalian, 116024, China Department of Mathematics, National University of Singapore 10 Lower Kent Ridge Road, Singapore 119076, Singapore Department of Mathematics, Harbin Institute of Technology Harbin, 150001, China [email protected], [email protected], [email protected]

Received: 8 April 2015

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