An infinite presentation for the mapping class group of a non-orientable surface

0.3 MB·

by  Genki Omori

AN INFINITE PRESENTATION FOR THE MAPPING CLASS
GROUP OF A NON-ORIENTABLE SURFACE
6
1 GENKIOMORI
0
2

l
u
J Abstract. Wegiveaninfinitepresentationforthemappingclassgroupofa

1 non-orientable surface. Thegenerating set consists of all Dehn twists and all
1 crosscappushingmapsalongsimpleloops.

] 1. Introduction
T
G Let Σ be a compact connected orientable surface of genus g ≥ 0 with n ≥ 0
g,n
. boundary components. The mapping class group M(Σg,n) of Σg,n is the group
h
of isotopy classes of orientation preserving self-diffeomorphisms on Σ fixing the
t g,n
a boundary pointwise. A finite presentation for M(Σ ) was given by Hatcher-
g,n
m
Thurston[6],Wajnryb[17],Harer[5],Gervais[4]andLabru`ere-Paris[9]. Gervais[3]
[ obtained an infinite presentation for M(Σ ) by using Wajnryb’s finite presenta-
g,n
tionforM(Σ ),andLuo[12]rewroteGervais’presentationintoasimplerinfinite
2 g,n
v presentation (see Theorem 2.5).
6 Let N be a compact connected non-orientable surface of genus g ≥ 1 with
g,n
1 n ≥ 0 boundary components. The surface N = N is a connected sum of g real
g g,0
4
projectiveplanes. ThemappingclassgroupM(N )ofN isthegroupofisotopy
1 g,n g,n
classesofself-diffeomorphismsonN fixingtheboundarypointwise. Forg ≥2and
0 g,n
. n ∈{0,1}, a finite presentation for M(Ng,n) was given by Lickorish [10], Birman-
1
Chillingworth [1], Stukow [14] and Paris-Szepietowski [13]. Note that M(N ) and
0 1
6 M(N1,1)aretrivial(see[2,Theorem3.4])andM(N2)isfinite(see[10,Lemma5]).
1 Stukow[15]rewroteParis-Szepietowski’spresentationintoafinitepresentationwith
: Dehn twists and a “Y-homeomorphism” as generators (see Theorem 2.11).
v
i In this paper, we give a simple infinite presentation for M(Ng,n) (Theorem 3.1)
X
when g ≥ 1 and n ∈ {0,1}. The generating set consists of all Dehn twits and all
r “crosscappushing maps” alongsimple loops. We review the crosscappushing map
a
in Section 2. We prove Theorem 3.1 by applying Gervais’ argument to Stukow’s
finite presentation.
2. Preliminaries
2.1. Relations among Dehn twists and Gervais’ presentation. Let S be
either N or Σ . We denote by N (A) a regular neighborhood of a subset A
g,n g,n S
in S . For every simple closed curve c on S, we choose an orientation of c and
fix it throughout this paper. However, for simple closed curves c , c on S and
1 2
f ∈M(S), f(c )=c means f(c ) is isotopic to c or the inverse curve of c . If S
1 2 1 2 2
is a non-orientable surface, we also fix an orientation of N (c) for each two-sided
S
simple closed curve c. For a two-sidedsimple closed curve c on S, denote by t the
c
Date:July12,2016.
1
2 G.OMORI
right-handed Dehn twist along c on S. In particular, for a given explicit two-sided
simple closed curve, an arrow on a side of the simple closed curve indicates the
direction of the Dehn twist (see Figure 1).
Figure 1. Theright-handedDehntwistt alongatwo-sidedsim-
c
ple closed curve c on S.
Recallthe followingrelationsonM(S) amongDehn twists along two-sidedsim-
ple closed curves on S.
