arXiv:1610.04999v1 [math.GT] 17 Oct 2016

AN INFINITE PRESENTATION FOR THE MAPPING CLASS
GROUP OF A NON-ORIENTABLE SURFACE WITH BOUNDARY
RYOMA KOBAYASHI AND GENKI OMORI

Abstract. We give an infinite presentation for the mapping class group of
a non-orientable surface with boundary components. The presentation is a
generalization of the presentation given by the second author [15].

1. Introduction
Let Σg,n be a compact connected oriented surface of genus g ≥ 0 with n ≥ 0
boundary components. The mapping class group M(Σg,n ) of Σg,n is the group
of isotopy classes of orientation preserving self-diffeomorphisms on Σg,n fixing the
boundary pointwise. A finite presentation for M(Σg,n ) was given by HatcherThurston [7], Harer [6], Wajnryb [21], Gervais [5] and Labruère-Paris [11]. Gervais [4] obtained an infinite presentation for M(Σg,n ) by using Wajnryb’s finite
presentation for M(Σg,n ), and Luo [14] rewrote Gervais’ presentation into a simpler infinite presentation (see Theorem 2.5).
Let Ng,n be a compact connected non-orientable surface of genus g ≥ 1 with
n ≥ 0 boundary components. The surface Ng = Ng,0 is a connected sum of g real
projective planes. The mapping class group M(Ng,n ) of Ng,n is the group of isotopy
classes of self-diffeomorphisms on Ng,n fixing the boundary pointwise. For g ≥ 2 and
n ∈ {0, 1}, a finite presentation for M(Ng,n ) was given by Lickorish [12], BirmanChillingworth [2], Stukow [17] and Paris-Szepietowski [16]. Note that M(N1 ) and
M(N1,1 ) are trivial (see [3, Theorem 3.4]) and M(N2 ) is finite (see [12, Lemma 5]).
Stukow [19] rewrote Paris-Szepietowski’s presentation into a finite presentation with
Dehn twists and a “Y-homeomorphism” as generators (see Theorem 3.1). The
second author [15] gave a simple infinite presentation for M(Ng,n ) for g ≥ 1 and
n ∈ {0, 1}. The generating set consists of all Dehn twits and all “crosscap pushing
maps” along simple loops. We review the crosscap pushing map in Section 2.
In this paper, we give a simple infinite presentation for M(Ng,n ) for arbitrary g ≥
0 and n ≥ 0 (Theorem 4.1). The presentation is a generalization of the presentation
given by the second author [15]. We will prove Theorem 4.1 by applying Gervais’
argument to a finite presentation for M(Ng,n ) in Proposition 3.2.
Contents of this paper are as follows. In Section 2, we prepare some elements of
M(Ng,n ) and some relations among their elements in M(Ng,n ), and review the infinite presentation for M(Σg,n ) (Theorem 2.5) which is an improvement by Luo [14]
of Gervais’ presentation in [4]. In Section 3, we review Stukow’s finite presentation for M(Ng,n ) when n ∈ {0, 1} (Theorem 3.1) and give a finite presentation for
M(Ng,n ) when n ≥ 2 (Proposition 3.2). In the proof of the main theorem in Section 4, we use their finite presentations for M(Ng,n ). In Section 4, we give the main
Date: January 2, 2018.
1

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R. KOBAYASHI AND G. OMORI

theorem (Theorem 4.1) in this paper and a proof of the main theorem. Finally, in
Section 5, we give a proof of Proposition 3.2.
2. Preliminaries
2.1. Relations among Dehn twists and Gervais’ presentation. Let S be
either Ng,n or Σg,n . We denote by NS (A) a regular neighborhood of a subset A
in S. We assume that every simple closed curve on S is oriented throughout this
paper, and for simple closed curves c1 , c2 on S, c1 = c2 means c1 is isotopic to c2
in consideration of their orientations. Denote by c−1 the inverse curve of a simple
closed curve c on S. Note that (c−1 )−1 = c. For a two-sided simple closed curve
c on S, we can take two orientations +c and −c of NS (c). When S is orientable,
we take +c as the orientation of NS (c) which is induced by the orientation of S.
Then for a two-sided simple closed curve c on S and an orientation θ ∈ {+c, −c } of
NS (c), denote by tc;θ the right-handed Dehn twist along c on S with respect to θ.
Note that tc;+c = tc−1 ;+c = t−1
c;−c . For some convenience, we write tc = tc;+c for a
two-sided simple closed curve c, where orientation of NS (c) is given explicitly (for
instance, S is an orientable surface). 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). For elements f = [ϕ], h = [ψ] ∈ M(S),
we define f h := [ϕ ◦ ψ] ∈ M(S).

Figure 1. The right-handed Dehn twist tc = tc;θ along a twosided simple closed curve c on S with respect to the orientation
θ ∈ {+c , −c } of NS (c) as in the figure.
Recall the following relations in M(S) among Dehn twists along two-sided simple
closed curves on S.
Lemma 2.1. Let c be a two-sided simple closed curve c on S and θ ∈ {+c , −c }
an orientation of NS (c). If c bounds a disk or a Möbius band in S, then we have
tc;θ = 1 in M(S).
For a two-sided simple closed curve c on S and f ∈ M(S), we have a bijection
f∗ = (f |NS (c) )∗ : {+c, −c } → {+f (c) , −f (c) }.
Lemma 2.2 (The braid relation (i)). For a two-sided simple closed curve c on S
and f ∈ M(S), we have
tf (c);θ′

= f tc;θ f −1

when f∗ (θ) = θ′ ,

t−1
f (c);θ ′

= f tc;θ f −1

when f∗ (θ) 6= θ′ .

When f in Lemma 2.2 is a Dehn twist td along a two-sided simple closed curve
d and the geometric intersection number |c ∩ d| of c and d is m, we denote by Tm
the braid relation.

PRESENTATION FOR MAPPING CLASS GROUP

3

Let c1 , c2 , . . . , ck be two-sided simple closed curves on S. The sequence c1 , c2 ,
. . . , ck is a k-chain on S if c1 , c2 , . . . , ck satisfy |ci ∩ ci+1 | = 1 for each i = 1, 2,
. . . , k − 1 and |ci ∩ cj | = 0 for |j − i| > 1.
Lemma 2.3 (The k-chain relation). Let c1 , c2 , . . . , ck be a k-chain on S and let
δ, δ ′ (resp. δ) be distinct boundary components (resp. the boundary component)
of NS (c1 ∪ c2 ∪ · · · ∪ ck ) when k is odd (resp. even). We give an orientation
θ ∈ {+δ , −δ } of NS (δ) and the orientation of NS (c1 ∪ c2 ∪ · · · ∪ ck ) which is induced
by θ. Then we have
εc

εc

εc

ε

(tc11;θc tc22;θc · · · tckk;θc )k+1

=

′
tδ;θ tδδ′ ;θ
δ′

εc
· · · tckk;θc )2k+2
k

=

tδ;θ

2

1

ε
ε
(tc1c1;θc tc2c2;θc
2
1

k

when k is odd,

when k is even,

where for c ∈ {c1 , c2 , . . . , ck , δ ′ }, εc = 1 if θc coincides with the orientation of
NS (c1 ∪ c2 ∪ · · · ∪ ck ), and εc = −1 otherwise.
Lemma 2.4 (The lantern relation). Let Σ be a subsurface of S which is diffeomorphic to Σ0,4 and let δ12 , δ23 , δ13 , δ1 , δ2 , δ3 and δ4 be simple closed curves on Σ as
in Figure 2. We give an orientation θ ∈ {+δ4 , −δ4 } of NS (δ4 ) and the orientation
of Σ which is induced by θ. Then we have
ε

ε

ε

δ12
δ23
δ13
tδ12
;θδ tδ23 ;θδ tδ13 ;θδ
12

23

13

ε

ε

ε

= tδ1δ1;θδ tδ2δ2;θδ tδ3δ3;θδ tδ4 ;θ ,
1

2

3

where for c ∈ {δ12 , δ23 , δ13 , δ1 , δ2 , δ3 }, εc = 1 if θc coincides with the orientation of
Σ, and εc = −1 otherwise.

Figure 2. The simple closed curves δ12 , δ23 , δ13 , δ1 , δ2 , δ3 and δ4
on Σ.
Luo’s presentation for M(Σg,n ), which is an improvement of Gervais’ one, is as
follows.
Theorem 2.5 ([4], [14]). For g ≥ 0 and n ≥ 0, M(Σg,n ) has the following presentation:
generators: {tc | c : s.c.c. on Σg,n }, where s.c.c. means simple closed curve.
relations:
(0′ )
(I′ )
(II)
(III)

tc = 1 when c bounds a disk in Σg,n ,
All the braid relations T0 and T1 ,
All the 2-chain relations,
All the lantern relations.

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R. KOBAYASHI AND G. OMORI

2.2. Relations among the crosscap pushing maps and Dehn twists. Let µ
be a one-sided simple closed curve on Ng,n and let α be a simple closed curve on
Ng,n such that µ and α intersect transversely at one point. Recall that α is oriented.
For these simple closed curves µ and α, we denote by Yµ,α a self-diffeomorphism on
Ng,n which is described as the result of pushing the Möbius band NNg,n (µ) once
along α. We call Yµ,α a crosscap pushing map. In particular, if α is two-sided,
we call Yµ,α a Y-homeomorphism (or a crosscap slide), where a crosscap means a
−1
Möbius band in the interior of a surface. Note that Yµ,α = Yµ,α
−1 = Yµ−1 ,α . The
Y-homeomorphism was originally defined by Lickorish [12]. We have the following
fundamental relation in M(Ng,n ) and we also call the relation the braid relation.
Lemma 2.6 (The braid relation (ii)). Let µ be a one-sided simple closed curve on
Ng,n and let α and β be simple closed curves on Ng,n such that µ and α intersect
transversely at one point. For f ∈ M(Ng,n ), suppose that f (α) = β or f (α) = β −1 .
Then we have
Yf (µ),β

= f Yµ,α f −1

Yf−1
(µ),β

−1

= f Yµ,α f

when f (α) = β,
when f (α) = β −1 .

