The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

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The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

Daciberg Gonçalves, John Guaschi

To cite this version:

Daciberg Gonçalves, John Guaschi. The braid groups of the projective plane and the Fadell-Neuwirth

short exact sequence. Geometriae Dedicata, Springer Verlag, 2007, 130, pp.93-107. �10.1007/s10711-

007-9207-z�. �hal-00160464�

hal-00160464, version 1 - 6 Jul 2007

The braid groups of the projective plane and the Fadell-Neuwirth short exact sequence

Daciberg Lima Gonc¸alves

Departamento de Matem´atica - IME-USP, Caixa Postal 66281 - Ag. Cidade de S˜ao Paulo,

CEP: 05311-970 - S˜ao Paulo - SP - Brazil.

e-mail: dlgoncal@ime.usp.br John Guaschi

Laboratoire de Math´ematiques Emile Picard,

Institut de Math´ematiques de Toulouse UMR CNRS 5219, UFR-MIG, Universit´e Toulouse III,

31062 Toulouse Cedex 9, France.

e-mail: guaschi@picard.ups-tlse.fr 12

April 2007

Abstract

We study the pure braid groups P

( R _P

₎ of the real projective plane R _P

_{, and in} particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 −→

P

( R _P

_{\ { x}

_{, . . . , x}

_}) ֒ −→ P

( R _P

_{) −→}

_P

₍ R _P

_{) −→ 1, where n ≥ 2 and m ≥ 1,} and p

is the homomorphism which corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p : F

( R _P

_{) −→ F}

₍ R _P

₎ of configuration spaces. Van Buskirk proved in 1966 that p and p

admit a section if n = 2 and m = 1. Our main result in this paper is to prove that there is no section if n ≥ 3. As a corollary, it follows that n = 2 and m = 1 are the only values for which a section exists. As part of the proof, we derive a presentation of P

( R _P

): this appears to be the first time that such a presentation has been given in the literature.

1 Introduction

Braid groups of the plane were defined by Artin in 1925 [A1], and further studied in [A2, A3]. They were later generalised using the following definition due to Fox [FoN].

Let M be a compact, connected surface, and let n ∈ N . We denote the set of all ordered n-tuples of distinct points of M, known as the n

configuration space of M, by:

F

( M) =

(p

, . . . , p

)

p

∈ M and p

6= p

if i 6= j .

Configuration spaces play an important r ˆole in several branches of mathematics and have been extensively studied, see [CG, FH] for example.

The symmetric group S

on n letters acts freely on F

(M) by permuting coordinates.

The corresponding quotient will be denoted by D

(M). Notice that F

(M) is a regular covering of D

(M). The n

pure braid group P

(M) (respectively the n

braid group B

(M)) is defined to be the fundamental group of F

(M) (respectively of D

(M)). If m ∈ N , then we may define a homomorphism p

: P

(M) −→ P

(M) induced by the projection p : F

( M) −→ F

( M) defined by p((x

, . . . , x

, . . . , x

)) = (x

, . . . , x

).

Representing P

(M) geometrically as a collection of n + m strings, p

corresponds to forgetting the last m strings. We adopt the convention, that unless explicitly stated, all homomorphisms P

(M) −→ P

(M) in the text will be this one.

If M is without boundary, Fadell and Neuwirth study the map p, and show ([FaN, Theorem 3]) that it is a locally-trivial fibration. The fibre over a point (x

, . . . , x

) of the base space is F

(M \ {x

, . . . , x

}) which we consider to be a subspace of the to- tal space via the map i : F

( M \ {x

, . . . , x

}) −→ F

(M) defined by i((y

, . . . , y

)) = (x

, . . . , x

, y

, . . . , y

). Applying the associated long exact sequence in homotopy, we obtain the pure braid group short exact sequence of Fadell and Neuwirth:

1 −→ P

(M \ {x

, . . . , x

}) −→

P

(M) −→

P

( M) −→ 1, (PBS) where n ≥ 3 if M is the sphere S

_{[Fa, FVB], n ≥ 2 if M} is the real projective plane R _P

_{[VB], and n ≥} 1 otherwise [FaN], and where i

and p

are the homomorphisms induced by the maps i and p respectively. The sequence also exists for the classical pure braid group P

, where M is the 2-disc D

(or the plane). The short exact sequence (PBS) has been widely studied, and may be employed for example to determine presentations of P

(M) (see Section 2), its centre, and possible torsion. It was also used in recent work on the structure of the mapping class groups [PR] and on Vassiliev invariants for surface braids [GMP].