Lemma 2.1. For a two-sided simple closed curve c on S which bounds a disk or a
M¨obius band in S, we have t =1 on M(S).
c
Lemma 2.2 (The braid relation (i)). For a two-sided simple closed curve c on S
and f ∈M(S), we have
tεf(c) =ft f−1,
f(c) c
where εf(c) = 1 if the restriction f|NS(c) : NS(c) → NS(f(c)) is orientation pre-
serving and εf(c) =−1 if the restriction f|NS(c) :NS(c)→NS(f(c)) is orientation
reversing.
When f in Lemma 2.2 is a Dehn twist t along a two-sided simple closed curve
d
d and the geometric intersection number |c∩d| of c and d is m, we denote by T
m
the braid relation.
Let c , c , ..., c be two-sided simple closed curves on S. The sequence c ,
1 2 k 1
c , ..., c of simple closed curves on S is a k-chain on S if c , c , ..., c satisfy
2 k 1 2 k
|c ∩c |=1 for each i=1, 2, ..., k−1 and |c ∩c |=0 for |j−i|>1.
i i+1 i j
Lemma 2.3 (The k-chain relation). Let c , c , ..., c be a k-chain on S and let
1 2 k
δ , δ (resp. δ) be distinct boundary components (resp. the boundary component)
1 2
of N (c ∪c ∪···∪c ) when k is odd (resp. even). Then we have
S 1 2 k
(tcε1c1tcε2c2 ···tcεkck)k+1 = tδε1δ1tδε2δ2 when k is odd,
(tcε1c1tcε2c2 ···tcεkck)2k+2 = tεδδ when k is even,
awnhderteεδε2c1(,reεscp2., .tε.δ.), aεrcek,rεigδh1,t-εhδa2ndaenddDεeahrnet1wiosrts−fo1r, saonmdetcεo1cr1i,entcεt2ca2t,io.n..o,ftNcεkck,(ctδε1δ∪1
δ2 δ S 1
c ∪···∪c ).
2 k
Lemma 2.4 (The lanternrelation). Let Σ be a subsurface of S which is diffeomor-
phic to Σ and let δ , δ , δ , δ , δ , δ and δ be simple closed curves on Σ as
0,4 12 23 13 1 2 3 4
in Figure 2. Then we have
tεδ12tεδ23tεδ13 =tεδ1tεδ2tεδ3tεδ4,
δ12 δ23 δ13 δ1 δ2 δ3 δ4
where ε , ε , ε , ε , ε , ε and ε are 1 or −1, and tεδ12, tεδ23, tεδ13, tεδ1,
tεδ2, tεδ3δ12andδ2t3εδ4 δa1r3e riδg1ht-δh2andδe3d Dehnδ4twists for some orienδ1t2ationδ23of Σδ.13 δ1
δ2 δ3 δ4
PRESENTATION FOR MAPPING CLASS GROUP 3
Figure 2. Simple closed curves δ , δ , δ , δ , δ , δ and δ on Σ.
12 23 13 1 2 3 4
Luo’s presentation for M(Σ ), which is an improvement of Gervais’ one, is as
g,n
follows.
Theorem 2.5 ([3], [12]). For g ≥0 and n≥0, M(Σ ) has the following presen-
g,n
tation:
generators: {t |c: s.c.c. on Σ }.
c g,n
relations:
(0′) t =1 when c bounds a disk in Σ ,
c g,n
(I′) All the braid relations T and T ,
0 1
(II) All the 2-chain relations,
(III) All the lantern relations.
2.2. Relations among the crosscap pushing maps and Dehn twists. Let µ
be a one-sided simple closed curve on N and let α be a simple closed curve on
g,n
N suchthatµandαintersecttransverselyatonepoint. Recallthatαisoriented.
g,n
Forthesesimpleclosedcurvesµ andα, wedenotebyY aself-diffeomorphismon
µ,α
N which is described as the result of pushing the Mo¨bius band N (µ) once
g,n Ng,n
along α. We call Y a crosscap pushing map. In particular, if α is two-sided, we
µ,α
callY aY-homeomorphism (orcrosscap slide),whereacrosscapmeansaMo¨bius
µ,α
band in the interior of a surface. The Y-homeomorphism was originally defined
by Lickorish[10]. We have the following fundamental relationon M(N ) and we
g,n
also call the relation the braid relation.
