a Xi :ma h/9910020 1 [ma h.GT] 5 Oc 1999
New p esen a ions o su ace b aid g oups
Juan Gonz´alez-Meneses
Oc obe , 1999
Abs ac
In his pape we gi e new p esen a ions o he b aid g oups and he pu e b aid
g oups o a closed su ace. We also gi e an algo i hm o sol e he wo d p oblem in
hese g oups, using he gi en p esen a ions.
1 In oduc ion
Le Mbe a closed su ace, no necessa ily o ien able, and le P={P1,...,Pn}be a se o
ndis inc poin s o M. A geome ic b aid o e Mbased a Pis an n- uple Γ = (γ1,...,γn)
o pa hs, γi: [0,1] −→ M, such ha
(1) γi(0) = Pi o all i= 1,...,n,
(2) γi(1) ∈ P o all i= 1,...,n,
(3) {γ1( ),...,γn( )}a e ndis inc poin s o M o all ∈[0,1].
Fo all i= 1,...,n, we will call γi he i- h s ing o Γ.
Two geome ic b aids based a Pa e said o be equi alen i he e exis s a homo opy
which de o ms one o hem in o he o he , p o ided ha a any ime we always ha e a
geome ic b aid based a P. We can na u ally de ine he p oduc o wo b aids as induced
by he usual p oduc o pa hs: o e e y i= 1,...,n, we compose he s ing o he i s
b aid which ends a Pi, wi h he i- h s ing o he second b aid. This p oduc is clea ly
well de ined, and i endows he se o equi alence classes o b aids wi h a g oup s uc u e.
This g oup is called he b aid g oup on ns ings o e Mbased a P, and is deno ed by
Bn(M, P). This g oup does no depend, up o isomo phism, on he choice o P, bu only
on he numbe o s ings, so we may w i e Bn(M) ins ead o Bn(M, P).
A b aid Γ = (γ1,...,γn) is said o be pu e i γi(1) = Pi o all i= 1,...,n, ha is, i
all i s s ings a e loops. The se o equi alence classes o pu e b aids o ms a subg oup o
Keywo ds: B aid - Su ace - P esen a ion - Wo d P oblem.
Ma hema ics Subjec Classi ica ion: P ima y: 20F36. Seconda y: 57N05.
Pa ially suppo ed by DGES-PB97-0723 and by he eu opean ne wo k TMR Sing. Eq. Di . e Feuill.
1
Bn(M, P) called pu e b aid g oup on ns ings o e Mbased a P, and deno ed P Bn(M, P).
Again, we may w i e PBn(M) since i does no depend on he choice o P. No e ha i
n= 1, hen B1(M) = PB1(M) = π1(M), he undamen al g oup o M.
The e exis s an in e p e a ion o b aid g oups as undamen al g oups o some spaces,
called con igu a ion spaces. Le FnMdeno e he space o n- uples o dis inc poin s o M,
ha is, FnM=Mn ∆, whe e
∆ = {(x1,...,xn)∈Mn/ xi=xj o some i6=j}.
I is clea ha P Bn(M)≃π1(FnM). Now conside he symme ic g oup on nelemen s,
Σn. This g oup ac s na u ally on FnMby pe mu ing coo dina es, so we can conside he
con igu a ion space:
b
FnM=FnM/Σn,
which can be seen as he space o embeddings o npoin s in M. We clea ly ha e Bn(M)≃
π1(b
FnM).
This way o look a b aids p o ides some use ul exac sequences, de i ed om ib a ions.
The i s one comes om he co e ing space map
FnM−→ b
FnM,
wi h ibe Σn. I induces he ollowing exac sequence:
1−→ P Bn(M)e
−→ Bn(M)
−→ Σn−→ 1.(1)
The homomo phism eis he na u al inclusion, and maps a gi en b aid o he pe mu a ion
ha i induces on P.
Now we conside he Fadell-Neuwi h ib a ion ([FN]): gi en 1 ≤m < n, he map
p:FnM−→ FmM
(x1,...,xn)7−→ (xn−m+1,...,xn)
is a locally i ial ib a ion wi h ibe Fn−m(M {Q1,...,Qm}), o any choice o he poin s
{Q1,...,Qm}. Se P′={P2,...,Pn}, ake m=n−1, and conside Mdi e en om
he sphe e and om he p ojec i e plane (so π2(M) = 1). By he long exac sequence o
homo opy g oups o his ib a ion, we ob ain
1−→ π1(M P′, P1)u
−→ P Bn(M, P)
−→ P Bn−1(M, P′)−→ 1.(2)
I γ∈π1(M P′, P1), hen u(γ) = (γ, eP2,...,ePn), whe e ePideno es he cons an pa h on
Pi, and, o Γ = (γ1,...,γn)∈P Bn(M, P), one has (Γ) = (γ2,...,γn).
The goal o his pape is o de e mine new p esen a ions o he b aid g oups o closed
su aces di e en om he sphe e and om he p ojec i e plane. These p esen a ions a e
2
much simple han hose which we e known be o e ([S]). Mo eo e , he gene a o s and
he ela ions ha e an easy geome ic in e p e a ion. We also show ha hese p esen a ions
u nish an algo i hm o sol e he wo d p oblem o su ace b aid g oups. No ice ha
simila p esen a ions o he b aid g oups o he sphe e and o he p ojec i e plane can be
ound in [F B] and in [ B], espec i ely.