We describe crosscap pushing maps from a different point of view. Let e : D′ ֒→
intS be a smooth embedding of the unit disk D′ ⊂ C. Put D := e(D′ ). Let S ′ be
the surface obtained from S − intD by the identification of antipodal points of ∂D.
We call the manipulation that gives S ′ from S the blowup of S on D. Note that the
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
that gives S from S ′ . We paste a disk on the boundary obtained by cutting S along
the center line µ of M . The blowdown of S ′ on M is the inverse manipulation of
the blowup of S on D.
Let µ be a one-sided simple closed curve on Ng,n and let S be the surface which
is obtained from Ng,n by the blowdown of Ng,n on NNg,n (µ). Note that S is
diffeomorphic to Ng−1,n or Σh,n for g = 2h + 1. Denote by xµ the center point
of a disk Dµ that is pasted on the boundary obtained by cutting S along µ. Let
e : D′ ֒→ Dµ ⊂ S be a smooth embedding of the unit disk D′ ⊂ C to S such
that Dµ = e(D′ ) and e(0) = xµ . Let M(S, xµ ) be the group of isotopy classes of
self-diffeomorphisms on S fixing the boundary ∂S and the point xµ , where isotopies
also fix the boundary ∂S and xµ . Then we have the blowup homomorphism
ϕµ : M(S, xµ ) → M(Ng,n )
that is defined as follows. For h ∈ M(S, xµ ), we take a representative h′ of h
which satisfies either of the following conditions: (a) h′ |Dµ is the identity map on
Dµ , (b) h′ (x) = e(e−1 (x)) for x ∈ Dµ , where e−1 (x) is the complex conjugation of
e−1 (x) ∈ C. Such h′ is compatible with the blowup of S on Dµ , thus ϕµ (h) ∈ M(S)
is induced and well defined (c.f. [20, Subsection 2.3]).
The point pushing map
jxµ : π1 (S, xµ ) → M(S, xµ )
is a homomorphism that is defined as follows. For γ ∈ π1 (S, xµ ), jxµ (γ) ∈ M(S, 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 γ1 , γ2 ∈ π1 (S, xµ ), γ1 γ2
means γ1 γ2 (t) = γ2 (2t) for 0 ≤ t ≤ 21 and γ1 γ2 (t) = γ1 (2t − 1) for 12 ≤ t ≤ 1.

PRESENTATION FOR MAPPING CLASS GROUP

5

Following Szepietowski [20] we define the composition of the homomorphisms:
ψxµ := ϕµ ◦ jxµ : π1 (S, xµ ) → M(Ng,n ).
For each closed curve α on Ng,n which transversely intersects with µ at one point,
we take a loop α on S 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 S 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 [10, Lemma 2.2, Lemma 2.3]).
Lemma 2.7. For a simple closed curve α on Ng,n which transversely intersects
with a one-sided simple closed curve µ on Ng,n at one point, we have
ψxµ (α) = Yµ,α .
Lemma 2.8. For a one-sided simple closed curve α on Ng,n which transversely intersects with a one-sided simple closed curve µ on Ng,n at one point, we take NS (α)
such that the interior of NS (α) contains Dµ and an orientation θα ∈ {+α , −α } of
NS (α). Denote by δ1 (resp. δ2 ) the boundary component of NS (α) on the right
(resp. left) side of α, and by δi (i = 1, 2) the two-sided simple closed curve on Ng,n
which is the image of δi with respect to the blowup of S on Dµ . Let θi ∈ {+δi , −δi }
(i = 1, 2) be the orientation of NS (δi ) which is induced by θα and θi ∈ {+δi , −δi }
(i = 1, 2) the orientation of NNg,n (δi ) which is induced by θi (see Figure 3). Then
we have
ε
−ε
Yµ,α = tδ1δ1;θδ tδ2 ;θδ2δ ,
1

2

where εδi = 1 if θδi = θi , and εδi = −1 otherwise.

Figure 3. Simple closed curves δ1 , δ2 , δ1 and δ2 , and orientations
θ1 , θ2 , θ1 and θ2 of their regular neighborhoods. The x-mark means
that antipodal points of ∂Dµ are identified.
By the definition of the homomorphism ψxµ and Lemma 2.7, we have the following lemma.
Lemma 2.9. Let α and β be simple closed curves on Ng,n which transversely
intersect with a one-sided simple closed curve µ on Ng,n at one point each. Suppose
that the product αβ of α and β in π1 (S, xµ ) is represented by a simple loop on S,
and αβ is a simple closed curve on Ng,n which is the image of the representative of
αβ with respect to the blowup of S on Dµ . Then we have
Yµ,αβ = Yµ,α Yµ,β .

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R. KOBAYASHI AND G. OMORI

Finally, we recall the following relation between a Dehn twist and a Yhomeomorphism.
Lemma 2.10. Let α be a two-sided simple closed curve on Ng,n which transversely
intersect with a one-sided simple closed curve µ on Ng,n at one point and let δ be
the boundary of NNg,n (α ∪ µ). Since α2 ∈ π1 (S, xµ ) is represented by a two-sided
simple loop, we take δi and θi (i = 1, 2) as in Lemma 2.8 when α in Lemma 2.8 is
replaced by α2 (see Figure 4). Then we have
2
Yµ,α
2
Yµ,α

ε

=

δ1
tδ;θ
δ

when δ = δ1 ,

=

−ε
tδ;θδδ2
2

when δ = δ2 ,

1

where εδi = 1 for i = 1, 2 if θδi = θi , and εδi = −1 otherwise.
Lemma 2.10 follows from relations in Lemma 2.1, Lemma 2.8 and Lemma 2.9.

Figure 4. The orientation θ1 of NNg,n (δ) when δ = δ1 or the
orientation θ2 of NNg,n (δ) when δ = δ2 .

3. Finite presentation for M(Ng,n )
In this section, we review Stukow’s finite presentation for M(Ng,n ) when n ∈
{0, 1} and give a finite presentation for M(Ng,n ) when n ≥ 2. We use their finite
presentations for M(Ng,n ) in the proof of the main theorem in Section 4.
Let ei : D′ ֒→ intΣ0,1 for i = 1, 2, . . . , g + n − 1 be smooth embeddings of the
unit disk D′ ⊂ C to a disk Σ0,1 such that Di := ei (D′ ) and Dj are disjoint for
distinct 1 ≤ i, j ≤ g + n − 1. For n ≥ 1, we take a model of Ng,n as the surface
obtained from Σ0,1 − (intDg+1 ⊔ · · · ⊔ intDn−1 ) by the blowups on D1 , . . . , Dg and
we describe the identification of ∂Di by the x-mark as in Figures 5. We denote by
δ1 , . . . , δn−1 and δ boundary components of Ng,n as in Figure 5 which are obtained
from ∂Dg+1 , . . . , ∂Dg+n−1 and ∂Σ0,1 , respectively. Let α1 , . . . , αg−1 , β and µ1 be
simple closed curves on Ng,n as in Figure 5 and let αi;j for 1 ≤ i ≤ g − 1 and
1 ≤ j ≤ n−1, ρi;j for 1 ≤ i ≤ g and 1 ≤ j ≤ n−1 and σi,j , σ̄i,j for 1 ≤ i < j ≤ n−1
be simple closed curves on Ng,n as in Figure 6. We give orientations of regular
neighborhoods of their simple closed curves as in Figure 5, 6. Then we define the
mapping classes
ai
b
y

:= tαi
:= tβ ,
:= Yµ1 ,α1 ,

for 1 ≤ i ≤ g − 1,

PRESENTATION FOR MAPPING CLASS GROUP

di

:= tδi

for 1 ≤ i ≤ n − 1,

ai;j
ri;j

:= tαi;j
:= tρi;j

for 1 ≤ i ≤ g − 1 and 1 ≤ j ≤ n − 1,
for 1 ≤ i ≤ g and 1 ≤ j ≤ n − 1,

si,j
s̄i,j

:= tσi,j
:= tσ̄i,j

for 1 ≤ i < j ≤ n − 1,
for 1 ≤ i < j ≤ n − 1,

s̄j,k;i

7

−1
−1
:= {(a1;k a−1
r2;k · · · (ai−1;k a−1
ri;k }−1 s̄j,k
1 )
i−1 )
−1
−1
{(a1;k a−1
r2;k · · · (ai−1;k a−1
ri;k }
1 )
i−1 )

for 2 ≤ i ≤ g and 1 ≤ j < k ≤ n − 1.
Remark that, for 2 ≤ i ≤ g and 1 ≤ j < k ≤ n − 1, s̄j,k;i is the Dehn twist along
the simple closed curve σ̄j,k;i on Ng,n as in Figure 7.

Figure 5. A model of Ng,n and simple closed curves α1 , . . . , αg−1 ,
β and µ1 on Ng,n .

Figure 6. The simple closed curves αi;j , ρi;j , σi,j and σ̄i,j on Ng,n .
−1
Epstein [3] show that M(N1,1 ) is trivial. We define [x1 , x2 ] = x1 x2 x−1
1 x2 .
Stukow gave the following finite presentation for M(Ng,1 ) when g = 2 in [17], and
when g ≥ 3 in [19] by rewriting the finite presentation in [16].

