Characteristics of Graph Braid Groups
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Abstract
We give formulae for the first homology of the nbraid group and the pure 2braid group over a finite graph in terms of graphtheoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the nbraid group over the graph is torsionfree and the conjectures about the first homology of the pure 2braid groups over graphs in Farber and Hanbury (arXiv:1005.2300 [math.AT]) can be verified. We discover more characteristics of graph braid groups: the nbraid group over a planar graph and the pure 2braid group over any graph have a presentation whose relators are words of commutators, and the 2braid group and the pure 2braid group over a planar graph have a presentation whose relators are commutators. The latter was a conjecture in Farley and Sabalka (J. Pure Appl. Algebra, 2012) and so we propose a similar conjecture for higher braid indices.
Keywords
Braid group Configuration space Graph Homology Presentation1 Introduction
In this article, we assume Γ is a finite connected graph regarded as a Euclidean subspace and we study topological characteristics, in particular their homologies and fundamental groups, of C _{ n } Γ and UC _{ n } Γ via graphtheoretical characteristics of Γ.
Conceiving applications to robotics, Abrams and Ghrist [2] around 2000 began to study configuration spaces over graphs and graph braid groups from the topological point of view. Research on graph braid groups has mainly been concentrated on characteristics of their presentations. An outstanding question was which graph braid group is a rightangled Artin group. The precise characterization of such graphs was given in [12] for n≥5 by extending the result in [8] for trees and n≥4. So it is natural to consider two other classes of groups defined by relaxing the requirement of rightangled Artin groups that have a presentation whose relators are commutators of generators. A simplecommutatorrelated group has a presentation whose relators are commutators, and a commutatorrelated group has a presentation whose relators are words of commutators. Farley and Sabalka proved in [9] that B _{2} Γ is simplecommutatorrelated if every pair of cycles in Γ are disjoint and they conjectured that B _{2} Γ are simplecommutatorrelated whose relators are related to two disjoints cycles if Γ is planar.
On the other hand, Farley showed in [6] that the homology groups of the unordered configuration space UC _{ n } T for a tree T are torsionfree and computed their ranks. Kim, Ko and Park proved that if Γ is nonplanar, H _{1}(UC _{ n } Γ) has a 2torsion and the converse holds for n=2, and they conjectured that H _{1}(B _{ n } Γ) is torsionfree iff Γ is planar [12]. Barnett and Farber showed in [3] that for a planar graph Γ satisfying a certain condition (which implies that Γ is either the Θshape graph or a simple and triconnected graph), β _{1}(C _{2} Γ)=2β _{1}(Γ)+1. Furthermore, Farber and Hanbury showed in [10] that for a nonplanar graph Γ satisfying a certain condition (which also implies that Γ is a simple and triconnected graph), β _{1}(C _{2} Γ)=2β _{1}(Γ). They also conjectured that H _{1}(C _{2} Γ) is always torsionfree and that β _{1}(C _{2} Γ)=2β _{1}(Γ) iff Γ is nonplanar, simple and triconnected (this is equivalent to their hypothesis).
In this article, we express H _{1}(UC _{ n } Γ) and H _{1}(C _{2} Γ) for an finite connected graph Γ in terms of graphtheoretic invariants (see Theorems 3.16 and 3.25). All the results and the conjectures, mentioned above, on the first homologies of configuration spaces over graphs are immediate consequences of these expressions. In addition, we prove that B _{ n } Γ is commutatorrelated for a planar graph Γ and P _{2} Γ is always commutatorrelated (see Theorem 4.6). By combining with a result of [3], we finally prove that for a planar graph Γ, B _{2} Γ and P _{2} Γ are simplecommutatorrelated whose relators are commutators of words corresponding to pairs of disjoint cycles on Γ (see Theorem 4.8).
The major tool for computing H _{1}(UC _{ n } Γ) is to use a Morse complex of UD _{ n } Γ obtained via discrete Morse theory. In Sect. 2, we first give an example that illustrates how to use the Morse complex to compute H _{1}(UC _{ n } Γ). Then we choose a nice maximal tree of Γ and its planar embedding, the second boundary map of the Morse complex induced from these choices becomes so manageable that a description of the second boundary map can be given.
In Sect. 3, the matrix for the second boundary map is systematically simplified (see Theorem 3.5) via row operations after giving certain orders on generating 1cells and 2cells (called critical cells) of the Morse complex. Then we decompose Γ into biconnected graphs and further decompose each biconnected graph into triconnected graphs and compute the contribution from critical 1cells that disappear under these decompositions. Then we show all critical 1cells except those coming from deleted edges are homologous up to signs for a given triconnected graph and generate a summand ℤ or ℤ_{2} depending on whether the graph is planar or not. Finally we collect results from all decompositions to have a formula for H _{1}(UC _{ n } Γ). For n=2, the second boundary map of the Morse complex of D _{ n } Γ is not any harder than the Morse complex of UD _{ n } Γ. Thus the formula for H _{1}(C _{2} Γ) is obtained by a similar argument.
In Sect. 4, we develop noncommutative versions of some of technique in the previous section to obtain optimized presentations of (pure) graph braid groups so that they have certain desired properties via Tietze transformation. In fact, the orders on critical 1cells and 2cells play crucial roles in systematic eliminations of canceling pairs of a 2cell and a 1cell. And we show that (pure) graph braid groups have presentations with special characteristics mentioned above. We finish the paper with the conjecture about a graph Γ such that B _{ n } Γ and P _{3} Γ are simplecommutatorrelated groups.
2 Discrete Configuration Spaces and Discrete Morse Theory
Given a finite graph Γ, the unordered discrete configuration space UD _{ n } Γ is collapsed to a complex called a Morse complex by using discrete Morse theory developed by Forman [11]. In Sect. 2.1, we briefly review this technology following [7, 12] and use it to compute H _{1}(UD _{2} K _{3,3}) as a warmup that demonstrates what is ahead for us. In Sect. 2.2, we extend the technique to the discrete configuration space D _{ n } Γ and compute H _{1}(D _{2} K _{3,3}) as an example. In Sect. 2.3, we show how to choose a nice maximal tree and its embedding so that the second boundary map of the induced Morse complex can be described in the fewest possible cases. Then we list up all of these cases in a few lemmas.
2.1 Discrete Morse Theory on UD _{ n } Γ
A vertex v in an icell c is said to be blocked if for the edge e in T such that ι(e)=v, τ(e) is in c or is an end vertex of another edge in c. Let K _{ i } denote the set of all icells of UD _{ n } Γ and K _{−1}=∅. Define W _{ i }:K _{ i }→K _{ i+1}∪{void} for i≥−1 by induction on i. Let c={c _{1},c _{2},…,c _{ n }} be an icell. If \(c\notin\operatorname{im}(W_{i1})\) and there are unblocked vertices in c and, say, c _{1} is the smallest unblocked vertex, then W _{ i }(c)={vc _{1},c _{2},…,c _{ n }} where the edge vc _{1} is in T. Otherwise, W _{ i }(c)=void. Let K _{∗}=⋃K _{ i }. Define W:K _{∗}→K _{∗}∪{void} by W(c)=W _{ i }(c) for an icell c. Then it is not hard to see that W is welldefined, and each cell in W(K _{∗})−{void} has the unique preimage under W, and there is no cell in K _{∗} that is both an image and a preimage of other cells under W. For example, each arrow on the right of Fig. 2 points from c to W(c) in UD _{2} K _{3,3} and the dashed lines represent 1cells sent to void under W.
Farley and Sabalka in [7] gave an alternative description for these three kinds of cells in UD _{ n } Γ as follows: An edge e in a cell c={c _{1},…,c _{ n−1},e} is orderrespecting if e is not a deleted edge and there is no vertex v in c such that v is adjacent to τ(e) in T and τ(e)<v<ι(e). A cell is critical if it contains neither orderrespecting edges nor unblocked vertices. A cell is collapsible if it contains at least one orderrespecting edge and each unblocked vertex is larger than the initial vertex of some orderrespecting edge. A cell is redundant if it contains at least one unblocked vertex that is smaller than the initial vertices of all orderrespecting edges. Notice that there is exactly one critical 0cell {0,1,…,n−1} by the assumption that there are at least n−1 edges between 0 and the nearest vertex with valency ≥3 in the maximal tree.
A choice of a maximal tree of Γ and its planar embedding determine an order on vertices and in turn a Morse complex UM _{ n } Γ that is homotopy equivalent to UD _{ n } Γ. We wish to compute its homology groups via the cellular structure of UM _{ n } Γ.
Let R:C _{ i }(UD _{ n } Γ)→C _{ i }(UD _{ n } Γ) be a homomorphism defined by R(c)=0 if c is a collapsible icell, by R(c)=c if c is critical, and by R(c)=±∂W(c)+c if c is redundant where the sign is chosen so that the coefficient of c in ∂W(c) is −1. By [11], there is a nonnegative integer m such that R ^{ m }=R ^{ m+1}, and let \(\widetilde{R}=R^{m}\). Then \(\widetilde{R}(c)\) is in M _{ i }(UD _{ n } Γ) and we have a homomorphism \(\widetilde{R} : C_{i}(UD_{n}\varGamma)\to M_{i}(UD_{n}\varGamma)\). Define a map \(\widetilde{\partial}:M_{i}(UD_{n}\varGamma)\to M_{i1}(UD_{n}\varGamma)\) by \(\widetilde{\partial}(c) = \widetilde{R}\partial(c)\). Then \((M_{i}(UD_{n}\varGamma),\widetilde{\partial})\) forms a chain complex. However, the inclusion M _{∗}(UD _{ n } Γ)↪C _{∗}(UD _{ n } Γ) is not a chain map. Instead, consider a homomorphism ε:M _{ i }(UD _{ n } Γ)→C _{ i }(UD _{ n } Γ) defined as follows: For a (critical) icell c, ε(c) is obtained from c by minimally adding collapsible icells until it becomes closed in the sense that for each redundant (i−1)cell c′ in the boundary of every icell summand in ε(c), W(c′) already appears in ε(c). Then ε is a chain map that is a chain homotopy inverse of \(\widetilde{R}\). Thus \((M_{i}(UD_{n}\varGamma ),\widetilde{\partial})\) and (C _{ i }(UD _{ n } Γ),∂) have the same chain homotopy type.
Example 2.1
Since UM _{2} K _{3,3} is a nonorientable surface of nonorientable genus 5 as seen in Fig. 2, we easily see that H _{1}(B _{2} K _{3,3})≅ℤ^{4}⊕ℤ_{2}. However, we want to compute it directly from the chain complex \((M_{i}(UD_{n}K_{3,3}),\widetilde{\partial})\) to demonstrate discrete Morse theory. In fact, H _{1}(B _{ n } K _{3,3})≅H _{1}(B _{2} K _{3,3}) for any braid index n (see Lemma 3.12) and the existence of a 2torsion will be needed later.
Since there is only one critical 0cell, the first boundary map is zero. So the cokernel of the second boundary map is isomorphic to H _{1}(B _{2} K _{3,3}). The free part of H _{1}(B _{2} K _{3,3}) is generated by critical 1cells corresponding to columns that do not contain a pivot (the first nonzero entry in a row). The torsion part of H _{1}(B _{2} K _{3,3}) generated by critical 1cells corresponding to a column contains a pivot that is not ±1. Thus H _{1}(B _{2} K _{3,3})≅ℤ^{4}⊕ℤ_{2}.