Lemma 2.6 (The braid relation (ii)). Let µ be a one-sided simple closed curve
on N and let α be a simple closed curve on N such that µ and α intersect
g,n g,n
transversely at one point. For f ∈M(N ), we have
g,n
Yεf(α) =fY f−1,
f(µ),f(α) µ,α
where ε = 1 if the fixed orientation of f(α) coincides with that induced by the
f(α)
orientation of α, and ε =−1 otherwise.
f(α)
We describe crosscappushing maps froma differentpoint ofview. Let e:D′ ֒→
intS be a smooth embedding of the unit disk D′ ⊂ C. Put D := e(D′). Let S′ be
the surface obtainedfromS−intD by the identificationofantipodalpoints of ∂D.
WecallthemanipulationthatgivesS′ fromS theblowup ofS onD. Notethatthe
image M ⊂ S′ of NS−intD(∂D)⊂ S−intD with respect to the blowup of S on D
is a crosscap. Conversely, the blowdown of S′ on M is the following manipulation
thatgivesS fromS′. WepasteadiskontheboundaryobtainedbycuttingS along
the center line µ of M. The blowdown of S′ on M is the inverse manipulation of
the blowup of S on D.
4 G.OMORI
Let µ be a one-sided simple closed curve on Ng,n. Note that we obtain Ng−1,n
from N by the blowdown of N on N (µ). Denote by x the center point
g,n g,n Ng,n µ
of a disk D that is pasted on the boundary obtained by cutting S along µ. Let
µ
e:D′ ֒→Dµ ⊂Ng−1,n be a smooth embedding of the unit disk D′ ⊂C to Ng−1,n
such that Dµ = e(D′) and e(0) = xµ. Let M(Ng−1,n,xµ) be the group of isotopy
classes of self-diffeomorphisms on Ng−1,n fixing the boundary ∂Ng−1,n and the
pointxµ,where isotopiesalsofix the boundary∂Ng−1,n andxµ. Thenwehavethe
blowup homomorphism
ϕµ :M(Ng−1,n,xµ)→M(Ng,n)
that is defined as follows. For h ∈ M(Ng−1,n,xµ), we take a representative h′ of
h which satisfies either of the following conditions: (a) h′| is the identity map
Dµ
on D , (b) h′(x)=e(e−1(x)) for x∈D , where e−1(x) is the complex conjugation
µ µ
of e−1(x) ∈ C. Such h′ is compatible with the blowup of Ng−1,n on Dµ, thus
ϕ (h)∈M(N ) is induced and well defined (c.f. [16, Subsection 2.3]).
µ g,n
The point pushing map
jxµ :π1(Ng−1,n,xµ)→M(Ng−1,n,xµ)
is a homomorphism that is defined as follows. For γ ∈ π1(Ng−1,n,xµ), jxµ(γ) ∈
M(Ng−1,n,xµ) is described as the result of pushing the point xµ once along γ.
The point pushing map comes from the Birman exact sequence. Note that for γ ,
1
γ2 ∈π1(Ng−1,n),γ1γ2meansγ1γ2(t)=γ2(2t)for0≤t≤ 21 andγ1γ2(t)=γ1(2t−1)
for 1 ≤t≤1.
2
Following Szepietowski [16] we define the composition of the homomorphisms:
ψxµ :=ϕµ◦jxµ :π1(Ng−1,n,xµ)→M(Ng,n).
For each closed curve α on N which transversely intersects with µ at one point,
g,n
we take a loop α on Ng−1,n based at xµ such that α has no self-intersection points
on Dµ and α is the image of α with respect to the blowup of Ng−1,n on Dµ. If
α is simple, we take α as a simple loop. The next two lemmas follow from the
description of the point pushing map (see [8, Lemma 2.2, Lemma 2.3]).