Ou wo k is o ganized as ollows. In Sec ion 2 we s a e he esul s, in oducing he
gene a o s and ela ions o ou new p esen a ions. Then we explain in Sec ion 3 he
me hod ollowed in he p oo s, which we apply h oughou Sec ions 4 and 5, o o ien able
and non-o ien able su aces, espec i ely. Finally, we desc ibe in Sec ion 6 an algo i hm o
sol e he wo d p oblem in su ace b aid g oups.
I would like o hank Luis Pa is o gi ing me he idea o applying Lemma 3.1 o su ace
b aid g oups, and also o i s aluable help in he w i ing o his pape .
2 S a emen s
The aim o his sec ion is o s a e ou p esen a ions o su ace b aid g oups, de ining he
gene a o s and showing ha he p oposed ela ions a e sa is ied. We s a wi h he case
o an o ien ed su ace di e en om he sphe e.
Le Mbe a closed, o ien able su ace o genus g≥1. The i s hing we wan is o
ha e a geome ical ep esen a ion o a b aid o e M. We ep esen Mas a polygon Lo
4gsides, iden i ied in he way o Figu e 1 (See [M], page 34, ex. 8.9).
α1
α2g
α2g−1
α1
α2
α2g−1
α2g
α2
Figu e 1: The polygon L ep esen ing M.
We could now ake he cylinde L×I(I= [0,1]), and ep esen a b aid Γ o e Mas i
is usually done o he open disc, ha is, in L× { }we d aw he npoin s γ1( ),...,γn( ).
Bu in his case a s ing could “go h ough a wall” o he cylinde and appea om he
o he side. Hence, i we look a he cylinde om he usual iewpoin , i would no be
clea which a e he “c ossed walls” (see he le hand side o Figu e 2).
3
L
I
Figu e 2: A b aid o e a su ace o genus 2: wo di e en iewpoin s.
The solu ion we p opose is o look a he cylinde om abo e, as in he igh hand side
o Figu e 2. In his way, we ge id o he ambigui y, and mo eo e we see he s ings again
as pa hs in he su ace. When wo s ings c oss, we see passing abo e he one ha eaches
be o e he c ossing poin . Anyway, i is good o keep in mind he idea ha we a e looking
o a cylinde , and o conside he pa hs as s ings: in his manne , one can see mo e easily
when wo geome ic b aids a e equi alen .
Now we can de ine he gene a o s o Bn(M). We choose he nbase poin s along he
ho izon al diame e o L, as in Figu e 3. Now gi en , 1 ≤ ≤2gwe de ine he b aid a
as ollows: i s only non i ial s ing is he i s one, which goes h ough he - h wall, in
he way o Figu e 3. Tha is, he i s s ing will go upwa ds i is odd, and downwa ds
o he wise.
We also de ine, o all i= 1,...,n−1, he b aid σias in Figu e 3. No e ha σ1,...,σn−1
a e he classical gene a o s o he b aid g oup Bno he disc.
a2k+1
α2k+1
α2k+1
Pn
P1P1
α2k
α2k
Pn
a2kσi
Pn
Pi+1
P1Pi
Figu e 3: The gene a o s o BnM.
4
We will see la e ha he se {a1,...,a2g, σ1,...,σn−1}is a se o gene a o s o Bn(M).
The e a e wo ela ions be ween hese gene a o s ha we can deduce as ollows. Conside
he in e io o L. I is a subsu ace Do Mhomeomo phic o a disc, so clea ly e e y
ela ion sa is ied in he b aid g oup Bn=Bn(D) will be sa is ied as well in Bn(M) ( he
same homo opy can be used in bo h cases). In ac , since g≥1, i is known ha Bnis
a subg oup o Bn(M) (see [PR]). Hence, om he classical p esen a ion o Bn, we ob ain
wo ela ions in Bn(M):
σiσj=σjσi(|i−j| ≥ 2),
σiσi+1σi=σi+1σiσi+1 (1 ≤i≤n−2).
No e also ha i i∈ {2,...,n−1}and ∈ {1,...,2g}, hen he non- i ial s ings o
σiand he one o a may be aken o be disjoin . This clea ly implies ha hese wo b aids
commu e. Hence we ha e
a σi=σia (1 ≤ ≤2g;i≥2).
Now, in o de o ind mo e ela ions be ween he se o gene a o s, we do he ollowing
cons uc ion. Deno e by s he i s s ing o a , o all = 1,...,2g, and conside all he
pa hs s1,...,s2g. We can “cu ” he polygon Lalong hem, and “glue” he pieces along he
pa hs α1,...,α2g. We ob ain ano he polygon o 4gsides which a e labeled by s1,...,s2g
(see in Figu e 4 he case o a su ace o genus 2; he gene al case is analogous). We will call
his new polygon he P1-polygon o M, since all o i s e ices a e iden i ied o P1, while L
will be called he ini ial polygon. We ob ain in his way a new ep esen a ion o he su ace
M.
P1
α3
α3α2
α2
α4
α1α4
α1
Pn
s4
s2
s3
s1
P1
s2
s3
s1
s4
α1α4
P1
Pn
P1
α2α3
s2
s1
s4
s3
P1
P1
P1
P1P1
Figu e 4: The ini ial and he P1-polygon o a su ace o genus 2.