2.2 Discrete Morse Theory on D _{ n } Γ
The discrete Morse theory on D _{ n } Γ is similar to that on UD _{ n } Γ except the fact that it uses ordered ntuples instead unordered ntuples.
Let \(\widetilde{K}_{i}\) denote the set of all icells of D _{ n } Γ and \(\widetilde{K}_{1}=\emptyset\). Define \(\widetilde{W}_{i}:\widetilde{K}_{i}\to \widetilde{K}_{i+1}\cup\{\mathrm{void}\}\) for i≥−1 by induction on i. Let o=(c _{1},c _{2},…,c _{ n }) be an icell. If \(o\notin \operatorname{im} (\widetilde{W}_{i1})\) and there are unblocked vertices in o as an entry and, say, c _{ j } is the smallest unblocked vertex, then \(\widetilde{W}_{i}(o)=(c_{1},c_{2},\ldots,v\mbox{}c_{j},\ldots,c_{n})\) where the edge vc _{ j } is in T. Otherwise, \(\widetilde{W}_{i}(o)= \mathrm{void}\). Let \(\widetilde{K}_{*} =\bigcup\widetilde{K}_{i}\). Define \(\widetilde{W}:\widetilde{K}_{*} \to\widetilde{K}_{*}\cup\{\mathrm{void}\}\) by \(\widetilde{W}(o)=\widetilde{W}_{i}(o)\) for an icell o. Then \(\widetilde{W}\) is welldefined and each cell in \(\widetilde{W}(\widetilde{K}_{*})\{\mathrm{void}\}\) has the unique preimage under \(\widetilde{W}\), and there is no cell in \(\widetilde{K}_{*} \) that is both an image and a preimage of other cells under \(\widetilde{W}\).
In order to carry over some of computational results on UD _{ n } Γ to D _{ n } Γ, we introduce a bookkeeping notation. Give an order among vertices and edges of Γ by comparing the number assigned to vertices or terminal vertices of edges. Define a projection ϕ:D _{ n } Γ→S _{ n } by sending o=(c _{1},…,c _{ n }) to the permutation σ such that c _{ σ(1)}<⋯<c _{ σ(n)}. And define a bijection Φ:D _{ n } Γ→UD _{ n } Γ×S _{ n } by Φ(o)=(ρ(o),ϕ(o)). For example, Φ((13,2))=({13,2},id) and Φ((4,35))=({4,35},(1,2)) where id is the identity permutation. The maps \(\widetilde{W}\), ∂, \(\widetilde{R}\), and \(\widetilde{\partial}\) are carried over to K ^{∗}×S _{ n }, C _{∗}(UD _{ n } Γ)×S _{ n }, and M _{∗}(UD _{ n } Γ)×S _{ n } by conjugating with Φ. For example, the ith boundary homomorphism on M _{∗}(UD _{ n } Γ)×S _{ n } is given by \(\varPhi\circ\widetilde{\partial}\circ\varPhi^{1}\). To make the notation more compact, an element (c,σ)∈K ^{∗}×S _{ n } will be denoted by c _{ σ }.
Example 2.2
Let Γ be K _{3,3} and a maximal tree and an order be given as in Fig. 1. We want to compute H _{1}(P _{2} K _{3,3}) which will be used later.
2.3 The Second Boundary Homomorphism
To give a general computation of the second boundary homomorphism \(\widetilde{\partial}\) on a Morse complex, we first exhibit redundant 1cells whose reductions are straightforward and then explain how to choose a maximal tree of a given graph to take advantage of these simple reductions.
Let Γ be a graph and T be a maximal tree of Γ. Let c be a redundant icell in UD _{ n } Γ, v be an unblocked vertex in c and e be the edge in T starting from v. Let V _{ e }(c) denote the icell obtained from c by replacing v by τ(e). Define a function V:K _{ i }→K _{ i } by V(c)=V _{ e }(c) if c is redundant and ι(e) is the smallest unblocked vertex in c, and by V(c)=c otherwise. The function V should stabilize to a function \(\widetilde{V}:K_{i}\to K_{i}\) under iteration, that is, \(\widetilde{V}=V^{m}\) for some nonnegative integer m such that V ^{ m }=V ^{ m+1}.
Lemma 2.3
(Kim–Ko–Park [12])
Let c be a redundant cell and v be an unblocked vertex. Suppose that for the edge e starting from v, there is no vertex w that is either in c or an end vertex of an edge in c and satisfies τ(e)<w<ι(e). Then \(\widetilde{R}(c)=\widetilde{R}V_{e}(c)\).
We continue to define more notation and terminology. For each vertex v in Γ, there is a unique edge path γ _{ v } from v to the base vertex 0 in T. For vertices v, w in Γ, v∧w denotes the vertex that is the first intersection between γ _{ v } and γ _{ w }. Obviously, v∧w≤v and v∧w≤w. The number assigned to the branch of v occupied by the path from v to w in T is denoted by g(v,w). If v=w∧v, g(v,w)≥1 and if v>w∧v, g(v,w)=0. An edge e in Γ is said to be separated by a vertex v if ι(e) and τ(e) lie in two distinct components of T−{v}. It is clear that only a deleted edge can be separated by a vertex. If a deleted edge d is not separated by v, then ι(d), τ(d), and ι(d)∧τ(d) are all in the same component of T−{v}.
For redundant 1cells, we can strengthen the above lemma as follows.
Lemma 2.4
(Special Reduction)
 (a)
Every vertex w in c satisfying τ(e)<w<ι(e) is blocked.
 (b)
If an end vertex w of p satisfies τ(e)<w<ι(e) then p is not separated by τ(e).
Proof
Assume that both ends of p are not between τ(e) and ι(e). Since p is the only edge in c that can initiate a blockage, it is impossible to have a vertex between τ(e) and ι(e) due to the condition (a). Then we are done by Lemma 2.3.
Let W be the set of all 1cells obtained from c _{ ι(p)} replacing vertices in T _{ p } by vertices that are also in T _{ p }. If c′∈W has no unblocked vertex in T _{ p } then c′ is unique because Γ is suitably subdivided. This 1cell is denoted by c _{ p }. If c′∈W has an unblocked vertex in T _{ p }, let u be the smallest unblocked vertex in T _{ p } and e′ be the edge starting from u. Then c′ and u satisfy the hypothesis of Lemma 2.3 since e is the only edge in c′ and every vertex in T−T _{ p } is not between τ(e) and ι(e). So \(\widetilde{R}(c')=\widetilde{R}V_{e'}(c')\). By iterating this argument, we have \(\widetilde{R} (c_{\iota(p)})=\widetilde{R} (c_{p})=\widetilde{R} (c_{\tau(p)})\) because V _{ e′}(c′) is also in the finite set W.
If p is not a deleted edge, then the condition (b) always holds and so c and the smallest unblocked vertex in c satisfy the hypothesis of this lemma. So \(\widetilde{R}(c)=\widetilde{R}V(c)\). By repeating the argument, we have \(\widetilde{R}(c)=\widetilde{R}\widetilde{V}(c)\). □
For an oriented discrete configuration space D _{ n } Γ, the statement corresponding to Lemma 2.3 holds at least for n=2 (see Lemma 3.18), but the statement corresponding to Lemma 2.4 is false in general.
Discrete Morse theory can be powerful in discrete situations but we need to reduce the number of instances to be investigated and the amount of computation involved for each instance. In our situation, it is important to choose a nice maximal tree and its planar embedding. The following lemma make such choices which will be used throughout the article. For example, the Morse complex induced from such choices has the second boundary map describable by using Lemma 2.4.
From now on, we assume that every graph is suitably subdivided, finite, and connected unless stated otherwise. When n=2, it is convenient to additionally assume that each path between two vertices of valency ≠2 in a suitably subdivided graph contains at least two edges.
Lemma 2.5
(Maximal Tree and Order)
 (T1)
The initial vertices of all deleted edges are vertices of valency 2.
 (T2)
Every deleted edge d is not separated by any vertex v such that v<τ(d).
 (T3)
If the kth branch of a vertex v has the property that v separates a deleted edge d and g(v,ι(d))=k, and the jth branch of v does not have the property, then j<k.
Proof
 (I)
Choice of a base vertex 0 on Γ We assign 0 to a vertex v such that v is of valency 1 in Γ or Γ−{v} is connected if there is no vertex of valency 1. This is necessary to make the base vertex have valency 1 in a maximal tree so that there is one critical 0cell.
 (II)Choice of deleted edges We consider a metric on Γ such that each edge is of length 1.Then the order on vertices obtained by any planar embedding p of T satisfies the conditions (T1) and (T2) since the terminal vertices of all deleted edges are of valency ≥3 in Γ.
 (1)
Delete an edge nearest from 0 on a circuit nearest from 0.
 (2)
Repeat (1) until the remainder is a tree T.
 (1)
 (III)Modification of a planar embedding If the order on vertices obtained by p does not satisfy the condition (T3), then there are a vertex A with valency ≥3 on T and branches j of A that violate (T3). The base vertex 0 and branches j do not lie on the same component of Γ−{A}. We slide the components containing branches j over other branches so that every branch of A satisfies (T3) (see Fig. 6). We repeat this process until the induced order satisfies (T3). □
From now on, we assume that we always choose a maximal tree and its embedding as given in Lemma 2.5.
Example 2.6
Proposition 2.7
 (i)
If a _{ m }=b _{ m } for all m>ℓ, then \(\mathbf{A}(\vec{a},\ell )=\mathbf{A}(\vec{b},\ell)\).
 (ii)
If \(p(\vec{a})>\ell\), then \(\mathbf{A}(\vec{a},\ell )\mathbf {A}(\vec{a} 1,\ell)=R(A_{p(\vec{a})}(\vec{a}\vec{\delta}_{p(\vec{a})} + \vec{\delta}_{\ell}))\).
As mentioned above, there are three types of critical 2cells. We will describe the images of each of these three types under \(\widetilde{\partial}\). Since an edge A _{ k } is never separated by any vertex, Lemma 2.5 implies \(\widetilde{\partial}(A_{k}(\vec{a})\cup B_{\ell}(\vec{b}))=0\), which was first proved by Farley and Sabalka in [7]. So we consider the remaining two types. To help grasp the idea behind, examples are followed by general formulae.
Example 2.8
Lemma 2.9
(Boundary Formula I)
Proof
Since A _{ k } is not separated by any vertex, Lemma 2.4 implies \(\widetilde{R}(A_{k}(\vec{a})\cup\dot{B}(\vec{b}))=\widetilde{R}\circ\widetilde{V}(A_{k}(\vec{a})\cup\dot{B}(\vec{b}))\) and \(\widetilde{R}(A_{k}(\vec{a})\cup B(\vec{b})\cup\{\iota(d)\} )=\widetilde{R}\circ\widetilde{V}(A_{k}(\vec{a})\cup B(\vec{b})\cup\{\iota(d)\} )\).