Theorem 3.1 ([3], [17], [19]). M(N1,1 ) is the trivial group. M(N2,1 ) has the
presentation
M(N2,1 ) =

a1 , y | ya1 y −1 = a−1
.
1

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R. KOBAYASHI AND G. OMORI

Figure 7. The simple closed curve σ̄j,k;i on Ng,n .
If g ≥ 3, then M(Ng,1 ) admits a presentation with generators a1 , . . . , ag−1 , y,
and b for g ≥ 4. The defining relations are
(A1) [ai , aj ] = 1
for g ≥ 4, |i − j| > 1,
(A2) ai ai+1 ai = ai+1 ai ai+1
for i = 1, . . . , g − 2,
(A3) [ai , b] = 1
for g ≥ 4, i 6= 4,
(A4) a4 ba4 = ba4 b
for g ≥ 5,
(A5) (a2 a3 a4 b)10 = (a1 a2 a3 a4 b)6
for g ≥ 5,
(A6) (a2 a3 a4 a5 a6 b)12 = (a1 a2 a3 a4 a5 a6 b)9
for g ≥ 7,
(A9a) [b2 , b] = 1
for g = 6,
(A9b) [ag−5 , b g−2 ] = 1
for g ≥ 8 even,
2
where b0 = a1 , b1 = b and
bi+1 = (bi−1 a2i a2i+1 a2i+2 a2i+3 bi )5 (bi−1 a2i a2i+1 a2i+2 a2i+3 )−6
for 1 ≤ i ≤ g−4
2 ,
−1 −1 −1
−1 −1 −1 −1
(B1) y(a2 a3 a1 a2 ya−1
for g ≥ 4,
2 a1 a3 a2 ) = (a2 a3 a1 a2 ya2 a1 a3 a2 )y
−1
−1
(B2) y(a2 a1 y −1 a2 ya1 a2 )y = a1 (a2 a1 y −1 a2 ya1 a2 )a1 ,
(B3) [ai , y] = 1
for g ≥ 4, i = 3, . . . , g − 1,
(B4) a2 (ya2 y −1 ) = (ya2 y −1 )a2 ,
(B5) ya1 = a−1
1 y,
−1 −1
−1 −1
−1
(B6) byby −1 = {a1 a2 a3 (y −1 a2 y)a−1
)a3 a2 }
for g ≥
3 a2 a1 }{a2 a3 (ya2 y
4,
−1 −1 −1 −1 −1 −1 −1
(B7) [(a4 a5 a3 a4 a2 a3 a1 a2 ya−1
for g ≥ 6,
2 a1 a3 a2 a4 a3 a5 a4 ), b] = 1
−1 −1 −1 −1
−1 −1 −1 −1
(B8) {(ya1 a2 a3 a4 )b(a4 a3 a2 a1 y −1 )}{(a−1
(a4 a3 a2 a1 )}
1 a2 a3 a4 )b
−1 −1
−1 −1 −1
−1
= {(a−1
a2 a3 }{a−1
for g ≥ 5.
4 a3 a2 )y(a2 a3 a4 )}{a3 a2 y
2 ya2 }y
For n ≥ 2, we have the following finite presentation for M(Ng,n ) and give the
proof in Section 5.4.
Proposition 3.2. For g ≥ 1 and n ≥ 2, M(Ng,n ) has the presentation which obtained from the finite presentation for M(Ng,1 ) in Theorem 3.1 by adding generators
di (i = 1, . . . , n−1), ai;j (1 ≤ i ≤ g−1, 1 ≤ j ≤ n−1), ri,j (1 ≤ i ≤ g, 1 ≤ j ≤ n−1),
si,j (1 ≤ i < j ≤ n − 1) and s̄i,j (1 ≤ i < j ≤ n − 1) and the following relations for
1 ≤ i, m ≤ g, 1 ≤ j < k ≤ n − 1 and 1 ≤ l < t < k:
(D0) [dj , ai ] = [dj , y] = [dj , b] = [dj , dl ] = [dj , ai;l ] = [dj , ri;l ] = [dj , sl,t ] =
[dj , s̄l,t ] = 1,

−1
−1
 (ai;k ai )(ai−1;k ai−1 ) for m = i − 1,
−1 −1
−1 −1
(D1a) am (ai;k ai )am =
(a
ai+1 ) (ai;k a−1
for m = i + 1,
i )
 i+1;k
−1
ai;k ai
for m 6= i − 1, i + 1,

−1
−2
 (a1;k a1 )−1 r2;k r1;k dn−1 for i = 1,
−1 −1
−1
−1
(D1b) y(ai;k ai )y =
(a a2 )r1;k dk
for i = 2,
 2;k −1
ai;k ai
for i ≥ 3,

PRESENTATION FOR MAPPING CLASS GROUP
−1
(D1c) 
b(ai;k a−1
=
i )b
−1
−1 −1
−1
−1
{(a
a
)(a
(a1;k a−1
for i = 1,

3;k 3
1;k a1 )}
1 ){(a3;k a3 )(a1;k a1 )}


−1
−1
−1
−1
−1
−1

 {(a3;k a3 )(a1;k a1 )} (a2;k a2 ){(a3;k a3 )(a1;k a1 )} for i = 2,
−1
−1
(a1;k a−1
(a3;k a−1
for i = 3,
1 )
3 )(a1;k a1 )

−1
−1
−1
−1


(a
a
)(a
a
)(a
a
)d
for i = 4,
3;k 3
1;k 1
4

k
 4;k −1
ai;k ai
for i ≥ 5,
−1
(D1d) am;l (ai;k a−1
)a
i
m;l =

−1 −1
−1
−1 −1
−1 −1 −1
−1 −1

 [(sl,k dl ) , (am;k am ) ] (ai;k ai )[(sl,k dl ) , (am;k am ) ]


for m ≤ i − 2,



−1
−1
−1
−1
−1

 [(ai−1;k a−1
, (sl,k d−1
](ai;k a−1
i−1 )
i )(sl,k dl )(ai−1;k ai−1 )dk
l )
for m = i − 1,

−1
−1 −1
−1
−1


{(s
(ai;k a−1
for m = i,
l,k dl )(ai;k ai )}

i ){(sl,k dl )(ai;k ai )}

−1
−1
−1

−1
−1

(a
a
)
(s
d
)
(a
a
)d
for
m
=
i
+
1,
i+1;k
l,k
i;k
k

i+1
i
l

ai;k a−1
for m ≥ i + 2,
i
−1
(D1e) rm;l (ai;k a−1
i )rm;l =

−1 −1 −1
−1 −1 −1

[(sl,k d−1
, rm;k ] (ai;k a−1
, rm;k ] for m ≤ i − 1,

i )[(sl,k dl )
l )


−1
−1
−1
−1

(sl,k d−1
)
r
}(s
d
)(a
a
)
 {ri;k
i;k
l,k
i;k
i
l
l
−1 −1
−1
(s̄l,k;i d−1
{ri;k (sl,k d−1
ri;k }−1 d−2
for m = i,
l )
l )
k


−1
−1 −1
−1
−1 2

r
(s
d
)
r
(s̄
d
)(a
a
for m = i + 1,
 i+1;k l,k l
i+1;k l,k;i+1 l
i;k i )dk


−1
ai;k ai
for m ≥ i + 2,
−1
−1
(D1f) sl,t (ai;k a−1
)s
i
l,t = ai;k ai ,
−1 −1
(D1g) s̄l,t (ai;k ai )s̄l,t =

−1 −1
−1 −1
−1
−1
−1 −1
 [(s̄l,k dl ) , st,k dt ] (sl,k dl )(a1;k a1 )r1;k (s̄t,k dt )r1;k

 (s d−1 )−1 r (s̄ d−1 )−1 r−1 [(s̄ d−1 )−1 , (s d−1 )] for i = 1,
l,k l
1;k t,k t
l,k l
t,k t
1;k
−1
−1 −1
, (st,k d−1
][sl,k d−1
, r1;k (s̄t,k d−1
r1;k ]}(ai;k a−1
 {[s̄l,k d−1
t )
t )
i )
l
l


−1
−1 −1
−1
−1 −1 −1 −1
{[s̄l,k dl , (st,k dt ) ][sl,k dl , r1;k (s̄t,k dt ) r1;k ]}
for i ≥ 2,

−1 −1
−1
r
r
(a
a
)
r
(a
a
)d

i;k i−1;k i−1;k i−1
i;k i−1;k i−1 k


for m = i − 1,
−1
(D2a) am ri;k am =
−1 −1
ri+1;k (ai;k a−1
for m = i,
 (ai;k a−1
i )
i )


ri;k
for m 6= i − 1, i,

−1
−1
−1
{(a1;k a−1
r2;k r1;k }−1 r1;k
{(a1;k a−1
r2;k r1;k }

1 )
1 )


for
i
=
1,
(D2b) yri;k y −1 =
−1

(a a−1 )r1;k (a1;k a−1
r2;k r1;k dn−1 for i = 2,
1 )

 1;k 1
ri;k
for i ≥ 3,

9

10

R. KOBAYASHI AND G. OMORI

(D2c) bri;k b−1 =















































−1
−1
(a1;k a−1
(a3;k a3−1 )−1 (a2;k a−1
1 )
2 )
−1
−1 −1
−1 −1
r4;k (a3;k a3 )r3;k (a2;k a2 )r2;k (a1;k a1−1 ) for i = 1,
−1 −1
−1
(a1;k a−1
{(a3;k a−1
3 )(a1;k a1 )}
1 )(a3;k a3 )r2;k
−1 −1
−1 −1
−1
r1;k (a1;k a1 ) r2;k (a2;k a2 ) r3;k (a3;k a−1
3 )
−1
−1
−1
r4;k (a2;k a2 ){(a3;k a3 )(a1;k a1 )} for i = 2,
−1 −1 −1
{(a3;k a−1
r4;k
3 )(a1;k a1 )}
−1 −1
−1
−1 −1
(a3;k a3 )r3;k (a2;k a2 )r2;k (a1;k a1−1 )r1;k
−1 −1
(a2;k a2 ) r3;k
−1
(a3;k a−1
(a1;k a1−1 )−1 {(a3;k a3−1 )(a1;k a−1
for i = 3,
3 )
1 )}
−1
r4;k (a2;k a2 )
−1
−1
r1;k (a1;k a−1
r2;k (a2;k a2−1 )−1 r3;k (a3;k a−1
1 )
3 )
−1
−1
r4;k (a3;k a3 )(a1;k a1 ) for i = 4,
ri;k
for i ≥ 5,

(D2d) am;l ri;k a−1
m;l =

−1
−1 −1
−1
−1
[(sl,k d−1
, (am;k a−1
] ri;k [(sl,k d−1
, (am;k a−1
]

m )
m )
l )
l )



for m ≤ i − 2,



−1
−1
−1

 {(sl,k d−1
(ai−1;k a−1
i−1 )(sl,k dl )ri;k
l )(ai−1;k ai−1 )}
−1 −1
−1
−1
ri−1;k (ai−1;k ai−1 ) ri;k (s̄l,k;i dl ){(sl,k dl )(ai−1;k a−1
i−1 )}



for m = i − 1,



−1 −1
−1

(a a−1 )−1 (sl,k d−1
ri+1;k (ai;k ai−1 )(s̄l,k;i d−1
for m = i,

l )
l )
 i;k i
ri;k for m ≥ i + 1,

−1 −1 −1
−1 −1

[(sl,k d−1
, rm;k ] ri;k [(sl,k d−1
, rm;k ]

l )
l )

for
m
≤
i
−
1,
−1
(D2e) rm;l ri;k rm;l =
−1
−1
−1


 {(sl,k dl )ri;k } ri;k {(sl,k dl )ri;k } for m = i,
ri;k
for m ≥ i + 1,
(D2f) sl,t ri;k s−1
=
r
,
i;k
l,t
(D2g) s̄l,t ri;k s̄−1
=
l,t

