On the structure of braid groups on complexes
OON THE STRUCTURE OF BRAID GROUPS ON COMPLEXES
BYUNG HEE AN AND HYO WON PARK
Abstract.
We consider the braid groups B n ( X ) on finite simplicial complexes X , which are generalizations of those on both manifolds and graphs that havebeen studied already by many authors. We figure out the relationships betweengeometric decompositions for X and their effects on braid groups, and providean algorithmic way to compute the group presentations for B n ( X ) with theaid of them.As applications, we give complete criteria for both the surface embeddabil-ity and planarity for X , which are the torsion-freeness of the braid group B n ( X ) and its abelianization H ( B n ( X )), respectively. Contents
1. Introduction 22. Braid groups on complexes 42.1. Appending a point 63. Modifications and simple complexes 73.1. Edge contraction 73.2. Attaching higher cells 93.3. Simple complex 134. Decompositions and Elementary complexes 145. k -Closure of a complex 166. k -Connected sum of a pair of complexes 206.1. Complex-of-groups 206.2. k -connected sum 217. Applications – Embeddabilities and connectivities 277.1. Surface embeddability 287.2. The first homology groups 287.3. Connectivities 30Appendix A. More deformations 35A.1. Attaching a disk at a trivalent vertex 35A.2. Edge-to-band replacement 38A.3. 2-dimensional capping-off 39References 40 Mathematics Subject Classification.
Primary 20F36; Secondary 05E45, 57M20.
Key words and phrases. braid group, simplicial complex, surface embeddability, planarity.This work was supported by IBS-R003-D1. a r X i v : . [ m a t h . G T ] A ug BYUNG HEE AN AND HYO WON PARK Introduction
The braid group B n ( D ) on a 2-disk D was firstly introduced by E. Artin in1920’s, and Fox and Neuwirth generalized it to braid groups B n ( X ) on arbitrarytopological spaces X via configuration spaces , which are defined as follows. For acompact, connected topological space X , the ordered configuration space F n ( X ) isthe set of n -tuples of distinct points in X , and the orbit space B n ( X ) under theaction of the symmetric group S n on F n ( X ) permutting coordinates is called the unordered configuration space on X . F n ( X ) = X n \ ∆ , B n ( X ) = F n ( X ) / S n , where ∆ = { ( x , . . . , x n ) | x i = x j for some i (cid:54) = j } ⊂ X n . Let ¯ ∗ n and ∗ n be basepoints for F n ( X ) and B n ( X ), respectively. Then the pure n -braid group P n ( X, ¯ ∗ n ) and (full) n -braid group B n ( X, ∗ n ) are defined to be thefundamental groups of the configuration spaces F n ( X ) and B n ( X ), respectively.We will suppress basepoints and denote these groups by P n ( X ) and B n ( X ) unlessany ambiguity occurs.However, most of research on braid groups has been focused on braid groups onmanifolds, more specifically, on surfaces, until the end of 20th century when Ghristpresented a pioneering paper [11] about braid groups on graphs Γ which are finite,1-dimensional simplicial complexes. In 2000, Abrams defined in his Ph. D. the-sis [1] a combinatorial version of configuration space, called a discrete configurationspace , consisting of n open cells in Γ having pairwise no common boundaries. A dis-crete configuration space has the benefit that it admits a cubical complex structuremaking the description of paths of points easier. However it depends not only onhomeomoprhic type but also the cell structure of the underlying graph Γ. Abramsovercame this problem by proving stability up to homotopy under the subdivisionof edges once Γ is sufficiently subdivided.Crisp and Wiest showed the embeddabilities between braid groups on graphsand surface groups into right-angled Artin groups, which is one of the most impor-tant subjects in geometric group theory. Farley and Sabalka in [9] used Forman’s Discrete Morse theory [8] on discrete configuration spaces to provide an algorithmicway to compute a presentation of B n (Γ), and furthermore they figured out the rela-tion between braid groups on trees and right-angled Artin groups. On the extensionof these works, Kim-Ko-Park in [3] and Ko-Park in [4] provided geometric criteriafor the braid group on a given graph to be right-angled Artin, and moreover a newalgebraic criterion for the planarity of a graph, and answered some open questionsas well.On the contrary, for a simplicial complex, not manifold, of dimension 2 or higher,braid theory is still unexplored. We will focus on the braid groups on finite, con-nected simplicial complex X of arbitrary dimension, which are generalizations ofboth graphs and surfaces. We consider modifications—attaching or removing highercells, edge contraction or inverses, and so on— and how these modifications changethe braid groups. Indeed, via suitable modifications we may obtain a simple com-plex X (cid:48) of dimension 2 whose vertices have very obvious links. Furthermore, thiscan be done without changing the braid group. Theorem 1.1.
Let X be a complex. Then there is a simple complex X (cid:48) of dimen-sion such that B n ( X ) (cid:39) B n ( X (cid:48) ) for all n ≥ . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 3
Once we have a simple complex X , then it can be decomposed by cuts into muchsimpler pieces, and eventually into elementary complexes, where an elementarycomplex plays the role of a building block and can be thought as either a star graphor a manifold of dimension at least 2. For the build-up process, we provide two typesof combination theorem which are generalizations of capping-off and connected sum.Furthermore, the combination theorems ensure that the build-up process preservessome geometry of the given pieces. In other words, the braid group B n ( X ) capturessome geometric properties of X as observed before.More precisely, we start with the obvious observations about the various embed-dabilities of X into manifolds as follows. For two complexes X and Y , we denoteby Y ⊂ X and say that X contains Y if there is an simplicial embedding betweenthem after sufficient subdivisions. Then a complex X embeds into (i) a circle iff T (cid:54)⊂ X ; (ii) a surface iff S (cid:54)⊂ X ; and (iii) a plane iff K , K , , S (cid:54)⊂ X .The complexes T and S are the tripod and the cone C ( S (cid:116) {∗} ) of the unionof a circle and a point, respectively. See Figure 1. The graphs K n and K m,n arecomplete and complete bipartite graphs, respectively. T = 1 0 23 S = 01 Figure 1.
A tripod T and a complex S Then it can be formulated as follows.
Theorem 1.2.
Let X be a finite, connected simplicial complex different from S and R P . Then X embeds into (1) a circle if and only if B n ( X ) is abelian for any n ≥ ; (2) a surface if and only if B n ( X ) is torsion-free for any n ≥ ; (3) a plane if and only if H ( B n ( X )) is torsion-free for any n ≥ .Moreover, if X does not embed into any surface, then B n ( X ) contains S n for any n ≥ . Remark that we exclude the cases for S and R P since their braid groups havetorsion even though they are braid groups on surfaces , 2-dimensional manifolds.However, by the complementary statement, they are classified as B n ( X ) containsa torsion but not the whole S n for any n ≥ unwrapping and connected-sum decomposition , and look at the shapes of the elementary complexes.The effects of the inverses, called closure and connected sum , of these two operationson braid groups will be discussed separately in Section 5 and Section 6, whichthey let us know how the build-up process is working. Finally, in Section 7, asapplications we prove the criteria, Theorem 1.2, for the embeddability of givencomplex X into a surface and a plane. BYUNG HEE AN AND HYO WON PARK Braid groups on complexes
Throughout this paper, a complex denoted by X means a finite, connected,simplicial complex of dimension at least 1. Especially, a complex of dimension 1is usually denoted by Γ and called a graph . Since the braid group on X dependsonly on the homeomorphism type of X , we sometimes assume that X is sufficientlysubdivided , which can be achieved via the barycentric subdivision twice. The star st( K ) is the union of all open simplices whose closure intersects K , and the link lk( K ) of K is the complement of the star st( K ) of the closure K in its closurest( K ). That is, lk( K ) = st( K ) \ st( K ), as usual.Note that both F n ( X ) and B n ( X ) can be regarded as finite simplicial complexesup to homotopy as follows. Since X is a finite simplicial complex, so is the n -foldproduct X n , and after barycentric subdivisions if necessary, the diagonal ∆ becomesa simplicial subcomplex of X n . Hence the further subdivision makes X n \ st(∆)a strong deformation retract of X n \ ∆ = F n ( X ). Therefore if we endow a metric d on X , we may assume that there exists a constant (cid:15) = (cid:15) ( X ) > x ∈ F n ( X ) never approach within (cid:15) of each other with respect to themetric d , and the same holds for B n ( X ).From the definitions of configuration spaces, we have the following exact se-quence.(1) 1 −→ P n ( X, ¯ ∗ n ) −→ B n ( X, ∗ n ) ρ −→ S ( ∗ n )Here the group S ( ∗ n ) is the symmetric group on the set ∗ n , usually denoted by S n ,and the map ρ is called the induced permutation . It is easy to see that P n ( I ) = B n ( I ) = { e } , and P n ( S ) = n Z ⊂ Z = B n ( S ).On the other hand, it is known that ρ for B n ( T ) on a tripod T is surjective foreach n ≥
2. Hence whenever X contains T , then ρ is surjective as well. Proposition 2.1.
Let X be a complex. Then X embeds into a circle if and only if B n ( X ) is abelian for any n ≥ .Proof. The only if part is obvious.Suppose B n ( X ) is abelian. Then since S n is non-abelian for any n ≥ ρ neverbe surjective. Hence X is either I or S , and therefore it embeds into a circle. (cid:3) We call X trivial if X is either I or S . Then T can be thought as the obstructioncomplex for given complex to be trivial. From now on we assume that X is non-trivial. Definition 2.2.
For any x ∈ X , there is a trichotemy as follows.(1) x is in the interior ˚ X if lk( x ) (cid:39) S k for some k ≥ x is in the boundary ∂X if lk( x ) (cid:39) D k for some k ≥ x is in the branch set br( X ) of X otherwise. Definition 2.3.
Let X be a complex.(1) A 0-cell v is called a vertex , whose valency val( v ) is defined by the numberof connected components of lk( v ).(2) A 1-cell e = ( v, w ) is called an edge if there is no 2-cell containing e in itsboundary.(3) For a subset K of X , a deletion X K of K in X is defined by the complement X \ st( K ) of st( K ). N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 5 = ⇒ Figure 2.
A decomposition of X into sets of interior, boundary,and branch pointsThe Figure 2 shows an example. The thin lines and dots are in br( X ), and thethick lines are in ∂X . Note that br( X ) is a closed subcomplex of X , and X is amanifold if and only if br( X ) = ∅ . Theorem 2.4. [6]
Let M be a manifold of dimension at least , not necessarilycompact and possibly with boundary. Then the pure and full braid groups are asfollows. P n ( M ) = n (cid:89) π ( M ) , B n ( M ) = n (cid:89) π ( M ) (cid:111) S n , where the symmetric group S n acts on the product (cid:81) n π ( M ) by permuting factors. Hence there is no braid theory for manifolds of dimension 3 or higher. On theother hand, for a surface Σ, then there is a fiber bundle structures between theordered configuration spaces F n (Σ)’s which can be used to compute and analyzebraid groups on Σ. Note that since compact surfaces are completely characterizedby a few parameters, so are their braid groups. Indeed, for a given surface Σ, onecan extract geometric information from its braid group as follows. The proof isobvious by the group presentation for B n (Σ), see [5], and we omit the proof. Theorem 2.5. [5, 6, 10]
Let Σ be a surface. Then the following holds. (1) B n (Σ) has torsion if and only if Σ is either S or R P . (2) The abelianization H ( B n (Σ)) has torsion if and only if Σ is nonplanar. On the contrary, if X is not a manifold, the global topology for a complex X canhardly be determined by a few parameters in general, even if X is 1-dimensional.Therefore one might not expect that similar results hold for X , but surprisingly,the braid group still detects some of the global geometry of the complex X when X is a graph Γ as follows. Theorem 2.6. [3, 11]
Let Γ be a graph. Then B n (Γ) is always torsion free, andmoreover H ( B n (Γ)) has torsion if and only if Γ is nonplanar. Hence Theorem 1.2 is the generalization of the two theorems above, and toprove our theorem, we adopt the notion from graph theory which is the mimic ofthe minor relation, that is, edge contraction and deletion. Note that the minorrelations reduce the number of edges and so the result is usually considered assimpler than the original one. However, they may increase valencies, and the highervalency tends to imply a more complicated situation in the computation of the braidgroup. Therefore we may think that a complex having lower valencies is simpler.Rigorously speaking, we define a simple complex as follows.
BYUNG HEE AN AND HYO WON PARK
Definition 2.7.
A vertex v is said to be simple if lk( v ) is either connected, adisjoint union of a connected complex and a point, or 0-dimensional. A complex X is said to be simple if all vertices in X are simple.Hence a simple complex is really easy to handle, but is too special and far fromthe generic ones. However, we claim that any complex can be transformed intoa simple complex by a sequence of certain modifications such as attaching andremoving (higher) cells, where each step induces an isomorphism between braidgroups.For convenience’s sake, we say that an embedding f : X → Y is a braid equiva-lence if it induces an isomorphism f ∗ : B n ( X ) → B n ( Y ) for each n ≥
1, respectively.Moreover, we simply say that X and Y are braid equivalent if they can be joinedby a sequence of (possibly inverse of) braid equivalences, denoted by X ≡ B Y ,respectively.Then the above claim can be reformulated as for any X , there is a simple repre-sentative in the braid equivalence class of X as presented in Theorem 1.1. We willprove this proposition later.2.1. Appending a point.
Let v ∈ ∂X , and i v : B n − ( X \ { v } ) → B n ( X ) for n ≥ i v ( x ) = { v } ∪ x for x ∈ B n − ( X \ { v } ). Note that i v ( ∅ ) = { v } if n = 1.Then it induces a homomorphism( i v ) ∗ : B n − ( X \ { v } , ∗ n − ) → B n ( X, i v ( ∗ n − )) , where ∗ n − is a basepoint for B n − ( X \ { v } ).Note that B n ( X \{ v } ) is homotopy equivalent to B n ( X ) via the inclusion, whosehomotopy inverse is a map h resizing cells incident to v . Hence we can consider acomposition ¯ i v : B n − ( X ) h −→ B n − ( X \ { v } ) i v −→ B n ( X ) , which induces (¯ i v ) ∗ : B n − ( X, ∗ n − ) → B n ( X, ¯ i v ( ∗ n − )) . We can use safely i v and ( i v ) ∗ instead of ¯ i v and (¯ i v ) ∗ since there is no ambiguityup to homotopy. Proposition 2.8.