The decomposition of P

as a repeated semi-direct product of free groups (known as the ‘combing’ operation) is the principal result of Artin’s classical theory of braid groups [A2], and allows one to obtain normal forms and to solve the word problem.

More recently, it was used by Falk and Randell to study the lower central series and the residual nilpotence of P

[FR], and by Rolfsen and Zhu to prove that P

is bi- orderable [RZ].

The problem of deciding whether such a decomposition exists for surface braid groups is thus fundamental. This was indeed a recurrent and central question dur- ing the foundation of the theory and its subsequent development during the 1960’s [Fa, FaN, FVB, VB, Bi]. If the fibre of the fibration is an Eilenberg-MacLane space then the existence of a section for p

is equivalent to that of a cross-section for p [Ba, Wh]

(cf. [GG2]). But with the exception of the construction of sections in certain cases (for the sphere [Fa] and the torus [Bi]), no progress on the possible splitting of (PBS) was recorded for nearly forty years. In the case of orientable surfaces without boundary of genus at least two, the question of the splitting of (PBS) which was posed explicitly by Birman in 1969 [Bi], was finally resolved by the authors, the answer being positive if and only if n = 1 [GG1].

In this paper, we study the braid groups of R _P

, in particular the splitting of the

sequence (PBS), and the existence of a section for the fibration p. These groups were

first studied by Van Buskirk [VB], and more recently by Wang [Wa]. Clearly P

( R _P

_{) =}

B

( R _P

₎ ^∼ ₌ Z

. Van Buskirk showed that P

( R _P

₎ is isomorphic to the quaternion group

Q

, B

( R _P

₎ is a generalised quaternion group of order 16, and for n > 2, P

( R _P

_{) and} B

( R _P

₎ are infinite. He also proved that these groups have elements of finite order (including one of order 2n in B

( R _P

)). The torsion elements (although not their or- ders) of B

( R _P

₎ were characterised by Murasugi [M]. In [GG2], we showed that for n ≥ 2, B

( R _P

₎ has an element of order ℓ if and only if ℓ divides 4n or 4(n − 1), and that P

( R _P

₎ has torsion exactly 2 and 4. With respect to the splitting problem, Van Buskirk showed that for all n ≥ 2, neither the fibration p : F

( R _P

_{) −→ F}

₍ R _P

₎ nor the homo- morphism p

: P

( R _P

_{) −→ P}

₍ R _P

₎ admit a cross-section (for p, this is a manifestation of the fixed point property of R _P

), but that the fibration p : F

( R _P

_{) −→ F}

₍ R _P

_{) ad-} mits a cross-section, and hence so does the corresponding homomorphism. It follows from (PBS) that P

( R _P

₎ is isomorphic to a semi-direct product of π

( R _P

_{\ {x}

_{, x}

_}), which is a free group F

of rank 2, by P

( R _P

₎ which as we mentioned, is isomorphic to Q

(see [GG2] for an explicit algebraic section). This fact will be used in the proof of Proposition 5 (see Section 3). Although there is no relation with the braid groups of the sphere, it is a curious fact that the commutator subgroup of B

( S

₎ is isomorphic to a semi-direct product of Q

by F

[GG4]. In fact B

( S

₎ possesses subgroups isomorphic to Q

if and only if n ≥ 4 is even [GG3].

In [GG2], we determined the homotopy type of the universal covering space of F

( R _P

). From this, we were able to deduce the higher homotopy groups of F

( R _P

_).