Assume that d is not separated by A. Then \(\widetilde{V}(A_{k}(\vec{a})\cup\dot{B}(\vec{b}))=\widetilde{V}(A_{k}(\vec{a})\cup B(\vec{b})\cup\{ \iota (d)\})\). So we only consider \(\widetilde{R}(\dot{A}(\vec{a})\cup d(\vec{b})A(\vec{a}+\vec{\delta}_{k})\cup d(\vec{b}))\). Let C be the unique largest vertex of valency ≥3 such that C<A. Since d is not separated by any vertex between C and A, Lemma 2.3 implies \(\widetilde{R}(\dot{A}(\vec{a})\cup d(\vec{b}))=\widetilde{R}(C((\vec{a}+1)\vec{\delta}_{g(C,A)})\cup d(\vec{b}))=\widetilde{R}(A(\vec{a}+\vec{\delta}_{k})\cup d(\vec{b}))\). Thus \(\widetilde{\partial}(c)=0\).
Example 2.10
Lemma 2.11
(Boundary Formula II)
Proof
Now assume that d′ is separated by A. If k≠ℓ, then d′ (and d, respectively) is not separated by any vertex other than A on the path between A and ι(d) (and ι(d′)). So we see that \(\widetilde{R}(A(\vec{a})\cup d'(\vec{b})\cup\{\iota(d)\})=\widetilde{R}(A(\vec{a} +\vec{\delta}_{k})\cup d'(\vec{b}))\) and \(\widetilde{R}(d(\vec{a})\cup B(\vec{b})\cup\{\iota(d')\})=\widetilde{R}(d(\vec{a}+\vec{\delta}_{\ell})\cup B(\vec{b}))\).
The remaining part can be proved by the same argument as in the proof of Lemma 2.9. □
To prove that for planar graphs the first homologies of graph braid groups are torsionfree, we need an additional requirement. So we modify Lemma 2.5 for planar graphs as follows.
Lemma 2.12
(Maximal Tree and Order for Planar Graph)
 (T4)
If τ(d′)<τ(d) and g(τ(d),ι(d))=g(τ(d),ι(d′)) then ι(d)<ι(d′).
Proof
 (I)
Choices of a base vertex 0 and a planar embedding We assign 0 to a vertex v such that v is of valency 1 in Γ or Γ−{v} is connected if there is no vertex of valency 1. Choose a planar embedding of Γ such that the base vertex 0 lies in the outmost region. Let T=Γ. Go to Step II.
 (II)
Choice of deleted edges Take a regular neighborhood R of T. As traveling the outmost component of ∂R clockwise from the base vertex until either coming back to 0 or meeting an edge that is on a circuit. If the former is the case, we are done. If the latter is the case, delete the edge and let T be the rest. Repeat Step II.
Suppose that there is a vertex A with valency ≥3 on T and branches of A that violate condition (T3). Step II guarantees that the base vertex 0 and the branches do not lie on the same component of Γ−{A}. So we can modify the planar embedding of Γ as in Lemma 2.5 to make A satisfy condition (T3) while maintaining condition (T4) because the modification neither changes the maximal tree T nor creates the unwanted possibility of Fig. 11(b). □
Example 2.13
Condition (T4) implies that there are no critical 2cells whose boundary images correspond to the case ε=1 in Lemma 2.11. Note that condition (T4) implies that the given graph is planar. Thus a given graph has a maximal tree and an order on vertices satisfies conditions (T1)–(T4) if and only if the graph is planar.
3 First Homologies
We will derive formulae for H _{1}(B _{ n } Γ) and H _{1}(P _{2} Γ) in terms of graphtheoretical quantities. We will characterize presentation matrices for H _{1}(B _{ n } Γ) over bases given by critical 2cells and critical 1cells in Sect. 3.1 and will count the number of relevant critical 1cells in terms of graphtheoretical quantities in Sect. 3.2. A parallel discussion for H _{1}(P _{2} Γ) will be presented in Sect. 3.3.
3.1 Presentation Matrices
A presentation matrix of H _{1}(B _{ n } Γ) is determined by the second boundary homomorphism over bases given by critical 2cells and critical 1cells. We will give orders on critical 1cells and critical 2cells to easily locate pivots and zero rows in the presentation matrices.
The number of critical cells enormously grows in both the size of graph and the braid index. For example, consider K _{5} with braid index 4 and its maximal tree and an order given in Example 2.6. The numbers of critical 1cells of the form \(A_{k}(\vec{a})\) and \(d(\vec{a})\) are 58 and 21. And the numbers of critical 2cells of the form \(A_{k}(\vec{a})\cup B_{\ell}(\vec{b})\), \(A_{k}(\vec{a})\cup d(\vec{b})\) and \(d(\vec{a})\cup d'(\vec{b})\) are 15, 167 and 56. So we have a presentation matrix of the size 238×79. Fortunately, rows of the matrix are highly dependent. The following lemmas illustrates some points of this phenomenon.
Lemma 3.1
(Dependence Among Boundary Images I)
 (1)
\(\widetilde{\partial}(A_{k}(\vec{a})\cup d'(\vec{b}))=\widetilde{\partial}(A_{k}(\vec{a})\cup d')\).
 (2)
\(\widetilde{\partial}(d(\vec{a})\cup d'(\vec{b}))=\widetilde{\partial}(d(\vec{a})\cup d')\) for τ(d)>τ(d′).
Proof
We can observe that the boundary images in Lemmas 2.9 and 2.11 are independent of \(\vec{b}\) and depend only on the initial vertex of the first edge whose terminal vertex is less than ends of the second edge. □
Lemma 3.2
(Dependence Among Boundary Images II)
 (1)If A separates d′ and d″ and g(A,ι(d′))=g(A,ι(d″)), then$$\widetilde{\partial}\bigl(A_k(\vec{a})\cup d'\bigr)= \widetilde{\partial}\bigl(A_k(\vec{a})\cup d'' \bigr). $$
 (2)If τ(d) separates d′ and d″ and g(τ(d),ι(d))≠g(τ(d),ι(d′))=g(τ(d),ι(d″)), then$$\widetilde{\partial}\bigl(d(\vec{a})\cup d'\bigr)=\widetilde{\partial}\bigl(d(\vec{a})\cup d''\bigr). $$
 (3)If τ(d) separates d′ and d″ and g(τ(d),ι(d))=g(τ(d),ι(d′))=g(τ(d),ι(d″)), then$$\widetilde{\partial}\bigl(d(\vec{a})\cup d'd(\vec{a})\cup d''\bigr)=\pm\Bigl(\bigwedge \bigl(d,d'\bigr)\pm\bigwedge \bigl(d,d'' \bigr)\Bigr). $$
Proof
Immediate from Lemmas 2.9 and 2.11. □
Using the lemmas, we can reduce the size of the presentation matrix of H _{1}(B _{4} K _{5}) to 91×79 by ignoring zero rows. We will see that the number of rows is still large compared to the number of pivots. In order to find pivots systematically, we need to order critical cells.
Define the size s(c) of a critical 1cell c to be the number of vertices blocked by the edge in c; more precisely, define \(s(c)=\vec{a}\) for \(c=A_{k}(\vec{a})\) or \(c=d(\vec{a})\). Define the size s(c) of a critical 2cell c to be the number of vertices blocked by the edge in c that has the larger terminal vertex.
We assume that a set of mtuples is always lexicographically ordered in the discussion below. For edges e,e′, Declare e>e′ if e is a deleted edge and e′ is an edge on T or if both are either deleted edges or edges on T and (τ(e),ι(e))>(τ(e′),ι(e′)). The set of critical 1cells c is linearly ordered by triples \((s(c),e,\vec{a})\) where c is given by either \(A_{k}(\vec{a})\) or \(d(\vec{a})\). The following lemma motivates this order.
Lemma 3.3
(Leading Coefficient)
Let c be a critical 2cell containing two edges e and e′ such that τ(e)>τ(e′). Assume that \(\vec{a}\) represent vertices blocked by τ(e) in c. If \(\widetilde{\partial}(c)\ne0\) then the largest summand in \(\widetilde{\partial}(c)\) has the triple \((s(c)+1,e,\vec{a}+\vec{\delta}_{g(\tau(e),\iota(e'))})\). Furthermore, if e is a deleted edge d, then the largest summand is \( d(\vec{a}+\vec{\delta}_{g(\tau(e),\iota(e'))})\) and if e is on T, then the largest summand is \( A_{k}(\vec{a}+\vec{\delta}_{g(\tau(e),\iota(e'))})\) where A=τ(e) and k=g(A,ι(e)).
Proof
By Lemmas 2.9 and 2.11 we see that \(\widetilde{\partial}(c)\) is determined by e, \(\vec{a}\) and τ(e′). Using the order on critical 1cells, it is easy to verify the lemma. □
Lemma 3.3 implies that the second boundary homomorphism \(\widetilde{\partial}\) is represented by a blockuppertriangular matrix over bases of critical 2cells and critical 1cells ordered reversely. In fact, the presentation matrix is divided into blocks by s(c) and each block is further divided into smaller blocks by the value e of 6tuples. The first column of each diagonal block is a vector of −1. The −1 entry at the lower left corner of each diagonal block will be called a pivot and a critical 2cell corresponding to a pivotal row is said to be pivotal. In other word, a pivotal 2cell is the smallest one among all critical 2cells that have the same (up to sign) largest summand in their boundary images. The following lemma says that nonpivotal rows turn into a zero row with few exceptions under row operations.
Lemma 3.4
(Nonpivotal Rows)
Let c be a nonpivotal critical 2cell such that \(\widetilde{\partial}(c)\ne0\). If s(c)≥1, then the row corresponding to c is a linear combination of rows below. If s(c)=0, then the row corresponding to c is either a linear combination of rows below or made into a row consisting of only two nonzero entries that are ±1 by row operations.
Proof
 (I)Assume s(c)≥1 and \(c=d(\vec{a})\cup d'\) Set A=τ(d). We consider the following two cases separately:For case (a), we consider the following boundary image of a linear combination:
 (a)
There is a deleted edge d″ separated by A such that a _{ m }≠0 and m<ℓ for m=g(A,ι(d″));
 (b)
There is no such a deleted edge.
