a Xi :ma h/0112313 1 [ma h.GT] 31 Dec 2001
P esen a ions o he monoids o singula b aids
on closed su aces
Juan Gonz´alez-Meneses
No embe , 2000
Abs ac
We gi e p esen a ions, in e ms o gene a o s and ela ions, o he monoids SBn(M) o
singula b aids on closed su aces. The p oo o he alidi y o hese p esen a ions can also be
applied o e i y, in a new way, he p esen a ions gi en by Bi man o he monoids o Singula
A in b aids.
1 In oduc ion
In his pape we deal wi h he b aid g oups o a closed su ace M. These g oups a e a na u al
gene aliza ion o A in b aid g oups [A] and o he undamen al g oup o M. They a e also
subg oups o some Mapping Class g oups o M, and inally hey a e undamen al g oups o he so
called Con igu a ion spaces o M(see [B] o a gene al exposi ion).
They can be de ined as ollows. Fix n(n≥1) dis inc poin s {P1,...,Pn} ∈ M. A n-b aid on
Mis an n- uple b= (b1, . . . , bn) o disjoin smoo h pa hs biin M×[0,1], such ha o all i, he pa h
bi uns, mono onically on ∈[0,1], om (Pi,0) o some (Pj,1). These n-b aids a e conside ed
modulo iso opy (de o ma ion o b aids ixing he ends), and he e exis s a mul iplica ion o b aids,
gi en by conca ena ion o pa hs. The se o iso opy classes o n-b aids on M, along wi h his
mul iplica ion, o ms he b aid g oup wi h ns ings on M, deno ed by Bn(M).
The ollowing is a simple p esen a ion o Bn(M), in e ms o gene a o s and ela ions, whe e
Mis a closed, o ien able su ace o genus g[G-M]:
•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.
The gene a o s a e ep esen ed in Figu e 1, whe e we ha e d awn he he canonical p ojec ions
on Mo he conside ed b aids, and Mis ep esen ed as a polygon o 4gsides, pai wise iden i ied.
Keywo ds: B aid - Singula B aid - Su ace - Monoid - P esen a ion.
Ma hema ics Subjec Classi ica ion: P ima y: 20F36. Seconda y: 20F05.
Pa ially suppo ed by DGESIC-PB97-0723 and by he eu opean ne wo k TMR Sing. Eq. Di . e Feuill.
1
a2k
σi
α2k+1
α2k+1
α2k
α2k
P1Pn
P1Pn
PiPi+1 P1
α1
α2
α2g
α2g
α1
α2
a2k+1
Pn
Figu e 1: The gene a o s o Bn(M).
We can also ind in [G-M] a simila p esen a ion, when Mis a non-o ien able, closed su ace.
In he same way ha singula A in b aids we e de ined (see [B2]) o s udy Vassilie in a ian s
o hese b aids, we can de ine singula b aids on M. Thei de ini ion is he same ha he
one o non-singula b aids, bu his ime we allow a ini e numbe o singula poin s ( ans e se
in e sec ion o wo s ings). The iso opy classes o hese singula b aids, wi h he analogous
mul iplica ion, o m he monoid o singula b aids wi h ns ings on M, deno ed by SBn(M).
This monoid is used in [G-MP] o de ine he Vassilie in a ian s o b aids on closed, o ien able
su aces, p o ing, among o he esul s, ha hese in a ian s classi y hese b aids.
In [B2] we can ind p esen a ions o SBn, he monoids o singula A in b aids, in e ms
o gene a o s and ela ions. The main esul o his pape is o gi e p esen a ions o SBn(M).
We will see, as well, ha he p oo o his esul s u nishes a new p oo o he alidi y o he
p esen a ions in [B2].
2 P esen a ion o SBn(M)
We shall now gi e a p esen a ion o Bn(M), when Mis a closed, o ien able su ace o genus g≥0.
The non-o ien able case is comple ely analogous, and is ea ed in a inal ema k a he end o
his pape . We de ine, o all i= 1,...,n−1, he singula b aid τias in Figu e 2, whe e he only
non- i ial s ings a e he i- h and he (i+ 1)- h ones, which in e sec o o m a singula poin .
The esul is he ollowing:
Theo em 2.1. The monoid SBn(M)admi s he ollowing p esen a ion:
•Gene a o s: σ1, . . . , σn−1, a1,...,a2g, τ1,...,τn−1.
•Rela ions:
(R1-R6) Rela ions o Bn(M)
(R7) σiτj=τjσi(|i−j| ≥ 2)
(R8) τiτj=τjτi(|i−j| ≥ 2)
(R9) σiτi=τiσi(i= 1,...,n−1)
(R10) σiσjτi=τjσiσj(|i−j|= 1)
(R11) (ai, ai+1, )τi(a−1
i+1, a−1
i, ) = τi(i= 1, . . . , n −1; = 1,...,2g)
(R12) τiaj, =aj, τi(j6=i, i + 1; = 1,...,2g)
whe e
ai, =(σ−1
i−1···σ−1
1)a (σ−1
1···σ−1
i−1)i is odd,
(σi−1···σ1)a (σ1···σi−1)i is e en.