Let X be a complex and v ∈ ∂X . Then the homomorphism ( i v ) ∗ : B n − ( X ) → B n ( X ) is injective for all n ≥ .Proof. Let B ,n − ( X ) = F n ( X ) / S n − by considering S n − as a subgroup of S n which consists of permutations on { , . . . , n } fixing 1. Then B ,n − ( X ) = { ( x , { x , . . . , x n } ) | x i (cid:54) = x j if i (cid:54) = j } , and the quotient map p : B ,n − ( X ) → B n ( X ) forgetting the order is a coveringmap which is non-regular in general. Moreover, i v lifts to ˜ i v : B n − ( X \ v ) → B ,n − ( X ) defined by ˜ i v ( x ) = ( v, x ) and there is a map π : B ,n − ( X ) → B n − ( X )forgetting the first coordinate, namely, π ( x , { x , . . . , x n } ) = { x , . . . , x n } , satisfying that π ◦ ˜ i v is homotopic to the identity, and it induces the isomorphism π ∗ ◦ (˜ i v ) ∗ . Hence (˜ i v ) ∗ is injective and therefore so is ( i v ) ∗ = p ∗ ◦ (˜ i v ) ∗ . (cid:3) N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 7 Modifications and simple complexes
Edge contraction.
The first nontrivial observation is about edge contractionas follows.
Proposition 3.1.
Let X be a complex and e be an edge. Then the quotient map q : X → X/ ¯ e induces a map q ∗ : B n ( X/ ¯ e ) → B n ( X ) , which is surjective if none of ∂e is of valency 1.Proof. We first consider a subspace B n ; ≤ ( X ; ¯ e ) of B n ( X ) consisting of configura-tions x = { x , . . . , x n } such that at most 1 of x i ’s is lying on ¯ e , or equivalently, for x ∈ B n ( X ), x ∈ B n ; ≤ ( X ; ¯ e ) ⇐⇒ x ∩ ¯ e ) ≤ . Then the map q induces q | : B n ; ≤ ( X ; ¯ e ) → B n ( X/ ¯ e ).Let ∗ n ⊂ X \ ¯ e be a basepoint for both B n ; ≤ ( X ; ¯ e ) and B n ( X/ ¯ e ), and let apath γ : ( I, ∂I ) → ( B n ( X/ ¯ e ) , ∗ n ) be given. Then it is not hard to prove that thereexists a lift ˜ γ : ( I, ∂I ) → B n ; ≤ ( X ; ¯ e ) so that q ◦ ˜ γ = γ by regarding ¯ e as a path.Moreover, the lift is unique up to homotopy since ¯ e is contractible. Therefore, q | induces an isomorphism ( q | ) ∗ between fundamental groups. Then the map q ∗ isdefined by a composition ι ∗ ◦ ( q | ) − ∗ , where ι ∗ is the map induced from the obviousinclusion ι : B n ; ≤ ( X ; ¯ e ) → B n ( X ), and is well-defined as desired. X = e q (cid:47) (cid:47) = X/ ¯ e ˜ γ = = γ (cid:31) (cid:111) (cid:111) Figure 3.
Local pictures of X/ ¯ e and X , and the lift ˜ γ of a path γ Suppose that none of ∂e is of valency 1. Then for the surjectivity, it is enoughto show that ι ∗ is surjective. In other words, any δ : ( I, ∂I ) → ( B n ( X ) , ∗ n ) ishomotoped to δ (cid:48) relative to the boundary such that δ (cid:48) ( t ) ∩ ¯ e ) ≤ t ∈ [0 , δ into several pieces according to the change of m ( t ) = δ ( t ) ∩ ¯ e ),and use induction on m . Then since both val( v ) and val( w ) ≥
2, we have enoughroom for a given configuration to be evacuated from e . This can be done easily andwe omit the detail. (cid:3) If none of ∂e is of valency 1, then we may say that X is simpler than X/ ¯ e according to the definition of simplicity. Note that q does not directly induce themap between braid groups since it is not an embedding. Under some conditions,one can find an embedding which plays a similar role to q so that it induces preciselythe inverse of q ∗ , and therefore an isomorphism. We will see this later.On the other hand, if one of ∂e is of valency 1, then q can be considered as astrong deformation retract and therefore q ∗ is actually induced from the obvious BYUNG HEE AN AND HYO WON PARK embedding X/ ¯ e → X which is a homotopy inverse of q . However, q ∗ is neitherinjective nor surjective in general. It depends on the structure of st( e ). Example 3.2 (2-braid group on a tree) . We denote by T k a labelled tree homeo-morphic to the cone of k points as depicted in Figure 4.1 2 . . . k Figure 4.
A labelled tree T k with only one vertex of valency k ≥ B n ( T k ) is always a free group as follows. Lemma 3.3. [3]
The braid group B n ( T k ) is a free group of rank r = r ( n, k, k ) ,where (2) r ( n, ν, µ ) = ( ν − (cid:18) n + µ − n − (cid:19) − (cid:18) n + µ − n (cid:19) − ( ν − µ − . Especially, the 2-braid group B ( T k ) is of rank (cid:0) k − (cid:1) , indexed by { ( i, j ) | ≤ i 0) and moreover b is closer to 0 than a .Then we move b to the second leaf and a to the third leaf, and back b to the initialposition of a and back a to that of b . See Figure 5 and we will rigorously definethis loop later in detail. Then the loop s defined in this way generates the infinitecyclic group which is actually B ( T ). s = · · · Figure 5. A loop s in B ( T )For each pair ( i, j ) with i (cid:54) = j , there exists a unique embedding T → T k suchthat it maps ∂T = { , , } to { , i, j } ⊂ ∂T k in order. Then s i,j is nothing butthe image of s under the induced homomorphism B ( T ) → B ( T k ). Note that s j,i is the inverse of s i,j .Now let T be a tree with k = ∂T ). We first label on ∂T arbitrarily. Thenthere exists a unique label-preserving map q : T → T k which takes a quotient by allinternal edges and it induces a surjective homomorphism q ∗ : B ( T k ) → B ( T ) byProposition 3.1. By the definition of s i,j above, the image q ∗ ( s i,j ) coincides withthe image of s under the unique embedding T → T sending { , , } to { , i, j } as before. We mean by the center c ( i, j ) of i and j in T that the image of 0 ∈ T under this embedding. N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 9 T = 12 34 5 678 12 34 5 678 c (2 , Figure 6. A tree with labelled leaves and an embedding of T corresponding to s , Suppose that there is an isotopy H : T × I → T such that H t (1) = 1 forall t and H ( { , } ) = { i, j } , H ( { , } ) = { i (cid:48) , j (cid:48) } . Then it defines a homotopybetween the images of s i,j and s i (cid:48) ,j (cid:48) in B ( T ), hence they are considered as thesame in B ( T ). More precisely, the given tree T defines an equivalence relation on { ( i, j ) | ≤ i < j ≤ k } as follows. Definition 3.4 (Equivalence relation coming from a tree T with ordered leaves) . Suppose the set ∂T of leaves are indexed by { , . . . , k } and let (cid:0) ∂T − (cid:1) denote theset { ( i, j ) | ≤ i < j ≤ k } .Then we define an equivalence relation ∼ T on (cid:0) ∂T − (cid:1) as ( i, j ) ∼ T ( i (cid:48) , j (cid:48) ) if andonly if(1) c ( i, j ) = c ( i (cid:48) , j (cid:48) ) ∈ T ;(2) [ i ] = [ i (cid:48) ] and [ j ] = [ j (cid:48) ] in π ( T \ { c ( i, j ) } ).Therefore, the equivalence classes depend not on the whole tree T but only onthe local shape, namely the tangent space , of each vetex of valency ≥ 3. Hence eachgenerator s i,j corresponds to a triple ( v, e , e ) of a vertex v with val( v ) ≥ 3, andtwo half-edges e and e emitting from v which are not heading to the chosen pointin the boundary, 1 ∈ ∂T in our example.One can prove that this equivalence gives the complete set of defining relatorsfor B ( T ), and therefore B ( T ) is free as well. B ( T ) = (cid:104) s , , . . . , s k − ,k | s i,j = s i (cid:48) ,j (cid:48) if ( i, j ) ∼ T ( i (cid:48) , j (cid:48) ) (cid:105) . The rank is given by(3) r ( T ) = (cid:88) v ∈ V ( T ) (cid:18) val( v ) − (cid:19) , where V ( T ) is the set of vertices of T . Remark 3.5. Recall the point appending map B n − ( T ) → B n ( T ) defined inProposition 2.8, and consider all possible compositions which yield B ( T ) → B n ( T ).Then the images of s i,j ’s under these compositions generate B n ( T ).More precisely, each generator is characterized by a vertex of valency ≥ 3, twoedges as before, and in addition the number of points in each component of thecomplement of that vertex in T . See [3] for detail.3.2. Attaching higher cells. We first consider the generalized capping-off X ,which is to attach a k -simplex along a ( k − Y . Weexclude the cases when ( X, Y ) = ( D , S ) or ( D , S ) because their braid groupsare already known, and moreover they are extremal in the sense of that the braidgroups change dramatically before and after attaching simplices. Proposition 3.6. Let X = Y (cid:116) φ D k via the embedding φ : ∂D k = S k − → Y forsome k ≥ and a complex Y different from S . Then the embedding Y → X is abraid equivalence.Proof. We identify ∂D k with the subspace of Y via φ from now on.Let ∗ ∈ ˚ D k be a point, and consider the subspace B n − ( X ; ∗ ) of B n ( X ) con-sisting of configurations containing ∗ . Then this is of codimension k ≥ 3, in thesense that B n − ( X ; ∗ ) × R can be embedded into B n ( X ). Hence we may assumethat all paths and homotopies in B n ( X ) are in general position with respect to {∗} and therefore they avoid ∗ . In other words, the inclusion X \ {∗} → X is a braidequivalence.Let D k \ {∗} → ∂D k be the radial projection, or the strong deformation retract,which naturally extends to r : X \ {∗} → Y , the homotopy inverse of the inclusion Y → X \ {∗} . D k S k − p ∗ p ∗ [ ∗ , p ][ p, p + (cid:15) ] Figure 7. A radial projection on X \ {∗} and an extended ray[ p, p + (cid:15) ]Consider B r -fail n = { x ∈ B n ( X \ {∗} ) | r ( x )) < n } consisting of configurations x such that at least two points in x are lying in a ray emitting from ∗ in D k . Roughlyspeaking, it is the set of failures for r to be extended to ¯ r : B n ( X \ {∗} ) → B n ( Y ).Then B r -fail n is of codimension at least 2 in B n ( X \ {∗} ) as follows.codim (cid:0) B r -fail n ⊂ B n ( X \ {∗} ) (cid:1) ≥ codim(ray ⊂ D k \ {∗} ) = ( k − ≥ . Hence by assuming the general position with respect to B r -fail n , we may assumethat any loop misses B r -fail n for all k ≥ 3, and so does any disk for k ≥ 4. Note thatwhen k = 3, a disk in B n ( X \ {∗} ) may intersect finitely many times with B r -fail n .Therefore, the map r induces the surjective homomorphism r ∗ : B n ( Y ) → B n ( X \ {∗} ) (cid:39) B n ( X ) , which is also injective if k ≥ r ∗ is an isomorphism for k = 3 as well.Suppose that ∂D ⊂ ˚ X , or equivalently, ∂D is a component ∂ Y of ∂Y . Then X is an ordinary capping-off of the 2-sphere ∂ Y in Y , and so Y \ ∂ Y (cid:39) X \{∗} . Sincethe homotopy equivalence between Y and Y \ ∂ Y induces the braid equivalence,the inclusions Y → X \ {∗} → X induce B n ( Y ) (cid:39) B n ( X \ {∗} ) (cid:39) B n ( X ) . Indeed, the strong deformation retract pushing X \{∗} into Y \ ∂ Y , slightly smallerthan r , induces the isomorphism r ∗ .Suppose that ∂D (cid:54)⊂ ˚ X . Then since Y (cid:54) = S by the hypothesis, ∂D (cid:54)⊂ ∂X ,and therefore ∂D must intersect br( X ). The existence of a branch point p ∈ N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 11 ∂D ∩ br( X ) implies that we can extend the ray [ ∗ , p ] emitting from ∗ passingthrough p a little bit more. We denote the extended ray by [ p, p + (cid:15) ], where p + (cid:15) is a point lying in st( p ) \ D . U f = { f i } (cid:47) (cid:47) ∗ f ( U ) f ( U ) ⊂ B n ( X \ {∗} ) Figure 8. A homotopy disk and a small neighborhood U Let f = { f ( z ) , . . . , f n ( z ) } : ( D , ∂D ) → ( B n ( X \ {∗} ) , B n ( Y )) be given. Toprove the claim, it suffices to show that f can be homotoped into B n ( Y ). Since f is in general position with respect to B r -fail n , without loss of generality, we mayassume that f ( D ) intersects B r -fail n exactly once at 0 ∈ D , and furthermore thatthere exists only one ray [ ∗ , p (cid:48) ] emitting from ∗ , which contains exactly two points,say f (0) and f (0), among f (0). Here p (cid:48) = r ( f (0)) = r ( f (0)).Then we can further homotope f by keeping f in the general position so that p (cid:48) becomes p , and one of f and f , say f , is constantly p on a neighborhood U ⊂ D of 0. The last comes from that in a small enough neighborhood, each f i can behomotoped separately.Finally, we pull down f on U by using [ p, p + (cid:15) ] so that f (0) ⊂ ( p, p + (cid:15) ) asdepicted in Figure 9, and then r ◦ f : D → B n ( Y ) is well-defined and homotopicto f relative to ∂D , as desired. (cid:3) We call the subcomplex f − ( B r -fail n ) of D a failure locus . U = (cid:39) f (cid:47) (cid:47) S k − ⊂ Y Figure 9. Pulling down a homotopy disk along the extended ray Remark 3.7. The effect of capping-off as above on a link lk( v ) for v ∈ S k − ⊂ Y is again a capping-off of ( k − v ) since lk( v ) ∩ S k − = S k − and lk ( v ) ∩ D k = D k − in X .Conversely, for any v ∈ X and embedded sphere S in lk( v ), there exists acapping-off on X which caps lk( v ) off along S .The direct consequence of the above proposition is as follows. Corollary 3.8. Let X be a complex. Then the embedding X (2) → X of -skeleton X (2) is a braid equivalence unless X = D and X (2) = S . Moreover, we can obtain the same result for the 2-cell attaching under the certaincondition. Corollary 3.9. Let X = Y (cid:116) φ D via the embedding φ : ∂D → Y . Suppose φ ( ∂D ) bounds a disc D (cid:48) in Y , and ˚ D (cid:48) ∩ br( Y ) (cid:54) = ∅ . Then the embedding Y → X is a braid equivalence.Proof. Note that there exists an embedded sphere S = D ∪ D (cid:48) in X , such that˚ D (cid:48) ∩ br( Y ) ⊂ S ∩ br( X ) (cid:54) = ∅ . Hence X satisfies the assumption of Proposition 3.6, and so X → X (cid:116) S D is abraid equivalence.Then the embedding Y → X induces a surjection B n ( Y ) → B n ( X ) (cid:39) B n ( X (cid:116) S D ) as before, and moreover, in this case, we can think a strong deformation re-traction r (cid:48) from X (cid:116) S D to Y , which is nothing but an elementary collapsing .Then essentially the same argument as before with this elementary collapsingimplies the braid equivalence of Y → X . We omit the detail. (cid:3) In the last part of the proof, we are using the existence of a branch point in ˚ D (cid:48) again. Remark 3.10. Similar to the previous remark, the capping-off along S ⊂ Y affectsas the capping-off on lk( v ) for any v ∈ S along the 0-sphere S = lk( v ) ∩ S , and vice versa .On the other hand, we can consider another type of embedding as follows. Let e = ( v, w ) be an edge of X such that the closure st( v ) is homeomorphic to theboundary wedge sum D k ∨ ∂ ¯ e , or equivalently, lk( v ) = D k − (cid:116) { w } .Let Y be a space obtained from X by replacing st( v ) with C w ( D k − ), where C w ( D k − ) is a cone of D k − ⊂ lk( v ) with the cone point corresponding to w . SeeFigure 10. Then there is an obvious embedding f : X → Y . Note that Y ishomeomorphic to the quotient X/ ¯ e but f is different from the quotient map. X f (cid:47) (cid:47) X/ ¯ e st( v ) = v w (cid:47) (cid:47) (cid:79) (cid:79) w = C w ( D k − ) (cid:79) (cid:79) Figure 10. An embedding having the same effect as the edge contraction Proposition 3.11. Let X be given and e = ( v, w ) be an edge in X with lk( v ) = D k − (cid:116) { w } for some k ≥ . Then the embedding f : X → X/ ¯ e defined as above isa braid equivalence.Proof. We first endow a metric d on X , and the induced metric d (cid:48) on X/ ¯ e . Assumethat d ( v, w ) = diam (¯ e ) = 1. Let f (cid:15) : X → X/ ¯ e for 0 ≤ (cid:15) < diam (cid:48) ( f (cid:15) (¯ e )) = 1 − (cid:15) and for any 0 < (cid:15) < (cid:15) (cid:48) < f ( X ) = f ( X ) (cid:40) f (cid:15) ( X ) (cid:40) f (cid:15) (cid:48) ( X ) (cid:40) X/ ¯ e N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 13 as depicted in Figure 11.For convenience sake, we define f by the quotient map X → X/ ¯ e , and so X/ ¯ e = lim −→ (cid:15) ∈ [0 , f (cid:15) ( X ). This also implies that(4) B n ( X/ ¯ e ) = lim −→ (cid:15) ∈ [0 , f (cid:15) ( B n ( X )) . Moreover, since all f (cid:15) ( X )’s are ambient isotopic in X/ ¯ e , so are f (cid:15) ( B n ( X ))’sin B n ( X/ ¯ e ). Especially, the inclusion f (cid:15) ( B n ( X )) ⊂ f (cid:15) (cid:48) ( B n ( X )) is a homotopyequivalence for any (cid:15) < (cid:15) (cid:48) < 1. Hence by this fact and (4), it suffices to show thatany given c : ( D m , ∂D m ) → ( B n ( X/ ¯ e ) , f ( B n ( X ))) factors through the inclusion f (cid:15) ( B n ( X )) → B n ( X/ ¯ e ) for some (cid:15) < c is compact, we can choose a constant (cid:15) such that0 < − (cid:15) < 13 min x ∈ D m min { d (cid:48) ( x i , x j ) | x i (cid:54) = x j ∈ c ( x ) } . Now let r (cid:15) : X/ ¯ e × [0 , → X/ ¯ e be a strong deformation retraction of X/ ¯ e onto f (cid:15) ( X ). Then the composition r (cid:15) ◦ c is a well-defined homotopy between c and amap into f (cid:15) ( B n ( X )). This completes the proof. (cid:3) f ( X ) = (cid:40) (cid:40) (cid:40) = f ( X ) Figure 11. Local pictures of f ( X ) , f (cid:15) ( X ) , f (cid:15) (cid:48) ( X ) and f ( X ) for0 < (cid:15) < (cid:15) (cid:48) < Simple complex. For given X , we want to find a simple complex X (cid:48) whosebraid groups are isomorphic to those on X . To do this, we need the followingproposition which is the 2-dimensional analogue of Proposition 3.11. Proposition 3.12. Let X be a complex of dimension and e = ( v, w ) be an edgein X with lk( v ) = Γ (cid:116) { w } for some connected graph Γ . If Γ = S , we assumefurthermore that w (cid:54)∈ ∂X . Then there exist braid equivalences X → X → X/ ¯ e ← X/ ¯ e for some complex X , and therefore X ≡ B X/ ¯ e .Proof. Suppose Γ is trivial. Then we set X to be X itself. If Γ = I , then this is aspecial case of Proposition 3.11.We assume that Γ (cid:54) = I . Then the strategy is as follows.We first attach cells to X near v without touching e to obtain X (cid:48) so that X → X is a braid equivalence and lk( v ) = D k (cid:116) { w } in X for some k . This can be done byProposition 3.6 and Corollary 3.9 since v is a branch point and it always satisfiesthe assumption of Corollary 3.9. Then there exists a braid equivalence X → X/ ¯ e by Proposition 3.11.Notice that ¯ X/ ¯ e can be obtained from X/ ¯ e by attaching cells in the exactly sameways as we did for X to obtain X . Hence we have a braid equivalence X/ ¯ e → X/ ¯ e ,and therefore there exist braid equivalences X → X → X/ ¯ e ← X/ ¯ e as claimed. Recall the effects of attaching cells on lk( v ) as mentioned in Remark 3.7 and3.10, which are capping-off along embedded spheres in lk( v ). Hence, it suffices toshow that lk( v ) can be transformed to D k by the iterated capping-off process, andthis is actually equivalent to showing that lk( v ) is a subset of the 1-skeleton K (1) for some simplicial complex K homeomorphic to D k .Since any graph embeds into R , it is always possible for k = 3 and so is it for k = 2 when Γ is planar. This completes the proof. (cid:3) Proof of Theorem 1.1. Let X be the 2-skeleton of a sufficiently subdivided complex X . Then by Corollary 3.8, X ≡ B X .Let w ∈ br( X ) be a non-simple vertex. That is, lk( w ) has at least 2 graphcomponents or only one graph component with val( w ) ≥ 3. Let Γ be a graphcomponent of lk( w ) and X be a complex having an edge e = ( v, w ) such that X = X / ¯ e and lk( v ) = Γ (cid:116) { w } . Then X ≡ B X by Proposition 3.12.Since v is simple, and val( w ) in X is equal to val( w ) in X but the number ofgraph components of lk( w ) in X is strictly less than that in X . Therefore by theinduction on the number of graph components in nonsimple vertices, we eventualyobtain a simple complex X (cid:48) which is braid equivalent to X . (cid:3) Decompositions and Elementary complexes Let X and Y be simple complexes. We define two operations on X and a pair X, Y as follows. Definition 4.1 ( k -closure) . Let X be a connected complex and v = { v , . . . , v k } ⊂ ∂X . A k -closure (cid:92) ( X, v ) of X along v is a complex obtained by the mapping coneof the embedding v → X , and called trivial if st( v ) is trivial, or equivalently, k = 1and st( v ) = I .Notice that (cid:92) ( X, v ) can be also obtained by gluing T k to X along v , and (cid:92) ( X, v ) = X if and only if it is a trivial closure. Moreover, if st( v ) = T k for some k ≥ X v is connected, then the closure of X v along lk( v ) becomes X itself by definitionof X v . Hence X v is a kind of the inverse of the closure operation, usually called unwrapping in graph theory. Especially, we denote by Θ k the closure (cid:92) ( T k , ∂T k ) of T k along ∂T k , which is the union of two T k ’s and has two distinguished vertices 0and 0 (cid:48) of valency k .We will suppress v unless any ambiguity occurs.Let v ∈ X be a vertex with st( v ) = T k for some k ≥ 1. Then we denote by (cid:126)v a vertex v together with an ordering on lk( v ). In order words, we may identifyst( v ) with the labelled T k , and we regard lk( v ) = { v , . . . , v k } as an ordered k -tuple( v , . . . , v k ) if (cid:126)v is given. We call (cid:126)v a vertex with ordering . Definition 4.2 ( k -connected sum) . Let X and Y be complexes, and (cid:126)v and (cid:126)w bevertices of orderings of valency k ≥ X and Y . We further assume that both X v and Y w are connected.A k -connected sum X Y of ( X, (cid:126)v ) and ( Y, (cid:126)w ) is a complex obtained by con-necting each v i and w i via an interval e i , and is called trivial if one of X and Y isΘ k .See Figure 12 for a pictorial definition for connected sum. Note that ( X, (cid:126)v ) Y, (cid:126)w )is a boundary wedge sum X ∨ ∂ Y if k = 1, and is an ordinary connected sum if k = 2. N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 15 e e e k v v v k v w w w w k v v v k w w w k ... ... ... ...= Figure 12. A k -connected sumSince Θ k is a k -closure of T k , for any vertex v ∈ X with st( v ) = T k for some k ≥ 1, a k -connected sum ( X, (cid:126)v ) k ,(cid:126) 0) is nothing but a k -closure of X v alonglk( v ) and so it is X itself whatever the orderings on v and 0 are. Therefore Θ k plays the role of the identity under the k -connected sum.We sometimes suppress (cid:126)v and (cid:126)w unless any ambiguity occurs, and also say that X is decomposed into Y and Z via k -connected sum if X = Y Z .Furthermore, it is easy to see that both unwrapping and connected sum de-composition reduce the first Betti number or the number of vertices of connectedcomponents. Therefore by continuing these operation, we eventually have compo-nents which are elementary in the sense of the following definition. Definition 4.3 (Elementary complex) . Let X be a simple complex of dimension 2.We say that X is elementary if X can be expressed as neither a nontrivial k -closurenor a nontrivial k -connected sum.Let X be an elementary complex. Suppose dim X = 1. Then elementarinessforces X to be a tree having at most 1 vertex of valency k ≥ 3. Therefore X is homeomorphic to T k , which admits a trivial closure structure only, and so iselementary.Suppose dim X = 2 and f : X → M is a simplicial embedding into a piece-wise linear manifold M . Assume that dim M is minimal among all possible suchembeddings. Then dim M ≤ X = 2.Let N ( X ) be the closed regular neighborhood of X in M , or equivalently, st( X )in M after sufficient barycentric subdivisions. Hence N ( X ) can be obtained by at-taching 2 or higher dimensional cells to X . Roughly speaking, N ( X ) is a thickening of X . Note that N ( X ) depends on f but dim N ( X ) does not.If dim N ( X ) = 2, or equivalently, X can be embedded into a surface, then weclaim that there is no branch point in X and so X itself is a surface Σ = N ( X ).Suppose v ∈ br( X ). If lk( v ) is 0-dimensional, then st( v ) (cid:39) T k for k = val( v ) ≥ X can be decomposed further via a nontrivial k -closure or a nontrivial (cid:96) -connected sum for some (cid:96) < k . This contradicts to elementariness of X and sost( v ) is homeomorphic to a boundary wedge sum D ∨ ∂ [ v, w ) of a disk and anhalf-open edge [ v, w ) since X is simple and embeds into a surface. Hence either w is a point of 2-closure of X w when X w is connected, or v is a point of 1-closure or1-connected sum of X otherwise. This does not happen by the elementariness of X , and so st( v ) = D . Therefore v (cid:54)∈ br( X ), and this contradiction implies thatbr( X ) = ∅ .For a surface Σ, we omit the group presentation of B n (Σ) which is well-known,and actually we already introduced the result we need in Theorem 2.5.On the other hand, if dim N ( X ) ≥ 3, then by the same argument as above,all vertices of br( X ) are of valency 1. In this case, we call X a branched surface . Moreover, we can attach cells of dimension at least 2 to X to obtain N ( X ) sothat the inclusion X → N ( X ) becomes a braid equivalence by Proposition 3.6 andCorollary 3.9. This is essentially