We will use he P1-polygon o show h ee mo e ela ions in Bn(M). Fo ins ance,
conside he b aid a1···a2ga−1
1···a−1
2g. I we look a i in he P1-polygon, i is clea ha i
5

is equi alen o he b aid o Figu e 5. Bu his one can be seen in o he ini ial polygon as a
b aid ha does no go h ough he walls, namely, an elemen o Bn, he b aid g oup o he
disc. Then we can easily show ha i is equi alen o he b aid σ1···σn−2σ2
n−1σn−2···σ1.
So we ha e:
a1···a2ga−1
1···a−1
2g=σ1···σn−2σ2
n−1σn−2···σ1.
Pn
s1
s2g
P1
P1
s2g
s1
Figu e 5: The b aid a1···a2ga−1
1···a−1
2g.
Now we de ine, o each = 1,...,2g, he b aid
A2, =σ−1
1a1···a −1a−1
+1 ···a−1
2gσ−1
1.
We will use he P1-polygon o see how i looks like. In he le hand side o Figu e 6, we can
see a b aid which is clea ly equi alen o A2, (i is odd, he o he case being analogous).
I we “cu ” and “glue” o see his b aid in he P1-polygon, we ob ain he si ua ion o he
igh hand side o Figu e 6. Tha is, A2, can be seen as a b aid whose only non i ial
s ing is he second one, which goes upwa ds and c osses once he - h wall s . No e ha ,
unlike he case o a ,A2, always poin s upwa ds in he P1-polygon, no ma e he pa i y
o .
The e o e we ha e seen ha he b aid A2, can be ep esen ed by a geome ic b aid,
whose only non i ial s ing can be aken disjoin om all he pa hs s , 6= . This
clea ly implies ha
a A2, =A2, a (1 ≤ , ≤2g; 6= ).
Now we inish ou se o ela ions by conside ing he commu a o o he b aids (a1···a )
and A2, , o all = 1,...,2g. In Figu e 7 we can see a ske ch o he homo opy which
s a s wi h his commu a o and de o ms i o a b aid clea ly equi alen o σ2
1. The e o e,
we ob ain he ela ion:
(a1···a )A2, =σ2
1A2, (a1···a ) (1 ≤ ≤2g).
6
P2α1
α2
α +1
α −1
α +1 α −1
α2g
α2g−2
α3
P1
α1
α2
α3
α2g
α2g−1
α2g−1
α2g−2
α
α
P2
s
s2g
s2g−1
s1
s2
s2
s1
s2g
s2g−1
s −1
s +1 s
P1
s −1s +1
Figu e 6: The b aid A2, : In he P1-polygon and in he ini ial one.
Now we claim ha he six ela ions ha we ha e conside ed o m a comple e se o
de ining ela ions o Bn(M). In o he wo ds, we ha e he ollowing esul .
Theo em 2.1. I Mis a closed, o ien able su ace o genus g≥1, hen Bn(M)admi s
he ollowing p esen a ion:
•Gene a o s:
σ1,...,σn−1, a1,...,a2g.
•Rela ions:
(R1) σiσj=σjσi(|i−j| ≥ 2)
(R2) σiσi+1σi=σi+1σiσi+1 (1 ≤i≤n−2)
(R3) a1···a2ga−1
1···a−1
2g=σ1···σn−2σ2
n−1σn−2···σ1
(R4) a A2,s =A2,sa (1 ≤ , s ≤2g; 6=s)
(R5) (a1···a )A2, =σ2
1A2, (a1···a ) (1 ≤ ≤2g)
(R6) a σi=σia (1 ≤ ≤2g;i≥2)
whe e
A2, =σ−1
1a1···a −1a−1
+1 ···a−1
2gσ−1
1.
Now we u n o he non-o ien able case. Le Mbe a closed non-o ien able su ace o
genus g≥2. To ep esen a b aid in Mwe will also p esen he su ace as a polygon, his
ime o 2gsides, as in Figu e 8, and we make an addi ional cu : de ine he pa h eas in
7
P1P2
s
s
s1s1
P2
P1
s
s s
P1P2
s1
s
Figu e 7: The b aid [a1···a , A2, ].
he le hand side o Figu e 8, and cu he polygon along i . We ge M ep esen ed as in
he igh hand side o he same igu e, whe e we can also see how we choose he poin s
P1,...,Pn.
α2
α1
e
α1
αg−1
αg−1
αg
αg
α2α1
ee
Pn
P1
αg
αg
α1
Figu e 8: Rep esen a ion o a non-o ien able su ace M.
We de ine now he gene a o s o Bn(M). They will be simila o hose o he o ien able
su ace b aid g oups. Fo all i∈ {1,...n−1}, he b aid σiwill be he same as in he
o ien able case. Fo all ∈ {1,...,g}, he b aid a consis s on he i s s ing passing
h ough he - h wall, in he way o Figu e 9, while he o he s ings a e i ial pa hs.