Using coincidence theory, we then showed that for n = 2, 3 and m ≥ 4 − n, neither the fibration nor the short exact sequence (PBS) admit a section. More precisely:

Theorem 1 ([GG2]). Let r ≥ 4 and n = 2, 3. Then:

(a) the fibration p : F

( R _P

_{) −→ F}

₍ R _P

₎ does not admit a cross-section.

(b) the Fadell-Neuwirth pure braid group short exact sequence :

1 −→ P

( R _P

_{\ {x}

_{, . . . , x}

_{}) −→}

_P

₍ R _P

_{) −→}

_P

₍ R _P

_{) −→ 1} does not split.

Apart from Van Buskirk’s results for F

( R _P

_{) −→ F}

₍ R _P

_{) and F}

₍ R _P

_{) −→ F}

₍ R _P

₎ (published in 1966), no other results are known concerning the splitting of (PBS) for the pure braid groups of R _P

. The question is posed explicitly in the case r = n + 1 on page 97 of [VB]. In this paper, we give a complete answer. The main theorem is:

Theorem 2. For all n ≥ 3 and m ≥ 1, the Fadell-Neuwirth pure braid group short exact sequence (PBS):

1 −→ P

( R _P

_{\ {x}

_{, . . . , x}

_{}) −→ P}

₍ R _P

_{) −→}

_P

₍ R _P

_{) −→ 1} does not split, and the fibration p : F

( R _P

_{) −→ F}

₍ R _P

₎ does not admit a section.

Taking into account Van Buskirk’s results and Theorem 1, we deduce immediately the following corollary:

Corollary 3. If m, n ∈ N , the homomorphism p

: P

( R _P

_{) −→ P}

₍ R _P

₎ and the fibration p : F

( R _P

_{) −→ F}

₍ R _P

₎ admit a section if and only if n = 2 and m = 1.

In other words, Van Buskirk’s values (n = 2 and m = 1) are the only ones for which a

section exists (both on the geometric and the algebraic level). The splitting problem for

non-orientable surfaces without boundary and of higher genus is the subject of work in

progress [GG5]. In the case of the Klein bottle, the existence of a non-vanishing vector field implies that there always exists a section, both geometric and algebraic (cf. [FaN]).

This paper is organised as follows. In Section 2, we start by determining a presen- tation of P

( R _P

₎ (Theorem 4). To the best of our knowledge, surprisingly this appears to be the first such presentation in the literature (although Van Buskirk gave a presen- tation of B

( R _P

_)).

In order to prove Theorem 2, we argue by contradiction, and suppose that there exists some n ≥ 3 for which a section occurs. As we indicate in Section 4, it then suffices to study the case m = 1. The general strategy of the proof of Theorem 2 is based on the following remark: if H is any normal subgroup of P

( R _P

₎ contained in Ker (p

), the quotiented short exact sequence 1 −→ Ker (p

) /H ֒ −→ P

( R _P

_{)/H −→}

P

( R _P

_{) −→} 1 must also split. In order to reach a contradiction, we seek such a sub- group H for which this short exact sequence does not split. However the choice of H needed to achieve this is extremely delicate: if H is too ‘small’, the structure of the quotient P

( R _P

_)/H remains complicated; on the other hand, if H is too ‘big’, we lose too much information and cannot reach a conclusion. Taking a variety of possi- ble candidates for H, we observed in preliminary calculations that the line between the two is somewhat fine. If n is odd, we were able to show that the problem may be solved by taking the quotient Ker (p

) /H to be Abelianisation of Ker (p

) (which is a free Abelian group of rank n) modulo 2, which is isomorphic to the direct sum of n copies of Z

. However, this insufficient for n even.

With this in mind, in Section 3, we study the quotient of P

( R _P

₎ by a certain normal subgroup L which is contained in Ker ( p

) in the case m = 1. A key step in the proof of Theorem 2 is Proposition 5 where we show that Ker ( p

) /L is isomorphic to Z

⋊ Z , the action being given by multiplication by −1. This facilitates the cal- culations in P

( R _P

)/L, whilst leaving just enough room for a contradiction. This is accomplished in Section 4 where we show that the following quotiented short exact sequence:

1 −→ Ker (p

) /L −→ P

( R _P

_{)/L −→ P}

₍ R _P

_{) −→ 1} does not split.