The three terms other than c on the left side of the equation are critical 2cells less than c. So it is sufficient to show that the right side, which will be denoted by R, is a linear combination of boundary images of critical 2cells less than c. The sum R depends on the order among k, ℓ and m. If m≥k then \(\mathbf{A}(\vec{a} +\vec{\delta}_{\ell}\vec{\delta}_{m},m)=\mathbf{A}(\vec{a} +\vec{\delta}_{\ell}\vec{\delta}_{m}+ \vec{\delta}_{k},m)\) and \(\mathbf{A}(\vec{a}\vec{\delta}_{m},m)=\mathbf{A}(\vec{a}\vec{\delta}_{m}+\vec{\delta}_{k},m)\) by Proposition 2.7 and so R=0.$$\begin{aligned} &\widetilde{\partial}\bigl(d(\vec{a})\cup d'd(\vec{a}+\vec{\delta}_\ell\vec{\delta}_m)\cup d''  d(\vec{a} \vec{\delta}_m)\cup d' +d(\vec{a} \vec{\delta}_m)\cup d''\bigr) \\ &\quad =\mathbf{A}(\vec{a} +\vec{\delta}_\ell\vec{\delta}_m,m)\mathbf {A}(\vec{a}\vec{\delta}_m,m) \\ &\qquad {}\bigl\{\mathbf{A}(\vec{a} +\vec{\delta}_\ell\vec{\delta}_m+ \vec{\delta}_k,m) \mathbf{A}(\vec{a}\vec{\delta}_m+\vec{\delta}_k,m)\bigr\}. \end{aligned} $$Since m<ℓ, Proposition 2.7 and Lemma 2.9 imply that for any \(\vec{x}\), To shorten formulae, let \(\vec{b}=\vec{a}\vec{\delta}_{m}+\vec{\delta}_{k}\) and \(\vec{c}=\vec{a}\vec{\delta}_{m}\). If m<k. then If m<k<ℓ, \(\vec{b}\vec{b}_{\ell}=\vec{c} \vec{c}_{\ell}\) by Lemma 2.9. If m<ℓ≤k,since there is a deleted edge d‴ separated by A such that k=g(A,ι(d‴)) by (T3) of Lemma 2.5.$$\widetilde{\partial}\bigl(A_\ell\bigl(\bigl(\vec{c}\vec{c}_\ell\bigr)+ \vec{\delta}_m\bigr)\cup d''' \bigr)=A_\ell\bigl(\bigl(\vec{b} \vec{b}_\ell\bigr)+\vec{\delta}_m\bigr)A_\ell\bigl(\bigl(\vec{c} \vec{c}_\ell\bigr)+\vec{\delta}_m\bigr) $$In case (b), by the assumption there is no deleted edge d″ separated by τ(d) such that g(A,ι(d″))=m<ℓ and x _{ m }≠0 for \(\vec{x}=\vec{a}+\vec{\delta}_{\ell}\) and so there is no critical 2cell with the 6tuple \((s(c),d,\vec{x},m,d'',0)\) such that m<ℓ and A separates d″. If k≠ℓ, c would be pivotal by the assumption on c. So k=ℓ. By Lemma 3.2(3), \(\widetilde{\partial}(d(\vec{a})\cup d'd(\vec{a})\cup d''')=\widetilde{\partial}(d\cup d'd\cup d''')\) where d‴ is the smallest deleted edge such that A separates d‴ and g(A,ι(d‴))=ℓ. Note that \(\vec{a}\ge1\) since s(c)≥1. And d‴<d′ since c is pivotal. Thus we have a desired linear combination.
 (a)
 (II)Assume s(c)≥1 and \(c=A_{k}(\vec{a})\cup d'\) Consider the following cases separately:For cases (a)(i)–(iii), we consider the following boundary image of the linear combination:
 (a)There is a deleted edge d″ separated by A such that g(A,ι(d″))=m<ℓ and one of the following conditions holds:
 (i)
a _{ m }≥1 if k≤m,
 (ii)
a _{ m }≥1 and \(\vec{a}_{k}\ge2\) if m<k≤ℓ,
 (iii)
a _{ m }≥1 and \(\vec{a}_{k}\ge2\) if m<ℓ<k, and
 (iv)
a _{ m }≥1 and \(\vec{a}_{k}=1\) if m<ℓ<k;
 (i)
 (b)
There is no such a deleted edge.
The three terms other than c on the left side of the equation are critical 2cells less than c. Then it is sufficient to show that the right side is a linear combination of boundary images of critical 2cells less than c. We omit the proof since it is similar to case (I)(a).$$\begin{aligned} &\widetilde{\partial}\bigl(A_k(\vec{a})\cup d'A_k( \vec{a} +\vec{\delta}_\ell \vec{\delta}_m)\cup d''  A_k(\vec{a} \vec{\delta}_m)\cup d' +A_k(\vec{a}  \vec{\delta}_m)\cup d''\bigr) \\ &\quad =\mathbf{A}(\vec{a} +\vec{\delta}_\ell\vec{\delta}_m,m)\mathbf {A}(\vec{a}\vec{\delta}_m,m) \\ &\qquad {}\bigl\{\mathbf{A}(\vec{a} +\vec{\delta}_\ell\vec{\delta}_m +\vec{\delta}_k,m) \mathbf{A}(\vec{a}\vec{\delta}_m +\vec{\delta}_k,m)\bigr\}. \end{aligned}$$For case (a)(iv), we consider the following boundary image of the linear combination:where d‴ is a deleted edge separated by A and g(A,ι(d‴))=k. Note that the existence of d‴ is guaranteed by condition (T3) of Lemma 2.5.$$\widetilde{\partial}\bigl(A_k(\vec{a})\cup d' A_k(\vec{a}+\vec{\delta}_\ell \vec{\delta}_m)\cup d'' A_\ell(\vec{a}+\vec{\delta}_\ell\vec{\delta}_k)\cup d''' \bigr)=0 $$We will show that case (b) does not happen. Suppose that there is no deleted edge d″ separated by A such that g(A,ι(d″))=m<ℓ and \(A_{k}(\vec{x}\vec{\delta}_{m})\) is critical for \(\vec{x}=\vec{a}+\vec{\delta}_{\ell}\). So there is no critical 2cell with the 6tuple \((s(c),A_{k}(\vec{\delta}_{k}),\vec{x},m,d'',0)\) such that m<ℓ and A separates d″. Then c would be pivotal since d′ is the smallest among deleted edges d″ separated by A such that g(A,ι(d″))=ℓ.
 (a)
 (III)
Assume s(c)=0 and c=d∪d′ Let k=g(τ(d),ι(d)). Since c is nonpivotal, k=ℓ. By Lemma 3.2(3), \(\widetilde{\partial}(d\cup d'd\cup d'')=\pm(\bigwedge (d,d')\pm\bigwedge (d,d''))\) where d″ is the smallest deleted edge separated by A such that g(A,ι(d″))=ℓ. Note that if d′=d″ then c would be pivotal. This completes the proof. □
We are ready to see the main theorem of this section.
Theorem 3.5
 (1)
it consists of all zeros;
 (2)
there is a ±1 entry that is the only nonzero entry in the column it belongs to;
 (3)
there are only two nonzero entries which are ±1.
Proof
This theorem is merely a restatement of Lemma 3.4 via row operations. A pivotal row corresponding to a pivotal 2cell satisfies (2). A row of the type (3) is produced from the relation \(\widetilde{\partial}(d\cup d''d\cup d')=\pm(\bigwedge (d,d'')\pm\bigwedge (d,d'))\) in the last part of the proof of the previous lemma. Note that neither ⋀(d,d′) nor ⋀(d,d″) correspond to a pivotal column due to Lemma 3.3 and the order among critical 1cells.
Obviously the number of these relations does not depend on braid indices. If Γ is planar, the relation becomes \(\widetilde{\partial}(d\cup d''d\cup d')=\pm(\bigwedge (d,d'')\bigwedge (d,d'))\) by Lemmas 2.11 and 2.12. Therefore two nonzero entries in (3) have opposite signs. □
Further row operations among rows of the type (3) in the theorem may produce new pivots ±2 but if two nonzero entries have opposite signs, all of new pivots are ±1 and so we have the following corollary.
Corollary 3.6
If H _{1}(B _{ n } Γ) has a torsion, it is a 2torsion and the number of 2torsions does not depend on braid indices. For a planar graph Γ, H _{1}(B _{ n } Γ) is torsionfree.
 (i)
pivotal if it corresponds to pivotal columns, which is related to (2);
 (ii)
separating if it corresponds to columns of nonzero entries of (3);
 (iii)
free otherwise.
Clearly, a pivotal 1cell has no contribution to H _{1}(B _{ n } Γ) and a free 1cell contributes a free summand to H _{1}(B _{ n } Γ). To complete the computation of H _{1}(B _{ n } Γ), it is enough to consider the submatrix obtained by deleting pivotal rows and zero rows and deleting pivotal columns and columns of free 1cells. This submatrix will be referred as an undetermined block for H _{1}(B _{ n } Γ) and will be studied in Sect. 3.2. Rows of an undetermined block are of the type (3) and columns correspond to separating 1cells. It will be useful later to have a geometric characterization of pivotal 1cells.
Lemma 3.7
(Pivotal 1Cell)
A critical 1cell c is pivotal if and only if c is either \(A_{k}(\vec{a})\) or \(d(\vec{a})\) such that there is a deleted edge d′ separated by A or τ(d) and a _{ m }≥1 for m=g(A,ι(d′)), and in addition s(c)≥2 when \(c=A_{k}(\vec{a})\).
Proof
By the definition of pivotal 1cell and Lemma 3.3, c is a pivotal 1cell iff there is a critical 2cell whose boundary image has the largest summand c iff s(c)≥2 for \(A_{k}(\vec{a})\) (s(c)≥1 for \(d(\vec{a})\), respectively) and there is a deleted edge d′ separated by A such that the 1cell \(A_{k}(\vec{a}\vec{\delta}_{m})\) (\(d(\vec{a}\vec{\delta}_{m})\), respectively) exits and is critical for m=g(A,ι(d′)). A critical 1cell \(d(\vec{a}\vec{\delta}_{m})\) exits iff a _{ m }≥1. So we are done.
Assume that \(c=A_{k}(\vec{a})\). The “only if” part is now clear. To show the “if” part, consider \(\vec{a}_{k}\) and m. If \(\vec{a}_{k}\ge2\) or \(\vec{a}_{k}=1\) and m≥k, then \(A_{k}(\vec{a}\vec{\delta}_{m})\) is a critical 1cell and we are done. If \(\vec{a}_{k}=1\) and m≤k−1, then a _{ j }≥1 for some j≥k since s(c)≥2. By condition (T3) in Lemma 2.5, there is a deleted edge d″ separated by A such that g(A,ι(d″))=j. Then the largest summand of \(\widetilde{\partial}(A_{k}(\vec{a}\vec{\delta}_{j})\cup d'')\) is c and so c is pivotal. □
We can also have a geometric characterization for a separating 1cell which is clear from the definition of separating 1cells and Lemma 3.2(3).
Lemma 3.8
(Separating 1Cell)
A critical 1cell c is separating if and only if there are three deleted edges such that c is a summand of \(\widetilde{\partial}(d\cup d'd\cup d'')\) such that τ(d)>τ(d′), τ(d)>τ(d″) and g(τ(d),ι(d))=g(τ(d),ι(d′))=g(τ(d),ι(d″)). In fact, c is of the form \(A_{k}(\vec{\delta}_{m})\) such that c=⋀(d,d′) (or ⋀(d,d″), respectively) and deleted edges d and d′ (or d″) are separated by A.
It is now easy to recognize free 1cells. So we can compute H _{1}(B _{ n } Γ) by using the undetermined block after counting the number of free 1cells.
Example 3.9
Suppose a maximal tree and an order is given as in Example 2.6 for the complete graph K _{5}. We want to compute H _{1}(B _{4} K _{5}) which will be needed later.
After putting the undetermined block into a row echelon form, we see that all separating 1cells but A _{2}(1,0) are null homologous and A _{2}(1,0) represents a 2torsion homology class. Thus H _{1}(B _{4} K _{5})≅ℤ^{6}⊕ℤ_{2} and the free part is generated by [d _{ i }] for i=1,…,6.
3.2 First Homologies of Graph Braid Groups
In this section we will discuss how to compute the first integral homology of a graph braid group in terms of graphtheoretic invariants. Our strategy is to decompose a given graph into simpler graphs and to compute the contribution from simpler pieces and from the cost of decomposition. The following example illustrates this strategy.