Lemma 2.7. For a simple closed curve α on N which transversely intersects
g,n
with a one-sided simple closed curve µ on N at one point, we have
g,n
ψ (α)=Y .
xµ µ,α
Lemma 2.8. For a one-sided simple closed curve α on N which transversely
g,n
intersects with a one-sided simple closed curve µ on N at one point, we take
g,n
N (α) such that the interior of N (α) contains D . Suppose that δ and
Ng−1,n Ng−1,n µ 1
δ are distinct boundary components of N (α), and δ and δ are two-sided
2 Ng−1,n 1 2
simple closed curves on N which are image of δ , δ with respect to the blowup
g,n 1 2
of Ng−1,n on Dµ, respectively. Then we have
Y =tεδ1tεδ2,
µ,α δ1 δ2
where ε and ε are 1 or −1, and ε and ε depend on the orientations of α,
δ1 δ2 δ1 δ2
N (δ ) and N (δ ) (see Figure 3).
Ng,n 1 Ng,n 2
By the definition of the homomorphism ψ and Lemma 2.7, we have the fol-
xµ
lowing lemma.
PRESENTATION FOR MAPPING CLASS GROUP 5
Figure 3. If the orientations of α, N (δ ) and N (δ ) are
Ng,n 1 Ng,n 2
as above, then we have Y = t t−1. The x-mark means that
µ,α δ1 δ2
antipodal points of ∂D are identified.
µ
Lemma2.9. Letαandβ besimple closedcurves onN which transverselyinter-
g,n
sect with a one-sided simple closed curve µ on N at one point each. Suppose the
g,n
product αβ of α and β in π1(Ng−1,n,xµ) is represented by a simple loop on Ng−1,n,
and αβ is a simple closed curve on N which is the image of the representative of
g,n
αβ with respect to the blowup of Ng−1,n on Dµ. Then we have
Y =Y Y .
µ,αβ µ,α µ,β
Finally, we recall the following relation between a Dehn twist and a Y-
homeomorphism.
Lemma 2.10. Let α be a two-sided simple closed curve on N which transversely
g,n
intersect with a one-sided simple closed curve µ on N at one point and let δ be
g,n
the boundary of N (α∪µ). Then we have
Ng,n
Y2 =tε,
µ,α δ
where ε is 1 or −1, and ε depends on the orientations of α and N (δ) (see
Ng,n
Figure 4).
Lemma 2.10 follows from relations in Lemma 2.1, Lemma 2.8 and Lemma 2.9.
Figure 4. IftheorientationsofαandN (δ)areasabove,then
Ng,n
we have Y2 =t .
µ,α δ1
6 G.OMORI
2.3. Stukow’s finite presentation for M(N ). Let e : D′ ֒→ Σ for i = 1,
g,n i 0
2,..., g+1 be smooth embeddings of the unit disk D′ ⊂C to a 2-sphere Σ such
0
that D :=e (D′) and D are disjoint for distinct 1≤i,j ≤g+1. Then we take a
i i j
model of N (resp. N ) as the surface obtained from Σ (resp. Σ −intD ) by
g g,1 0 0 g+1
theblowupsonD ,...,D andwedescribetheidentificationof∂D bythex-mark
1 g i
asinFigures5and6. Whenn∈{0,1},for1≤i <i <···<i ≤g,letγ
1 2 k i1,i2,...,ik
bethesimpleclosedcurveonN asinFigure5. Thenwedefinethesimpleclosed
g,n
curves α := γ for i = 1, ..., g−1, β := γ and µ := γ (see Figure 6),
i i,i+1 1,2,3,4 1 1
and the mapping classes a := t for i = 1, ..., g−1, b := t and y := Y .
i αi β µ1,α1
Then the following finite presentation for M(N ) is obtained by Lickorish [10]
g,n
for (g,n) = (2,0), Stukow [14] for (g,n) = (2,1), Birman-Chillingworth [1] for
(g,n)=(3,0)andTheorem3.1andProposition3.3in[15]forthe other(g,n)such
that g ≥3 and n∈{0,1}.