Acknowledgements

This work took place during the visit of the second author to the Departmento de Matem´atica do IME-Universidade de S˜ao Paulo during the period 18

June – 18

July 2006, and of the visit of the first author to the Laboratoire de Math´ematiques Emile Pi- card, Universit´e Paul Sabatier during the period 15

November – 16

December 2006.

This project was supported by the international Cooperation USP/Cofecub project num- ber 105/06.

2 A presentation of P _n ( R _P ² ₎

If n ∈ N _and D

_⊆ R _P

is a topological disc, the inclusion induces a (non-injective) homomorphism ι : B

( D

_{) −→ B}

₍ R _P

_{). If β ∈ B}

₍ D

₎ then we shall denote its image ι(β) simply by β. For 1 ≤ i < j ≤ n, we consider the following elements of P

( R _P

_):

B

= σ

· · · σ

σ

σ

· · · σ

,

where σ

, . . . , σ

are the standard generators of B

( D

). The geometric braid corre- sponding to B

takes the i

string once around the j

string in the positive sense, with all other strings remaining vertical. For each 1 ≤ k ≤ n, we define a generator ρ

which is represented geometrically by a loop based at the k

point and which goes round the twisted handle. These elements are illustrated in Figure 1 ( R _P

minus a disc may be thought of as the union of a disc and a twisted handle).

B

ρ

k i j

Figure 1: The generators B

and ρ

of P

( R _P

_).

A presentation of B

( R _P

₎ was first given by Van Buskirk in [VB]. Although presen- tations of braid groups of orientable and non-orientable surfaces have been the focus of several papers [Bi, S, GM, Be], we were not able to find an explicit presentation of P

( R _P

₎ in the literature, so we derive one here.

Theorem 4. Let n ∈ N . The following constitutes a presentation of pure braid group P

( R _P

_):

generators: B

, 1 ≤ i < j ≤ n, and ρ

, 1 ≤ k ≤ n.

relations:

(a) the Artin relations between the B

emanating from those of P

( D

_):

B

B

B

=



 

 



 

 



B

if i < r < s < j or r < s < i < j B

B

B

B

B

if r < i = s < j

B

B

B

if i = r < s < j B

B

B

B

B

B

B

B

B

if r < i < s < j.

(b) for all 1 ≤ i < j ≤ n, ρ

ρ

ρ

= ρ

B

ρ

.

(c) for all 1 ≤ i ≤ n, the ‘surface relations’ ρ

= B

· · · B

B

· · · B

.

(d) for all 1 ≤ i < j ≤ n and 1 ≤ k ≤ n, k 6= j,

ρ

B

ρ

=



 

 

B

if j < k or k < i ρ

B

ρ

if k = i

ρ

B

ρ

B

B

B

ρ

B

ρ

if i < k < j.

Proof. We apply induction and standard results concerning the presentation of an ex- tension (see Theorem 1, Chapter 13 of [J]).

First note that the given presentation is correct for n = 1 (P

( R _P

_{) = π}

₍ R _P

₎ ∼ = Z

_), and n = 2 (P

( R _P

₎ ∼ = Q

). So let n ≥ 2, and suppose that P

( R _P

₎ has the given presentation. Consider the corresponding Fadell-Neuwirth short exact sequence:

1 −→ π

( R _P

_{\ {x}

_{, . . . , x}

_{}) −→ P}

₍ R _P

_{) −→}

_P

₍ R _P

_{) −→ 1. (1)} In order to retain the symmetry of the presentation, we take the free group Ker ( p

) to have the following one-relator presentation:

D

ρ

, B

, . . . , B

ρ

= B

· · · B

E .

Together with these generators of Ker (p

), the elements B

, 1 ≤ i < j ≤ n, and ρ

, 1 ≤ k ≤ n, of P

( R _P

₎ (which are coset representatives of the generators of P

( R _P

₎₎ form the required generating set of P

( R _P

_).