Example 3.10
Give an order on vertices obtained by traveling a regular neighborhood of the maximal tree T clockwise from 0. There are no pairs of critical 2cells that induce a row satisfying (3) in Theorem 3.5. So there are no separating 1cells. Thus there is no torsion and the rank of H _{1}(B _{3} Γ) is equal to the number of free 1cells. There are 28 free 1cells as follows:
d _{ i } for i=1,2,3,4; \(d_{i}(\vec{a})\) for i=1,2 and \(\vec{a}=(1,0,0,0), (0,1,0,0), (2,0,0,0), (1,1,0,0), (0,2,0,0)\); \(A_{2}(\vec{a})\) for \(\vec{a}=(1,0,0,0), (2,0,0,0), (1,1,0,0)\); \(A_{3}(\vec{a})\) for \(\vec{a}=(1,0,0,0), (0,1,0,0), (2,0,0,0), (1,1,0,0), (0,2,0,0)\); and \(A_{4}(\vec{a})\) for \(\vec{a}=(1,0,0,0), (0,1,0,0), (0,0,1,0), (2,0,0,0), (1,1,0,0), (0,2,0,0)\).
Consequently, H _{1}(B _{3} Γ)≅ℤ^{28}.
In order to formalize this idea, we need some notions and facts from graph theory. A cut of a connected graph is a set of vertices whose removal separates at least a pair of vertices. A graph is kvertexconnected if the size of a smallest cut is ≥k. If a graph has no cut (for example, complete graphs) and the number m of vertices is ≥2, then the graph is defined to be (m−1)vertexconnected. The graph of one vertex is defined to be 1vertexconnected. The “2vertexconnected” and “3vertexconnected” will be referred to as biconnected and triconnected. Let C be a cut of Γ. A Ccomponent is the closure of a connected component of Γ−C in Γ viewed as topological spaces. So a Ccomponent is a subgraph of Γ.
Recall that we are assuming that every graph is suitably subdivided, finite, and connected. A suitably subdivided graph is always simple, i.e has neither multiple edges nor loops, and moreover it has no edge between vertices of valency ≥3. A cut is called a kcut if it contains k vertices. The set of 1cuts of a graph Γ is welldefined and we can decompose Γ into components that are either biconnected or the complete graph K _{2} by iteratively taking Ccomponents for all 1cuts C. This decomposition is unique. The topological types of biconnected components of a given graph do not depend on subdivision. In fact, a subdivision merely affects the number of K _{2} components.
Let C be a 2cut {x,y} of a biconnected graph Γ. We find it convenient to modify each Ccomponent by adding an extra edge between x and y. We refer to this modified Ccomponent as a marked Ccomponent. If a marked Ccomponent has a 2cut C′, we take all marked C′components of the marked Ccomponent. By iterating this procedure, we can decompose a biconnected graph into components that are either triconnected or the complete graph K _{3}. This decomposition is unique for a biconnected suitably subdivided graph (for example, see [5]) and will be called a marked decomposition. The topological types of triconnected components of a given graph do not depend on subdivision. In fact, a subdivision merely affects the number of K _{3} components.
A graph is said to have topologically a certain property if it has the property after ignoring vertices of valency 2. We assume that each component in the above two decompositions is always suitably subdivided by subdividing it if necessary. Then triconnected components in the above decompositions are topologically triconnected. Note that a subdivision of a biconnected graph is again biconnected.
Lemma 3.11
(Decomposition of Connected Graph)
Proof
Assume that Γ has a maximal tree T and an order on vertices as Lemma 2.5. Except the xcomponent containing the base vertex 0, each xcomponent Γ _{ x,i } has new base point x and we maintain the numbering on vertices. Then x is the smallest vertex on each xcomponent not containing the original base vertex 0. Unless A=x, every critical 1cell of the type \(A_{k}(\vec{a})\) can be thought of as a critical 1cell in one of xcomponents by regarding vertices blocked by 0 as vertices blocked by x. Similarly, unless ι(d)=x or τ(d)=x, a deleted edge d does not join distinct xcomponents and so a critical 1cell of the type \(d(\vec{a})\) can be regarded as a critical 1cell in one of xcomponents. Therefore a critical 1cell in UD _{ n } Γ that belongs to none of xcomponents must contain an edge incident to x.
We first claim that the undetermined block for H _{1}(B _{ n } Γ) is a block sum of the undetermined blocks for H _{1}(B _{ n } Γ _{ x,i })’s. A row of an undetermined block is obtained by the boundary image of a critical 2cell of the form d∪d′ (see Lemma 3.8). If two deleted edges d and d′ are in distinct xcomponent, the boundary image is trivial since the terminal vertex of one edge cannot separate the other. Thus both d and d′ are in the same xcomponent and so each separating 1cell for UD _{ n } Γ must be a separating 1cell for exactly one of xcomponents.
The proof is completed by counting the number of free 1cells that cannot be regarded as those in any one of xcomponents. Let m be the valency of x in the maximal tree. Then μ≤m. Recall that branches incident to x are numbered by 0,1,…,m−1 clockwise starting from the 0th branch pointing the base vertex 0. The ith and the jth branches do not belong to the same xcomponent for 1≤i, j≤μ−1 by (T2) of Lemma 2.5. When μ≤m−1, the ith and the 0th branches belong to the same xcomponent for μ≤i≤m−1 by condition (T3) of Lemma 2.5. For 1≤i≤μ, let Γ _{ x,i } denote the xcomponent containing the ibranch. Then the xcomponent Γ _{ x,μ } contains the μth to the (m−1)st branches and the 0th branch.
 (a)
1≤k≤μ−1 and \(\vec{a}=\vec{a}_{\mu}\),
 (b)
1≤k≤μ−1 and \(\vec{a}>\vec{a}_{\mu}\),
 (c)
μ≤k≤m−1 and \(\vec{a}=\vec{a}_{\mu}\),
 (d)
μ≤k≤m−1 and \(\vec{a}>\vec{a}_{\mu}\).
The above lemma decomposes the first homology of a graph braid group into the first homologies of graph braid groups on biconnected components together with a free part determined by the valency and the number of xcomponent of each 1cut x. Since N(n,Γ,x)=0 for a 1cut x of valency 2 and UD _{ n }(Γ) is contractible if Γ is topologically a line segment, this decomposition of H _{1}(B _{ n } Γ) is independent of subdivision. Farley obtained a similar decomposition in [6] when Γ is a tree.
Lemma 3.12
For a biconnected graph Γ and n≥2, H _{1}(B _{ n } Γ)≅H _{1}(B _{2} Γ).
Proof
A sequence of vertices starting from the base vertex in a critical cell can be ignored to give a corresponding critical cell for a lower braid index. So a critical 1cell with s(c)≤1 in UD _{ n } Γ can be regarded as a critical 1cell in UD _{2} Γ. An undetermined block involves only critical 2cells with s(c)=0 and critical 1cells with s(c)=1 and so it is welldefined independently of braid indices ≥2.
It is now sufficient to show that every critical 1cell c with s(c)≥2 is pivotal. To show that a critical 1cell \(A_{k}(\vec{a})\) with \(\vec{a}\ge2\) is pivotal, we need to find a deleted edge satisfying Lemma 3.7. Suppose there is no deleted edge d′ such that A separate d′ and g(A,ι(d′))=g(A,v) for the second smallest vertex v blocked by A. By Lemma 2.5 (T2), τ(d′)<A. This means that the vertex A disconnects the g(A,v)th branch of A from the rest of Γ. This contradicts the biconnectivity of Γ.
For a critical 1cell \(d(\vec{a})\) with \(\vec{a}\ge2\), let v be the smallest vertex blocked by τ(d). Then we can argue similarly to show that \(d(\vec{a})\) is pivotal. □
For the sake of the previous lemma, it is enough to consider 2braid groups for biconnected graphs in order to compute nbraid groups.
Lemma 3.13
Proof
There is a natural graph embedding \(f':\widehat{\varGamma}'\to \varGamma\) by sending the extra edge to a path from y to 0 via the ν′th branch of y after suitable subdivision. Then the delete edge d _{0} is sent to one of d _{ i }’s. Also there is a natural graph embedding \(f'':\widehat{\varGamma}''\to\varGamma\) by sending the extra edge to the path from 0 to y in the maximal tree of Γ after subdivision. Both f′ and f″ are orderpreserving. It is easy to see that f″ induces a bijection between critical 1cells of \(UD_{2}\widehat{\varGamma}''\) and those of UD _{2} Γ and it preserves the types of critical 1cells: pivotal, free or separating. Thus the induced homomorphism \(f_{*}'':H_{1}(B_{2}\widehat{\varGamma}'')\to H_{1}(B_{2}\varGamma)\) is injective. Every critical 2cell in UD _{2} Γ is of the form d∪d′. If a critical 2cell d∪d′ is in neither UD _{2}(f′(Γ′)) nor UD _{2}(f″(Γ″)) then both deleted edges are not simultaneously in the same image under f′ or f″ and so \(\widetilde{\partial}(d\cup d')=0\) by Lemma 2.11. Thus the induced homomorphisms \(f_{*}':H_{1}(B_{2}\widehat{\varGamma}')\to H_{1}(B_{2}\varGamma)\) and \(f_{*}'':H_{1}(B_{2}\widehat{\varGamma}'')\to H_{1}(B_{2}\varGamma)\) are injective. Moreover, it is clear that \(\operatorname{im}(f_{*}')\cap\operatorname{im}(f_{*}'')\) is isomorphic to ℤ generated by \(f_{*}'([d_{0}])\).
We are done if we show \(\operatorname{im}(f_{*}')+ \operatorname{im}(f_{*}'')=H_{1}(B_{2}\varGamma )\). Set A=y. There are the following two types of 1cells in UD _{2} Γ that are neither in UD _{2}(f′(Γ′)) nor in UD _{2}(f″(Γ″)): \(d(\vec{\delta}_{m})\) for τ(d)=A and 1≤m<ν′≤g(A,ι(d))≤ν−1 or \(A_{k}(\vec{\delta}_{m})\) for 1≤m<ν′≤k≤ν−1. Since Γ′ is an {x,y}component, for each mth branch of A such that 1≤m<ν′ there is a deleted edge d′ separated by A satisfying τ(d′)>0 and g(A,ι(d′))=m and so \(d(\vec{\delta}_{m})\) are pivotal and so it vanishes in H _{1}(B _{2} Γ).
Let Θ _{ m } be the graph consisting of two vertices and m edges between them. For example, Θ _{3} is the letter shape of Θ.
Lemma 3.14
(Decomposition of Biconnected Graph)
Proof
Note that Θ _{ m } for m≥3 only occurs as a marked complementary graph and it never appears in a marked decomposition of a simple biconnected graph by 2cuts. We can repeatedly apply Lemma 3.14 to each marked 2cut component unless it is topologically a circle and end up with the problem how to compute H _{1}(B _{2} Γ) for a topologically triconnected graph Γ. Note that topologically triconnected components of a given biconnected graph are topologically simple since we assume that graphs are suitably subdivided.
Lemma 3.15
(Topologically Simple Triconnected Graph)
Proof
We use induction on the number s of vertices of valency ≥3. To check for the smallest triconnected graph K _{4}, consider the maximal tree of K _{4} and the order on vertices given in Fig. 12. Then it is easy to see that the lemma is true and in fact H _{1}(B _{2} K _{4})≅ℤ^{4}.
Let X be the second smallest vertex of valency ≥3 in Γ _{ r−1}. If there are topologically double edges between 0 and X in Γ _{ r−1}, then X has valency 3 in T _{ r−1}. Other vertices with topological double edges are situated in Γ _{ r−1} like Y in Fig. 17. Vertices of the types X or Y behave in the same way in both Γ _{ r−1} and Γ. So there is a onetoone correspondence between the set of all critical 1cells of the form \(A_{k}(\vec{\delta}_{\ell})\) in UD _{2} Γ _{ r−1} and the set of those in UD _{2} Γ.