the same process as described in the proof ofTheorem 1.1. Therefore B n ( X ) (cid:39) B n ( N ( X )) (cid:39) n (cid:89) π ( N ( X )) (cid:111) S n (cid:39) n (cid:89) π ( X ) (cid:111) S n . Lemma 4.4. Let X be an elementary complex. Then X is either T k , a surface Σ ,or a branched surface. Moreover, if X is a branched surface, then B n ( X ) (cid:39) n (cid:89) π ( X ) (cid:111) S n . Example 4.5 (An elementary, non-manifold complex S ) . Let S be the complexobtained by gluing a disk and an interval, depicted in Figure 1. S = { ( x, y, | − ≤ x, y ≤ } ∪ { (0 , , z ) | ≤ z ≤ } ⊂ R . Then it is obvious that S is elementary and dim N ( S ) = 3 since S can not beembedded into any surface. Hence N ( S ) (cid:39) D and therefore B n ( S ) is isomorphicto S n via the induced permutation ρ .It is not hard to see that an elementary complex X embeds into a surface if andonly if it does not contain S . Furthermore, the same holds for non elementarycomplexes. This is easy and we will see later. In this sense, S is the obstructioncomplex for given complex to be embedded into a surface.In the following two sections, we will present the braid groups on (cid:98) X and X Y in terms of the braid groups on X and both X and Y , respectively.Let X and Y be connected, disjoint subspaces of Z . Then for convenience’s sake,we denote by B r ; s ( X ; Y ) the subspace of B r + s ( Z ) defined as B r ; s ( X ; Y ) = { x ∈ B r + s ( Z ) | x ∩ X ) = r, x ∩ Y ) = s } (cid:39) B r ( X ) × B s ( Y ) . Hence π ( B r ; s ( X ; Y )) = B r ( X ) × B s ( Y ) and is denoted by B r ; s ( X ; Y ).5. k -Closure of a complex Let X be a complex and v = { v , . . . , v k } ⊂ ∂X . Let v be the cone point of the k -closure (cid:98) X of X along v , which is the mapping cone of v → X . We denote by e i the oriented edge ( v i , v ) from v i to v in (cid:98) X . Then for each 1 ≤ i ≤ k , the concatenation e e − i defines a path δ i in B n ( (cid:98) X ) from i v ( x ) to i v i ( x ) for any x ∈ B n − ( X ) inthe obvious way, and we denote this path by δ i again unless any ambiguity occurs.Obviously, δ defines a path homotopic to a constant path.Now we endow a metric d on (cid:98) X . Then there is a constant (cid:15) = (cid:15) ( (cid:98) X ) as discussedearlier so that d ( x i , x j ) ≥ (cid:15) for any x i , x j ∈ x ∈ B n ( (cid:98) X ). Then by subdividing alledges adjcent to v , we may assume that the diameter of st( v ) is less than (cid:15) . Inother words, we may assume that any configuration x = B n ( (cid:98) X ) intersects st( v ) atmost once, that is, (cid:16) x ∩ st( v ) (cid:17) ≤ B n ( (cid:98) X ) is separated into two subsets according to the presence of apoint in st( v ), and is the union of two subspaces B n − (cid:16) X \ v ; st( v ) (cid:17) and B n ( X ) N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 17 whose intersection is(5) B n − ( X \ v ; v ) = k (cid:71) i =1 ( B n − ( X \ v ) × { v i } ) . Notice that the intersection is not connected by the assumption that there is atmost 1 point in st( v ). Hence we need to choose paths joining components to makeit connected, and make the Seifert-van Kampen theorem applicable.To this end, we fix a basepoint ∗ (cid:96) of B (cid:96) ( X ) for each (cid:96) ≤ n such that i v ( ∗ n − ) = ∗ n . Then we choose a path γ i in B n ( X ) for each 1 ≤ i ≤ k between ∗ n and i v i ( ∗ n − )such that γ i ( t ) avoids v for 0 < t < (cid:83) ki =1 γ i ⊂ B n ( X ) is homotopy equivalentto T k . In other words, if γ i and γ j intersect at some point, then the images of γ i and γ j must coincide from the beginning. Since i v ( ∗ n − ) = ∗ n , we may assumethat γ is a constant path at ∗ n for convenience sake.In practice, the most convenient way to choose γ i ’s is as follows. At first, we fixa set { γ , . . . , γ k } of paths in B n ( T k ) as depicted in Figure 13. ∗ n = 1 2 ki γ i = −−−−−−−−−−−−−−−→ ki = i v i ( ∗ n − ) , Figure 13. A path γ i for T k joining ∗ n and i v i ( ∗ n − )Since X is connected, there exists a tree T ⊂ X with ∂T = v . Then there is amap q : T → T k which contracts all internal edges of T and induces a homotopyequivalence. Hence similar to Proposition 3.1, we can find a lift γ i for each γ i . Inthis case, points in ∗ n are lying near v . Lemma 5.1. Let X, v as above. Then there exists a homomorphism Ψ (cid:98) X : B n (Θ k ) → B n ( (cid:98) X ) . Proof. Let T be as above. Then by gluing T k to both T and T k , we have a mapˆ q : (cid:98) T → Θ k , where (cid:98) T is an obtained graph homotopy equivalent to Θ k . By Proposition 3.1, itinduces a surjective map ˆ q ∗ : B n (Θ k ) → B n ( (cid:98) T ). Then the desired map Ψ (cid:98) X is justa composition of ˆ q ∗ and the map induced by the inclusion (cid:98) T → (cid:98) X . (cid:3) It is important to remark that Ψ (cid:98) X is neither injective nor surjective in generalby the same reason as stated in the discussion after Proposition 3.1.Let (cid:98) B n − (cid:16) X \ v ; st( v ) (cid:17) = ( (cid:83) i γ i ) ∪ B n − (cid:16) X \ v ; st( v ) (cid:17) . Then for each i ,since γ i ∩ B n − (cid:16) X \ v ; st( v ) (cid:17) = {∗ n , i v i ( ∗ n − ) } and ∗ n and i v i ( ∗ n − ) is connected via the path δ − i δ in st( v ), and so γ i togetherwith γ defines a loop γ i δ − i δ γ − . Hence (cid:83) i γ i contributes ( k − 1) loops and so( k − 1) free letters in the fundamental group. More precisely, let t i = [ γ i δ − i δ γ − ] denote the homotopy class, and we set t = e . Then π (cid:16) (cid:98) B n − (cid:16) X \ v ; st( v ) (cid:17) , ∗ n (cid:17) (cid:39) B n − (cid:16) X \ v ; st( v ) (cid:17) ∗ (cid:104) t , . . . , t k (cid:105)(cid:39) (cid:16) B n − ( X \ v ) × π (cid:16) st( v ) (cid:17)(cid:17) ∗ (cid:104) t , . . . , t k (cid:105)(cid:39) B n − ( X ) ∗ (cid:104) t , . . . , t k (cid:105) . On the other hand, the intersection between (cid:98) B n − (cid:16) X \ v ; st( v ) (cid:17) and B n ( X )is precisely ( (cid:83) i γ i ) ∪ B n − ( X ; v ), and homotopy equivalent to a wedge sum of k -copies of B n − ( X ) indexed by v i ’s as shown in (5). Hence its fundamental group isisomorphic to ∗ ki =1 B n − ( X ). Moreover, for each i , there are two inclusions (cid:98) φ i and ψ i from B n − ( X ) to B n − ( X ) ∗(cid:104) t , . . . , t k (cid:105) and B n ( X ) defined as for all β ∈ B n − ( X ),as paths (cid:98) φ i ( β ) = ψ i ( β ) = γ i · i v i ( β ) · γ − i . However, as group elements, (cid:98) φ i ( β ) = t i βt − i , ψ i ( β ) = v i ∗ ( β ) , where v i ∗ = ( γ i ) − ∗ ( i v i ) ∗ and ( γ i ) ∗ is the automorphism changing the basepointfrom ∗ n to i v i ( ∗ n − ).Therefore we have a diagram below whose push-out defines B n ( (cid:98) X ) by the Seifert-van Kampen theorem.(6) B n − ( X ) ∗ (cid:104) t , . . . , t k (cid:105) B n ( X ) ∗ ki =1 B n − ( X ) ∗ ki =1 (cid:98) φ i (cid:105) (cid:105) ∗ ki =1 ψ i (cid:56) (cid:56) Hence, B n ( (cid:98) X ) = ( B n − ( X ) ∗ (cid:104) t , . . . , t k (cid:105) ) ∗ B n ( X ) (cid:68)(cid:68) (cid:98) φ i ( β ) = ψ i ( β ) , ∀ β ∈ B n − ( X ) , ≤ i ≤ k (cid:69)(cid:69) = B n ( X ) ∗ (cid:104) t , . . . , t k (cid:105)(cid:104)(cid:104) t i βt − i = v i ∗ ( β ) , ∀ β ∈ B n − ( X ) , ≤ i ≤ k (cid:105)(cid:105) . The last equality follows by identifying B n − as a subgroup of B n ( X ) via (cid:98) φ and v ∗ . Note that when k = 2, then B n ( (cid:98) X ) is an ordinary HNN extension of B n ( X )with the associated group B n − ( X ). Theorem 5.2. Let X be a complex and v = { v , . . . , v k } ⊂ ∂X . Then the braidgroup B n ( (cid:98) X ) on the k -closure (cid:98) X of X along v is as follows. For n ≥ , B n ( (cid:98) X ) = B n ( X ) ∗ (cid:104) t , . . . , t k (cid:105)(cid:104)(cid:104) t i βt − i = v i ∗ ( β ) , ∀ β ∈ B n − ( X ) , ≤ i ≤ k (cid:105)(cid:105) , where v i ∗ = ( γ i ) − ∗ ( i v i ) ∗ : B n − ( X ) → B n ( X ) , and γ i is a chosen path joining basepoints of B n ( X ) and i v i ( B n − ( X )) . Moreover, B n − ( X ) is identified with a subgroup of B n ( X ) via v ∗ . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 19 Let ∂ X be a connected component of ∂X of dimension at least 1, and suppose { v , . . . , v k } ⊂ ∂ X . Then since st( ∂ X ) is of dimension at least 2, we may choose γ i ’s in a small enough collar neighborhood of ∂ X as depicted in Figure 14 so thatthey intersect pairwise only at v . Then each path γ i can be regarded as disjointfrom B n − ( X ). This implies the triviality of the action ( γ i ) ∗ on ( i v i ) ∗ ( B n − ( X )).Hence v i ∗ ( β ) = β for all β ∈ B n − ( X ), and the defining relator is nothing but thecommutativity between t i and any β ∈ B n − ( X ). Corollary 5.3. Let X, v be as above. Suppose dim X ≥ and all v i ’s are lying inthe same component of ∂X . Then B n ( (cid:98) X ) (cid:39) B n ( X ) ∗ (cid:104) t , . . . , t k (cid:105) / (cid:104)(cid:104) [ B n − ( X ) , t i ] , ≤ i ≤ k (cid:105)(cid:105) . (cid:98) X = vv v v i v k γ γ i γ k (cid:124) (cid:123)(cid:122) (cid:125) ∗ n − (cid:41) = X } = ∂ X (cid:41) = st( v ) Figure 14. A choice of paths { γ i } when { v , . . . , v k } ⊂ ∂ X Remark 5.4. On the other hand, for a surface Σ, if we take a closure (cid:98) Σ along pointsnot contained in a single boundary component of Σ, then (cid:98) Σ is always nonplanar.Moreover it contains a nonplanar graph. Example 5.5 (2 braid group on the closure of a tree) . Let T be a tree with k = ∂T ) and (cid:98) T be the k -closure of T along ∂T . Since B ( T ) is trivial, B ( (cid:98) T ) isa free group and admits the following presentation. B ( (cid:98) T ) = B ( T ) ∗ π (Θ k )= (cid:104) s , , . . . , s k − ,k , t , . . . , t k | s i,j = s i (cid:48) ,j (cid:48) if ( i, j ) ∼ T ( i (cid:48) , j (cid:48) ) (cid:105) . Hence the rank is r ( T ) + ( k − r ( T ) is the rank of B ( T ) given by theformula (3) in Example 3.2. Example 5.6 (The braid group on Θ k ) . Since Θ k = (cid:98) T k , this is a special case ofthe previous example. Recall that T k produces only the trivial equivalence relationon (cid:0) ∂T k − (cid:1) . Therefore B (Θ k ) is a free group of rank (cid:0) k − (cid:1) + ( k − 1) = (cid:0) k (cid:1) .Now we consider B n (Θ k ) which is generated by t i ’s and B n ( T k ). More specifi-cally, we have a diagram B ( T k ) ( i v ) n − ∗ (cid:47) (cid:47) (cid:99) ( · ) ∗ (cid:15) (cid:15) B n ( T k ) (cid:99) ( · ) ∗ (cid:15) (cid:15) B (Θ k ) ξ (cid:47) (cid:47) B n (Θ k ) , where the vertical arrows are induced by the inclusion (cid:99) ( · ) : T k → Θ k , and ξ isdefined by ξ ( σ i,j ) = (cid:16)(cid:99) ( · ) ∗ ◦ ( i v ) n − ∗ (cid:17) ( σ i,j ) , ξ ( t i ) = t i ∈ B n (Θ k ) . Note that ξ is well-defined since B (Θ k ) is free, and commutativity is obviousby the definition of ξ . Lemma 5.7. [4] The map ξ is surjective. It is not hard to prove this lemma. Indeed, one can prove this by drawingcarefully the paths representing the generators for B n ( T k ) and t i ’s. In general, forany tree T with k = ∂T ), there is a surjective homomorphism B ( (cid:98) T ) → B n ( (cid:98) T ). Remark 5.8. The decomposition defined above is nothing but a graph-of-groups structure for B n ( X ) over the graph Θ k as follows. Note that this is essentially sameas the push-out diagram in (6). B n − ( X ) Id (cid:8) (cid:8) v ∗ (cid:21) (cid:21) B n − ( X ) Id (cid:115) (cid:115) v ∗ (cid:42) (cid:42) B n − ( X ) B n ( X ) B n − ( X ) Id (cid:107) (cid:107) v ∗ (cid:52) (cid:52) ... B n − ( X ) Id (cid:86) (cid:86) v k ∗ (cid:73) (cid:73) Here each cycle involving v ∗ and v i ∗ corresponds to the generator t i , and we call t i ’s stable letters for B n ( X ) as the ordinary HNN extension.6. k -Connected sum of a pair of complexes We will use the generalized notion, called a complex-of-groups , to consider thebraid group on X Y of two given complexes X and Y , and we briefly review aboutcomplex-of-groups. See [7] for details.6.1. Complex-of-groups. For two cells σ and τ of a regular CW-complex, wedenote by σ (cid:31) τ if τ is a face of σ . A face τ of σ is principal if it is of codimension1. By a directed corner α of σ we mean a triple ( τ , σ, τ ) where τ i are two differentprincipal faces of σ having a