There are three classes of relations of P

( R _P

₎ which are obtained as follows. The first consists of the single relation ρ

= B

· · · B

of Ker (p

). The second class is obtained by rewriting the relators of the quotient in terms of the coset representatives, and expressing the corresponding element as a word in the generators of Ker (p

). In this way, all of the relations of P

( R _P

₎ lift directly to relations of P

( R _P

), with the exception of the surface relations which become ρ

= B

· · · B

B

· · · B

B

for all 1 ≤ i ≤ n. Together with the relation of Ker (p

), we obtain the complete set of surface relations (relations (c)) for P

( R _P

_).

The third class of relations is obtained by rewriting the conjugates of the generators of Ker (p

) by the coset representatives in terms of the generators of Ker (p

):

(i) For all 1 ≤ i < j ≤ n and 1 ≤ l ≤ n,

B

B

B

=



 

 



 

 



B

if l < i or j < l

B

B

B

B

B

if l = j

B

B

B

if l = i

B

B

B

B

B

B

B

B

B

if i < l < j.

(ii) B

ρ

B

= ρ

for all 1 ≤ i < j ≤ n.

(iii) ρ

ρ

ρ

= ρ

B

ρ

for all 1 ≤ k ≤ n.

(iv) For all 1 ≤ k, l ≤ n,

ρ

B

ρ

=



 

 

B

if k < l

ρ

B

ρ

if k = l

ρ

B

ρ

B

B

B

ρ

B

ρ

if l < k.

Then relations (a) for P

( R _P

₎ are obtained from relations (a) for P

( R _P

_{) and rela-} tions (i), relations (b) for P

( R _P

₎ are obtained from relations (b) for P

( R _P

_{) and re-} lations (iii), and relations (d) for P

( R _P

₎ are obtained from relations (d) for P

( R _P

_), relations (iv) and (ii).

For future use, it will be convenient at this point to record the following supplemen- tary relations in P

( R _P

₎ which are consequences of the presentation of Theorem 4. Let 1 ≤ i < j ≤ n.

(I) The action of the ρ

on the ρ

may be deduced from that of ρ

: ρ

ρ

ρ

= B

ρ

. (II) By relations (b) and (d), we have:

ρ

(B

ρ

B

ρ

B

)ρ

= ρ

B

ρ

· ρ

B

ρ

· ρ

B

ρ

· ρ

B

ρ

· ρ

B

ρ

= B

. Hence ρ

B

ρ

= B

ρ

B

ρ

B

.

(III) From relations (b) and (I), we see that:

ρ

ρ

ρ

= ρ

ρ

B

ρ

ρ

· ρ

= ρ

B

· ρ

B

ρ

· B

ρ

· ρ

= B

, so ρ

ρ

ρ

= ρ

B

.

(IV) From relations (I) and (d), we obtain:

ρ

ρ

ρ

= ρ

ρ

B

ρ

= ρ

B

ρ

.

3 A presentation of the quotient P _n+1 ( R _P ² _)/L

For n ≥ 2, we have the Fadell-Neuwirth short exact sequence (1) whose kernel K = Ker (p

) is a free group of rank n with basis ρ

, B

, B

, . . . , B

. We first introduce a subgroup L of K which is normal in P

( R _P

), from which we shall be able to prove Theorem 2.

We define L to be the normal closure in P

( R _P

₎ of the following elements:

(i) [B

, B

], where 1 ≤ i < j ≤ n − 1, and (ii) [B

, ρ

], where 1 ≤ i ≤ n − 1 and 1 ≤ k ≤ n.

The elements [B

, B

] clearly belong to K. The presentation of P

( R _P

_{) given by} Theorem 4 implies that:

[B

, ρ

] =



 

 

1 if k < i

B

ρ

B

ρ

if k = i

B

ρ

B

ρ

B

B

B

ρ

B

ρ

if i < k ≤ n.

Thus L is a (normal) subgroup of K.