Figure 5. Simple closed curve γ on N .
i1,i2,...,ik g,n
Figure 6. Simple closed curves α1, ..., αg−1, β and µ1 on Ng,n.
Theorem 2.11 ([10], [1], [14], [15]). For (g,n) = (2,0), (2,1) and (3,0), we have
the following presentation for M(N ):
g,n
M(N ) = a ,y |a2 =y2 =(a y)2 =1 ∼=Z ⊕Z ,
2 1 1 1 2 2
M(N2,1) = (cid:10)a1,y |ya1y−1 =a−11 , (cid:11)
M(N3) = (cid:10)a1,a2,y |a1a2a1 =a(cid:11)2a1a2,y2 =(a1y)2 =(a2y)2 =(a1a2)6 =1 .
If g ≥ 4 and (cid:10)n ∈ {0,1} or (g,n) = (3,1), then M(Ng,n) admits a presentati(cid:11)on
with generators a1,...,ag−1,y, and b for g ≥4. The defining relations are
(A1) [a ,a ]=1 for g ≥4, |i−j|>1,
i j
(A2) a a a =a a a for i=1,...,g−2,
i i+1 i i+1 i i+1
(A3) [a ,b]=1 for g ≥4, i6=4,
i
(A4) a ba =ba b for g ≥5,
4 4 4
(A5) (a a a b)10 =(a a a a b)6 for g ≥5,
2 3 4 1 2 3 4
PRESENTATION FOR MAPPING CLASS GROUP 7
(A6) (a a a a a b)12 =(a a a a a a b)9 for g ≥7,
2 3 4 5 6 1 2 3 4 5 6
(A9a) [b ,b]=1 for g =6,
2
(A9b) [ag−5,bg−2]=1 for g ≥8 even,
2
where b =a , b =b and
0 1 1
bi+1 =(bi−1a2ia2i+1a2i+2a2i+3bi)5(bi−1a2ia2i+1a2i+2a2i+3)−6
for 1≤i≤ g−4,
2
(B1) y(a a a a ya−1a−1a−1a−1) = (a a a a ya−1a−1a−1a−1)y for g ≥
2 3 1 2 2 1 3 2 2 3 1 2 2 1 3 2
4,
(B2) y(a a y−1a−1ya a )y =a (a a y−1a−1ya a )a ,
2 1 2 1 2 1 2 1 2 1 2 1
(B3) [a ,y]=1 for g ≥4, i=3,...,g−1,
i
(B4) a (ya y−1)=(ya y−1)a ,
2 2 2 2
(B5) ya =a−1y,
1 1
(B6) byby−1 = {a a a (y−1a y)a−1a−1a−1}{a−1a−1(ya y−1)a a } for
1 2 3 2 3 2 1 2 3 2 3 2
g ≥4,
(B7) [(a a a a a a a a ya−1a−1a−1a−1a−1a−1a−1a−1),b]=1 for g ≥6,
4 5 3 4 2 3 1 2 2 1 3 2 4 3 5 4
(B8) {(ya−1a−1a−1a−1)b(a a a a y−1)}{(a−1a−1a−1a−1)b−1(a a a a )}
1 2 3 4 4 3 2 1 1 2 3 4 4 3 2 1
={(a−1a−1a−1)y(a a a )}{a−1a−1y−1a a }{a−1ya }y−1 for g ≥5,
4 3 2 2 3 4 3 2 2 3 2 2
(C1) (a1a2···ag−1)g =1 for g ≥4 even and n=0,
(C2) [a ,ρ]=1 for g ≥4 and n=0,
1
where ρ=(a1a2···ag−1)g for g odd and
ρ=(y−1a2a3···ag−1ya2a3···ag−1)g−22y−1a2a3···ag−1 for g even,
(C3) ρ2 =1 for g ≥4 and n=0,
(C4) (y−1a2a3···ag−1ya2a3···ag−1)g−21 =1 for g ≥4 odd and n=0,
where [x ,x ]=x x x−1x−1.