By the induction hypothesis, all critical 1cells of the form \(A_{k}(\vec{\delta}_{\ell})\) in \(UD_{2}\varGamma'_{r1}\) are separating and homologous up to signs. We first find out which critical 1cell of the form \(A_{k}(\vec{\delta}_{\ell})\) in UD _{2} Γ _{ r−1} is not separating. It is enough to check for the vertices of the type either X or Y since \(A_{k}(\vec{\delta}_{\ell})\) can be regarded as a critical 1cell in \(UD_{2}\varGamma'_{r1}\) for all other vertices A and UD _{2} Γ _{ r−1} has more critical 2cells than \(UD_{2}\varGamma'_{r1}\). For X, there is only one critical 1cell \(X_{2}(\vec{\delta}_{1})\) and it is not separating by Lemma 3.8. Suppose the pth and the (p+1)st branches of Y are topological double edges from Y to 0. Then \(Y_{p+1}(\vec{\delta}_{p})\) is not separating either by Lemma 3.8. Unless k=p+1 and ℓ=q, \(Y_{k}(\vec{\delta}_{\ell})\) is separating because one of the pth and the (p+1)st branches lies on \(\varGamma'_{r1}\) and so \(Y_{k}(\vec{\delta}_{\ell})\) is homologous up to signs to other separating 1cells by the induction hypothesis.
Finally we show that critical 1cells of UD _{2} Γ, \(X_{2}(\vec{\delta}_{1})\) and \(Y_{p}(\vec{\delta}_{q})\) are separating and homologous up to signs to other separating 1cells. Let d _{1} (d _{2}, respectively) be a deleted edge lying on the topological edge between 0 and X (Y, respectively) in Γ, d _{3} be a deleted edge lying on the topological edge between x _{1} and Y in Γ. Then g(Y,ι(d _{3})) and g(Y,ι(d _{2})) correspond to p and p+1. Since Γ is topologically triconnected, there is a deleted edge d other than d _{1}, d _{2} and d _{3} such that τ(d) is either 0 or x _{1}. Otherwise, {X,Y} would be a 2cut in Γ. In fact, Fig. 17 shows examples of d _{1}, d _{2}, d _{3}, and d. Consider the following boundary images on the Morse chain complex of UD _{2} Γ:
For a connected graph Γ, define \(N_{2}(\varGamma)=\sum_{i=1}^{k} N_{2}(\varGamma_{i})\) where Γ _{1},…,Γ _{ k } are biconnected components of Γ.
For a connected graph Γ, let N _{3}(Γ) (\(N'_{3}(\varGamma)\), respectively) be the number of triconnected components of Γ that are planar (nonplanar, respectively).
Theorem 3.16
Proof
It seems difficult to compute higher homology groups of B _{ n } Γ in general. However, UD _{2} Γ is a 2dimensional complex and so H _{2}(B _{2} Γ) is torsionfree. And the second Betti number of B _{2} Γ is given as follows:
Corollary 3.17
Proof
3.3 The Homologies of Pure Graph 2Braid Groups
In Sect. 2.2, we describe a Morse chain complex M _{ n } Γ of D _{ n } Γ. The technology developed for UD _{ n } Γ in this article is not enough to compute H _{1}(P _{ n } Γ). For example, the boundary image of \((A_{k}(\vec{a})\cup B_{\ell}(\vec{b}))_{\sigma}\) never vanishes in M _{ n } Γ for n≥4. However, for braid index 2 the second boundary map behaves in the way similar to unordered cases. This is because there are only one type critical 2cells (d∪d′)_{ σ }.
In general, the image of c _{ σ } under \(\widetilde{R}\) or \(\widetilde{\partial}\) is obtained by right multiplication by σ on the permutation subscript of each term in the image of c _{id}. For example, if \(\widetilde{R}(c_{\mathrm{id}})=\sum_{i} (c_{i})_{\tau_{i}}\) then \(\widetilde{R}(c_{\sigma})=\sum_{i} (c_{i})_{\tau_{i}\sigma}\). Thus we only consider c _{id}. We will discuss 2braid groups in this section and ρ denotes the nontrivial permutation in S _{2}.
We have the following lemma for D _{2} Γ that is similar to Lemma 2.3 for UD _{ n } Γ but it is hard to have a lemma corresponding to Lemma 2.4.
Lemma 3.18
(Special Reduction)
Suppose a redundant 1cell c _{id} in D _{2} Γ has a simple unblocked vertex. Then \(\widetilde{R}(c_{\mathrm{id}})=\widetilde{R}(V(c)_{\mathrm{id}})\).
Proof
We use induction on i such that R ^{ i }(c _{id})=R ^{ i+1}(c _{id}). Since c _{id} is redundant, i≥2. Since \(V(\{e_{v},\iota(e)\}_{\rho^{m}})=\{e_{v},\tau(e)\}_{\rho^{2m}}\), \(\widetilde{R}V(\{e_{v},\iota(e)\}_{\rho^{m}})=\widetilde{R}(\{e_{v},\tau (e)\}_{\rho^{2m}})\) by induction hypothesis. Thus \(\widetilde{R}(\{e_{v},\iota(e)\}_{\rho^{m}}\{e_{v},\tau(e)\}_{\mathrm{id}})=\widetilde{R}V(\{e_{v},\iota(e)\}_{\rho^{m}})\widetilde{R}(\{e_{v},\tau (e)\}_{\mathrm{id}})=0\). Thus \(\widetilde{R}(c_{\mathrm{id}})=\widetilde{R}(V(c)_{\mathrm{id}})\). □
Since all critical 2cells in D _{ Γ } is of the form (d∪d′)_{ σ }, we only need the following:
Lemma 3.19
(Boundary Formulae)
 (a)If d′ is separated by τ(d), k≠ℓ and ι(d)<ι(d′) then$$\widetilde{\partial}(c_{\mathrm{id}})=d_{\mathrm{id}}d(\vec{\delta}_\ell)_\rho. $$
 (b)If d′ is separated by τ(d), k=ℓ and ι(d)<ι(d′) then$$\widetilde{\partial}(c_{\mathrm{id}})=d_{\mathrm{id}}d(\vec{\delta}_\ell)_\rho \bigwedge\bigl(d,d' \bigr)_{\mathrm{id}}. $$
 (c)If d′ is separated by τ(d) and ι(d′)<ι(d) then$$\widetilde{\partial}(c_{\mathrm{id}})=d_{\mathrm{id}}d(\vec{\delta}_\ell)_\rho +\bigwedge\bigl(d,d' \bigr)_\rho. $$
 (d)
Otherwise \(\widetilde{\partial}(c_{\mathrm{id}})=0\).
Proof
It is sufficient to compute images under \(\widetilde{R}\) for each boundary 1cell after obtaining the boundary of c _{id} in D _{2} Γ.
Using the above lemma, we have the following lemma similar to Lemma 3.2.
Lemma 3.20
(Dependence Among Boundary Images)
 (1)
If g(τ(d),ι(d))≠g(τ(d),ι(d _{1})), then \(\widetilde{\partial}((d\cup d_{1})_{\mathrm{id}})=\widetilde{\partial}((d\cup d_{2})_{\mathrm{id}})\).
 (2)If g(τ(d),ι(d))=g(τ(d),ι(d _{1})), then$$\widetilde{\partial}\bigl((d\cup d_1)_{\mathrm{id}}(d\cup d_2)_{\mathrm{id}}\bigr)=(1)^i \wedge(d,d_1)_{\rho^i}+(1)^j\wedge (d,d_2)_{\rho^j}. $$
Note that the second formula of the above lemma contains i,j only for the parity purpose and plays an important role of showing that H _{1}(P _{2} Γ) is torsionfree.
Declare an order on S _{2} by id>ρ. Recall the orders on critical 1cells and critical 2cells of UD _{ n } Γ from Sect. 3.1. By adding a permutation as the last component of the orders, we obtain orders given by 4tuples \((s(c),e,\vec{a},\sigma)\) for critical 1cells in D _{2} Γ and by 7tuples \((s(c),e,\vec{a}+\vec{\delta}_{g(\tau(e),\iota(e'))},g(\tau(e),\iota(e')), e',\vec{b},\sigma)\) for critical 2cells.
The second boundary homomorphism \(\widetilde{\partial}\) is represented by a matrix over bases of critical 2cells and critical 1cells ordered reversely. We go through the exactly same arguments as in Sect. 3.1 by using Lemmas 3.20 and 3.19 and obtain the following theorem:
Theorem 3.21
 (1)
it consists of all zeros;
 (2)
there is a ±1 entry that is the only nonzero entry in the column it belongs to;
 (3)
there are only two nonzero entries which are ±1.
 (3′)

there are two nonzero entries which are ±1 and have opposite signs.
Proof
Lemma 3.20 (2) implies that (3′) can be achieved by choosing a basis of critical 1cells in which \((1)^{m} c_{\rho^{m}}\) is used instead of just c _{id} or c _{ ρ }. □
Since there are exactly two critical 0cells, the 0th skeleton (M _{2} Γ)^{0} of a Morse complex of M _{2} Γ of D _{2} Γ consists of two points. Then the second boundary homomorphism gives a presentation matrix for H _{1}(M _{2} Γ,(M _{2} Γ)^{0}). And H _{1}(M _{2} Γ,(M _{2} Γ)^{0})≅H _{1}(M _{2} Γ)⊕ℤ≅H _{1}(P _{2} Γ)⊕ℤ.
Critical 1cells of D _{2} Γ can be classified to be pivotal, free, or separating as before. The undetermined block of separating 1cells produces no torsion due to the property (3′) and so we have the following:
Corollary 3.22
For a finite connected graph Γ, H _{1}(P _{2} Γ) is torsionfree.
Using free 1cells and the undetermined block for H _{1}(M _{2} Γ,(M _{2} Γ)^{0}), we can compute H _{1}(P _{2} Γ).
Example 3.23
Let Γ be K _{5} and a maximal tree and an order be given as in Example 2.6. We want to compute H _{1}(P _{2} Γ).
For a free 1cell c in UD _{2} Γ, c _{id} and c _{ ρ } are free 1cells in D _{2} Γ. So it is easy to modify Lemmas 3.11 and 3.12 for H _{1}(M _{2} Γ,(M _{2} Γ)^{0}) accordingly and one can verify that the contribution by N _{1}(2,Γ) and N _{2}(Γ) doubles because the number of free 1cells doubles. However, the proof of Lemma 3.15 deals with the undetermined block and it is safe to redo.