−1
−1 −1
−1 −1
−1 −1
 [st,k dt , (s̄l,k dl ) ][r1;k (s̄t,k dt )r1;k , (sl,k dl ) ]r1;k for i = 1,
−1 −1
−1 −1 −1
{[s̄l,k d−1
][sl,k dl−1 , r1;k (s̄t,k d−1
r1;k ]} r1;k
t )
l , (st,k dt )

−1
−1 −1
−1
−1 −1 −1
{[s̄l,k dl , (st,k dt ) ][sl,k dl , r1;k (s̄t,k dt ) r1;k ]} for i ≥ 2,
−1
−1
(D3a) am (sj,k d−1
j )am = sj,k dj ,
−1 −1
−1
(D3b) y(sj,k dj )y = sj,k dj ,
−1
(D3c) b(sj,k d−1
= sj,k d−1
j )b
j ,
−1
(D3d) am;l (sj,k d−1
)a
=
j
m;l

−1 −1
−1 −1
−1 −1
−1
[(s
d
)
, (am;k a−1
] (sj,k d−1
, (am;k a−1
]
l,k

m )
m )
j )[(sl,k dl )
l

 for l > j,
−1
−1
for l = j,
(am;k a−1
(sj,k d−1

m )
j )(am;k am )


−1
sj,k dj
for l = j,
−1
(D3e) rm;l (sj,k d−1
j )rm;l =

−1
−1
−1
−1 −1
−1

 [(sl,k d )−1 , r ] (sj,k dj )[(sl,k d )−1 , r ] for l > j,
l

m;k

−1
for l = j,
rm;k
(sj,k d−1
j )rm;k

 s d−1 for l < j,
j,k j

l

m;k

PRESENTATION FOR MAPPING CLASS GROUP

11

−1
(D3f) sl,t (sj,k d−1
j )sl,t =

−1
−1
−1
−1
 {(st,k dt )(sj,k d−1
(sj,k d−1
for l = j,
j )}
j ){(st,k dt )(sj,k dj )}



 [(st,k d−1 )−1 , (sl,k d−1 )−1 ]−1 (sj,k d−1 )[(st,k d−1 )−1 , (sl,k d−1 )−1 ]

t
t
j
l
l
for l < j < t,

−1

−1

(sl,k d−1
(sj,k d−1
for t = j,

j )(sl,k dl )
l )

 s d−1 for the other cases,
j,k j
−1
(D3g) s̄l,t (sj,k d−1
j )s̄l,t =

−1 −1
−1 −1 −1 −1
{[s̄l,k d−1
][sl,k d−1
r1;k ]} (sj,k d−1

j )
l , (st,k dt )
l , r1;k (s̄t,k dt )


−1
−1 −1
−1
−1 −1 −1

{[s̄
d
,
(s
d
)
][s
d
,
r
(s̄
d
)
r
]}
for
l
> j,

l,k
t,k
l,k
1;k
t,k
t
t
l
l
1;k


−1
−1 −1
−1
−1 −1 −1 −1
−1


{[s̄
d
,
(s
d
)
](s
d
)r
(s̄
d
)
r
}
(s
d
j,k j
t,k t
j,k j
1;k t,k t
j,k j )

1;k
−1
−1 −1
−1
−1 −1 −1
{[s̄j,k dj , (st,k dt ) ](sj,k dj )r1;k (s̄t,k dt ) r1;k } for l = j,

−1

−1 −1
−1 −1

[(s̄
, st,k d−1
(sj,k d−1
, st,k d−1
for l < j < t,

l,k dl )
t ]
t ]
j )[(s̄l,k dl )


−1
−1 −1 −1
−1
−1
−1 −1

{(s̄l,k dl )(sj,k dj ) } (sj,k dj ){(s̄l,k dl )(sj,k dj ) } for t = j,



sj,k d−1
for t < j,
j
−1
(D4a) am (s̄j,k d−1
)a
=
 −1 j−1 m −1 −1
−1 −1
−1
{r1;k r2;k (a1;k a1 )} (s̄j,k d−1
for m = 1,
j ){r1;k r2;k (a1;k a1 )}
−1
s̄j,k dj
for m ≥ 2,
−1
(D4b) y(s̄j,k d−1
=
j )y
−1
−1 −2
−1
−1 −2
{r1;k (a1;k a1 ) r2;k r1;k }−1 (s̄j,k d−1
r2;k r1;k },
j ){r1;k (a1;k a1 )
−1 −1
(D4c) b(s̄j,k dj )b =
−1
−1
−1 −1
−1 −1
−1 −1
{r1;k
(a2;k a−1
r4;k (a3;k a−1
(s̄j,k d−1
2 )
3 )r3;k (a2;k a2 )r2;k (a1;k a1 )}
j )
−1
−1 −1 −1
−1 −1
−1 −1
−1
{r1;k (a2;k a2 ) r4;k (a3;k a3 )r3;k (a2;k a2 )r2;k (a1;k a1 )},
−1
(D4d) am;l (s̄j,k d−1
j )am;l =
 −1 −1
−1 −1 −1
−1 −1
−1 −1
{r1;k r2;k (a1;k a−1
} (s̄j,k dj−1 ){r1;k
r2;k (a1;k a−1
}

1 )(s̄l,k dl )
1 )(s̄l,k dl )



for
m
=
1,
l
<
j,



−1 −1 −1 −1
−1 −1 −1
−1

{(s̄l,k d−1
r1;k r2;k (a1;k a−1
(s̄j,k d−1
r1;k r2;k (a1;k a−1

1 )}
1 )}
j ){(s̄l,k dl )
l )



for
m
=
1,
l
>
j,



−1
−1 −1
−1 −1
−1


 {(am−1;k am−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )} (s̄j,k;m+1 dj )
−1
−1 −1
−1
{(am−1;k am−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )} for m ≥ 2, l = j,

−1 −1
−1 −1

{(am−1;k a−1

m−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )}


−1 −1 −1
−1 −1
−1


 {(s̄l,k;m dl ) rm;k (s̄l,k;m+1 dl )} (s̄j,k;m dj )


−1
−1
−1

 {(s̄l,k;m dl )−1 rm;k (s̄l,k;m+1 dl )}


−1
−1
−1

for m ≥ 2, l > j,
 {(am−1;k am−1 )rm−1;k · · · (a2;k a−1
2 )r2;k (a1;k a1 )}


−1
s̄j,k dj
for the other cases,

12

R. KOBAYASHI AND G. OMORI

−1
(D4e) rm;l (s̄j,k d−1
)rm;l
=
 −1 j −1
−1
−1
{r1;k (s̄l,k dl ) (sl,k d−1
(s̄j,k dj−1 )

l )r1;k }


−1
−1

{r1;k
(s̄l,k d−1
(sl,k d−1
for m = 1, l < j,

l )
l )r1;k }


−1
−1
−1
−1


{(s
d
)r
}
(s̄
d
){(s
d
for m = 1, l = j,
j,k
1;k
j,k
j,k

j
j
j )r1;k }


−1
−1
−1
−1
−1
−1

{(s̄
d
)
r
(s
d
)r
}
(s̄
d
)
l,k l
1;k
j,k j

1;k l,k l


−1

{(s̄l,k d−1
)−1 r1;k
(sl,k d−1
)r1;k } for m = 1, l > j,

l
l


−1 −1
−1 −1


{(am−1;k a−1

m−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )}

 (s̄
−1
d )
j,k;m j

−1 −1
−1
{(am−1;k a−1
for m ≥ 2, l = j,
m−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )}
−1
−1 −1
−1 −1
{(am−1;k am−1 )rm−1;k · · · (a2;k a2 )r2;k (a1;k a1 )}
−1 −1
{(s̄l,k;m d−1
rm;k (s̄l,k;m dl−1 )}−1
l )
−1
(s̄j,k;m dj )
−1 −1
{(s̄l,k;m d−1
rm;k (s̄l,k;m d−1
l )
l )}
−1
−1
{(am−1;k am−1 )rm−1;k · · · (a2;k a2−1 )r2;k
(a1;k a−1
for m ≥ 2, l > j,
1 )}
−1
s̄j,k dj
for m ≥ 2, l < j,
−1
(D4f) sl,t (s̄j,k d−1
j )sl,t =

−1
−1
(s̄l,k d−1
(s̄j,k d−1
for t = j,

j )(s̄l,k dl )
l )


−1

−1 −1
−1 −1

 [(s̄t,k d ) , (s̄l,k d ) ] (s̄j,k d−1 )



























t

l

j

−1
−1
[(s̄t,k d−1
, (s̄l,k d−1
] for l < j < t,
t )
l )

−1
−1
−1
−1
−1

−1

{(s̄
d
)(s̄
d
)}
(s̄
for l = j,
t,k
j,k
j,k dj ){(s̄t,k dt )(s̄j,k dj )}

t
j

 s̄ d−1 for the other cases,
j,k j
−1
(D4g) s̄l,t (s̄j,k d−1
j )s̄l,t =

−1
−1
−1
−1 −1
−1

 [(s̄t,k dt ), r1;k (sl,k dl ) r1;k ] (s̄j,k dj )

−1
−1
−1

−1

[(s̄t,k dt ), r1;k (sl,k dl ) r1;k ] for t < j,


 {r−1 (s d−1 )−1 r }(s̄ d−1 ){r−1 (s d−1 )−1 r }−1 for t = j,
1;k
j,k j
1;k
1;k l,k l
1;k l,k l
 (st,k d−1
)(s̄j,k d−1
)(st,k dt−1 )−1 for l = j,
t

j

−1

−1 −1
−1


[(s̄l,k d−1
, (st,k d−1
(s̄j,k d−1
, (st,k d−1
for l > j,
t )]
t )]
j )[(s̄l,k dl )

l )

−1
s̄j,k dj
for l < j < t.