The e a e six ela ions in he b aid g oup o M ha a e analogous o hose conside ed
o an o ien able su ace. They can be shown o hold in he same way as in he o ien able
case; he only di e ence is he cons uc ion o he P1-polygon. We deno e by s1,...,sg he
i s s ing o a1,...,ag, espec i ely, and in his case we de ine ano he pa h, e1, which goes
om P1 o he inal poin o e(see Figu e 9). Then we cu along he pa hs s1,...,sg, e1and
glue along α1,...,αg, e. The esul is he P1-polygon o Mwhose sides, eading clockwise,
8
a
ee
α α
Pn
e1
P1Pn
Pi+1
P1Pi
σi
ee
αg
αg
α1
α1
Figu e 9: The gene a o s o Bn(M).
a e labeled by s1, s1, s2, s2,...,sg, sg, e1, e−1
1.
We claim ha he six men ioned ela ions o m a se o de ining ela ions o Bn(M).
To be mo e p ecise, we claim he ollowing.
Theo em 2.2. I Mis a closed, non-o ien able su ace o genus g≥2, hen Bn(M)admi s
he ollowing p esen a ion:
•Gene a o s:
σ1,...,σn−1, a1,...,ag.
•Rela ions:
( 1) σiσj=σjσi(|i−j| ≥ 2)
( 2) σiσi+1σi=σi+1σiσi+1 (1 ≤i≤n−2)
( 3) a2
1···a2
g=σ1···σn−2σ2
n−1σn−2···σ1
( 4) a A2,s =A2,sa (1 ≤ , s ≤g; 6=s)
( 5) a2
1···a2
−1a A2, =σ2
1A2, a2
1···a2
−1a (1 ≤ ≤g)
( 6) a σj=σja (1 ≤ ≤g;j≥2)
whe e
A2, =σ−1
1a2
1···a2
−1a−1
a−2
−1···a−2
1σ1.
9
No e ha he le hand side maps by ϕ o ˜ , whe e is a ela o o PBn−1(M) co esponding
o (PR1), while he igh hand side maps by ϕ o a wo d o e GA. Hence, Uis equal o he
igh hand side o he equa ion, and his yields he i s ela ion o Type 2. The emaining
ela ions o Type 2 a e also images by ϕo ela ions in P esen a ion 1; namely (PR2),
(PR3), (PR4) and (PR5) when i≥2, (PR6) when i≥2 and j≥2, and (PR7), (PR8)
when j≥2. Fo all hese ela ions, he wo d Uis jus he i ial wo d, excep o (PR8),
o which U=a−1
1,2g···a−1
1,1T1,j−1T−1
1,j a1,1···a1,2g.
Finally, we ind he ela ions o Type 3. Fo i= 1, (PR2) becomes
Aj,sa1, A−1
j,s =a1, ( 6=s),
so V=a1, . Nex , using (PR2), Rela ion (PR3) u ns o be equi alen o
Aj, a1, A−1
j, =a−1
1, −1···a−1
1,1T1,j−1T−1
1,j (a1,1···a1, ),
so Vequals he igh hand side o his equa ion. Rela ions o he o m Tk,lT1,jT−1
k,l =V,
whe e Vis a wo d o e GA, ollow om (PR4)-(PR5), while hose o he o m Tk,la1, T−1
k,l =
V ollow om (PR6), when i= 1. Also, i j > k, we ob ain om (PR6) he ela ions
Aj, T1,kA−1
j, =V, whe e Vis a wo d o e GA.
The only emaining ela ions a e hose o he o m Aj, T1,kA−1
j, =V, when 1 < j ≤k,
which a e deduced as ollows: By (PR7), we know ha aj,s commu es wi h he elemen
a−1
1,2g···a−1
1,1T1,ka1,2g···a1,1
o s= 1,...,2g. This implies ha Aj, commu es wi h he same elemen , so
a−1
1,2g···a−1
1,1T1,ka1,2g···a1,1=Aj, a−1
1,2g···a−1
1,1T1,ka1,2g···a1,1A−1
j,
=Aj, a−1
1,2gA−1
j, ···Aj, a−1
1,1A−1
j, Aj, T1,kA−1
j, Aj, a1,2gA−1
j, ···Aj, a1,1A−1
j, .
Bu using (PR2) and (PR3) we know how o w i e all he e ms in he abo e p oduc
(excep he middle one) as wo ds o e GA, so we a e done.
Hence, we ha e shown ha PBn(M)ϕ
≃PBn(M) and he e o e, we ha e p o ed:
Theo em 4.2. I Mis a closed, o ien able su ace o genus g≥1, hen PBn(M)admi s
P esen a ion 1 (and also P esen a ion 2) as p esen a ion.
S ep 3. Now we wan o ind a p esen a ion o Bn(M), o g≥1. We de ine hen he
g oup Bn(M), gi en by he p esen a ion in Theo em 2.1.
This is he mos educed p esen a ion we ha e ound. Bu o show i s alidi y we will
modi y i , ob aining a new one wi h mo e gene a o s and ela ions, bu equi alen o he
i s one.
16

Fi s , we change ou no a ion, and call a1, he gene a o s a , o = 1,...,2g. Then
we mus simply add o he gi en p esen a ion he gene a o s
−ai, i= 2,...,n; = 1,...,2g,
−Tj,k 1≤j < k ≤n,
and he ela ions
(R7) aj+1, =σjaj, σj( 1 ≤j≤n−1; 1 ≤ ≤2g; e en).