unique principal face τ ∩ τ in common. For a directedcorner α = ( τ , σ, τ ), we denote by ¯ α the inverse ( τ , σ, τ ) of α . Definition 6.1 (Complex-of-spaces) . [7] A (good) complex-of-spaces K over K isa CW-complex with a cellular map p : K → K satisfying the conditions as follows:(1) for each cell σ of K , there is a connected CW-complex K σ with p − ( σ ) (cid:39)K σ × σ ;(2) for each cell σ of K , the inclusion-induced map π ( K σ ) → π ( K ) is injective.A k -skeleton K ( k ) is defined as p − ( K ( k ) ). Then the fundamental group π ( K )is isomorphic to π ( K (2) ). Moreover, there is a surjection π ( K (1) ) → π ( K ) whosekernel is generated by elements corresponding to ∂ ˜ σ where ˜ σ is a lift of 2-cell σ ∈ K [7, § N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 21 Definition 6.2 (Complex-of-groups) . [7] A complex-of-groups G is a triple ( K, G, φ )where(1) K is a regular CW-complex;(2) G assigns to each cell σ of K a group G σ and each pair ( σ, τ ) with σ (cid:31) τ an injective homomorphism i σ,τ : G σ → G τ ;(3) φ is a corner labeling function that assigns to each direct corner α =( τ , σ, τ ) an element φ ( α ) ∈ G τ ∩ τ satisfying the condition as follows:(a) φ (¯ α ) = φ ( α ) − for each directed corner α ;(b) If α = ( τ , σ, τ ), then the two compositions G σ → G τ → G τ ∩ τ and G σ → G τ → G τ ∩ τ differ by conjugation by φ ( α ).For a complex-of-spaces K over K , an associated complex-of-groups G over K can be defined by taking G σ = π ( K σ ). Note that for each σ (cid:31) τ , the inclusion G σ → G τ depends on the choice of basepoints of K σ and K τ and the choice of apath joining them. Hence, it is uniquely determined only up to inner automorphismon G τ .A k -skeleton G ( k ) of a complex-of-groups is nothing but a restriction on K ( k ) .We say that G is a graph-of-groups when K is 1-dimensional.Let G be a graph-of-groups and T be a maximal tree of K . We identify thegenerators for π (Γ) with the set of oriented edges in Γ \ T . Let ˜ π ( G ) be the freeproduct of π ( K ) and colim T K of K over T . Indeed, colim T K is obtained fromthe free product with amalgamation of vertex groups along all edge groups. Thenwe define a group π ( G ) by HNN extension with all edge groups corresponding tothe generators of π ( K ). More precisely, π ( G ) is obtained by declaring i e,v ( g ) = e − i e,w ( g ) e for all g ∈ G e in ˜ π ( G ) for each edge e = ( v, w ) ∈ Γ \ T .Similar to before, the fundamental group π ( G ) is the same as π ( G (2) ) which isa quotient of π ( G (1) ) by elements coming from corner and edge reading for each2-cell σ of K .For a 2-cell σ , let ∂σ = e e . . . e m . Then the label φ ( σ ) on σ is defined up tocyclic permutation as φ ( σ ) = e φ ( α ) e φ ( α ) . . . e m φ ( α m ) , where φ ( α i ) is a corner label for α i = ( e i , σ, e i +1 ), and e i ∈ π ( K (1) ) is either trivialwhen it belongs to the maximal tree T or a corresponding generator otherwise.Remark that if a complex-of-groups G is associated with a complex-of-spaces K , then π ( K ) (cid:39) π ( G ), and so one may identify these two concepts only for thefundamental group.6.2. k -connected sum. Let X and Y be complexes, and (cid:126)v ∈ X and (cid:126)w ∈ Y be vertices of valency k ≥ (cid:126)v ) = ( v , · · · , v k ), and lk( (cid:126)w ) =( w , · · · , w k ), respectively. We denote ( X, (cid:126)v ) Y, (cid:126)w ) by X Y , edges ( v i , w i ) by e i and E = ∪ e i for simplicity.For a given metric d on X Y , by rescaling X Y near E after sufficient subdi-visions, we may assume that for any configuration x ∈ B n ( X Y ), the closure ¯ e i ofeach e i contains at most 1 point of x . Then according to whether each ¯ e i contains apoint, B n ( X Y ) can be split as the disjoint union of subspaces indexed by (subsetof) the power set of E as follows.Let F ⊂ E be a subset of edges with F ) = a ≤ n , and let (cid:81) F denote theproduct of closures of edges contained in F , so that it is homeomorphic to a closed a -cube D a . Then the spaces B r ; s ( X v ; Y w ) × (cid:89) F indexed by F and r, s with r + s = n − a decompose B n ( X Y ) as desired. Wesimply denote these pieces by B r ; s ( F ) where r + s = n − F ), and regard B r ; s ( F )as F )-dimensional cube. Then for each e i = ( v i , w i ) ∈ F , the two maps definedby i v i : B r ; s ( F ) → B r +1; s ( F \ { e i } ) , i w i : B r ; s ( F ) → B r ; s +1 ( F \ { e i } )correspond to two face maps of the a -dimensional cube (cid:81) F . Moreover, by Proposi-tion 2.8, these induce injective homomorphisms on fundamental groups. We denotethese maps by v i ∗ and w i ∗ for simplicity.Hence all this information defines a complex-of-spaces K for B n ( X Y ) over thecube complex K ( n, k ) depending on n and k . Let G be an associated complex-of-groups with K . Then π ( K ) = B n ( X Y ) = π ( G ). Remark 6.3. The dimension dim K of the cube complex K is the minimum be-tween n and k , and moreover K can be defined inductively as K ( n, k ) = K ( n, k − (cid:116) K ( n − , k − × I, but this is not necessary in this paper and we omit the detail.Since π ( G ) depends only on the 2-skeleton of K as mentioned before, we considera 2-complex-of-groups G (2) over the 2-skeleton K (2) . (0-cells) B r +1; s − ( ∅ ) B r ; s ( ∅ ) B r − ,s +1 ( ∅ )... ... B r ; s − ( { e i } ) wi ∗ (cid:55) (cid:55) vi ∗ (cid:103) (cid:103) B r − s ( { e i } ) vi ∗ (cid:104) (cid:104) wi ∗ (cid:55) (cid:55) (1-cells) . . . ... ... . . . B r ; s − ( { e j } ) wj ∗ (cid:62) (cid:62) vj ∗ (cid:94) (cid:94) B r − s ( { e j } ) vj ∗ (cid:64) (cid:64) vj ∗ (cid:96) (cid:96) ... ...... ... ...(2-cells) B r ; s − ( { e i , e j } ) wj ∗ (cid:67) (cid:67) wi ∗ (cid:60) (cid:60) B r − s − ( { e i , e j } ) vj ∗ (cid:92) (cid:92) vi ∗ (cid:99) (cid:99) wj ∗ (cid:66) (cid:66) wi ∗ (cid:59) (cid:59) B r − s ( { e i , e j } ) vj ∗ (cid:91) (cid:91) vi ∗ (cid:98) (cid:98) ... ... ... Figure 15. A complex-of-groups G The way to compute π ( G (2) ) is described earlier, and before doing the com-putation, we fix basepoints ∗ r and ∗ s for B r ( X v ) and B s ( Y w ) for each r and s ,respectively. We denote ∗ r (cid:116) ∗ s by ∗ r ; s .Let F ⊂ E with F ) ≤ n and e (cid:54)∈ F . For each r , we glue B r ; s ( F ) and B r − s +1 ( F ) via B r − s ( F ∪ { e } ) by using paths e r − s from ∗ r ; s to ∗ r − s +1 . Thenthe distinction on basepoints ∗ r ; s for B r ; s ( F ) disappears, and we denote them by ∗ m if m = r + s . Moreover, we may assume that all points of each ∗ m are lyingon e , and label them by { , . . . , m } with respect to the order coming from theorientation v → w .For i ≥ 2, we choose paths γ mi and δ mi from ∗ m to i v i ( ∗ m − ) and i w i ( ∗ m − ) suchthat γ mi moves only the first point in X v , and δ mi moves only the last point in Y w . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 23 { v i , v j } e i ∪ { v j } ivi (cid:111) (cid:111) iwi (cid:47) (cid:47) { w i , v j }{ v i } ∪ e jivj (cid:79) (cid:79) iwj (cid:15) (cid:15) e i × e jivi (cid:111) (cid:111) iwi (cid:47) (cid:47) ivj (cid:79) (cid:79) iwj (cid:15) (cid:15) { w i } ∪ e jivj (cid:79) (cid:79) iwj (cid:15) (cid:15) { v i , w j } e i ∪ { w j } ivi (cid:111) (cid:111) iwi (cid:47) (cid:47) { w i , w j } B r +1; s − ( ∅ ) B r ; s − ( { e i } ) vi ∗ (cid:111) (cid:111) wi ∗ (cid:47) (cid:47) B r ; s ( ∅ ) B r ; s − ( { e j } ) vj ∗ (cid:79) (cid:79) wj ∗ (cid:15) (cid:15) B r − s − ( { e i , e j } ) vj ∗ (cid:79) (cid:79) wj ∗ (cid:15) (cid:15) vi ∗ (cid:111) (cid:111) wi ∗ (cid:47) (cid:47) B r − s ( { e j } ) vj ∗ (cid:79) (cid:79) wj ∗ (cid:15) (cid:15) B r ; s ( ∅ ) B r − s ( { e i } ) vi ∗ (cid:111) (cid:111) wi ∗ (cid:47) (cid:47) B r − s +1 ( ∅ ) Figure 16. A 2-cell e i × e j in K and corresponding complex-of-groups B r − s − ( { e i , e j } ) in K For convenience’s sake, we set γ m and δ m to be constant paths. Notice that e alsodefines a constant path but has the effect of changing the domain from B r ; s ( F ) to B r − s +1 ( F ). We will suppress the decorations for γ i and δ i unless any ambiguityoccurs. Note that two maps v i ∗ and w i ∗ are precisely ( γ i ) − ∗ ( i v i ) ∗ and ( δ i ) − ∗ ( i w i ) ∗ ,respectively. Lemma 6.4. Let ( X, v ) and ( Y, w ) be as above. Then there exist homomorphisms Φ X : B n ( X ) → B n ( X Y ) , Φ Y : B n ( Y ) → B n ( X Y ) . Proof. Since X v and Y w are connected, there are embedded trees T X and T Y in X v and Y w , respectively, such that ∂T X = ln( v ) and ∂T Y = ln( w ). We denote by q X : T Y → T k and q Y : T X → T k the label-preserving map defined in Example 3.2.Hence as described in Lemma 5.1, we haveˆ q X : X v ∪ T Y → X v ∪ T k = X and ˆ q Y : T X ∪ Y w → T k ∪ Y w = Y, which induce ˆ q ∗ X and ˆ q ∗ Y by Proposition 3.1. Hence we have the desired homomor-phisms Φ X : B n ( X ) ˆ q ∗ X −→ B n ( X v ∪ T Y ) → B n ( X Y )and Φ Y : B n ( Y ) ˆ q ∗ Y −→ B n ( T X ∪ Y w ) → B n ( X Y )by composing the maps induced by the inclusions X v ∪ T Y → X Y and T X ∪ Y w → X Y . (cid:3) Now we can do exactly the same business as before. Let t i,r = [ γ i e r − n − ri δ − i ]for 1 ≤ r ≤ n . More precisely, the action of t i,r on B r − s ( X v ; Y w ) is as follows.(7) t − i,r v i ∗ ( β ) t i,r = w i ∗ ( β ) , where for all β ∈ π ( B r − s ( { e i } )) (cid:39) B r − s ( X v ; Y w ) with r + s = n .Therefore, π ( G (1) ) = ( ∗ r + s = n B r ; s ( X v ; Y w )) ∗ (cid:104) t i,r | ≤ i ≤ k, ≤ r ≤ n (cid:105)(cid:104)(cid:104) t − i,r v i ∗ ( β ) t i,r = w i ∗ ( β ) β ∈ B r − s ( X v ; Y w ) (cid:105)(cid:105) As before, we may identify B r ( X v ) and B s ( Y w ) with subgroups of B n ( X v ) and B n ( Y w ) via v ∗ and w ∗ , respectively. Then for α ∈ B r − ( X v ), the supports of α and δ i are disjoint. Hence w i ∗ ( α ) = α , and similarly, v i ∗ ( β ) = β for any β ∈ B s ( Y w ).That is, π ( G (1) ) = ( ∗ r + s = n B r ; s ( X v ; Y w )) ∗ (cid:104) t i,r | ≤ i ≤ k, ≤ r ≤ n (cid:105) (cid:42)(cid:42)(cid:40) t − i,r v i ∗ ( α ) t i,r = α α ∈ B r − ( X v ) t − i,r βt i,r = w i ∗ ( β ) β ∈ B s ( Y w ) (cid:43)(cid:43) Moreover, this is a quotient of B n ( X v ) ∗ B n ( Y w ) ∗ (cid:10) t i,r , u i,r (cid:12)(cid:12) u i,r = t − i,n − r , ≤ i ≤ k, ≤ r ≤ n (cid:11) since B r ; s ( X v ; Y w ) = B r ( X v ) × B s ( Y w ) and both B r ( X v ) and B s ( Y w ) are subgroupsof B n ( X v ) and B n ( Y w ), respectively. Here, we furthermore add a set { u i,r } ofdummy generators by declaring u i,r = t − i,n − r . Then it becomes π ( G (1) ) = B n ( X v ) ∗ B n ( Y w ) ∗ (cid:10) t i,r , u i,r (cid:12)(cid:12) u i,r = t − i,n − r , ≤ i ≤ k, ≤ r ≤ n (cid:11)(cid:42)(cid:42) [ B r ( X v ) , B s ( Y w )] r + s = nt − i,r v i ∗ ( α ) t i,r = α α ∈ B r − ( X v ) u − i,s w i ∗ ( β ) u i,s = β β ∈ B s ( Y w ) (cid:43)(cid:43) . Notice that the second and third types of defining relators are those appearingin B n ( X ) and B n ( Y ).Suppose k = 1. Then X v ≡ B X , Y w ≡ B Y and X Y is just a boundary wedgesum X ∨ ∂ Y of X and Y . Since there is no t i,r , we have B n ( X ∨ ∂ Y ) (cid:39) B n ( X ) ∗ B n ( Y ) (cid:104)(cid:104) [ B r ( X ) , B s ( Y )] , r + s = n (cid:105)(cid:105) . In this case, we have a graph-of groups as well over the linear graph of length n .Hence B n ( X ∨ ∂ Y ) is obtained by the iterated amalgamated free product.On the other hand, if n = 1, then the decoration for t i is not necessary. Therefore π ( X Y ) (cid:39) π ( X v ) ∗ π ( Y w ) ∗ (cid:104) t , . . . , t k (cid:105) (cid:39) π ( X ) ∗ π ( Y ) (cid:104)(cid:104) t i u i | ≤ i ≤ k (cid:105)(cid:105) , where t i ’s and u i ’s in the right correspond to generators defined as before. This isnothing but the usual Seifert-van Kampen theorem.Suppose k, n ≥ F i,j = { e i , e j } with i < j . Then the boundary ∂ ( (cid:81) F )has 4 corners, and so we have to consider 8 maps, { LU, U L, U R, RU, RD, DR, DL, LD } corresponding to the ways of compositions as shown in Figure 16.In the northwest corner, we need consider two maps left-and-up LU and up-and-left U L . Then for r + s = n − 2, the maps LU and U L are the compositions LU : B r ; s ( X v ; Y w ) v i ∗ −−→ B r +1; s ( X v ; Y w ) v j ∗ −−→ B r +2; s ( X v ; Y w ) U L : B r ; s ( X v ; Y w ) v j ∗ −−→ B r +1; s ( X v ; Y w ) v i ∗ −−→ B r +2; s ( X v ; Y w ) , which are nothing but conjugates by two paths η and η from ∗ n to ∗ n − (cid:116) { v i , v j } ,where η moves the first point to v j and the second point to v i but η moves the firstto v i and the second to v j . More precisely, η = γ j · i v j ( γ i ) and η = γ i · i v i ( γ j ), andtherefore they differ by the loop η · η − which represents the element exactly thesame as s i,j defined in Example 3.2, which is the image of the generator of B ( T )via the map induced from the embedding ( T , (1 , , → ( T X , (1 , i, j )). Note thatwhen i = 1, then we set s ,j to be trivial for all 1 < j . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 25 In summary, U L ( · ) = s i,j LU ( · ) s − i,j . In the southeast corner, we have a similar result as above. That is, the two maps RD for right-and-down and DR for down-and-right are related as DR ( · ) = s (cid:48) i,j RD ( · ) s (cid:48)− i,j , where s (cid:48) i,j is the image of the generator of B ( T ) via the map induced from theembedding ( T , (1 , , → ( T Y , (1 , i, j )), and is set to be trivial if i = 1.In the northeast corner, two maps U R for up-and-right and RU for right-and-upcoincide since the supports of γ j and δ i are disjoint, and so we need not conjugatefor transport.In the southwest corner, this is the exactly same situation as above and so thetwo maps LD and DL for left-and-down and down-and-left coincide.On the other hand, the four edges of ∂ ( (cid:81) F ) are as follows. U : v i ∗ ( β ) (cid:55)→ w i ∗ ( β ) , L : w j ∗ ( β ) → v j ∗ ( β ) ∀ β ∈ B r +1; s ( X v ; Y w ) ,R : v j ∗ ( β ) (cid:55)→ w j ∗ ( β ) , D : w i ∗ ( β ) → v i ∗ ( β ) ∀ β ∈ B r ; s +1 ( X v ; Y w ) . Then as seen in the computation of π ( G (1) ), the maps U, R, D and L satisfyrelations similar to (7), which are given by U ( · ) = t − i,r +2 ( · ) t i,r +2 , L ( · ) = u − j,s ( · ) u j,s ,R ( · ) = t − j,r +1 ( · ) t j,r +1 , D ( · ) = u − i,s +1 ( · ) u i,s +1 . Recall that we set t ,r = u ,s to be trivial.Finally, the reading of the edges and corners of ∂ [ e i , e j ] gives us a word H i,j = s − i,j t i,r +2 t j,r +1 s (cid:48)− i,j u i,s +1 u j,s , and π ( G (2) ) is obtained by declaring H i,j = e for all 1 ≤ i < j ≤ k .Suppose i = 1. Then since r + s = n − ≤ r ≤ n − H ,j = t j,r +1 u j,s = t j,r +1 t − j,r +2 . Therefore the defining relator H ,j = e removes all decorations on t j ’s and on u j ’sas well.If i > 1, then H i,j = e gives s − i,j t i t j s (cid:48)− i,j u i u j = e , or equivalently, s i,j t j t i s (cid:48) i,j u j u i = e. In summary,(8) B n ( X Y ) = B n ( X ) ∗ B n ( Y ) (cid:42)(cid:42) [ B r ( X v ) , B s ( Y w )] r + s = nt i u i = e ≤ i ≤ ks i,j t j t i s (cid:48) i,j u j u i = e ≤ i < j ≤ k (cid:43)(cid:43) . Before we state the theorem, we will discuss the s i,j ’s further. As mentionedabove, both s i,j and s (cid:48) i,j naturally come from B ( T k ) as follows.Let us break Θ k into T Lk and T Rk , which are left and right halves. That is, wemay assume that if br(Θ k ) = { , (cid:48) } , T Lk = Θ k \ st(0) , T Rk = Θ k \ st(0 (cid:48) ) . We also denote the closures of these halves by Θ Lk and Θ Rk , and denote the mapsby (cid:99) ( · ) L and (cid:99) ( · ) R . (cid:99) ( · ) L : T Lk → Θ Lk , (cid:99) ( · ) R : T Rk → Θ Rk . Then as seen in Example 5.6, B (Θ Lk ) = (cid:104) σ i,j , t (cid:96) (cid:105) , B (Θ Rk ) = (cid:104) σ (cid:48) i,j , u (cid:96) (cid:105) . Since Θ Lk Rk = Θ k , we can use (8) to compute B n (Θ k ). Notice that s i,j and s (cid:48) i,j correspond to σ i,j and σ (cid:48) i,j , respectively since they essentially come from B ( T Lk )and B ( T Rk ) by definition. Therefore, B (Θ k ) = B (Θ k k ) = (cid:104) σ i,j , t (cid:96) , σ (cid:48) i,j , u (cid:96) | t (cid:96) u (cid:96) = e, σ i,j t j t i σ (cid:48) i,j u j u i = e (cid:105) , which is obviously isomorphic to B (Θ Lk ) and B (Θ Rk ).Now we turn back to B n ( X Y ). Recall the surjective map ξ described in Ex-ample 5.6. In this situation, we have two surjections ξ L : B (Θ k ) → B n (Θ Lk ) , ξ R : B (Θ k ) → B n (Θ Rk )satisfying that (cid:99) ( · ) L, ∗ ◦ ( i v ) n − ∗ = ξ L ◦ (cid:99) ( · ) L, ∗ , (cid:99) ( · ) R, ∗ ◦ ( i w ) n − ∗ = ξ R ◦ (cid:99) ( · ) R, ∗ . Then s i,j and s (cid:48) i,j are nothing but the images of σ i,j and σ (cid:48) i,j in B (Θ k ) underthe compositions Ψ X ◦ ξ L and Ψ Y ◦ ξ R .(Ψ X ◦ ξ L )( σ i,j ) = s i,j ∈ B n ( X ) , (Ψ Y ◦ ξ R )( σ (cid:48) i,j ) = s (cid:48) i,j ∈ B n ( Y ) . Moreover, we have(Ψ X ◦ ξ L )( t (cid:96) ) = t (cid:96) ∈ B n ( X ) , (Ψ Y ◦ ξ R )( u (cid:96) ) = u (cid:96) ∈ B n ( Y ) , which correspond to the stable letters for B n ( X ) and B n ( Y ).Therefore, the second and third types of defining relations in (8) precisely declarethat the two images of B (Θ k ) under Ψ X ◦ ξ L and Ψ Y ◦ ξ R are the same. Moreover,both Ψ X ◦ ξ L and Ψ Y ◦ ξ R factor through ξ : B (Θ k ) → B n (Θ k ), so there exist ˜Ψ X and ˜Ψ Y satisfying the following commutative diagram, where the innermost squareinvolving ˜Ψ X and ˜Ψ Y is the push-out diagram. B n (Θ k ) Ψ X (cid:47) (cid:47) B n ( X ) (cid:37) (cid:37) Φ X (cid:42) (cid:42) B (Θ k ) ξ L (cid:57) (cid:57) (cid:57) (cid:57) ξ R (cid:37) (cid:37) (cid:37) (cid:37) ξ (cid:47) (cid:47) (cid:47) (cid:47) B n (Θ k ) (cid:39) (cid:79) (cid:79) (cid:39) (cid:15) (cid:15) ˜Ψ X (cid:58) (cid:58) ˜Ψ Y (cid:36) (cid:36) (cid:101) B n ( X ; Y ) ∃ Q (cid:47) (cid:47) B n ( X Y ) B n (Θ k ) Ψ Y (cid:47) (cid:47) B n ( Y ) (cid:57) (cid:57) Φ Y (cid:52) (cid:52) N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 27 Hence the group (cid:101) B n ( X ; Y ) is defined as the free product with amalgamation asfollows. (cid:101) B n ( X ; Y ) = B n ( X ) ∗ B n ( Y ) (cid:104)(cid:104) ˜Ψ X ( β ) = ˜Ψ Y ( β ) | β ∈ B n (Θ k ) (cid:105)(cid:105) = B n ( X ) ∗ B n ( Y ) (cid:42)(cid:42)(cid:40) t i u i = e ≤ i ≤ ks i,j t j t i s (cid:48) i,j u j u i = e ≤ i < j ≤ k (cid:43)(cid:43) . Finally, the map Q is obviously taking the quotient (cid:101) B n ( X ; Y ) by (cid:104)(cid:104) [ B r ( X v ) , B s ( X w )] , r + s = n (cid:105)(cid:105) . In summary, we have the following theorem. Theorem 6.5. Let X, Y be complexes, (cid:126)v ∈ X and (cid:126)w ∈ Y be vertices of valency k ≥ with orderings lk( (cid:126)v ) = ( v , . . . , v k ) and lk( (cid:126)w ) = ( w , . . . , w k ) , respectively.Then the braid group B n (( X, (cid:126)v ) Y, (cid:126)w )) for n ≥ is as follows. B n (( X, (cid:126)v ) Y, (cid:126)w )) = B n ( X ) ∗ B n (Θ k ) B n ( X ) (cid:14) (cid:104)(cid:104) [ B r ( X v ) , B s ( Y w )] , r + s = n (cid:105)(cid:105) for B n ( X ) ˜Ψ X ←−− B n (Θ k ) ˜Ψ Y −−→ B n ( Y ) , where ˜Ψ X and ˜Ψ Y are defined as ˜Ψ X ◦ ξ = Ψ X ◦ ξ L , ˜Ψ Y ◦ ξ = Ψ Y ◦ ξ R . Example 6.6 (2-braid group on the union of two trees) . Let T and T (cid:48) be treeswith k = ∂T ) = ∂T (cid:48) ), and (cid:98) T and (cid:98) T (cid:48) be k -closures whose closing vertices aredenoted by v and w , respectively. We fix orderings on lk( v ) and lk( w ), and consider2-braid group on ( (cid:98) T , (cid:126)v ) (cid:98) T (cid:48) , (cid:126)w ). Then by Example 5.5 and Theorem 6.5, B ( (cid:98) T (cid:98) T (cid:48) ) = (cid:42) s i,j , s (cid:48) i,j , t r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s (cid:48) i,j = t − i t − j s − i,j t i t j ≤ i < j ≤ ks i,j = s i (cid:48) ,j (cid:48) ( i, j ) ∼ T ( i (cid:48) , j (cid:48) ) s (cid:48) i,j = s (cid:48) i (cid:48) ,j (cid:48) ( i, j ) ∼ T (cid:48) ( i (cid:48) , j (cid:48) ) (cid:43) . Note that the generators s (cid:48) i,j are not necessary by the first type of definingrelators, moreover, the third reduces s i,j ’s as well. Indeed, under certain conditionit has a generating set consisting of t r ’s and only one s i,j . We will see this later. Remark 6.7. Both k -closures and k -connected sums for any k ≥ Applications – Embeddabilities and connectivities In this section, we will prove Theorem 1.2 (2) and (3), namely, how the braidgroup B n ( X ), or its abelianization H ( B n ( X )) is related with the geometry of X . Unless mentioned otherwise, we assume that X is sufficiently subdivided andsimple. Surface embeddability. We start with the following easy observation. Lemma 7.1. Let X and Y be simple complexes. Then X and Y embed into surfacesif and only if so do (cid:98) X and (cid:98) Y if and only if so does X Y , for arbitrary closuresand connected sums.Especially, X embeds into a surface if and only if so does any elementary sub-complex of X . Lemma 7.2. Let X and Y be complexes. Then the following are equivalent. (1) B n ( X ) and B n ( Y ) are torsion-free for all n ≥ . (2) B n ( (cid:98) X ) and B n ( (cid:98) Y ) are torsion-free for all n ≥ . (3) B n ( X Y ) is torsion-free for any n ≥ .Proof. As remarked in the end of the previous section, both k -closures and k -connected sums yield graph-of-groups structures, which correspond to HNN-extensionsand free products with amalgamations. The proof follows since these group opera-tions preserve both torsions and torsion-freeness. (cid:3) It is worth remarking that indeed the configuration space B n ( X ) or B n ( X Y ) is aspherical if and only if so is B n ( X v ) or so are B n ( X ) and B n ( Y ), respectively. Thisfollows easily by considering graphs-of-spaces structures on configuration spaces. Corollary 7.3. Let X be a complex, not necessarily elementary. Suppose X cannot be embedded into any surface. Then B n ( X ) contains S n as a subgroup.Proof. By Lemma 7.1, there exists an elementary subcomplex Y in X , which doesnot embed into any surface. Then as mentioned in Example 4.5, we suppose thatthere exists an embedding i : S → Y ⊂ X .Let ρ and ρ (cid:48) be the induced permutations from B n ( S ) and B n ( X ), respectively.Then it is obvious that ρ = ρ (cid:48) ◦ i ∗ . However, ρ is an isomorphism and therefore ρ (cid:48) ◦ i ∗ ◦ ρ − is the identity on S n . In other words, i ∗ ◦ ρ − : S n → B n ( X )is injective. (cid:3) Proof of Theorem 1.2 (2). Suppose X can be embedded into a surface Σ. Thenit can be modified to a simple complex X (cid:48) by using only the reverse process ofedge contractions as Proposition 3.12. Hence X (cid:48) ≡ B X and embeds into Σ, andmoreover, all the elementary complexes contained in X (cid:48) embed into surfaces byLemma 7.1, as well. Therefore by Lemma 4.4, all their braid groups are torsion-free.As seen in Lemma 7.2, since the build-up processes preserve torsion-freeness, B n ( X ) is torsion-free, as desired.The converse follows from Corollary 7.3. (cid:3) The first homology groups. Suppose n ≥ 2. Then the induced permutation ρ induces the map ¯ ρ : H ( B n ( X )) → H ( S n ) (cid:39) Z . Corollary 7.4. Let X be an elementary complex. Then for n ≥ , H ( B n ( X )) (cid:39) Z r X = T k ; H ( X ) ⊕ (cid:104) [ σ ] (cid:105) dim X = 2 , X is planar ; H ( X ) ⊕ (cid:104) [ σ ] (cid:105) / (cid:104) σ ] (cid:105) otherwise, N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 29 where ¯ ρ ([ σ ]) is nontrivial in H ( S n ) (cid:39) Z , and r = r ( n, k, k ) is given by the equation (2) .Especially, H ( B n ( X )) is torsion-free if and only if X is planar.Proof. This is a direct consequence of the discussion in Section 4. (cid:3) Hence Theorem 1.2 (3) holds for elementary complexes, and from now on weassume that X is nonelementary, and furthermore ∂X (cid:54) = ∅ . If ∂X = ∅ , then thereexists w ∈ br( X ) such that S ⊂ st( w ), and we obtain a boundary by attaching adisk in st( w ) as depicted in Figure 17.Therefore we can consider a map B ( X ) → B n ( X ) which is a composition of( n − 1) maps described in Proposition 2.8. This induces the map i X,n : H ( X ) → H ( B n ( X )), whose cokernel plays an important role in proving our theorem.For example, since ¯ ρ is a surjection and H ( X ) ⊂ ker(¯ ρ ), we have a surjectioncoker( i X,n ) → H ( S n ) (cid:39) Z . Therefore coker( i X,n ) can never be trivial for any n ≥ 2. On the other hand, if we have an embedding ι : X → Y , then it induces amap ι ∗ : coker( i X,n ) → coker( i Y,n ). Since ¯ ρ is equivariant, ι ∗ is nontrivial too. Corollary 7.5. [3] Suppose X contains a nonplanar graph. Then coker( i X,n ) has -torsion.Proof. For a nonplanar graph Γ, it is known that H ( B n (Γ)) has 2-torsion corre-sponding to the generator for H ( S n ). See [3] or Proposition 7.20 below. Hence forany complex X containing a nonplanar graph, coker( i X,n ) has 2-torsion as discussedabove. (cid:3) We observe how k -closure and k -connected sum affect the first homology of braidgroups as follows. These two lemma are direct consequences of Theorem 5.2 andTheorem 6.5. Lemma 7.6. Let X be a complex and (cid:98) X be a k -closure of X . Then there exists acommutative diagram with exact rows as follows. (cid:47) (cid:47) H ( X ) (cid:47) (cid:47) i X,n (cid:15) (cid:15) H ( (cid:98) X ) (cid:47) (cid:47) i (cid:99) X,n (cid:15) (cid:15) H (Θ k ) (cid:47) (cid:47) (cid:39) (cid:15) (cid:15) H ( B n ( X )) (cid:47) (cid:47) H ( B n ( (cid:98) X )) (cid:47) (cid:47) H (Θ k ) (cid:47) (cid:47) . Note that since H (Θ k ) (cid:39) Z k − is free abelian, the surjective map in each rowsplits. Corollary 7.7. Let Y be an elementary complex of dimension 2. Suppose X isobtained by taking closures several times of Y . Then H ( B n ( X )) = (cid:40) H ( X ) ⊕ Z X is planar ; H ( X ) ⊕ Z otherwise. Proof. By Lemma 7.6, the map coker( i Y,n ) → coker( i X,n ) is surjective, and Corol-lary 7.4 implies that coker( i Y,n ) is either Z or Z .If Y is nonplanar, then X is nonplanar and coker( i Y,n ) (cid:39) Z . Hence so iscoker( i X,n ) by above.Suppose Y is planar but X is nonplanar. Then since Y is a surface, X con-tains a nonplanar graph as mentioned in Remark 5.4. Hence coker( i X,n ) (cid:39) Z byCorollary 7.5. Suppose X is planar, and consider the map ι ∗ : coker( i X,n ) → coker( i R ,n ) (cid:39) Z induced by the embedding ι : X → R . Since ι ∗ is nontrivial and coker( i X,n ) isgenerated by a single element, ι ∗ must be an isomorphism. (cid:3) Lemma 7.8. Let X, Y and v, w be as given in Theorem 6.5. Then there exists anexact sequence as follows. (cid:47) (cid:47) H (Θ k ) (cid:47) (cid:47) i Θ k,n (cid:15) (cid:15) H ( X ) ⊕ H ( Y ) (cid:47) (cid:47) i X,n ⊕ i Y,n (cid:15) (cid:15) H ( X Y ) (cid:47) (cid:47) i X Y,n (cid:15) (cid:15) H ( B n (Θ k )) (cid:47) (cid:47) H ( B n ( X )) ⊕ H ( B n ( Y )) (cid:47) (cid:47) H ( B n ( X Y )) (cid:47) (cid:47) . Connectivities. We adopt another notion for a decomposition, called a cut .This notion originated in graphs and is extended to complexes in slightly differentways as follows. Definition 7.9. Let X be a sufficiently subdivided simple complex. A set v = { v , . . . , v k } of vertices in br( X ) is a k -cut if X v = X \ st( v ) is disconnected.We say that a k -cut v is trivial if there exists a subcomplex Y ⊂ X with v ⊂ ∂Y and X is homeomorphic to (cid:92) ( Y, v ), and that X is vertex- k -connected unless there isa nontrivial ( k − Remark 7.10. If v = { v , . . . , v k } is a trivial k -cut and (cid:98) Y = X , then val( v i ) = 1in Y since v ⊂ ∂Y . Therefore val( v i ) = 2 in X for all i . However, since all v i are inbr( X ) and X is simple, lk( v i ) = D m (cid:116) {∗} for some m ≥ 1. Consequently, a trivialcut may exist only when X is of dimension at least 2.Especially, if there is a trivial 1-cut v , then it satisfies the assumption of Propo-sition 3.11 and therefore we may assume that there is no trivial 1-cut without lossof any generality.For example, any nontrivial tree has a nontrivial 1-cut and so it is not vertex-2-connected, and Θ k for k ≥ S (cid:48) = ≡ B = S Figure 17. A complex S (cid:48) which is braid equivalent to S We introduce the famous result of Menger about the relationship between vertex- k -connectivity and the existence of embedded Θ k as follows. Lemma 7.11. [12] Let Γ be a graph without a vertex of valency 1. Then Γ is vertex- k -connected if and only if for any v, w ∈ br(Γ) , there is an embedding (Θ k , { , (cid:48) } ) → (Γ , { v, w } ) of pairs. Example 7.12 (Vertex-3-connectivity for the union of two trees) . Recall the graph (cid:98) T (cid:98) T (cid:48) defined in Example 6.6. Then it is vertex-3-connected only if there is anembedding (Θ , { , (cid:48) } ) → ( (cid:98) T (cid:98) T (cid:48) , { v, w } ) for any v ∈ br( T ) and w ∈ br( T (cid:48) ).Indeed, we may assume that such Θ k always passes the point 1 ∈ ∂T . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 31 Let T L and T R be two halves of Θ k as before. Then the restrictions of anembedding Θ k → (cid:98) T (cid:98) T to T L and T R give us a pair of equivalence classes [ s i,j ]and [ s (cid:48) i,j ] with respect to ∼ T and ∼ T (cid:48) , which are related as s (cid:48) i,j = t − i t − j s − i,j t i t j asdescribed in Example 3.2 and Example 6.6.Therefore the vertex-3-connectivity of (cid:98) T (cid:98) T (cid:48) implies that any pair of generatorsfor B ( T ) and B ( T (cid:48) ) are related, and so B ( (cid:98) T (cid:98) T (cid:48) ) is generated by t i ’s and onlyone s i,j .7.3.1. Assume that X has a 1-cut v of valency k ≥ 2, and X . . . , X m areconnected components of X v . Note that if k = 2, then m must be 2 and this is a1-connected sum decomposition of X . Therefore we assume that k ≥ ≤ i ≤ k , let k i be the number of vertices in X i which are adjacent to v .Let k = ( k , . . . , k m ), then (cid:80) mi =1 k i = k .Now we decompose X into ( m + 1) pieces, namely, (cid:98) X , . . . , (cid:98) X m and (cid:98) T k , via k i -connected sums for all 1 ≤ i ≤ m , where (cid:98) T k looks like a graph depicted in Figure 18.We apply Lemma 7.8 for X = (cid:98) T k (cid:98) X . . . (cid:98) X m as follows. m (cid:77) i =1 H ( B n (Θ k i )) ⊕ ˜Ψ i −−−→ H ( B n ( (cid:98) T k )) ⊕ m (cid:77) i =1 H ( B n ( (cid:98) X i )) −→ H ( B n ( X )) −→ . vX X X X X X = v (cid:98) X (cid:98) X (cid:98) X (cid:98) X (cid:98) X (cid:98) X Figure 18. A decomposition of X near 1-cut v and a graph (cid:98) T k with k = (3 , , , , , Lemma 7.13. [4, Lemma 3.11] The first homology group H ( B n ( (cid:98) T k )) is isomorphicto H ( B n ( (cid:98) T k )) = Z r ( n,k,m ) ⊕ m (cid:77) i =1 H ( B n (Θ k i )) . Hence the obvious embedding Θ k i → (cid:98) T k yields an injection˜Ψ i : H ( B n (Θ k i )) → H ( B n ( (cid:98) T k )) , and therefore the sequence above becomes a short exact sequence. Moreover, wehave the following lemma which is obvious by the decomposition in Lemma 7.13. Lemma 7.14. Let X, v and X i ’s be as before. Then H ( B n ( X )) = Z r ( n,k,m ) ⊕ m (cid:77) i =1 H ( B n ( (cid:98) X i )) . In summary, we can say that each nontrivial 1-cut v contributes to the first Bettinumber as much as r ( n, k, m ) where k = val( v ) and m = π ( X v )).From now on, we assume that X has no 1-cut. Lemma 7.15. [4, Lemma 3.12] Let Γ be a graph without a 1-cut. Then for all n ≥ , H ( B n (Γ)) (cid:39) H ( B (Γ)) .Therefore, H ( B n (Θ k )) (cid:39) H (Θ k ) ⊕ Z ( k − ) . Assume that X has no 1-cut but a nontrivial 2-cut v = { v , v } , and X , . . . , X m are the connected components of X v as before. Let k i,j be the numberof components of lk( v j ) in X i and k j = ( k ,j , . . . , k m,j ) for 1 ≤ i ≤ m, j = 1 , X into ( m + 1)-pieces via ( k i, + k i, )-connectedsums as depicted in Figure 19. The connected summands will be denoted by (cid:98) X , . . . , (cid:98) X m and Θ k , k . Then by Lemma 7.8, H ( B n ( X )) is isomorphic to thecokernel of m (cid:77) i =1 H ( B n (Θ k i, + k i, )) ⊕ ˜Ψ i (cid:47) (cid:47) H ( B n (Θ k , k )) ⊕ m (cid:77) i =1 H ( B n ( (cid:98) X i )) .X = v v X X X = (cid:98) X (cid:98) X (cid:98) X k , k Figure 19. A decomposition of X near a 2-cut v and a graphΘ k , k with k = (3 , , 3) and k = (2 , , q ←− = Θ , , Θ , , Θ Θ , , Figure 20. A decomposition of (cid:101) Θ k , k via 2-connected-sumsLet Θ a,b,c denote a (possibly subdivided) graph obtained by replacing respectiveedges of the triangle with a , b and c multiple edges. We take 2-connected sumsbetween Θ k i, ,k i, , and Θ m to obtain (cid:101) Θ k , k . Then Θ k , k comes from (cid:101) Θ k , k bycontracting all edges adjacent to vertices of Θ m . See Figure 20.We want to use (cid:101) Θ k , k instead of Θ k , k . That is, we define (cid:101) X by taking ( k i, + k i, )-connected-sums between (cid:98) X i and (cid:101) Θ k , k . The lemma below ensures that wecan safely do this. Lemma 7.16. [4, Lemma 3.14] Let q : (cid:101) Θ k , k → Θ k , k be the quotient map and q ∗ : B n (Θ k , k ) → B n ( (cid:101) Θ k , k ) be the map defined in Proposition 3.1. N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 33 Then q ∗ induces an isomorphism between abelianizations, that is, the first ho-mology group H ( B n (Θ k , k )) is isomorphic to H ( B n ( (cid:101) Θ k , k ))Therefore H ( B n ( X )) = H ( B n ( (cid:101) X )) and now we can decompose (cid:101) X by using2-connected-sums into (cid:101) X i ’s and Θ m , where (cid:101) X i = (cid:98) X i k i, ,k i, , . (cid:101) X = , , Θ , , Θ Θ , , (cid:98) X (cid:98) X (cid:98) X = (cid:101) X (cid:101) X (cid:101) X = Θ (cid:101) X (cid:101) X (cid:101) X . Figure 21. A 2-connected-sum decomposition of (cid:101) X near a 2-cutBy Lemma 7.8 again, we have a short exact sequence1 → m (cid:77) i =1 H ( B n (Θ )) → H ( B n (Θ m )) ⊕ m (cid:77) i =1 H ( B n ( (cid:101) X i )) → H ( B n ( (cid:101) X )) → . Lemma 7.17. Let X, v and X i ’s be as above. Then H ( B n ( X )) ⊕ Z m = H ( B n (Θ m )) ⊕ m (cid:77) i =1 H ( B n ( (cid:101) X i ))= Z ( m ) ⊕ m (cid:77) i =1 H ( B n ( (cid:101) X i )) . Proof. This follows easily from the above exact sequence and H ( B n (Θ )) = Z . (cid:3) Vertex-3-connected complexes. We claim the following. Proposition 7.18. Let X be a simple and vertex-3-connected complex. Then for n ≥ , H ( B n ( X )) = (cid:40) H ( X ) ⊕ (cid:104) [ σ ] (cid:105) X is planar ; H ( X ) ⊕ (cid:104) [ σ ] (cid:105) / (cid:104) σ ] (cid:105) X is nonplanar . This is a generalization of the result for vertex-3-connected graphs stated in [4,Lemma 3.15], and we can prove Theorem 1.2 (3) with the aid of this propositionas follows. Proof of Theorem 1.2 (3). Since vertex-1 and 2-connected sums and 1-cut and 2-cut decompositions preserve planarity, X is nonplanar if and only if X has a nonpla-nar vertex-3-connected component with respect to 1-cut and 2-cut decompositions.On the other hand, Lemma 7.13 implies that H ( B n ( X )) has torsion if and onlyif one of its vertex-2-connected components does. However, Lemma 7.16 does not imply directly the corresponding result because of Z summands. Each Z summandis mapped to a summand of the homology group H ( (cid:101) X i ) of a vertex-3-connectedcomponent (cid:101) X i via the map induced from the embedding Θ = S → (cid:101) X i . Hence H ( B n ( X )) has torsion for a vertex-2-connected complex X if and only if one ofits vertex-3-connected components does.Finally, Proposition 7.18 completes the proof. (cid:3) For the rest of the paper, we will prove Proposition 7.18. Hence from now on,we suppose that X is a simple and vertex-3-connected complex. Lemma 7.19. Let T and T (cid:48) be trees with k = ∂T ) = ∂T (cid:48) ) . Suppose (cid:98) T (cid:98) T (cid:48) is vertex-3-connected. Then Proposition 7.18 holds for (cid:98) T (cid:98) T (cid:48) .Proof. This is a special case of Lemma 3.15 in [4], and we introduce a new proof.By Lemma 7.15, it suffices to consider H ( B ( (cid:98) T (cid:98) T (cid:48) )). Recall the group presen-tation for B ( (cid:98) T (cid:98) T (cid:48) ) from Example 6.6. Then its abelianization has a presentationas follows. H ( B ( (cid:98) T (cid:98) T (cid:48) )) = Z k − ⊕ (cid:77) ≤ i Let X = Y Z . Suppose that Proposition 7.18 holds for allvertex-3-connected subcomplexes of Y and Z . Then it does for X as well.Proof. Let X = ( Y, (cid:126)v ) Z, (cid:126)w ) for lk( (cid:126)v ) = ( v , . . . , v k ) and lk( (cid:126)w ) = ( w , . . . , w k ).Then both Y v and Z z are connected by definition, and therefore both Y and Z arevertex-2-connected.However, in general, Y and Z are not necessarily vertex-3-connected, and if a2-cut exists in Y or Z , then one of the