4. Infinite presentation for M(Ng,n )
The main theorem in this paper is as follows:
Theorem 4.1. For g ≥ 1 and n ≥ 0, M(Ng,n ) has the following presentation:
generators: {tc;+c , tc;−c | c : two-sided s.c.c. on Ng,n }
∪{Yµ,α | µ : one-sided s.c.c. on Ng,n , α : s.c.c. on Ng,n , |µ ∩ α| = 1}.
Denote the generating set by X.
relations:
(0)

(i) tc;θc = 1 when θc ∈ {+c , −c } and c bounds a disk or a Möbius band in
Ng,n ,
−1
,
(ii) tc;+c = tc−1 ;+c = tc;−
c
−1
(iii) Yµ,α = Yµ,α−1 = Yµ−1 ,α ,

PRESENTATION FOR MAPPING CLASS GROUP

(I) All the braid relations

(i) tf (c);θ′ = f tc;θ f −1



−1
t−1
f (c);θ ′ = f tc;θ f
(ii) Yf (µ),β = f Yµ,α f −1



−1
Yf−1
(µ),β = f Yµ,α f

(II)
(III)
(IV)
(V)

All
All
All
All

the
the
the
the

when
when
when
when

13

f∗ (θ) = θ′ and f ∈ X,
f∗ (θ) 6= θ′ and f ∈ X,
f (α) = β and f ∈ X,
f (α) = β −1 and f ∈ X,

2-chain relations,
lantern relations,
relations in Lemma 2.9, i.e. Yµ,αβ = Yµ,α Yµ,β ,
ε
−ε
relations in Lemma 2.8, i.e. Yµ,α = tδ1δ1;θδ tδ2 ;θδ2δ .
1

2

The second author [15] proved Theorem 4.1 when g ≥ 1 and n ∈ {0, 1}. Since
we do not distinguish tc;+c , tc−1 ;+c and t−1
c;−c , and also do not distinguish Yµ,α ,
−1
Yµ,α−1 and Yµ−1 ,α in [15], the presentation in Theorem 3.1 of [15] is different from
the presentation in Theorem 4.1. However, these presentation are equivalent by
Relation (0)(ii) and (0)(iii). In fact, we can apply the proof of Theorem 3.1 in [15]
to the presentation in Theorem 4.1. In (I) and (IV) one can substitute the right
hand side of (V) for each generator Yµ,α with one-sided α. Then one can remove
the generators Yµ,α with one-sided α and relations (V) from the presentation.
We denote by G the group which has the presentation in Theorem 4.1. Let
ι : Σh,m ֒→ Ng,n be a smooth embedding and let G′ be the group whose presentation has all Dehn twists along simple closed curves on Σh,m as generators and
Relations (0′ ), (I′ ), (II) and (III) in Theorem 2.5. By Theorem 2.5, M(Σh,m ) is
isomorphic to G′ , and we have the homomorphism G′ → G defined by the correspondence of tc;+c to tι(c);ι∗ (+c ) . Then we remark the following.
Remark 4.2. The composition ι∗ : M(Σh,m ) → G of the isomorphism
M(Σh,m ) → G′ and the homomorphism G′ → G is a homomorphism. In particular,
if a product tεc11 ;θc tεc22 ;θc · · · tεckk ;θc of Dehn twists along simple closed curves c1 , c2 ,
1
2
k
. . . , ck on a connected compact orientable subsurface of Ng,n is equal to the identity map in the mapping class group of the subsurface, then tεc11 ;θc tεc22 ;θc · · · tεckk ;θc is
1
2
k
equal to 1 in G. That means such a relation tεc11 ;θc tεc22 ;θc · · · tεckk ;θc = 1 is obtained
1
2
k
from Relations (0), (I), (II) and (III).
Set X ± := X ∪ {x−1 | x ∈ X}. By Relation (I), we have the following lemma.
Lemma 4.3. For f ∈ G, suppose that f = f1 f2 . . . fk , where f1 , f2 , . . . , fk ∈ X ± .
Then we have

(i) tf (c);θ′ = f tc;θ f −1
when f∗ (θ) = θ′ ,


−1

−1
tf (c);θ′ = f tc;θ f
when f∗ (θ) 6= θ′ ,
(ii) Yf (µ),β = f Yµ,α f −1 when f (α) = β,



−1
Yf−1
when f (α) = β −1 .
(µ),β = f Yµ,α f

The next lemma follows from an argument of the combinatorial group theory
(for instance, see [8, Lemma 4.2.1, p42]).

Lemma 4.4. 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(e
x) for
x ∈ Γ, where x
e ∈ F is a lift of x with respect to π (see the diagram below).
Then if ker π ⊂ ker ν, ν ′ is well-defined and a homomorphism.

14

R. KOBAYASHI AND G. OMORI

F❅
❅❅
❅❅ν
π
❅❅
❅

Γ ❴ ❴ ❴/ Γ′
ν′

We start the proof of Theorem 4.1. When n ∈ {0, 1}, we proved Theorem 4.1
in [15]. Assume g ≥ 1 and n ≥ 2. Then we obtain Theorem 4.1 if M(Ng,n ) is
isomorphic to G. Let ϕ : G → M(Ng,n ) be the surjective homomorphism defined
by ϕ(tc;+c ) := tc;+c , ϕ(tc;−c ) := tc;−c and ϕ(Yµ,α ) := Yµ,α .
Denote by X0 ⊂ M(Ng,n ) the generating set of the finite presentation for
M(Ng,n ) in Proposition 3.2. Let F (X0 ) be the free group which is freely generated
by X0 and let π : F (X0 ) → M(Ng,n ) be the natural projection. We define the
homomorphism ν : F (X0 ) → G by ν(ai ) := ai , ν(b) := b, ν(y) := y, ν(ai;j ) := ai;j ,
ν(ri;j ) := ri;j , ν(si,j ) := si,j and ν(s̄i,j ) := s̄i,j , and a map ψ = ν ′ : M(Ng,n ) → G
±1
±1
±1
±1
by ψ(ai±1 ) := a±1
) := b±1 , ψ(y ±1 ) := y ±1 , ψ(a±1
i , ψ(b
i;j ) := ai;j , ψ(ri;j ) := ri;j ,
±1
±1
±1
e
ψ(s±1
i,j ) := si,j , ψ(s̄i,j ) := s̄i,j and ψ(f ) := ν(f ) for the other f ∈ M(Ng,n ), where
e
f ∈ F (X0 ) is a lift of f with respect to π (see the diagram below).
F (X0 )
●●
●● ν
●●
π
●●
●●

#
M(Ng,n ) ❴ ❴ ❴/ G
ψ

If ψ is a homomorphism, ϕ ◦ ψ = idM(Ng,n ) by the definition of ϕ and ψ. Thus
it is sufficient to show that ψ is a homomorphism and surjective for proving that ψ
is isomorphism.
4.1. Proof that ψ is a homomorphism. By Lemma 4.4, if the relations of the
presentation in Proposition 3.2 are obtained from Relations (0), (I), (II), (III), (IV)
and (V), then ψ is a homomorphism.
Let N be the subsurface of Ng,n as in Figure 8. N is diffeomorphic to Ng,1
and includes simple closed curves α1 , . . . , αg−1 , µ and β. We regard M(N ) as
a subgroup of M(Ng,n ). Relations (A1), . . . , (A9b) and (B1), . . . , (B8) of the
presentation for M(Ng,n ) in Proposition 3.2 are relations of M(N ) ∼
= M(Ng,1 ). By
Theorem 3.1 in [15], Relations (A1), . . . , (A9b) and (B1), . . . , (B8) are obtained
from Relations (0), (I), (II), (III), (IV) and (V).
By arguments in the last part of Section 5.4 and Section 5.5, Relations (D0)(D4g) are obtained from Relations (I) and (III). We have proved that ψ is a homomorphism.
4.2. Surjectivity of ψ. For some convenience, we write tc;+c = tc in this subsection. We show that there exist lifts of tc ’s and Yµ,α ’s with respect to ψ for cases
below, to prove the surjectivity of ψ.
(1) tc ; c is non-separating and Ng,n − c is non-orientable,
(2) tc ; c is non-separating and Ng,n − c is orientable,
(3) tc ; c is separating,
(4) Yµ,α ; α is two-sided and Ng,n − α is non-orientable,
(5) Yµ,α ; α is two-sided and Ng,n − α is orientable,

PRESENTATION FOR MAPPING CLASS GROUP

15

Figure 8. The subsurface N of Ng,n which is diffeomorphic to Ng,1 .
(6) Yµ,α ; α is one-sided.
Set X0± := X0 ∪ {x−1 | x ∈ X0 }. For a simple closed curve c on Ng,n , we denote
by (Ng,n )c the surface obtained from Ng,n by cutting Ng,n along c and denote by
Σ the component of (Ng,n )c which does not include δ.
For generators of type (1), (2), (4), (5), (6), by similar arguments in Section 3.2
of [15], there exist their lifts with respect to ψ. We note that we use the existence
for lifts of generators of type (3) for the proof of the existence for lifts of generators
of type (6).
Case (3) where Σ is diffeomorphic to Σ0,m+1 for m ≥ 0. We proceed by
induction on m ≥ 0. When m = 0, tc is trivial by Relation (0). When m = 1,
c = δi for some 1 ≤ i ≤ b − 1. Hence di is the lift of tc .
When m = 2, there exists a product f = f1 f2 · · · fk ∈ M(Ng,n ) of f1 , f2 , · · · ,
fk ∈ X0± which satisfy either c = f (σi,j ) or c = f (σ̄i,j ) for some 1 ≤ i < j ≤ b − 1.
Thus, if c = f (σi,j ), we have
ψ(f si,j f −1 )

=
Lem. 4.3

f1 f2 · · · fk si,j fk−1 · · · f2−1 f1−1

=

tεf (σi,j )

=

tεc ,

where ε is 1 or −1. Thus f sεi,j f −1 ∈ M(Ng,n ) is a lift of tc ∈ G with respect to ψ for
some ε ∈ {−1, 1}. By a similar argument, when c = f (σ̄i,j ), f s̄εi,j f −1 ∈ M(Ng,n )
is also a lift of tc ∈ G with respect to ψ for some ε ∈ {−1, 1}.
For m ≥ 3, there exists a simple closed curve c′ on Σ such that c′ separates Σ into
′
Σ and Σ′′ which are diffeomorphic to Σ0,4 and Σ0,m−1 , respectively, and c ⊂ Σ′ .
By using a lantern relation on Σ′ , there exist simple closed curves c1 = c′ , c2 , . . . , c6
on Σ′ such that tc = tεc11 tεc22 · · · tεc66 ∈ G for some ε1 , ε2 , . . . , ε6 ∈ {−1, 1}. Since each
ci (i = 1, 2, . . . , 6) bounds a subsurface of Ng,n which does not include c and is
diffeomorphic to Σ0,mi +1 for some mi < m, by the inductive assumption, there
exist lifts h1 , . . . , h6 ∈ M(Ng,n ) of tc1 , . . . , tc6 ∈ G with respect to ψ, respectively.
Thus hε11 hε22 · · · hε66 ∈ M(Ng,n ) is a lift of tc with respect to ψ.
Case (3) where Σ is diffeomorphic to Σh,m+1 for h ≥ 1 m ≥ 0. In this
case, there exists a simple closed curve c′ on Σ such that c′ separates Σ into Σ′ and
Σ′′ which are diffeomorphic to Σh,2 and Σ0,m+1 , respectively. Then there exists a
2h+1-chain c1 , c2 , . . . , c2h+1 on Σ′ such that NNg,n (c1 ∪c2 ∪· · ·∪c2h+1 ) = Σ′ . By the
′
ε2h+1 2h+2
chain relation, (tεc11 tεc22 · · · tc2h+1
)
= tc tεc′ for some ε1 , ε2 , . . . , ε2h+1 , ε′ ∈ {−1, 1}.