2
P1PiPi+1 Pn
M× {1}
M× {0}
Figu e 2: The singula b aid τi.
Pn
Pi
α
Pn
Pi
P1
α
α α
P1
Figu e 3: The b aid ai, = (σ−1
i−1···σ−1
1)a (σ−1
1···σ−1
i−1), when is odd.
Rema k ha ai, can be hough o as he i- h s ing c ossing he “wall” α , as we can see in
Figu e 3 o he case when is odd.
P oo o Theo em 2.1: Fi s , i is e iden ha {σ1,...,σn−1, a1,...,a2g, τ1,...,τn−1}is a se
o gene a o s o SBn(M), once ha we know (by [G-M]) ha {σ1,...,σn−1, a1,...,a2g}gene a es
Bn(M).
I is also easy o p o e ha he p oposed ela ions hold: (R1-R6) hold in Bn(M), which is a
sub-monoid o SBn(M). (R7-R10) a e known o hold in SBn, so hey hold in a cylinde D×[0,1],
whe e Dis a disk con aining he npoin s P1,...,Pn. We ha e jus o ex end he co esponding
iso opy o all M×[0,1] by he iden i y. (R11) can be seen o hold in Figu e 4, and inally (R12)
is clea , since he only non i ial s ings o τiand aj, can be iso oped o ha e disjoin p ojec ions
on M, so hese b aids commu e.
PiPi+1
P1Pi
P1Pi+1
PnPn
α
α
α α
Figu e 4: The b aids ai, ai+1, τiand τiai, ai+1, a e iso opic, when is odd.
In o de o show ha he ela ions a e su icien , we need he ollowing lemma:
Lemma 2.2. The monoid SBn(M)is le -cancela i e. Tha is, o all a, b, c ∈SBn(M), one has:
c a =c b ⇒a=b.
P oo : Since σ1,...,σn−1, a1,...,a2ga e in e ible, o hey belong o Bn(M), we jus need o
p o e ha τia=τib⇒a=b o all a, b ∈SBn(M) and all i= 1,...,n−1.
Thus, le us suppose ha he e exis s an iso opy H o M×[0,1], such ha H0= idM×[0,1]
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and H1(τia) = τib. Call p he i s singula poin o τia( he one co esponding o τi), and le
p =H (p). One has p0=p1=p.
Le Vbe he in e io o a sphe e o adius εcen e ed a p. We ake εsmall enough, such ha
V∩(τia) is as ollows:
V
p
Deno e s =H (τia) and V =H (V). We can suppose, wi hou loss o gene ali y, ha V is
he in e io o he sphe e o adius εcen e ed a p , and ha V ∪s is as in he abo e pic u e.
Now, o ∈[0,1], deno e by es he b aid which is ob ained by modi ying s , only inside V , as
ollows:
We obse e ha es0=aand es1=b, so H is an iso opy which ans o ms ain o b. The e o e,
a=b.
Le us hen show ha Rela ions (R1-R12) a e su icien . Le b, b′∈SBn(M) be wo iso opic
singula b aids, w i en in he gene a o s o Theo em 2.1. We mus show ha we can ans o m
bin o b′by using Rela ions (R1-R12).
Being iso opic, bo h b aids ha e he same numbe o singula poin s, say k. I k= 0, he esul
ollows om [G-M], since (R1-R6) a e su icien ela ions o Bn(M).
Suppose ha k > 0, and he esul holds o b aids wi h less han ksingula poin s. We can
assume ha he i s le e o bis τi, o some i(o he wise we can mul iply band b′on he le
by he g ea es nonsingula “p e ix” o b). We will show ha , using (R1-R12), we can ans o m
b′in o a b aid whose i s le e is τi. The esul hen ollows om Lemma 2.2, and by induc ion
hypo hesis.
Le pbe he poin o b′co esponding ( ia iso opy) o he i s singula poin o b. This poin p
mus co espond o some τj, le e o b′. By (R10) and he b aid ela ions (R1-R2), we can easily
deduce he ollowing:
τj=(σj−1σj−2···σi)(σjσj−1···σi+1)τi(σ−1
i+1 ···σ−1
j)(σ−1
i···σ−1
j+1) i i < j,
(σj+1σj+2 ···σi)(σjσj+1 ···σi−1)τi(σ−1
i−1···σ−1
j)(σ−1
i···σ−1
j+1) i i > j.
Hence, we can assume ha he le e co esponding o pis τi.
Le us hen w i e b′=u τi , whe e u, ∈SBn(M) and τiis he abo e le e . Since bis
iso opic o b′, we can assume, up o eplacing τiby σiτiσ−1
i(using (R9)), ha he i- h s ing o u
ends a he poin (Pi, s), o some s∈[0,1]. Hence, i s canonical p ojec ion on Mis a loop in M
based a Pi, which induces an elemen µ∈π1(M, P1). This elemen can be modi ied as desi ed:
i su ices o use (R11), eplacing τiby aε
i, aε
i+1, τia−ε
i+1, a−ε
i, (ε=±1), o ha e µ ans o med in o
µaε
i, , whe e aε
i, is he p ojec ion on Mo he i- h s ing o aε
i, . Since {ai,1, . . . , ai,2g}is a se o
gene a o s o π1(M, Pi), we can assume ha µ= 1.