(R8) aj+1, =σ−1
jaj, σ−1
j(1 ≤j≤n−1; 1 ≤ ≤2g; odd).
(R9) Tj,k =σjσj+1 ···σk−2σ2
k−1σk−2···σj(1 ≤j < k ≤n).
Clea ly, bo h p esen a ions de ine he same g oup, ha is, Bn(M). Now we de ine
ψ:Bn(M)→Bn(M) in he na u al way. I is an easy exe cise o show, using he
same me hods as be o e, ha Rela ions (R7), (R8) and (R9) map o ela ions in Bn(M).
The e o e, ψis a well de ined homomo phism.
Recall now he exac sequence (1):
1−→ P Bn(M)e
−→ Bn(M)
−→ Σn−→ 1.
We know by Theo em 4.2 a p esen a ion o P Bn(M) (say P esen a ion 1), and i is also
known ha a p esen a ion o Σnis
•Gene a o s: δ1,...,δn−1.
•Rela ions:
–δiδj=δjδi|i−j| ≥ 2,
–δiδi+1δi=δi+1δiδi+1 1≤i≤n−2,
–δ2
i= 1 1 ≤i≤n−1,
whe e δiis he pe mu a ion (i, i + 1), o any i.
Now σiis clea ly a p e-image by o δi, so by Lemma 3.1 Bn(M) and Bn(M) ha e he
same gene a o s, and ψis su jec i e.
Simila ly o wha we did in S ep 2, we show now ha ψis an isomo phism by he
ollowing p ocedu e.
Fi s , we deno e by GA he se o gene a o s o PBn(M), and by G he se o gene a o s
o Bn(M). Fo each ela ion in he p esen a ion o PBn(M), we conside i ia eas a
ela ion in Bn(M), and we show ha i also holds in Bn(M).
17
Nex , o each ela o o Σn, we conside i s canonical p e-image by , deno ed by ˜ .
Then we ind a wo d Uo e Gsuch ha he equali y ˜ =Uholds in Bn(M), and such
ha ψ(U) is a wo d o e GA.
Finally, o each x∈GAand each gene a o δio Σn, we ind a wo d Vo e Gsuch
ha he equali y σix σ−1
i=Vholds in Bn(M), and such ha ψ(V) is a wo d o e GA.
This gi es us he ela ions o Types 1, 2, and 3 o Lemma 3.1 and, he e o e, a p esen-
a ion o Bn(M), and, a he same ime, his shows ha ψis injec i e, and, consequen ly,
ha ψis an isomo phism.
Le us hen e i y in Bn(M) he ela ions o Type 1. In he case o (PR1), we s a
wi h (R3):
a1,1···a1,2ga−1
1,1···a−1
1,2g=σ1···σn−2σ2
n−1σn−2· · ·σ1.(3)
Using (R7) and (R8), we see ha he le hand side o Equa ion (3) becomes
σ1···σn−1(an,1···an,2g)σ−1
n−1···σ−2
1···σ−1
n−1a−1
n,1···a−1
n,2gσn−1···σ1.
On he o he hand, om (R1), (R2) (b aid ela ions) and (R9), we ge
T−1
i,n−1Ti,n =σ−1
iσ−1
i+1 ···σ−1
n−2σ2
n−1σn−2···σi=σn−1···σi+1σ2
iσ−1
i+1 ···σ−1
n−1,
so n−1
Y
i=1
T−1
i,n−1Ti,n =σn−1···σ2
1···σn−1.
The e o e, Equa ion (3) becomes
an,1···an,2g n−1
Y
i=1
T−1
i,n−1Ti,n!−1
a−1
n,1···a−1
n,2g= 1,
which is clea ly equi alen o (PR1).
We will use in wha ollows some ela ions o Bn(M) easily deduced om (R1)-(R9).
F om (R7) and (R8), we ge
ai, =σ−1
i−1···σ−1
1a1, σ−1
1···σ−1
i−1i is odd. (4)
ai, = (σi−1···σ1)a1, (σ1···σi−1) i is e en. (5)
Aj,s =σ−1
j−1···σ−1
2A2,s σ−1
2···σ−1
j−1=σ−1
j−1···σ−1
1A1,s σ−1
1···σ−1
j−1.(6)
Also, om (R1) and (R2), we ob ain
σj(σkσk−1···σi) = (σkσk−1···σi)σj+1 (i≤j < k).(7)
σjσ−1
kσ−1
k−1···σ−1
i=σ−1
kσ−1
k−1···σ−1
iσj+1 (i≤j < k).(8)
18
σi···σk−1σ2
kσ−1
k−1···σ−1
i=σ−1
k···σ−1
i+1σ2
iσi+1 ···σk.(9)
Now using (6), (7), (8) and (R6), we see ha i 1 ≤k≤j−2;
σkAj,s =σkσ−1
j−1···σ−1
1A1,s σ−1
1···σ−1
j−1
=σ−1
j−1···σ−1
1σk+1A1,s σ−1
1···σ−1
j−1
=σ−1
j−1···σ−1
1A1,sσk+1 σ−1
1···σ−1
j−1=Aj,sσk.(10)
In he same way, using (6), (R6) and (R4), we ge
a1, Aj,s =a1, σ−1
j−1···σ−1
2A2,s σ−1
2···σ−1
j−1=Aj,sa1, ,
i 6=sand 1 < j.