1 2 1 2 1 2
3. Presentation for M(N )
g,n
The main theorem in this paper is as follows:
Theorem 3.1. For g ≥1 and n∈{0,1}, M(N ) has the following presentation:
g,n
generators: {t |c: two-sided s.c.c. on N }
c g,n
∪{Y |µ: one-sided s.c.c. on N , α: s.c.c. on N , |µ∩α|=1}.
µ,α g,n g,n
Denote the generating set by X.
relations:
(0) t =1 when c bounds a disk or a M¨obius band in N ,
c g,n
(I) All the braid relations
(i) ft f−1 =tεf(c) for f ∈X,
c f(c)
(ii) fY f−1 =Yεf(α) for f ∈X,
( µ,α f(µ),f(α)
(II) All the 2-chain relations,
(III) All the lantern relations,
(IV) All the relations in Lemma 2.9, i.e. Y =Y Y ,
µ,αβ µ,α µ,β
(V) All the relations in Lemma 2.8, i.e. Y =tεδ1tεδ2.
µ,α δ1 δ2
In (I) and (IV) one can substitute the right hand side of (V) for each generator
Y with one-sided α. Then one can remove the generators Y with one-sided α
µ,α µ,α
and relations (V) from the presentation.
8 G.OMORI
We denote by G the group which has the presentation in Theorem 3.1. Let
ι : Σ ֒→ N be a smooth embedding and let G′ be the group whose pre-
h,m g,n
sentation has all Dehn twists along simple closed curves on Σ as genera-
h,m
tors and Relations (0′), (I′), (II) and (III) in Theorem 2.5. By Theorem 2.5,
M(Σ ) is isomorphic to G′, and we have the homomorphism G′ → G de-
h,m
fined by the correspondence of t to tει(c), where ε = 1 if the restriction
c ι(c) ι(c)
ι|NΣh,m(c) : NΣh,m(c) → NNg,n(ι(c)) is orientation preserving, and ει(c) = −1 if
the restriction ι|NΣh,m(c) : NΣh,m(c) → NNg,n(ι(c)) is orientation reversing. Then
we remark the following.
Remark 3.2. The composition ι∗ : M(Σh,m) → G of the isomorphism
M(Σ ) → G′ and the homomorphism G′ → G is a homomorphism. In par-
h,m
ticular, if a product tεc11tεc22···tεckk of Dehn twists along simple closed curves c1, c2,
...,c onaconnectedcompactorientablesubsurfaceofN isequaltotheidentity
k g,n
map in the mapping class group of the subsurface, then tε1tε2···tεk is equal to 1
c1 c2 ck
in G. That means such a relation tε1tε2···tεk = 1 is obtained from Relations (0),
c1 c2 ck
(I), (II) and (III).
Set X± :=X ∪{x−1 |x∈X}. By Relation (I), we have the following lemma.
Lemma 3.3. For f ∈G, suppose that f =f f ...f , where f , f , ..., f ∈X±.
1 2 k 1 2 k
Then we have
(i) ft f−1 =tεf(c),
c f(c)
(ii) fY f−1 =Yεf(α) .
( µ,α f(µ),f(α)
The next lemma follows from an argument of the combinatorial group theory
(for instance, see [7, Lemma 4.2.1, p42]).
Lemma 3.4. For groups Γ, Γ′ and F, a surjective homomorphism π :F →Γ and
a homomorphism ν : F → Γ′, we define a map ν′ : Γ → Γ′ by ν′(x) := ν(x) for
x∈Γ, where x∈F is a lift of x with respect to π (see the diagram below).