16

R. KOBAYASHI AND G. OMORI

Since tc1 , tc2 , . . . , tc2h+1 are Dehn twists of type (1) and c′ bounds Σ′′ , tc1 , tc2 ,
. . . , tc2h+1 , tc′ ∈ G have lifts h1 , h2 , . . . , h2h+1 , h′ ∈ M(Ng,n ) with respect to ψ,
respectively. Thus we have
ε

′

2h+1 2h+2
ψ((hε11 hε22 . . . h2h+1
)
(h′ )−ε )

=
(0),(I),(II),(III)

=

′

2h+1 2h+2 −ε
)
tc ′
(tεc11 tεc22 · · · tcε2h+1

tc .

We remark that the last relation is obtained from Theorem 2.5 and Remark 4.2.
ε2h+1 2h+2 ′ −ε′
Thus ((hε11 hε22 . . . h2h+1
)
(h )
∈ M(Ng,n ) is a lift of tc ∈ G with respect to
ψ.
Case (3) where Σ is diffeomorphic to Nh,m+1 for h ≥ 1 m ≥ 0. We proceed
by induction on m ≥ 0. When m = 0, by similar arguments in Section 3.2 in [15],
there exists their a lift of tc ∈ G with respect to ψ.
When m = 1, we proceed by induction on h ≥ 1. When h = 1, there exists a
product f = f1 f2 · · · fk ∈ M(Ng,n ) of f1 , f2 , · · · , fk ∈ X0± such that c = f (ρ1;j )
for some 1 ≤ j ≤ n − 1. By a similar argument in the case where Σ is diffeomorphic
to Σ0,m+1 , we can obtain a lift of tc with respect to ψ. Suppose h ≥ 2. Then there
exist simple closed curves c1 and c2 on Σ such that c1 ⊔ c2 separates Σ into Σ′ , Σ′′
and Σ′′′ which are diffeomorphic to Σ0,4 , N1,1 and Nh−1,1 , respectively, and c ⊂ Σ′ .
By using a lantern relation on Σ′ , there exist simple closed curves c3 , . . . , c6 on Σ′
such that tc = tεc11 tεc22 tεc33 · · · tεc66 ∈ G for some ε1 , ε2 , ε3 , . . . , ε6 ∈ {−1, 1}. Since each ci
(i = 1, . . . , 6) is a boundary component of a subsurface of Σ which is diffeomorphic
to an orientable surface, Nhi ,1 for some hi ≤ h or Nhi ,2 for some hi < h, by the
inductive assumption, there exist lifts h1 , . . . , h6 ∈ M(Ng,n ) of tc1 , . . . , tc6 ∈ G with
respect to ψ, respectively. Thus hε11 hε22 · · · hε66 ∈ M(Ng,n ) is a lift of tc with respect
to ψ.
Suppose m ≥ 2. Then there exist simple closed curves c1 and c2 on Σ such
that c1 ⊔ c2 separates Σ into Σ′ , Σ′′ and Σ′′′ which are diffeomorphic to Σ0,4 , Σ0,m
and Nh,1 , respectively, and c ⊂ Σ′ . By using a lantern relation on Σ′ , there exist
simple closed curves c3 , . . . , c6 on Σ′ such that tc = tεc11 tεc22 tεc33 · · · tεc66 ∈ G for some
ε1 , ε2 , ε3 , . . . , ε6 ∈ {−1, 1}. Since each ci (i = 1, . . . , 6) is a boundary component
of a subsurface of Σ which is diffeomorphic to an orientable surface or Nh,mi +1 for
some mi < m, by the inductive assumption, there exist lifts h1 , . . . , h6 ∈ M(Ng,n )
of tc1 , . . . , tc6 ∈ G with respect to ψ, respectively. Thus h1ε1 hε22 · · · hε66 ∈ M(Ng,n )
is a lift of tc with respect to ψ.
We have completed the proof of Theorem 4.1.
5. Proof of Proposition 3.2 and preliminaries for the proof
In this section, we give a proof of Proposition 3.2 which is used in the proof of
Theorem 4.1. The proof is given in Section 5.4 and 5.5. For giving the proof, we
prepare Section 5.1, 5.2 and 5.3.
5.1. Group presentations and short exact sequence. Let G be a group and
let H = X | R , Q = Y | S be presented groups which have the exact sequence
ι

ν

1 −→ H −→ G −→ Q −→ 1.
We take a lift ỹ ∈ G of y ∈ Q with respect to ν for each y ∈ Q. Then we put
e := {ι(x) | x ∈ X} ⊂ G and Ye := {ỹ | y ∈ Y } ⊂ G. Denote by r̃ the word in X
e
X
which obtained from r ∈ R by replacing each x ∈ X by ι(x) and denote by s̃ the

PRESENTATION FOR MAPPING CLASS GROUP

17

word in Ye which obtained from s ∈ S by replacing each y ∈ Y by ỹ. We note that
r̃ = 1 in G. For each s ∈ S, since s̃ ∈ G is an element of ker ν, there exists a word
e such that s̃ = vs in G. Since ι(H) is a normal subgroup of G, for each
vs in X
x ∈ X and y ∈ Y , ỹι(x)ỹ −1 is an element of ι(H). Hence there exists a word wx,y
e such that ỹι(x)ỹ −1 = wx,y in G. The next lemma follows from an argument
in X
of the combinatorial group theory (for instance, see [8, Proposition 10.2.1, p139]).

Lemma 5.1. In this situation above, the group G has the following presentation:
generators: {ι(x), ỹ | x ∈ X, y ∈ Y }.
relations:
(A) r̃ = 1
for r ∈ R,
(B) s̃ = vs
for s ∈ S,
(C) ỹι(x)ỹ −1 = wx,y
for x ∈ X, y ∈ Y .

5.2. Extended lantern relations. Let S be a connected compact surface and let
D be a disk on intS with the center point x0 . Then we have the point pushing map
(defined in Section 2.2) jx0 : π1 (S, x0 ) → M(S, x0 ). For a two-sided simple loop γ
on S based at x0 , we take an orientation θγ ∈ {+γ , −γ } of NS (γ). Denote by c1
(resp. c2 ) the boundary component of NS (γ) on the right (resp. left) side of γ, and
by θi ∈ {+ci , −ci } (i = 1, 2) the orientation of NS (ci ) which is induced by θγ . We
regard γ as an element of π1 (S, x0 ). Then we have a well-known relation
jx0 (γ) = tc1 ;θ1 t−1
c2 ;θ2 .
Let L+ = L+ (S, x0 ) be the subset of π1 (S, x0 ) which consists of elements represented by two-sided simple loops. Then we define a map
∆ = ∆x0 : L+ → M(S − intD)
as follows. For any two-sided simple loop γ on S based at x0 , we take NS (γ) whose
interior contains D. Then we take c1 , c2 , θ1 and θ2 as above. Define the inclusion
ι : S − intD → S and c̃i := ι−1 (ci ) for i = 1, 2. Then we define
∈ M(S − intD),
∆(γ) := tc̃1 ;θc̃1 tc̃−1
2 ;θc̃
2

where θc̃i := (ι|S−intD )−1
∗ (θci ) ∈ {+c̃i , −c̃i } for i = 1, 2.
The next two lemmas are obtained from an argument in Section 3 of [8].
Lemma 5.2. Let α and β be two-sided simple loops on S based at x0 such that
α tangentially intersects with β at x0 only and the composition αβ ∈ π1 (S, x0 ) is
represented by a simple loop. We take the orientation of NS (α∪β) which is induced
by the orientation of NS−intD (∂D). Then we have
∆(α)∆(β) = ∆(αβ)tε∂D ,
where ε = 1 if α and β are counterclockwise around x0 as on the left-hand side of
Figure 9 and ε = −1 if α and β are clockwise around x0 as on the right-hand side
of Figure 9.
The relations in Lemma 5.2 are original lantern relations. We call the relations
in Lemma 5.2 Relations (L+) when ε = 1 and Relations (L-) when ε = −1 (see
Figure 9).
Lemma 5.3. Let α and β be two-sided simple loops on S based at x0 such that α
transversely intersects with β at x0 only. Then we have
∆(α)∆(β) = ∆(αβ).

18

R. KOBAYASHI AND G. OMORI

Figure 9. Oriented subsurfaces NS (α ∪ β).
We call the relations in Lemma 5.3 Relations (L0). We have the following lemma.
Lemma 5.4. Relations (L0) are obtained from the braid relations (i).
Proof. Let α and β be two-sided simple loops on S based at x0 such that α transversely intersects with β at x0 only. We take a representative of αβ ∈ π1 (S, x0 ) by a
simple loop γ and also take the orientations of NS (α∪β) ⊂ S and NS (α∪β)−intD ⊂
S − intD which is induced by the orientation of NS−intD (∂D). Define boundary
components a1 ⊔ a2 = ∂NS (α), b1 ⊔ b2 = ∂NS (β) and c1 ⊔ c2 = ∂NS (γ) such that
a1 , b1 and c1 are on the right-hand side of α, β and γ, respectively. We consider
the case where the algebraic intersection number, with respect to the orientation
of NS (α ∪ β), of α and β is 1 and orientations of NS−intD (ãi ), NS−intD (b̃i ) and
NS−intD (c̃i ) are compatible with the orientation of NS (α ∪ β). Figure 10 expresses
this situation. Then we have ∆(α) = tã1 tã−1
, ∆(β) = tb̃1 tb̃−1 and ∆(γ) = tc̃1 tc̃−1
. For
2
2
2
the other cases, we can prove this lemma by an argument similar to the following
argument.
Since t−1
ã2 (b̃i ) = c̃i for i = 1, 2, we have
∆(αβ)

We have the lemma.