The e o e, i i < j and 6=s, by (4) and (5) ai, is a p oduc o elemen s which commu e
wi h Aj,s, so we ob ain
ai, Aj,s =Aj,sai, ,
which shows ha (PR2) holds in Bn(M).
Now we e i y Rela ion (PR3). We will do he case when is odd, he o he case being
analogous. I is clea which o he known ela ions o Bn(M) we a e using a each o he
ollowing equali ies:
(ai,1. . . ai, )Aj, =σ−1
i−1···σ−1
1(a1,1. . . a1, )σ−1
1···σ−1
i−1Aj,
=σ−1
i−1···σ−1
1(a1,1. . . a1, )Aj, σ−1
1···σ−1
i−1
=σ−1
i−1···σ−1
1(a1,1. . . a1, )σ−1
j−1···σ−1
2A2, σ−1
2···σ−1
j−1σ−1
1···σ−1
i−1
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2(a1,1. . . a1, )A2, σ−1
2···σ−1
j−1σ−1
1···σ−1
i−1
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2σ2
1A2, (a1,1. . . a1, )σ−1
2···σ−1
j−1σ−1
1···σ−1
i−1
=σi···σj−2σ2
j−1σ−1
j−2···σ−1
1σ−1
j−1···σ−1
2A2, σ−1
2···σ−1
j−1(a1,1. . . a1, )σ−1
1···σ−1
i−1
=σi···σj−2σ2
j−1σ−1
j−2···σ−1
1Aj, (a1,1. . . a1, )σ−1
1···σ−1
i−1
=σi···σj−2σ2
j−1σ−1
j−2···σ−1
iAj, (ai,1. . . ai, )
=Ti,jT−1
i,j−1Aj, (ai,1. . . ai, ).
19
This shows he case o (PR3). Rela ions (PR4) and (PR5) a e ac ually ela ions in
he b aid g oup o he disc, so hey a e a consequence o (R1) and (R2). (PR6) is ob-
ained easily om (R9), (4), (5) and he b aid ela ions (R1) and (R2). So we may
u n o (PR7): I is clea ha i su ices o show ha in Bn(M), Ai, commu es wi h
a−1
j,2g···a−1
j,1Tj,kaj,2g···aj,1 o 1 ≤j < i ≤k < n. This is shown as ollows ( emembe
ha we can al eady use (PB1)-(PB6)):
Ai, a−1
j,2g···a−1
j,1Tj,kaj,2g···aj,1
=a−1
j,2g···a−1
j, +1Ai, a−1
j, ···a−1
j,1Tj,kaj,2g···aj,1
=a−1
j,2g···a−1
j,1Tj,iT−1
j,i−1Ai, Tj,kaj,2g···aj,1
=a−1
j,2g···a−1
j,1Tj,iT−1
j,i−1Ai, σj···σ2
k−1···σjaj,2g···aj,1
=a−1
j,2g···a−1
j,1Tj,iT−1
j,i−1Ai, σj···σ2
k−1···σ1a1,2g···a1,1σ−1
1· · · σ−1
j−1
(using (R3))
=a−1
j,2g···a−1
j,1Tj,iT−1
j,i−1Ai, σj···σk−1σ−1
k···σ−2
n−1···σ−1
1a1,1···a1,2gσ−1
1···σ−1
j−1
(by (R9) and (10))
=a−1
j,2g···a−1
j,1σj···σ2
i−1Ai, σi−1···σk−1σ−1
k···σ−2
n−1···σ−1
1a1,1···a1,2gσ−1
1···σ−1
j−1
=a−1
j,2g···a−1
j,1σj···σi−1Ai−1, σi···σk−1σ−1
k···σ−2
n−1···σ−1
1a1,1···a1,2gσ−1
1···σ−1
j−1
=a−1
j,2g···a−1
j,1σj···σk−1σ−1
k···σ−2
n−1···σ−1
iσi−1σ−1
i−2···σ−1
1Ai, a1,1···a1,2gσ−1
1···σ−1
j−1
=a−1
j,2g···a−1
j,1σj···σk−1σ−1
k···σ−2
n−1···σ−1
1a1,1···a1,2gσ−1
1···σ−1
j−1Ai,
(by (R3) again)
=a−1
j,2g···a−1
j,1Tj,k σj−1···σ1a1,2g···a1,1σ−1
1···σ−1
j−1Ai,
=a−1
j,2g···a−1
j,1Tj,kaj,2g···aj,1Ai, .
Finally, Rela ion (PR8) is e i ied using some in e media y esul s. The i s is e iden :
by (R4) we see ha in Bn(M), A1,2gA2,2g=A2,2gA1,2g, and mo eo e his b aid commu es
20
wi h σ1, since
A1,2gA2,2gσ1=A1,2gσ−1
1A1,2g=σ1A2,2gA1,2g=σ1A1,2gA2,2g.