Now no ice ha uhas less han ksingula poin s, hence any b aid iso opic o ucan be ob ained
om i by applying (R1-R12), by induc ion hypo hesis. We can hen de o m uin such a way ha
i s i- h s ing will no go h ough he “walls” α1,...,α2g(we can do his since µ= 1).
Le us go back o b, and conside a “band” Γ, de e mined by he i- h and he (i+ 1)- h s ings
o b, and which goes om s= 0 o he i s singula poin o b, as in he igu e below.
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PiPi+1
b
Γ
Conside also an iso opy H which ans o ms bin o b′. Recall ha b′=u τi , whe e he i- h
s ing o udoes no go h ough he walls. We can now conside Γ1=H1(Γ), and de o m he
(i+ 1)- h s ing o ualong his band, in such a way ha i will be as close o he i- h s ing as
desi ed ( ecall ha we a e allowed o de o m u). We can hen assume ha nei he he i- h no he
(i+ 1)- h s ing o ugoes h ough he “walls” o he cylinde M×[0,1]. Mo eo e , using (R9) we
can modi y he numbe o c ossings o hese wo s ings, as desi ed (jus eplacing τiby σ
iτiσ−
i,
∈Z). The e o e, we can assume ha hey do no c oss, i.e. he e is no σjin uin ol ing he
i− h and he (i+ 1)- h s ings.
We can also assume ha hese wo s ings a e so close ha one has he ollowing p ope y:
i he e is a le e σε
j(ε=±1) o uwhich in ol es he i- h o he (i+ 1)- h s ing, hen his
le e , oge he wi h ei he he p e ious o he ollowing one, o ms a sub-wo d o uo one o he
ollowing ou ypes:
σjσj+1 σ−1
j+1σ−1
jσ−1
jσ−1
j+1 σj+1σj
Γ1Γ1
Γ1Γ1
Bu in his case i is easy o see ha , using ela ions (R7), (R8), (R10) and (R12), we can
“ aise” he poin p, un il we ge τias he i s le e o b′. So by Lemma 2.2 we can cancel τi, and
by induc ion hypo hesis he esul ing b aids a e equi alen by means o Rela ions (R1-R12). This
ends he p oo o Theo em 2.1
Rema k 2.3. The e is an analogous p esen a ion o SBn(M), when Mis a non-o ien able, closed
su ace. We jus need o conside he p esen a ion gi en in [G-M] o Bn(M). Then eplace, in he
p esen a ion o Theo em 2.1, he gene a o s a1,...,a2gby he co esponding gene a o s on he
non-o ien able su ace, and Rela ions (R1-R6) by he ela ions gi en in [G-M]. The same p oo
emains alid.
Rema k 2.4. The p esen a ion gi en in Theo em 2.1 can be easily simpli ied. I su ices o elim-
ina e he gene a o s τ2, . . . , τn−1, eplacing in he ela ions τ3by (σ2σ1σ3σ2)τ1(σ−1
2σ−1
3σ−1
1σ−1
2),
and elimina ing all ela ions con aining some τj(j6= 1,3), since hey a e ob ained om he e-
maining ones. We p oposed he p esen a ion abo e since i is mo e use ul o handling singula
b aids.
Rema k 2.5. We can also eplace (R1-R6) by any o he se o su icien ela ions o he gi en
gene a o s o Bn(M).
Rema k 2.6. The abo e p oo o Theo em 2.1, a e elimina ing e e y allusion o π1(M), is a
new p oo o he alidi y o he p esen a ion o SBnp oposed in [B2].
Acknowledgemen s: I wish o hank P o . O lando Ne o, o gi ing me he oppo uni y o enjoy
he excellen wo king condi ions I ound a he Cen o de Ma em´a ica e Aplica¸coes Fundamen ais
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o he Uni e si y o Lisbon, whe e his pape was w i en down.
Re e ences
[A] E. ARTIN, Theo y o b aids, Annals o Ma h. 48 (1946) 101-126.
[B] J. S. BIRMAN, “B aids, Links and Mapping Class G oups”, Annals o Ma h. S udies 82, P ince on
Uni e si y P ess, 1973.
[B2] J. S. BIRMAN New poin s o iew in kno heo y, Bull. Ame . Ma h. Soc. 28 (1993), no. 2, 253-287.
[G-M] J. GONZ ´
ALEZ-MENESES, New p esen a ions o su ace b aid g oups, J. o Kno Theo y and i s
Rami iac ions. To appea .
[G-MP] J. GONZ ´
ALEZ-MENESES and L. PARIS, Vassilie in a ian s o su ace b aid g oups, P ep in .
J. GONZ´
ALEZ-MENESES
Depa amen o de Ma em´a ica Aplicada I
Escuela T´ecnica Supe io de A qui ec u a
A da. Reina Me cedes, 2
41012-Se illa (Spain)
[email protected]
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