=

tc̃1 tc̃−1
2

(I)

=

tã−1
(tb̃1 tb̃−1 )tã2
2

=

tã−1
t t−1 tã2 · (tb̃1 tb̃−1 )−1 tb̃1 tb̃−1
2 b̃1 b̃

=

tã−1
(tb̃1 tb̃−1 )tã2 (tb̃1 tb̃−1 )−1 · tb̃1 tb̃−1
2

(I)

=

tã−1
t
2 ã1

=

∆(α)∆(β).

2

2

2

2

·

2

2

2

tb̃1 tb̃−1
2


We remark that Relations (L+), (L-) and (L0) are obtained from Relations (I)
and (III).
5.3. Generators for the subgroup of the fundamental group generated
by two-sided loops. Recall that we take a model of Ng,n as in Figure 5 for
n ≥ 1. Assume n ≥ 2. We regard Ng,n−1 as the surface obtained by regluing
Dg+n−1 and Ng,n . Put the center point x0 of Dg+n−1 . Let π1 (Ng,n−1 )+ be the
subgroup of π1 (Ng,n−1 , x0 ) which consists of elements that is represented by loops
such that the pushing maps along their loops preserve a local orientation of x0 .
Let x1 , . . . , xg , y1 , . . . , yn−2 be loops on Ng,n−1 based at x0 as in Figure 11 and
we regard Ng,n−1 as the surface in Figure 11 for some conveniences. Note that

PRESENTATION FOR MAPPING CLASS GROUP

19

Figure 10. Two-sided simple loops α and β on NS (α ∪ β) such
that their algebraic intersection number is 1 (on the left side), and
simple closed curves ãi , b̃i and c̃i on NS (α ∪ β) − intD for i = 1, 2
(on the center and the right side).
x2i for 1 ≤ i ≤ g, xi+1 xi for 1 ≤ i ≤ g − 1 and x−1
1 yi x1 for 1 ≤ i ≤ n − 2
are elements of π1 (Ng,n−1 )+ . x2i , xi+1 xi and x1−1 yi x1 are represented by loops as
in Figure 12. Since π1 (Ng,n−1 , x0 ) is the free group which is freely generated by
x1 , . . . , xg , y1 , . . . , yn−2 , π1 (Ng,n−1 )+ is also isomorphic to a free group. We have
the following lemma.
Lemma 5.5. For g ≥ 1 and n ≥ 2, π1 (Ng,n−1 )+ is the free group which is freely
generated by x21 , . . . , x2g , x2 x1 , . . . , xg xg−1 , y1 , . . . , yn−2 , x1−1 y1 x1 , . . . , x−1
1 yn−2 x1 .
Proof. We use the Reidemeister-Schreier method (for instance see [9]) for
π1 (Ng,n−1 )+ ⊂ π1 (Ng,n−1 , x0 ) to obtain the generators for π1 (Ng,n−1 )+ . Since
π1 (Ng,n−1 )+ is an index 2 subgroup of π1 (Ng,n−1 , x0 ) and the non-trivial element of the quotient group π1 (Ng,n−1 , x0 )/π1 (Ng,n−1 )+ is represented by x1 , the
set U := {1, x1 } ⊂ π1 (Ng,n−1 )+ is a Schreier transversal for π1 (Ng,n−1 )+ in
π1 (Ng,n−1 , x0 ). Set X := {x1 , . . . , xg , y1 , . . . , yn−2 }. For any word w in X, denote
by w the element of U whose equivalence class in π1 (Ng,n−1 , x0 )/π1 (Ng,n−1 )+ is
the same as that of w. Then π1 (Ng,n−1 )+ is the free group which is freely generated
by
B

=

{xu−1 xu | x ∈ X, u ∈ U, xu 6∈ U }

=

−1
{xi x1 , x−1
1 xj , yk , x1 yk x1 | i = 1, . . . , g, j = 2, . . . , g, k = 1, . . . , n − 2}.

Put z1 := x21 , zi := (xi x1 )(x−1
1 xi ) for i = 2, . . . , g, w1 := x2 x1 and wi :=
−1
(xi+1 x1 )(x1 xi ) for i = 2, . . . , g − 1 as words in B. By using the Tietze transformations (for instance see [9, Proposition 4.4.5, p46]) and relations (x−1
1 xi ) =
−1
(xi x1 )−1 zi and (xi+1 x1 ) = wi (x−1
for i ≥ 2, we have isomorphisms
1 xi )
B|
∼
=

B ∪ {zi , wj | i = 1, . . . g, j = 1, . . . , g − 1} |
−1
z1 = x21 , zi = (xi x1 )(x−1
1 xi ), w1 = x2 x1 , wj = (xj+1 x1 )(x1 xj )

∼
=

B ∪ {zi , wj | i = 1, . . . g, j = 1, . . . , g − 1} |
−1
z1 = x21 , w1 = x2 x1 , xi x1 = wi−1 zi−1
· · · w2 z2−1 w1 ,
−1
−1
−1
x−1
1 xj+1 = w1 z2 w2 · · · zj wj zj+1

∼
=

{zi , wj , yk , x−1
1 yk x1 | i = 1, . . . g, j = 1, . . . , g − 1, k = 1, . . . , n − 2} | .

20

R. KOBAYASHI AND G. OMORI

Note that zi = x2i and wi = xi+1 xi as elements of π1 (Ng,n−1 )+ . We get this
lemma.


Figure 11. Loops x1 , . . . , xg , y1 , . . . , yn−2 on Ng,n−1 based at x0 .

Figure 12. Loops x2i , xi+1 xi , x−1
1 yi x1 on Ng,n−1 based at x0 .
5.4. Proof of Proposition 3.2. Let M+ (Ng,n−1 , x0 ) be the subgroup of
M(Ng,n−1 , x0 ) whose elements preserve a local orientation of x0 . For n ≥ 2, the
forgetful homomorphism M(Ng,n−1 , x0 ) → M(Ng,n−1 ) induces the following exact
sequence
(5.1) 1 −→ π1 (Ng,n−1 )+

jx0 |π

+
1 (Ng,n−1 )

−→

M+ (Ng,n−1 , x0 ) −→ M(Ng,n−1 ) −→ 1.

Since π1 (Ng,n−1 , x0 ) is isomorphic to a free group, the center of π1 (Ng,n−1 , x0 ) is
trivial. Thus the homomorphism jx0 is injective by [1, Corollary 1.2].
The natural inclusion ι : Ng,n ֒→ Ng,n−1 induces the surjective homomorphism
ι∗ : M(Ng,n ) → M+ (Ng,n−1 , x0 ). By Theorem 3.6 in [18], we have the exact
sequence
(5.2)

ι

∗
M+ (Ng,n−1 , x0 ) −→ 1.
1 −→ Z[dn−1 ] −→ M(Ng,n ) −→

We remark that for a relation v1ε1 · · · vkεk = w1δ1 · · · wlδl of a group G, we call
· · · vkεk (w1δ1 · · · wlδl )−1 the relator of G (obtained from the relation).

v1ε1

Proof of Proposition 3.2. We proceed by induction on n ≥ 1. The finite presentation for M(Ng,1 ) is given by Theorem 3.1.
Assume n ≥ 2. For some conveniences, we define ι∗ (ai ) = ai ∈ M+ (Ng,n−1 , x0 ),
ι∗ (y) = y ∈ M+ (Ng,n−1 , x0 ), ι∗ (b) = b ∈ M+ (Ng,n−1 , x0 ), etc., and we take
lifts ai , y, b, . . . ∈ M+ (Ng,n−1 , x0 ) of ai , y, b, . . . ∈ M(Ng,n−1 ), respectively.
By Lemma 5.5, the inductive assumption and applying Lemma 5.1 to the exact

PRESENTATION FOR MAPPING CLASS GROUP

21

sequence 5.1, we obtain the following presentation for M+ (Ng,n−1 , x0 ):
generators:
(1) ai (1 ≤ i ≤ g − 1), y, b, di (1 ≤ i ≤ n− 2), ai;j (1 ≤ i ≤ g − 1, 1 ≤ j ≤ n− 2),
ri;j (1 ≤ i ≤ g, 1 ≤ j ≤ n − 2), si,j (1 ≤ i < j ≤ n − 2), s̄i,j
(1 ≤ i < j ≤ n − 2),
(1 ≤ i ≤ g − 1),
(2) jx0 (x2i ) = ri;n−1 (1 ≤ i ≤ g), jx0 (xi+1 xi ) = ai;n−1 a−1
i
−1
(1 ≤ i ≤ n − 2) and jx0 (x−1
jx0 (yi ) = si,n−1 d−1
1 yi x1 ) = s̄i,n−1 di
i
(1 ≤ i ≤ n − 2).
Denote by X1 the set of generators in (1) and by X2 the set of generators in (2).
relations:
(1) For any relator s of the presentation for M(Ng,n−1 ) and the lift s̃ of s to
M+ (Ng,n−1 , x0 ),
s̃ = vs
for some product vs of elements in X2 .
(2) For x ∈ X2 and f ∈ X1 ,
f xf −1 = wx,f
for some product wx,f of elements in X2 .
Note that the generators in (1) come from the generators for M(Ng,n−1 ) and the
generators in (2) come from the generators for π1 (Ng,n−1 )+ in Lemma 5.5.
We calculate each vs and wx,f in the relation (1) and (2) above. We take the
subsurface N ′ of Ng,n which is diffeomorphic to Ng,n−1 as in Figure 13. Since
every simple closed curve used in a generator of the presentation for M(Ng,n−1 ) is
isotopic to a simple closed curve on intN ′ , we regard generators of the presentation
for M(Ng,n−1 ) as elements of M(N ′ ). In particular, the inclusion ι′ : N ′ ֒→
Ng,n induces the injective homomorphism ι′∗ : M(N ′ ) → M(Ng,n ). By using the
composition ι∗ ◦ ι′∗ : M(N ′ ) → M+ (Ng,n−1 , x0 ), we can take vs = 1 for each relator
s of the presentation for M(Ng,n−1 ).
Recall that s̄j,k;i is the Dehn twist along the simple closed curve ι(σ̄j,k;i ) and
σ̄j,k;i is defined in Figure 7. For the relation (2) above, we can take wx,f as follows.
−1
Assume 1 ≤ i, m ≤ g, 1 ≤ j ≤ n − 2, 1 ≤ l < t ≤ n − 2 and [x1 , x2 ] = x1 x2 x−1
1 x2 .