Analogously, one shows ha (a1,2ga2,2g) commu es wi h σ1. The ollowing esul is a con-
sequence o he p e ious ones and o (R5):
a1,2gA2,2ga−1
1,2g=a−1
1,2g−1···a−1
1,1σ2
1A2,2g(a1,1···a1,2g−1)
=A−1
1,2gσ2
1A2,2gA1,2g=A−1
1,2gA2,2gA1,2gσ2
1=A2,2gσ2
1,
so we ob ain
a−1
1,2gA2,2g=A2,2ga−1
1,2gσ−2
1.(11)
Now we conside he ac o s in he igh hand side o (PR8), and we see ha
a−1
i,2g···a−1
i,1Ti,j−1T−1
i,j (ai,1···ai,2g)
=a−1
i,2g···a−1
i,1σi···σj−2σ2
j−1σ−1
j−2···σ−1
i(ai,1···ai,2g)
=σ−1
i−1···σ−1
1a−1
1,2g···a−1
1,1σ1···σj−2σ2
j−1σ−1
j−2···σ−1
1(a1,1· · ·a1,2g)σ1···σi−1
(by (9))
=σ−1
i−1···σ−1
1a−1
1,2g···a−1
1,1σ−1
j−1···σ−1
2σ−2
1σ2···σj−1(a1,1···a1,2g)σ1···σi−1
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2a−1
1,2g···a−1
1,1σ−2
1(a1,1···a1,2g)σ2···σj−1σ1···σi−1
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2a−1
1,2gA−1
1,2gσ−2
1A1,2ga1,2g(σ2···σj−1σ1···σi−1)
(since (A1,2gA2,2g) commu es wi h σ1)
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2a−1
1,2gA2,2gσ−2
1A−1
2,2ga1,2g(σ2···σj−1σ1···σi−1)
(by (11))
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2A2,2ga−1
1,2gσ−2
1a1,2gA−1
2,2g(σ2···σj−1σ1···σi−1)
(since, (a1,2ga2,2g) commu es wi h σ1)
=σ−1
i−1···σ−1
1σ−1
j−1···σ−1
2A2,2ga2,2gσ−2
1a−1
2,2gA−1
2,2g(σ2···σj−1σ1···σi−1)
21

=σ−1
i−1···σ−1
1Aj,2gaj,2gσ−1
j−1···σ−1
2σ−2
1σ2···σj−1a−1
j,2gA−1
j,2g(σ1···σi−1)
(by (10))
=Aj,2gaj,2gσ−1
i−1···σ−1
1σ−1
j−1···σ−1
2σ−2
1σ2···σj−1(σ1· · · σi−1)a−1
j,2gA−1
j,2g
(by (9))
=aj,1···aj,2gσ−1
j−1···σ−1
i+1σ−2
iσi+1 ···σj−1a−1
j,2g···a−1
j,1.
And his clea ly yields (PR8):
Qj−1
i=1 a−1
i,2g···a−1
i,1Ti,j−1T−1
i,j ai,1···ai,2gaj,1···aj,2ga−1
j,1···a−1
j,2g
=Qj−1
i=1 aj,1···aj,2gσ−1
j−1···σ−1
i+1σ−2
iσi+1 ···σj−1a−1
j,2g···a−1
j,1aj,1···aj,2ga−1
j,1···a−1
j,2g
=aj,1···aj,2gσ−1
j−1···σ−1
2σ−2
1σ−1
2···σ−1
j−1a−1
j,1···a−1
j,2g
= (σj···σn−1)an,1···an,2gσ−1
n−1···σ−2
1···σ−1
n−1a−1
n,1···a−1
n,2g(σn−1···σj)
(by (R9) and (PR1))
= (σj···σn−1) (σn−1···σj) = Tj,n.
We ha e hus inished wi h ela ions o Type 1.
Conside now hose o Type 2. Fo each ela o in he p esen a ion o Σn, we mus ind
he wo d Umen ioned abo e.
The i s ela o is δiδjδ−1
iδ−1
j, when |i−j| ≥ 2 which, by (R1), yields in Bn(M) he
ela ion
σiσjσ−1
iσ−1
j= 1 (|i−j| ≥ 2).
Clea ly, Uis he i ial wo d.
The second ela o , δiδi+1δiδ−1
i+1δ−1
iδ−1
i+1 gi es, by (R2),
σiσi+1σiσ−1
i+1σ−1
iσ−1
i+1 = 1 (i= 1,···, n −2),
so in his case Uis also he i ial wo d.
Finally, by he hi d ela o δ2
i, we ob ain, using (R9),
σ2
i=Ti,i+1 (i= 1,···, n −1),
22
hence U=Ti,i+1.
So we ha e ob ained he ela ions in Bn(M) mapped by ψ o he ela ions o Type 2.
We inish he p oo o Theo em 2.1 ob aining he ela ions o Type 3. They a e e y
easy o deduce, using (10), (R1), (R2), (R7), (R8) and (R9). They a e he ollowing:
σiaj, σ−1
i=aj, (j6=i, i + 1),
σiai, σ−1
i=ai+1, T−1
i,i+1 i is e en,
σiai, σ−1
i=Ti,i+1ai+1, i is odd,
σiai+1, σ−1
i=Ti,i+1ai, i is e en,
σiai+1, σ−1
i=ai, T−1
i,i+1 i is odd,
σiTj,kσ−1
i=Tj,k (i6=j−1, j, k),
σiTi+1,kσ−1
i=Ti,kT−1
i,i+1,
σiTi,kσ−1
i=Ti,i+1Ti+1,k,
σiTj,iσ−1
i=Tj,i−1T−1
j,i Tj,i+1.