Lemma 3.24
(Topologically Simple Triconnected Graph)
Proof
We need to show that \(H_{1}(M_{2}\varGamma,(M_{2}\varGamma)^{0})\cong\mathbb{Z}^{2\beta _{1}(\varGamma)+\epsilon+1}\). Critical 1cells are of the forms d _{ σ }, d(δ _{ ℓ })_{ σ } and \(A_{k}(\vec{\delta}_{\ell})_{\sigma}\) with k>ℓ. It is easy to see that every critical 1cell of the form d(δ _{ ℓ })_{ σ } is pivotal and the number of critical 1cells of the form d _{ σ } is equal to 2β _{1}(Γ). We consider the undetermined block. From the proof of Lemma 3.15, there are at most two homology classes of the form \([A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}]\) and \([A_{k}(\vec{\delta}_{\ell})_{\rho}]\). So it is sufficient to show that \([A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}]\ne[A_{k}(\vec{\delta}_{\ell})_{\rho}]\) if Γ is planar and \([A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}]=[A_{k}(\vec{\delta}_{\ell})_{\rho}]\) if Γ is nonplanar. If Γ is planar, then by condition (T4) in Lemma 2.12, there is no row representing \(\pm\{A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}+B_{k'}(\vec{\delta}_{\ell'})_{\rho}\}\). So \([A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}]\ne [A_{k}(\vec{\delta}_{\ell})_{\rho}]\). For nonplanar graphs, we only need to verify for K _{5} and K _{3,3} as was explained in the proof of Lemma 3.15. Examples 2.2 and 3.23 show that H _{1}(P _{2} K _{5}) and H _{1}(P _{2} K _{3,3}) satisfy the lemma. □
Theorem 3.25
4 Applications and More Characteristics of Graph Braid Groups
In this section we first discuss consequences of the formulae obtained in the previous section. Then we develop a technology for graph braid groups themselves that is parallel to the technology successfully applied for the first homologies of graph braid groups. And we discover more characteristics of graph braid groups and pure braid groups beyond their homologies. These characteristics are defined by weakening the requirement for rightangled Artin groups.
4.1 Planar and Nonplanar Triconnected Graphs
Since we are not interested in trivial graphs such as a topological line segment of a topological circle, we assume Γ has at least a vertex of valency ≥3 in this discussion. For any 1cut x of valency ≥3, N(n,Γ,x)>0. Thus N _{1}(n,Γ)=0 if and only if there is no 1cut of valency ≥3 if and only if Γ is biconnected. If N _{2}(Γ)=0 for a biconnected graph Γ, then μ({x,y})=2 for every 2cut {x,y}. If Γ has multiple edges between vertices x and y after ignoring vertices with valency 2, then μ({x,y})>2 for some 2cut {x,y} and so N _{2}(Γ)>0. Thus if N _{2}(Γ)=0 for a biconnected graph, then Γ is topologically simple. If \(N_{3}(\varGamma)=N_{3}'(\varGamma)=0\), then Γ does not topologically contain the complete graph K _{4} by the construction of triconnected graphs. If N _{1}(n,Γ)=N _{2}(Γ)=0 and \(N_{3}(\varGamma )+N_{3}'(\varGamma)=1\), then Γ is topologically simple and triconnected.
 (i)
the closure of every domain \(\bar{U}_{i>0}\) is contractible and \(\bar{U}_{0}\) is homotopy equivalent to S ^{1}, and
 (ii)
for every i,j∈{1,…,r}, \(\bar{U}_{i}\cap\bar{U}_{j}\) is connected,
Condition (i) implies that Γ has no 1cut. Condition (ii) implies that either Γ is the Θshape graph if V(Γ)=2 or Γ has neither multiple edges nor 2cuts if V(Γ)>2. So the hypotheses imply that Γ is either the Θshape graph or a planar simple triconnected graph. Thus Theorem 3.25 covers this result. Furthermore, for any planar graph Γ, β _{1}(P _{2} Γ)=2β _{1}(Γ)+1 if and only if N _{1}(2,Γ)+N _{2}(Γ)+N _{3}(Γ)=1 and \(N_{3}'(\varGamma)=0\). There are three nonnegative solutions: (N _{1}(2,Γ),N _{2}(Γ),N _{3}(Γ))=(1,0,0), (0,1,0) and (0,0,1).
 (i)
Γ _{1} is either K _{5} or K _{3,3} and Γ _{ r }=Γ;
 (ii)
for 1≤i≤r−1, Γ _{ i+1} is obtained by adding an edge with ends {x,y} to Γ _{ i } such that the complement Γ _{ i }−{x,y} is connected where x and y are points in Γ _{ i };
The above construction obviously produces a nonplanar, simple and triconnected graph Γ. Then N _{1}(2,Γ)=N _{2}(Γ)=N _{3}(Γ)=0 and \(N_{3}'(\varGamma)=1\) and so Theorem 3.25 contains this result. Moreover, they conjectured that Γ is nonplanar and triconnected (this is equivalent to their hypothesis) if and only if β _{1}(D _{2} Γ)=2β _{1}(Γ) and H _{1}(P _{2} Γ) is torsionfree. The same theorem also verifies this conjecture. Theorem 3.25 implies that β _{1}(P _{2} Γ)=2β _{1}(Γ) if and only if \(2N_{1}(2,\varGamma)+2N_{2}(\varGamma)+2N_{3}(\varGamma )+N_{3}'(\varGamma )=1\). There is only one nonnegative solution N _{1}(2,Γ)=N _{2}(Γ)=N _{3}(Γ)=0 and \(N_{3}'(\varGamma)=1\) for the equation. Thus β _{1}(P _{2} Γ)=2β _{1}(Γ) if and only if the graph Γ is nonplanar, topologically simple and topologically triconnected.
4.2 Graph Braid Groups and CommutatorRelated Groups
A group G is commutatorrelated if it has a finite presentation 〈x _{1},…,x _{ n }∣r _{1},…,r _{ m }〉 such that each relator r _{ j } belongs to the commutator subgroup [F,F] of the free group F generated by x _{1},…,x _{ n }. We will prove that planar graph braid groups and pure graph 2braid groups are commutatorrelated groups.
Since the abelianization of a given group G is the first homology of G, we have the following.
Proposition 4.1
Let G be a group such that H _{1}(G)≅ℤ^{ m }. If G has a finite presentation with mgenerators, then G is commutatorrelated.
Let Γ be a planar graph. Since UD _{ n } Γ is a finite complex, B _{ n } Γ has a finite presentation. To prove that B _{ n } Γ is a commutatorrelated group, it is sufficient to show that there is a finite presentation with m generators for B _{ n } Γ for m=β _{1}(UD _{ n } Γ).
The braid group B _{ n } Γ is given by the fundamental group of a Morse complex UM _{ n } Γ of UD _{ n } Γ. Thus B _{ n } Γ has a presentation whose generators are critical 1cells and whose relators are boundary words of critical 2cells in terms of critical 1cells. On the other hand, the computation using critical 1cells and critical 2cells in a Morse complex M _{ n } Γ of D _{ n } Γ does not give P _{ n } Γ since there are n! critical 0cells and critical 1cells between distinct critical 0cells are also treated as generators. Instead it gives π _{1}(M _{ n } Γ/∼) where M _{ n } Γ/∼ is the quotient obtained by identifying all critical 0cells.
Even though discrete Morse theory can apply to D _{ n } Γ for any braid index n, we have not reached a level of sophistication sufficient to make good use due to obstacles explained in Sect. 3.3. For n=2, π _{1}(M _{2} Γ/∼)=P _{2} Γ∗ℤ. In fact, M _{2} Γ/∼ is homotopy equivalent to the wedge product of M _{2} Γ and S ^{1} under a homotopy sliding one critical 0cell to the other along a critical 1cell and therefore a presentation of P _{2} Γ is obtained from that of π _{1}(M _{2} Γ/∼) by killing any one of critical 1cells joining two 0cells in the Morse complex M _{2} Γ, for example, a critical 1cell of the form \(A_{k}(\vec{\delta}_{\ell})_{\mathrm{id}}\). Thus it is enough to show that π _{1}(M _{2} Γ/∼) is a commutatorrelated group.
By rewriting the boundary word of a critical 2cell in terms of critical 1cells, it is possible to compute a presentation of B _{ n } Γ (or π _{1}(D _{ n } Γ/∼), respectively) using a Morse complex of UD _{ n } Γ (or D _{ n } Γ). However, the computation of \(\tilde{r}\) is usually tedious and the following lemma somewhat shortens it.
Lemma 4.2
(Kim–Ko–Park [12])
Let c be a redundant 1cell and v be an unblocked vertex. Suppose that for the edge e starting from v, there is no vertex w that is either in c or an end vertex of an edge in c and satisfies τ(e)<w<ι(e). Then \(\tilde{r}(c)=\tilde{r}(V_{e}(c))\) where V _{ e }(c) denotes the 1cell obtained from c by replacing ι(e) by τ(e).
Example 4.3
We show that B _{3} Θ _{4} and P _{3} Θ _{4} are surface groups. These will serve counterexamples later.
In fact, UD _{3} Θ _{4} is an orientable closed surface of genus 3. So we can see that its sixfold cover D _{3} Θ _{4} is an orientable surface of genus 13 by considering Euler characteristics.
The rewriting algorithm seems exponential in the size of graphs. Fortunately, we need not precisely compute the boundary word of a critical 2cell since we are only interested in the number of generators and how to eliminate generators via Tietze transformations. We use the technique developed in Sect. 3.1 for UD _{ n } Γ and the parallel technique developed in Sect. 3.3 for D _{2} Γ. Recall that the orders on critical 1cells an on critical 2cells were important ingredients for the techniques. Using the presentation matrices for H _{1}(B _{ n } Γ) or H _{1}(D _{2} Γ,(D _{2} Γ)^{0}) over bases of 2cells and 1cells ordered reversely, critical 1cells were classified into pivotal, free, and separating 1cells.
Lemma 4.4
(Elimination of Pivotal 1Cells)
Assume that Γ has a maximal tree and an order according to Lemma 2.5. Then B _{ n } Γ and π _{1}(D _{2} Γ/∼) are generated by free and separating 1cells.
Proof
There is no difference between B _{ n } Γ and π _{1}(D _{2} Γ/∼) in our argument. We discuss only B _{ n } Γ. The proof for π _{1}(D _{2} Γ/∼) is exactly the same except the fact that permutations are used as subscripts to express critical cells.
Consider pairs (c _{2},c _{1}) of a pivotal 2cell c _{2} that produces a pivotal 1cell c _{1}. Then either s(c _{1})≥2 or c _{1} is of the form \(d(\vec{\delta}_{\ell})\). In Sect. 3.1, a pivotal 1cell c _{1} is the largest summand of \(\widetilde{\partial}(c_{2})\) and so is not a summand of \(\widetilde{\partial}(c'_{2})\) for a pivotal 2cell \(c'_{2}<c_{2}\). We want to obtain the corresponding noncommutative version.
We need to slightly modify the order on critical 1cells when only pivotal 1cells are compared. For an edge e in Γ, set t(e)=0 if e is in the maximal tree T and t(e)=1 otherwise. Declare e>e′ if (τ(e),t(e),ι(e))>(τ(e′),t(e′),ι(e′)). The set of pivotal 1cells are linearly ordered by the triple \((s(c),e,\vec{a})\) under the modified order on edges. We modified the order on the set of pivotal 2cells accordingly, that is, \(c_{2}>c'_{2}\) for pivotal 2cells c _{2} and \(c'_{2}\) if \(c_{1}>c'_{1}\) when (c _{2},c _{1}) and \((c'_{2},c'_{1})\) are pairs of a pivotal 2cell and the corresponding pivotal 1cell.
 (a)
c _{1} appears in the word \(\tilde{r}\circ\partial_{w}(c_{2})\) exactly once (as a letter or the inverse of a letter).