two vertices is precisely v or w , respectively.Then from the 2-cut decompositions for Y and Z , we can obtain two trees T Y ⊂ Y v and T Z ⊂ Z w such that ∂T Y = lk( v ) and ∂T Z = lk( w ), where vertices in T Y and T Z correspond to vertex-3-connected components in Y and Z , respectively. Since X is vertex-3-connected, so is (cid:98) T Y (cid:98) T Z by construction.Moreover, by the assumption about Y and Z , coker( i Y,n ) and coker( i Z,n ) aregenerated by r ( T Y ) and r ( T Z ) elements, respectively.The commutative diagram in Lemma 7.8 produces an exact sequencecoker( i Θ k ,n ) → coker( i Y,n ) ⊕ coker( i Z,n ) → coker( i X,n ) → . However, the quotient of coker( i Y,n ) ⊕ coker( i Z,n ) by the image of coker( i Θ k ,n )is nothing but coker( i (cid:98) T Y (cid:98) T Z ,n ) and by Lemma 7.19, it is either Z or Z . N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 35 Finally, it is Z if and only if either one of vertex-3-connected components of Y and Z is nonplanar, or both Y and Z are planar but (cid:98) T Y (cid:98) T Z is nonplanar. Sincethese conditions are equivalent to the nonplanarity of X , we are done. (cid:3) Therefore we may assume furthermore that X is not decomposable in a nontrivialway via k -connected sum for all k ≥ 1. However, note that X might be expressibleas a closure.Indeed, there is no such complex of dimension 1, a graph, as follows. If it exists,then it has at least 3 vertices of valency ≥ ≥ V (Γ) of vertices of valency ≥ V and V , where both the full subgraphs Γ and Γ containing V and V are connected. Then for each i , consider the closure of the complement ofst(Γ i ) along its boundary. Then Γ is nothing but the connected sum of these twocomplexes. Hence it is decomposable, and therefore the only possibilities are Θ a,b,c defined above. Since vertex-3-connectedness implies the absence of multiple edges,two of a, b and c must be 1. This is a contradiction.Therefore X must be obtained by taking closures several times of an elementarycomplex of dimension 2. However, this case has been treated already in Corol-lary 7.7. This completes the proof of Proposition 7.18. Appendix A. More deformations All the attaching cells we consider previously induce braid equivalences. Howeverwe can consider more deformations which do not induce braid equivalences, butare helpful in computing braid groups. In this section, we introduce several 2-dimensional deformations ι : X → Y such that ι induces a surjection between braidgroups, and try to characterize their kernel. Hence we may use these deformationsto compute B n ( Y ) from B n ( X ) which is already known.For example, by using the deformations we will introduce below, one can deforma trivalent, vertex-3-connected graph into a surface by keeping the cokernel of i ( · ) ,n : H ( · ) → H ( B n ( · )) unchanged. Therefore one may deduce another proof forProposition 7.18. Definition A.1. An embedding ¯ ι : X → Y is a local deformation of type ι : U → V via i : U → X if i ( U ) = st( K ) for some K ⊂ X and Y is the push-out of ι and i . X ¯ ι (cid:47) (cid:47) YU i (cid:79) (cid:79) ι (cid:47) (cid:47) V ¯ i (cid:79) (cid:79) A.1. Attaching a disk at a trivalent vertex. Let v be a trivalent vertex in X which we attach a disk on, and let ¯ D be the result as depicted in Figure 22.Suppose v is not a 1-cut of X . Then attaching a disk is nothing but 3-connectedsum with (cid:98) D , where (cid:98) D is a closure of a disk along 3 points in the boundary. SoTheorem 6.5 and Lemma 7.8 are applicable. We denote X (cid:98) D by X . Proposition A.2. If v is not a 1-cut of X , then the 2-braid groups B ( X ) and B ( X ) are isomorphic via ι ∗ . T = ι −→ = ¯ D Figure 22. Attaching a disk on a trivalent vertex Proof. By Corollary 5.3, B ( (cid:98) D ) (cid:39) B ( D ) ∗ (cid:104) t , t (cid:105) , and B ( D ) (cid:39) B ( T ) (cid:39) Z .Hence the assertion follows from Theorem 6.5. (cid:3) In general, we consider the local deformation ¯ ι : X → Y of type ι : T → ¯ D via i : T → X , and let ¯ ι ∗ : B n ( X ) → B n ( Y ) be the induced homomorphism from ¯ ι . Proposition A.3. The map ¯ ι ∗ is surjective and its kernel ker ¯ ι ∗ is generated by i ∗ (ker ι ∗ ) .Proof. We first regard T and ¯ D as subspaces of R as follows. T = { ( x, || x | ≤ } ∪ { (0 , y ) | − ≤ y ≤ } ,D = { ( x, y ) || x | ≤ , ≤ y ≤ } , ¯ D = T ∪ D.T = ι (cid:47) (cid:47) = ¯ D r (cid:111) (cid:111) Figure 23. A complex ¯ D and the projection r We consider a projection (or strong deformation retract) r of ¯ D onto T thatprojects D to x -axis, and it extends to the projection of Y onto X . Then similarto the proof of Proposition 3.6, r does not induce a map between configurationspaces, and let B r -fail n be the subspace of B n ( Y ) consisting of configurations x that r can not act on.Since B r -fail n is of codimension 1, a path γ ( t ) in B n ( Y ) in general position inter-sects this space finitely many times. Indeed, it happens only if exactly two pointsin γ ( t ) ∈ B n ( Y ) are lying in the ray { ( x , y ) | ≤ y ≤ } for some x . However,it can be homotoped to γ (cid:48) so that x = 0, and homotoped into B n ( X ) by using( − y )-axis as desired.Let f = { f , . . . , f n } : ( D , ∂D ) → ( B n ( Y ) , B n ( X )) be a homotopy disk ingeneral position with respect to B r -fail n . Then a failure locus F = f − ( B r -fail n )is a 1-dimensional subcomplex of D away from ∂D . Indeed, F is a union ofarcs (including circles) which intersect pairwise transversely , and is not necessarilyclosed.Suppose that we travel along a path γ ( t ) in D which passes through one ofarcs of F at t . Then there are i and j such that f i ( γ ( t )) and f j ( γ ( t )) make afailure of r . Moreover, the sign of the difference of x -coordinates of these two paths N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 37 must change near t since f is in general position. That is, two paths f i ( γ ( t )) and f j ( γ ( t )) really make a crossing as usual.Note that there are two types of intersection points between arcs in F , that is,either two arcs or three arcs intersect at one point in D .The former case happens when the failure occurs at two different rays simulta-neously, therefore four points of a configuration are involved. On the other hand,the latter case happens when the failure occurs at a single ray but three points arelying on it simultaneously. See Figure 24. f : pδ p, δ p, qδ q, δ q, −→ B n ( Y ) f ( p ) = f ( δ p, ) = f ( δ p, ) = f ( q ) = f ( δ q, ) = f ( δ q, ) = Figure 24. A failure locus for the projection r and curves near by intersectionsEvidently, the small loops δ ’s around these two kinds of intersections corre-spond to the braid relations such that σ i σ j = σ j σ i for | i − j | > σ i σ i +1 σ i = σ i +1 σ i σ i +1 , which can be considered as images of ker( ι ∗ ).Now we perturb f near δ so that f ( δ ) belongs to B n ( X ) as before. Then r ( f ( δ )) islying in ker( ι ∗ ) by definition. Perturb f further so that all arcs without intersectionpoint disappear in a failure locus. Then the failure locus consists of those nearintersection points, which implies that f ∈ ker(¯ ι ∗ ) is generated by i ∗ (ker( ι ∗ )) bychoosing paths from the basepoint of ∂D to a point in each δ i . (cid:3) Remark A.4. Let ι n ∗ : B n ( T ) → B n ( ¯ D ). Since ker( ι n ∗ ) is generated by twobraid relations which can be considered as 3- or 4-braids, it is generated by theimages of ker( ι ∗ ) and ker( ι ∗ ) under the maps B m ( T ) → B n ( T ) i ∗ −→ B n ( ¯ D ) , for m = 3 or 4. Therefore ker(¯ ι ∗ ) is also generated by these images of them underthe compositions with i ∗ . A.2. Edge-to-band replacement. Let D i be oriented disks, and D ∨ ∂ D = D (cid:116) D / ∗ ∼ ∗ be the boundary wedge sum of D i ’s where ∗ i ∈ ∂D i . By Proposi-tion 3.11, this is braid equivalent to the 1-connected sum D D , joining two disksby an edge. For the sake of convenience, we use the wedge sum instead. Then byTheorem 6.5, B ( D ∨ ∂ D ) = B ( D ) ∗ B ( D ) = (cid:104) σ , , σ , (cid:105) , where σ ,i is the generator for B ( D i ) such that [ σ ,i ] ∈ H ( D i \ { } ) = H ( ∂D i ) (cid:39) H ( D i , ∂D i ) corresponds to the orientation of D i . D ∨ ∂ D = D D Figure 25. A boundary wedge sum of two disksNote that any orientation preserving embedding ι : D ∨ ∂ D → D induces anhomomorphism ι ∗ between braid groups which maps both σ ,i to σ . Thereforethe kernel ker ι ∗ is generated by σ , σ − , . Moreover, even for n ≥ 3, the kernel of B n ( D ∨ ∂ D ) → B n ( D ) is generated by the image of ker ι ∗ under the inclusionmap i ∗ : B ( D ∨ ∂ D ) → B n ( D ∨ ∂ D ) coming from Proposition 2.8. This followseasily from the group presentations for B n ( D ∨ ∂ D ) and B n ( D ).As before, let us consider a local deformation ¯ ι : X → Y of type ι : D ∨ ∂ D → D via i . Note that Y can be regarded intuitively as a blowing-up of X at v ,and there are two possibilities for constructing Y according to the choices of theorientations on D i ’s. Proposition A.5. The kernel ker ¯ ι ∗ is generated by i ∗ (ker ι ∗ ) .Proof. Let a = { v } × [0 , ⊂ X and let r : Y → Y /a = X be the quotientmap. Then we define a failure B r -fail n of r as a subspace of B n ( Y ) consisting ofconfigurations x which intersect a at least twice. Then the codimension of B r -fail n is 2 and therefore any path in B n ( Y ) can be homotoped into B n ( X ) as before.For a disk f , the failure locus F = f − ( B r -fail n ) is a set { z , . . . , z m } of finitepoints in D , and for each z ∈ F , there exists i and j so that { f i ( z ) , f j ( z ) } ⊂ a .Now we consider two subspaces F i = f − i ( a ) and F j = f − j ( a ), whose intersectioncontains z . Then a small loop δ around z can be separated into four pieces asdepicted in Figure 26.Hence by choosing δ close enough to z , we may assume that all but only f i and f j remains stationary. However, the element r ∗ ( { f i ( δ ) , f j ( δ ( t )) } ) in B ( D ∨ ∂ D )is a generator of ker( ι ∗ ). In other words, any z ∈ F corresponds to an element inthe image of ker( ι ∗ ) under B ( D ∨ ∂ D ) → B n ( D ∨ ∂ D ) i ∗ −→ B n ( X ) . (cid:3) N THE STRUCTURE OF BRAID GROUPS ON COMPLEXES 39 δ δ δ δ z zr ( f ( δ )) = r ( f ( δ )) = r ( f ( δ )) = r ( f ( δ )) = Figure 26. A null-homotopic disk and a loop δ near z in a failure locusA.3. Let γ be an embedded loop in ∂X , which isnot homotopically trivial and let Y = X (cid:116) γ D be a capping-off of X along γ .Then X → Y can never be a braid equivalence since their fundamental groups aredifferent.Since γ ⊂ ∂X , γ ∩ br( X ) = ∅ and therefore st( γ ) is a manifold with boundarycontaining an annulus A , where one of whose boundary is γ . We parametrize A as A = { ( x, y ) ∈ R | / ≤ x + y ≤ } , so that the inner boundary ∂ / is precisely γ , and let D be the unit disc in R .Then we have the following commutative diagram X ≡ B (cid:47) (cid:47) Y \ { } ¯ ι (cid:47) (cid:47) YA i (cid:79) (cid:79) ≡ B (cid:47) (cid:47) D \ { } i (cid:79) (cid:79) ι (cid:47) (cid:47) D i (cid:79) (cid:79) which satisfies X \ i ( A ) (cid:39) ( Y \{ } ) \ i ( D \{ } ) (cid:39) Y \ ¯ i ( D ). The braid equivalencesin the left come from Proposition 3.6 and Corollary 3.9. Proposition A.6. The kernel ker(¯ ι ∗ ) of ¯ ι ∗ is generated by the image of γ ∈ π ( A ) ⊂ B n ( A ) under i ∗ .Proof. We consider a subspace B n − ( Y \{ } ; { } ) = B n − ( Y \{ } ) ×{ } of B n ( Y ).Then it is obviously of codimsion 2. Hence we may assume that any path can behomotoped to a path avoiding 0 and any homotopy disk intersects { } finitely manytimes.Moreover, for a homotopy f and z ∈ D with 0 ∈ f ( z ), the image f ( δ ( t )) of asmall enough loop δ ( t ) enclosing z via f is lying in B n ( Y \ { } ) and we may assumethat all but 1 point, say f ( δ ( t )), remains stationary since δ ( t ) can be arbitrarilysmall.On the contrary, the image of f ( δ ( t )) is homotopic to γ or its inverse. Thiscompletes the proof. (cid:3) References [1] A. Abrams, Configuration space of braid groups of graphs , Ph.D. thesis in UC Berkeley,ProQuest LLC, Ann Arbor, MI, 2000.[2] E. 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