Then if kerπ ⊂kerν, ν′ is well-defined and a homomorphism.
e
e
F
π (cid:15)(cid:15)(cid:15)(cid:15) ❅❅❅❅❅❅ν❅❅
Γ❴ ❴ ❴//Γ′
ν′
Proof of Theorem 3.1. M(N ) and M are trivial (see [2]). Assume g ≥ 2 and
1
n ∈ {0,1}. Then we obtain Theorem 3.1 if M(N ) is isomorphic to G. Let
g,n
ϕ : G → M(N ) be the surjective homomorphism defined by ϕ(t ) := t and
g,n c c
ϕ(Y ):=Y .
µ,α µ,α
Set X0 := {a1,...,ag−1,b,y} ⊂ M(Ng,n) for g ≥ 4 and X0 :=
{a1,...,ag−1,y} ⊂ M(Ng,n) for g = 2, 3. Let F(X0) be the free group which
is freely generated by X and let π :F(X )→M(N ) be the natural projection
0 0 g,n
(by Theorem 2.11). We define the homomorphism ν : F(X ) → G by ν(a ) := a
0 i i
for i=1, ..., g−1, ν(b):=b and ν(y):=y, and a map ψ =ν′ :M(N )→G by
g,n
ψ(a±1):=a±1 fori=1,...,g−1,ψ(b±1):=b±1,ψ(y±1):=y±1 andψ(f):=ν(f)
i i
for the other f ∈ M(N ), where f ∈ F(X ) is a lift of f with respect to π (see
g,n 0
the diagram below). e
e
PRESENTATION FOR MAPPING CLASS GROUP 9
F(X )
0
π (cid:15)(cid:15)(cid:15)(cid:15) ●●●●●●ν●●●●##
M(N )❴ ❴ ❴//G
g,n
ψ
Ifψ isahomomorphism,ϕ◦ψ =idM(Ng,n) bythedefinitionofϕandψ. Thusit
is sufficient for proving that ψ is isomorphism to show that ψ is a homomorphism
and surjective.
3.1. Proof that ψ is a homomorphism. M(N ) and M(N ) are trivial (see
1 1,1
[2, Theorem3.4]). For (g,n)∈{(2,0),(2,1),(3,0)},relationsofthe presentationin
Theorem 2.11 are obtained from Relations (0), (I), (II), (III), (IV) and (V), clearly.
Thus by Lemma 3.4, ψ is a homomorphism.
Assume g ≥4 or (g,n)=(3,1). By Lemma 3.4, if the relations of the presenta-
tion in Theorem 2.11 are obtained from Relations (0), (I), (II), (III), (IV) and (V),
then ψ is a homomorphism.
Thegroupgeneratedbya1,...,ag−1andbwithRelations(A1)-(A9b)asdefining
relations is isomorphic to M(Σ ) (resp. M(Σ )) for g = 2h+1 (resp. g =
h,1 h,2
2h + 2) by Theorem 3.1 in [13], and Relations (A1)-(A9b) are relations on the
mapping class group of the orientable subsurface NNg,n(α1 ∪···∪αg−1) of Ng,n.
Hence Relations (A1)-(A9b) are obtained from Relations (0), (I), (II) and (III) by
Remark 3.2.
Stukow [15] gave geometric interpretations for Relations (B1)-(B8) in Section 4
in [15]. By the interpretation, Relations (B1), (B2), (B3), (B4), (B5), (B7) are
obtained from Relations (I) (use Lemma 3.3), Relation (B6) is obtained from Re-
lations (0), (I), (III), (IV) and (V) (use Lemma 2.10 and Lemma 3.3), and Rela-
tion (B8) is obtained from Relations (I), (IV) and (V) (use Lemma 3.3). Thus ψ is
a homomorphism when n=1.
We assume n = 0. By Remark 3.2, k-chain relations are obtained from Rela-
tions (0), (I), (II) and (III) for each k. Relation (C1) is interpreted in G as follows.
(a1a2···ag−1)g (0),(I)=,(II),(III)tγ1,2,...,gt−γ11,2,...,g =1.