 (b)
Under the order defined above, c _{1} is the largest of pivotal 1cells appeared in \(\tilde{r}\circ\partial_{w}(c_{2})\)
Note that (b) implies that c _{1} does not appear in \(\tilde{r}\circ \partial_{w}(c'_{2})\) for any pivotal 2cell \(c'_{2}<c_{2}\). Then, via Tietze transformations, we can inductively eliminate pivotal 1cells from the set of generators given by critical 1cells in UD _{ n } Γ. Thus B _{ n } Γ is generated by free and separating 1cells. Note that it is easy to perform inductive eliminations of pivotal 1cells in decreasing order because no substitution is required.
To show our claim, we have to analyze each term in ∂ _{ w }(c _{2}) due to the lack of luxury such as Lemmas 2.9 and 2.11. First consider the image of a redundant 1cell under \(\tilde{r}\). Let c={e,v _{1},…,v _{ n−1}} be a 1cell. Repeated applications of Lemma 4.2 imply that for any critical 1cell \(c'=\{e',v'_{1},\ldots,v'_{n1}\}\) appearing in \(\tilde{r}(c)\), the vertex τ(e′) is of valency ≥3 in Γ and of the form v _{ i }∧v _{ j } or v _{ i }∧τ(e) or v _{ i }∧ι(e) and, moreover, s(c′) is less than or equal to the number of vertices that do not lie on the 0th branch of τ(e′) among v _{1},…,v _{ n−1}, ι(e), and τ(e). By Lemma 3.3, if c′ contains a deleted edge, then c also contains a deleted edge and c′ appears only once in \(\tilde{r}(c)\). And the terminal vertex of the edge in c _{1} is the larger one between terminal vertices of two edges in c _{2}.
In the case of \(c_{2}=A_{k}(\vec{a})\cup d'\), the corresponding pivotal 1cell c _{1} is \(A_{k}(\vec{a}+\vec{\delta}_{g(A,\iota(d'))})\) by Lemma 3.3. Repeated application of Lemma 4.2 implies \(\tilde{r}(A_{k}(\vec{a})\cup\iota(d'))=c_{1}\). Consider other three redundant 1cells. Since s(c _{1})=s(c _{2})+1 and τ(d′)∧A=τ(d′)<A, the words \(\tilde{r}(d'\cup\dot{A}(\vec{a}))\) and \(\tilde{r}(A_{k}(\vec{a})\cup\tau(d'))\) contain no pivotal 1cells >c _{1}. Since \(d'\cup A(\vec{a}+\vec{\delta}_{k})\) contains a vertex <ι(A _{ k }), a _{ i }≥1 for some i<k, that is, \(p(\vec{a})<k\). If \(c=A_{p(\vec{a})}(\vec{a}+\vec{\delta}_{k}\vec{\delta}_{p(\vec{a})}+\vec{\delta}_{g(A,\iota (d'))})\) is pivotal then c<c _{1}. This implies that \(\tilde{r}(d'\cup A(\vec{a}))\) contains no pivotal 1cells >c _{1}. We are done.
In the case of \(c_{2}=d(\vec{a})\cup d'\), \(c_{1}=d(\vec{a}+\vec{\delta}_{g(\tau (d),\iota(d'))})\) by Lemma 3.3 and so (a) is true since c _{1} contains a deleted edge. Both words \(\tilde{r}(d'\cup\dot{A}(\vec{a}))\) and \(\tilde{r}(d(\vec{a})\cup\tau(d'))\) contain no pivotal 1cell >c _{1} by the same argument as for \(A_{k}(\vec{a})\cup d'\). For any vertex v>τ(d) of valency ≥3 in Γ, there is only one vertex in \(d(\vec{a})\cup\iota(d')\) that does not lie on the 0th branch of v. If c′ is a critical 1cell in \(\tilde{r}(d(\vec{a})\cup\iota(d'))\) such that the terminal vertex of the edge in c is larger than τ(d), then s(c′)=1 and so it is a critical 1cell of the form \(B_{k'}(\vec{\delta}_{\ell'})\) and so c′ is not pivotal. Similarly \(\tilde{r}(d'\cup A(\vec{a})\cup\iota(d))\) contains no pivotal 1cell >c _{1}. This completes the proof. □
Lemma 4.5
(Fewest Generators)
B _{ n } Γ (π _{1}(D _{2} Γ/∼), respectively) has a presentation over m generators for the rank m of H _{1}(UD _{ n } Γ;ℤ_{2}) (H _{1}(D _{2} Γ,(D _{2} Γ)^{0}), respectively).
Proof
By our setup, if two separating 1cells ⋀(d,d′) and ⋀(d,d″) are homologous, then (i) there is a relator w(d,d′,d″) that contains each of them once, (ii) any other separating 1cell in w(d,d′,d″) is less than them, and (iii) the exponent sum of each generator other than them is 0. Clearly the converse is also true. Consider a labeled graph G in which separating 1cells are vertices and there are edges labeled by w(d,d′,d″) between two vertices ⋀(d,d′) and ⋀(d,d″). The number of connected components of G is exactly the number of homology classes of separating 1cells. We are done if each connected component becomes a graph with one vertex (and loops) via inductive edge contractions.
Starting from the vertex ⋀(d,d′) that is the smallest separating 1cell in G, we eliminate ⋀(d,d′) via a Tietze transformation as follows: Choose the smallest vertex ⋀(d,d″) among all vertices adjacent to ⋀(d,d′), contract an edge w(d,d′,d″) (choose any edge if it is a multiedge) by throwing away the vertex ⋀(d,d′), solve w(d,d′,d″) for ⋀(d,d′), and assign new labels obtained by substitutions to all other edges that used to be incident to ⋀(d,d′). Then all edge labels except for loops again have the properties (i), (ii), and (iii) above. In particular, (i) follows from (ii) since ⋀(d,d′) was the smallest vertex in G. To iterate the process, let G be the modified graph. Go to the smallest vertex in G and start again. Since separating 1cells are linearly ordered, this iteration clearly turns each connected component of G into a graph with only the largest vertex together with loops. Note that the exponent sum of the vertex in the label of each loop is either 0 or ±2 due to the property of original labels. □
Theorem 4.6
If Γ is a finite connected planar graph (a finite connected graph, respectively), then B _{ n } Γ (P _{2} Γ, respectively) is a commutatorrelated group.
Proof
Note that if Γ is planar, H _{1}(B _{ n } Γ)≅ℤ^{ m } for the rank m of H _{1}(B _{ n } Γ;ℤ_{2}). Now the theorem is immediate from Proposition 4.1 and Lemma 4.5. In fact, a careful analysis of the proof of Lemma 4.5 can also prove the theorem without Proposition 4.1. □
4.3 Presentations of B _{2} Γ and P _{2} Γ
A group G is simplecommutatorrelated if G has a presentation whose relators are commutators. Clearly a rightangled Artin group is simplecommutatorrelated and a simplecommutatorrelated group is commutatorrelated.
In [9], Farley and Sabalka conjectured that B _{2} Γ is simplecommutatorrelated for a planar graph Γ and relators are commutators of two words that represent disjoint circuits on the planar graph. In a private correspondence, Abrams conjectured that P _{2} Γ is simplecommutatorrelated for a planar graph Γ. There has been some doubt on these conjectures (for example, see [13]). By combining our result with the result by Barnett and Farber in [3], we will prove that for a planar graph Γ, both B _{2} Γ and P _{2} Γ are simplecommutatorrelated and relators are commutators of disjoint circuits on Γ. So these conjectures are true.
First we need the following lemma proved by Barnett and Farber in [3].
Lemma 4.7
(Barnett and Farber [3])
Let Γ⊂ℝ^{2} be a planar graph and U _{0}, U _{1},…,U _{ r } be the connected components of ℝ^{2}−Γ with U _{0} denoting the unbounded component and S(i) denote \(\partial\overline{U}_{i}\subset\varGamma\). Then homology classes [S(i)×S(j)] with S(i)∩S(j)=∅ freely generate H _{2}(D _{2} Γ).
Theorem 4.8
For a planar graph Γ, both B _{2} Γ and P _{2} Γ are simplecommutatorrelated and relators are commutators of two disjoint circuits on Γ. In fact, there is a presentation of B _{2} Γ (P _{2} Γ, respectively) over β _{1} generators such that it has β _{2} relators that are all commutators, where β _{1} and β _{2} are the first and second Betti numbers of B _{2} Γ (P _{2} Γ, respectively).
Proof
There is no difference between B _{2} Γ and P _{2} Γ in our argument. We discuss only P _{2} Γ. Each torus S(i)×S(j) in Lemma 4.7 is embedded in the discrete configuration space D _{2} Γ. Since each circuit in Γ contains at least a deleted edge, so does each S(i). So the embedded torus S(i)×S(j) remains as an immersed torus T _{ ij } in a Morse complex M _{2} Γ since deleted edges gives critical 1cells. The immersed tori may intersect each other but are never identified since they generate H _{2}(M _{2} Γ).
Each Tietze transformation performed in the proofs of Lemmas 4.4 and 4.5 is an elimination of a pair of a generator and a relation. In the cell complex M _{2} Γ, this corresponds to collapsing of a canceling pair of a 1cell and a 2cell. Let \(M'_{2}\varGamma\) denote the cell complex obtained from M _{2}(Γ) by collapsing all canceling pairs corresponding to Tietze transformations performed in the proofs of the two lemmas. Each immersed torus T _{ ij } in M _{2} Γ remains as an immersed torus \(T'_{ij}\) in \(M'_{2}\varGamma\) after collapsing even though it may become complicated.
By Lemma 4.5, the cell complex \(M'_{2}\varGamma\) has two 0cells and (m−1) 1cells for the rank m of H _{1}(D _{2} Γ/∼) since the identification space D _{2} Γ/ can also be obtained by adding a 1cell between two base vertices which remains in \(M'_{2}\varGamma \). By consideration of Euler characteristics, the number of 2cells in \(M'_{2}\varGamma\) is equal to the rank of H _{2}(D _{2} Γ) and so equal to the number of (ordered) tori S(i)×S(j). Therefore each 2cell must form an immersed torus \(T'_{ij}\) and produces a relator that must be a commutator. □
Note that Theorem 4.8 is false for braid index n≥3. For example, B _{3} Θ _{4} and P _{3} Θ _{4} are surface groups (see Example 4.3). One can show that B _{3} Θ _{4}/(B _{3} Θ _{4})_{3} is isomorphic to B _{ n } Θ _{4}/(B _{ n } Θ _{4})_{3} for n≥4 where G _{3} denotes the third lower central subgroup of a group G. Thus B _{ n } Θ _{4} is not simplecommutatorrelated for n≥3. If Γ contains a subgraph Θ _{4}, B _{ n } Γ (P _{3} Γ, respectively) has a subgroup that is not simplecommutatorrelated since there is a local isometric embedding from UD _{ n } Θ _{4} (P _{3} Θ _{4}, respectively) to UD _{ n } Γ (P _{3} Γ, respectively). Thus it seems reasonable to propose the following conjecture:
Conjecture 4.9
For a planar graph Γ, B _{ n } Γ for n≥3 and P _{3} Γ are simplecommutatorrelated if and only if Γ does not contain a subgraph Θ _{4}.
For instance, Farley and Sabalka showed in [7] that every tree braid group is simplecommutatorrelated.
Notes
Acknowledgements
This work was supported by the National Research Foundation of Korea grant No. 20100027001.
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