Let g : P

( R _P

_{) −→ P}

₍ R _P

_)/L denote the canonical projection. For i = 1, . . . , n − 1, let A

= g(B

). Apart from these elements, if x is a generator of P

( R _P

_), we shall not distinguish notationally between x and g(x). The quotient P

( R _P

_)/L is generated by ρ

, . . . , ρ

, B

, 1 ≤ i < j ≤ n, and A

, A

, . . . , A

(we delete B

from the list using the surface relation ρ

= A

A

· · · A

B

, so B

= A

· · · A

A

ρ

).

A presentation of P

( R _P

_)/L may obtained from that of P

( R _P

₎ by adding the

relations arising from the elements of L. We list those relations which are relevant for

our description of P

( R _P

_)/L.

(a) The Artin relations between the B

, 1 ≤ i < j ≤ n.

(b) The relations of P

( R _P

_{) between ρ}

_{, ρ}

_{, 1 ≤ i} < j ≤ n.

(c) The relations of P

( R _P

_{) between B}

_{, 1 ≤ i} < j ≤ n and ρ

, 1 ≤ k ≤ n.

The following two sets of relations arise from the definition of L:

(d) A

⇋ _A

_{, 1 ≤ i} < j ≤ n − 1 (the symbol ⇋ is used to mean that the given elements commute).

(e) A

⇋ _ρ

_{, i = 1, . . . , n − 1 and j = 1, . . . , n.}

(f) The surface relations:

ρ

= B

· · · B

B

· · · B

A

for i = 1, . . . , n − 1 ρ

= B

B

· · · B

· A

· · · A

A

ρ

.

(g) For i = 1, . . . , n − 1, ρ

A

ρ

= A

(since ρ

A

ρ

= ρ

A

ρ

= A

).

The following relations are implied by the above relations:

– for 1 ≤ i < j ≤ n − 1, ρ

A

ρ

= ρ

A

ρ

A

A

A

ρ

A

ρ

(both are equal to A

).

– for 1 ≤ i ≤ n − 1, ρ

A

ρ

= A

ρ

A

ρ

A

(both are equal to A

) (h) For 1 ≤ j ≤ n − 1,

ρ

ρ

ρ

= ρ

A

ρ

= A

ρ

= ρ

A

. From these relations, it follows that ρ

⇋ _ρ

n+1

for i = 1, . . . , n − 1.

(i) ρ

ρ

ρ

= ρ

B

ρ

= ρ

ρ

A

· · · A

ρ

= A

· · · A

ρ

. From this relation, it follows that ρ

ρ

ρ

= ρ

.

For i = 1 . . . , n − 1, the following relations are implied by the above relations:

ρ

A

ρ

= ρ

B

ρ

B

A

B

ρ

B

ρ

(both are equal to A

).

Proposition 5. The quotient group K/L has a presentation of the form:

generators: A

, . . . , A

, ρ

.

relations: A

⇋ _A

_{for 1 ≤ i} < j ≤ n − 1, and ρ

A

ρ

= A

for 1 ≤ i ≤ n − 1.

In particular, K/L is isomorphic to Z

⋊ Z , the action being given by multiplication by −1.

Hence the other relations in P

( R _P

_)/L (which involve only elements from P

( R _P

₎₎ do not add any further relations to the quotient K/L.

Proof of Proposition 5. Clearly A

, . . . , A

, ρ

generate K/L, and from relations (d) and (g) of P

( R _P

)/L, they are subject to the given relations. Consider the following commutative diagram of short exact sequences:

1

K

g|_K

P

( R _P

₎

g

P

( R _P

₎

₁

1

K/L

P

( R _P

_)/L

_P

₍ R _P

₎

_1,

(2)

where ι is the inclusion of K/L in P

( R _P

_{)/L, and p}

∗

is the homomorphism induced by p

. Let Γ be the group with presentation:

Γ = D

α

, . . . , α

, ρ

α

⇋ _α

_{for 1 ≤ i} < j ≤ n − 1, and ρα

ρ

= α

E

.