5 The b aid g oups o a non-o ien able su ace
This sec ion is de o ed o p o e Theo em 2.2, using he same me hod as be o e. Thus, le
Mbe a closed, non-o ien able su ace o genus g≥2.
S ep 1. Deno e by PBn(M) he g oup de ined by he ollowing p esen a ion.
P esen a ion 3
•Gene a o s: {ai, ; 1 ≤i≤n, 1≤ ≤g} ∪ {Tj,k; 1 ≤j < k ≤n}.
•Rela ions:
(P 1) a2
n,1···a2
n,g =Qn−1
i=1 T−1
i,n−1Ti,n.
(P 2) ai, Aj,s =Aj,sai, (1 ≤i < j ≤n; 1 ≤ , s ≤g; 6=s).
(P 3) a2
i,1···a2
i, −1ai, Aj, a−1
i, a−2
i, −1···a−2
i,1A−1
j, =Ti,j T−1
i,j−1
(1 ≤i < j ≤n; 1 ≤ ≤g).
(P 4) Ti,jTk,l =Tk,lTi,j (1 ≤i < j < k < l ≤no 1 ≤i < k < l ≤j≤n).
(P 5) Tk,lTi,jT−1
k,l =Ti,k−1T−1
i,k Ti,jT−1
i,l Ti,kT−1
i,k−1Ti,l (1 ≤i < k ≤j < l ≤n).
23
(P 6) ai, Tj,k =Tj,kai, (1 ≤i < j < k ≤no 1 ≤j < k < i ≤n),(1 ≤ ≤g).
(P 7) ai, a−2
j,g ···a−2
j,1Tj,k=a−2
j,g ···a−2
j,1Tj,kai, (1 ≤j < i ≤k≤n).
(P 8) Tj,n =a2
j,1···a2
j,g Qj−1
i=1 T−1
j−i,jTj−i,j−1.
Whe e
Aj, =a2
j,1···a2
j, −1a−1
j, a−2
j, −1···a−2
j,1.
We shall need, as in he o ien able case, ano he p esen a ion o PBn(M), which is he
ollowing one.
P esen a ion 4
•Gene a o s: {Ai, ; 1 ≤i≤n, 1≤ ≤g} ∪ {Tj,k; 1 ≤j < k ≤n}.
•Rela ions: he same as in P esen a ion 3, whe e
ai, =A2
i,1···A2
i, −1A−1
i, A−2
i, −1A−2
i,1.
I is clea ha P esen a ion 3 and P esen a ion 4 a e equi alen , in he same way as
hey we e P esen a ion 1 and P esen a ion 2. We mus now de ine he homomo phism
PBn(M)ϕ
−→ P Bn(M),
by gi ing he image o he gene a o s. They will be simila o hose o he o ien able
su ace. Fo all iand jsuch ha 1 ≤i≤j≤n, he b aid Ti,j will be he same as in
Sec ion 4. Fo all i, , such ha 1 ≤i≤nand 1 ≤ ≤g, he b aid ai, will ep esen he
i- h s ing passing h ough he - h wall, in he way o Figu e 13. We de ine as well he
pa h ei(i= 1,...,n), which goes om Pi o he inal poin o e, as in Figu e 13.
Gi en i∈ {1,...,n}, deno e by si, he i- h s ing o ai, . We can p oceed as we did o
he P1-polygon in Sec ion 2 o ge he Pi-polygon: Cu along he pa hs eiand si,1,...,si,g,
and glue along eand α1, . . . , αg. The esul ing Pi-polygon is labeled by he pa hs
si,1, si,1, si,2, si,2,...,si,g, si,g, ei, e−1
i,
eading clockwise. Now we can epea he p ocess o Sec ion 2 o see ha o 1 ≤i < j,
he b aid
Aj, =a2
j,1···a2
j, −1a−1
j, a−2
j, −1···a−2
j,1
can be ep esen ed in he Pi-polygon in he way o Figu e 14.
The emainde o S ep 1, ha is o show ha ϕis a well de ined homomo phism,
is analogous o he o ien able case. Tha is, Rela ions (P 4), (P 5) and (P 6) a e ob i-
ous; Rela ions (P 1), (P 2) and (P 3) a e analogous o Rela ions ( 3), ( 4) and ( 5) o
Theo em 2.2; and we can easily check Rela ions (P 7) and (P 8) in he Pj-polygon.
24
ai,
ee
α α
ei
Pi
Pn
P1
P1Pn
Pi
Ti,j
Pj
Figu e 13: The gene a o s o PBn(M).
si,
ei
Pi
si,
Pn
Pj
P1
si,1si,g
ei
Figu e 14: The b aid Aj, in he Pi-polygon (i < j).
S ep 2. This s ep pa allels, up o e iden subs i u ions, he co esponding one in Sec ion 4,
showing he ollowing heo em:
Theo em 5.1. I Mis a closed, non-o ien able su ace o genus g≥2, hen P Bn(M)
admi s P esen a ion 3(and also P esen a ion 4) as p esen a ion.
S ep 3. Deno e by Bn(M) he g oup de ined by he p esen a ion o Theo em 2.2. Call
a1, he elemen s a o = 1,...,g, and hen add he gene a o s
−ai, i= 2,...,n; = 1,...,g,
−Tj,k 1≤j < k ≤n,
and he ela ions
25