Thus Relation (C1) is obtained from Relations (0), (I), (II) and (III).
Relation (C2) is obtained from Relations (I) by Lemma 3.3, clearly.
When g is odd, by using the (g−1)-chain relation, Relation (C3) is interpreted
in G as follows.
ρ2 =(a1a2···ag−1)2g (0),(I)=,(II),(III)tε∂NNg(γ1,2,...,g) (=0)1,
where ε is 1 or −1. Note that N (γ ) is a Mo¨bius band in N . Thus Rela-
Ng 1,2,...,g g
tion (C3) is obtained from Relations (0), (I), (II) and (III) when g is odd.
When g is even, we rewrite the left-hand side ρ2 of Relation (C3) by braid
relations. Set A:=a2a3···ag−1. Note that
Y A2Y A−2 =Y
µ1,γ1,2,3 µ1,γ1,2,...,2i−1 µ1,γ1,2,...,2i+1
for i=2, ..., g−2 by Relation (I), (IV) and then we have
2
ρ
= y−1A(yAy−1A)g−22
10 G.OMORI
(=I) y−1A(ya2y−1a3···ag−1A)g−22
= y−1A(y(a2y−1a−21)A2)g−22
(I)=,(IV) y−1A(Yµ1,γ1,2,3A2)g−22.
= y−1AY A2···Y A2Y A2Y A2
µ1,γ1,2,3 µ1,γ1,2,3 µ1,γ1,2,3 µ1,γ1,2,3
= y−1AY A2···Y A2Y A2Y A−2A4
µ1,γ1,2,3 µ1,γ1,2,3 µ1,γ1,2,3 µ1,γ1,2,3
(I)=,(IV) y−1AY A2···Y A2Y A4
µ1,γ1,2,3 µ1,γ1,2,3 µ,γ1,2,3,4,5
= y−1AY A2···Y A2Y A−2A6
µ1,γ1,2,3 µ1,γ1,2,3 µ,γ1,2,3,4,5
(I)=,(IV) y−1AY A2···Y A6
µ1,γ1,2,3 µ1,γ1,2,3,4,5,6,7
.
.
.
(I)=,(IV) y−1AY Ag−2
µ1,γ1,2,...,g−1
= y−1·AY A−1·Ag−1
µ1,γ1,2,...,g−1
(I)=,(IV) Y Ag−1.
µ1,γ1,2,...,g
Since Y commutes with a for i=2, ..., g−1,and∂N (µ ∪γ )=
µ1,γ1,2,...,g i Ng 1 1,2,...,g
∂NNg(α2∪···∪αg−1) (see Figure 7), we have
ρ2 = Y Ag−1Y Ag−1
µ1,γ1,2,...,g µ1,γ1,2,...,g
(=I) Y2 A2g−2
µ1,γ1,2,...,g
(0),(I)=,(II),(III) Yµ21,γ1,2,...,gt∂NNg(α2∪···∪αg−1)
Lem=.2.10 t−∂N1Ng(α2∪···∪αg−1)t∂NNg(α2∪···∪αg−1)
= 1.
Recall that the relations in Lemma 2.10 are obtained from Relations (0), (IV) and
(V). Thus Relation (C3) is obtained from Relations (0), (I), (II), (III), (IV) and (V)
when g is even.
Figure 7. Simple closed curve ∂NNg(α2∪···∪αg−1) on Ng.
Finally, we alsorewrite the left-hand side (y−1a2a3···ag−1ya2a3···ag−1)g−21 of
Relation (C4) by braid relations. Remark that g is odd. For 1 ≤ i < i < ··· <
1 2
i ≤g,wedenotebyγ′ thesimpleclosedcurveonN asinFigure8. Note
k i1,i2,...,ik g,n
that
Yµ1,γ1′,2,3A2Yµ1,γ1′,2,...,2i−1A−2 =Yµ1,γ1′,2,...,2i+1