So Γ is isomorphic to Z

⋊ Z , where the action is given by multiplication by −1. The map f : Γ −→ K/L defined on the generators of Γ by f (α

) = A

for i = 1, . . . , n − 1, and f (ρ) = ρ

, extends to a surjective homomorphism. We claim that f is an isomorphism, which will prove the proposition. To prove the claim, it suffices to show that ι ◦ f is injective. Let w ∈ Ker (ι ◦ f ). Then we may write w uniquely in the form w = ρ

α

· · · α

, where m

, m

, . . . , m

∈ Z _{, and so ι ◦ f (w) = ρ}

n+1

A

· · · A

= 1 in P

( R _P

_)/L.

Let z = ρ

B

· · · B

∈ P

( R _P

_{). Since g(z) = ι ◦ f (w) = 1, we must} have z ∈ L. Now L is the normal closure in P

( R _P

₎ of the following elements:

– c

= [B

, B

], where 1 ≤ j < k ≤ n − 1,

– d

= [B

, ρ

] = B

ρ

B

ρ

, where 1 ≤ j ≤ n − 1,

– e

= [B

, ρ

] = B

ρ

B

ρ

B

B

B

ρ

B

ρ

, where 1 ≤ j < k ≤ n.

Hence z may be written as a product of conjugates of the elements c

, d

, e

, and their inverses.

For i = 1, . . . , n − 1, let π

: P

( R _P

_{) −→ P}

₍ R _P

_{, (p}

_{, p}

_{, p}

₎₎ be the projection obtained geometrically by forgetting all of the strings, with the exception of the i

, n

and (n + 1)

strings (here P

( R _P

_{, (p}

_{, p}

_{, p}

₎₎ denotes the fundamental group of F

( R _P

₎ taking the basepoint to be (p

, p

, p

)). We interpret P

( R _P

_{, (p}

_{, p}

_{, p}

_{)) as} the semi-direct product F

_(B

_{, ρ}

₎ ⋊ _P

₍ R _P

_{, (p}

_{, p}

₎₎ [VB]. Under π

, the elements c

, d

, e

(for the allowed values of j and k) are all sent to the trivial element, with the exception of the two elements d

and e

. Set h

= π

(d

) = B

ρ

B

ρ

∈ F

_(B

_{, ρ}

_{). Since B}

_B

_{= ρ}

n+1

in P

( R _P

_{, ( p}

_{, p}

_{, p}

)), we have B

= B

ρ

. Hence:

π

(e

) =B

ρ

B

ρ

B

B

B

ρ

B

ρ

=B

ρ

ρ

B

ρ

ρ

B

B

B

ρ

ρ

B

ρ

ρ

=B

ρ

B

ρ

· ρ

B

ρ

B

ρ

B

B

ρ

· ρ

B

ρ

B

ρ

ρ

· ρ

ρ

B

ρ

B

ρ

=h

· ρ

B

h

B

ρ

· ρ

h

ρ

· ρ

h

ρ

.

Thus π

(z) may be written as a product of conjugates in P

( R _P

_{, ( p}

_{, p}

_{, p}

_{)) of h}

i

: π

(z) = ρ

B

=

l

∏

j=1

w

h

w

, (3) where l ∈ N _{, w}

_{∈ P}

₍ R _P

_{, (p}

_{, p}

_{, p}

_{)), and µ(j) ∈ {1,} −1}. We claim that each w

h

w

is in fact a conjugate in F

_(B

_{, ρ}

_{) of h}

i

. This follows by studying the action of the generators ρ

and ρ

of P

( R _P

_{, ( p}

_{, p}

₎₎ on the basis of F

_(B

_{, ρ}

_):

ρ

h

ρ

=ρ

B

ρ

B

ρ

ρ

=ρ

B

ρ

· ρ

B

ρ

· ρ

B

ρ

· ρ

B

ρ

=ρ

B

ρ

B

ρ

=ρ

B

· ρ

B

ρ

B

· B

ρ

= ρ

B

h

B

ρ

,