−1
−1
 (ai;n−1 ai )(ai−1;n−1 ai−1 ) for m = i − 1,
−1 −1
−1 −1
′
(D1a) am (ai;n−1 ai )am =
(a
ai+1 ) (ai;n−1 ai−1 ) for m = i + 1,
 i+1;n−1
−1
ai;n−1 ai
for m 6= i − 1, i + 1,

−1
 (a1;n−1 a1 )−1 r2;n−1 r1;n−1 for i = 1,
−1
(D1b)′ y(ai;n−1 a−1
=
(a
a−1
for i = 2,
2 )r1;n−1
i )y
 2;n−1 −1
ai;n−1 ai
for i ≥ 3,
−1
(D1c)′ 
b(ai;n−1 a−1
=
i )b
−1
−1 −1
−1
−1
{(a
a
)(a
(a1;n−1 a−1

3;n−1
1;n−1 a1 )}
3
1 ){(a3;n−1 a3 )(a1;n−1 a1 )}



for i = 1,



−1
−1
−1
−1
−1

 {(a3;n−1 a3 )(a1;n−1 a1 )}−1 (a2;n−1 a2 ){(a3;n−1 a3 )(a1;n−1 a1 )}
for i = 2,

−1 −1
−1

(a
(a3;n−1 a−1
for i = 3,

1 )
3 )(a1;n−1 a1 )
 1;n−1 a−1

−1
−1


)
for
i = 4,
)(a
a
)(a
a
(a
a
1;n−1
3;n−1
4;n−1
1
3
4


ai;n−1 a−1
for
i
≥
5,
i

22

R. KOBAYASHI AND G. OMORI

−1
(D1d)′ am;l (ai;n−1 a−1
i )am;l =

−1 −1
[(sl,n−1 d−1
)−1 , (am;n−1 a−1
] (ai;n−1 a−1

m )
i )
l


−1 −1
−1 −1


[(s
d
)
,
(a
a
)
]
for
m
≤
i − 2,
l,n−1 l
m;n−1 m


−1 −1
−1 −1
−1
−1
−1


[(a
a
)
,
(s
d
)
](a
a
i−1;n−1 i−1
l,n−1 l
i;n−1 i )(sl,n−1 dl )(ai−1;n−1 ai−1 )


for m = i − 1,
−1
−1 −1
−1
−1

{(s
(ai;n−1 a−1
l,n−1 dl )(ai;n−1 ai )}

i ){(sl,n−1 dl )(ai;n−1 ai )}



for m = i,



−1
−1

(a
a−1
(sl,n−1 d−1
(ai;n−1 a−1
for m = i + 1,

i+1 )
i )
l )
 i+1;n−1
−1
ai;n−1 ai
for m ≥ i + 2,
−1
(D1e)′ rm;l (ai;n−1 a−1
i )rm;l =

−1
−1 −1 −1
−1 −1

[(sl,n−1 d−1
, rm;n−1 ] (ai;n−1 a−1
, rm;n−1 ]

i )[(sl,n−1 dl )
l )



for
m
≤
i
−
1,



−1
−1
−1

(sl,n−1 d−1
ri;n−1 }(sl,n−1 d−1
 {ri;n−1
l )
l )(ai;n−1 ai )
−1 −1 −1
−1 −1
−1
(s̄l,n−1;i dl ) {ri;n−1 (sl,n−1 dl ) ri;n−1 }
for m = i,


−1
−1 −1
−1
−1

r
(s
d
)
r
(s̄
d
)(a
i+1;n−1 l,n−1;i+1 l
i;n−1 ai )

i+1;n−1 l,n−1 l



for m = i + 1,



ai;n−1 a−1
for m ≥ i + 2,
i
−1
−1
(D1f)′ sl,t (ai;n−1 a−1
)s
i
l,t = ai;n−1 ai ,
−1
−1
(D1g)′ s̄l,t (ai;n−1 ai )s̄l,t =

−1 −1
−1 −1
−1
−1
−1 −1
 [(s̄l,k dl ) , st,k dt ] (sl,k dl )(a1;k a1 )r1;k (s̄t,k dt )r1;k


−1
−1
−1
−1
−1
−1
−1
−1

 (sl,k dl ) r1;k (s̄t,k dt ) r1;k [(s̄l,k dl ) , (st,k dt )] for i = 1,


−1 −1
−1 −1 −1
{[s̄l,n−1 d−1
][sl,n−1 d−1
r1;n−1 ]}
l , (st,n−1 dt )
l , r1;n−1 (s̄t,n−1 dt )
−1
 (ai;n−1 ai )


−1
−1 −1
−1
−1 −1 −1

−1


 {[s̄l,n−1 dl , (st,n−1 dt ) ][sl,n−1 dl , r1;n−1 (s̄t,n−1 dt ) r1;n−1 ]}
for i ≥ 2,
−1
−1
(D1h)′ dl (ai;n−1 a−1
i )dl  = ai;n−1 ai ,
−1
−1
 ri;n−1 ri−1;n−1 (ai−1;n−1 ai−1 )−1 ri;n−1 (ai−1;n−1 ai−1 )


for m = i − 1,
(D2a)′ am ri;n−1 a−1
m =
−1 −1
ri+1;n−1 (ai;n−1 a−1
for m = i,
 (ai;n−1 a−1
i )
i )


ri;n−1 for m 6= i − 1, i,

−1
−1
{(a1;n−1 a−1
r2;n−1 r1;n−1 }−1 r1;n−1

1 )


−1 −1
{(a1;n−1 a1 ) r2;n−1 r1;n−1 } for i = 1,
(D2b)′ yri;n−1 y −1 =
−1

(a
a−1 )r1;n−1 (a1;n−1 a−1
r2;n−1 r1;n−1 for i = 2,
1 )

 1;n−1 1
ri;n−1 for i ≥ 3,

PRESENTATION FOR MAPPING CLASS GROUP

(D2c)′ bri;n−1 b−1 =























































23

−1
−1
(a1;n−1 a−1
(a3;n−1 a−1
(a2;n−1 a2−1 )−1
1 )
3 )
−1
−1 −1
−1
−1
r4;n−1 (a3;n−1 a3 )r3;n−1 (a2;n−1 a−1
2 )r2;n−1 (a1;n−1 a1 )
for i = 1,
−1 −1
−1
{(a3;n−1 a−1
(a1;n−1 a−1
3 )(a1;n−1 a1 )}
1 )(a3;n−1 a3 )r2;n−1
−1 −1
−1 −1
r1;n−1 (a1;n−1 a1 ) r2;n−1 (a2;n−1 a2 ) r3;n−1
−1
−1
(a3;n−1 a−1
r4;n−1 (a2;n−1 a2−1 ){(a3;n−1 a−1
3 )
3 )(a1;n−1 a1 )}
for i = 2,
−1 −1 −1
{(a3;n−1 a−1
r4;n−1
3 )(a1;n−1 a1 )}
−1 −1
−1
−1 −1
(a3;n−1 a3 )r3;n−1 (a2;n−1 a−1
2 )r2;n−1 (a1;n−1 a1 )r1;n−1
−1
(a2;n−1 a−1
r3;n−1 (a3;n−1 a3−1 )−1 (a1;n−1 a1−1 )−1
2 )
−1
{(a3;n−1 a3 )(a1;n−1 a−1
for i = 3,
1 )}
−1
r4;n−1 (a2;n−1 a−1
)r
(a
a−1
r2;n−1 (a2;n−1 a2−1 )−1
1;n−1
1;n−1
2
1 )
−1 −1
−1
r3;n−1 (a3;n−1 a3 ) r4;n−1 (a3;n−1 a3 )(a1;n−1 a1−1 )
for i = 4,
ri;n−1 for i ≥ 5,

(D2d)′ am;l ri;n−1 a−1
m;l =

−1
−1 −1
−1
−1
[(sl,n−1 d−1
, (am;n−1 a−1
] ri;n−1 [(sl,n−1 d−1
, (am;n−1 a−1
]

m )
m )
l )
l )



for
m
≤
i
−
2,



−1
−1
−1


{(sl,n−1 d−1
(ai−1;n−1 a−1
i−1 )(sl,n−1 dl )ri;n−1

l )(ai−1;n−1 ai−1 )}

−1 −1
−1
−1
ri−1;n−1 (ai−1;n−1 ai−1 ) ri;n−1 (s̄l,n−1;i dl ){(sl,n−1 dl )(ai−1;n−1 a−1
i−1 )}

for
m
=
i
−
1,


−1 −1

−1
−1 −1

(ai;n−1 a−1
(sl,n−1 d−1
ri+1;n−1 (ai;n−1 a−1

i )
i )(s̄l,n−1;i dl )
l )




 for m = i,
ri;n−1 for m ≥ i + 1,

−1
−1 −1
−1 −1

[(sl,n−1 d−1
, rm;n−1 ] ri;n−1 [(sl,n−1 d−1
, rm;n−1 ]

l )
l )


 for m ≤ i − 1,
−1
−1
(D2e)′ rm;l ri;n−1 rm;l
=
{(sl,n−1 d−1
ri;n−1 {(sl,n−1 d−1
l )ri;n−1 }
l )ri;n−1 }



for
m
=
i,


ri;n−1 for m ≥ i + 1,
(D2f)′ sl,t ri;n−1 s−1
=
r
,
i;n−1
l,t
−1
′
(D2g) s̄l,t ri;n−1 s̄l,t =

−1 −1
−1
−1 −1
[st,n−1 d−1
][r1;n−1 (s̄t,n−1 d−1
]r1;n−1

t , (s̄l,n−1 dl )
t )r1;n−1 , (sl,n−1 dl )



 for i = 1,
−1 −1
−1 −1 −1
{[s̄l,n−1 d−1
][sl,n−1 d−1
r1;n−1 ]}−1
l , (st,n−1 dt )
l , r1;n−1 (s̄t,n−1 dt )

−1
−1
−1
−1 −1


r
{[s̄
d , (st,n−1 dt )−1 ][sl,n−1 dl , r1;n−1 (s̄t,n−1 d−1
r1;n−1 ]}
t )

 1;n−1 l,n−1 l
for i ≥ 2,
(D2h)′ dl ri;n−1 d−1
= ri;n−1 ,
l
−1
−1
(D3a)′ am (sj,n−1 d−1
j )am = sj,n−1 dj ,
−1
−1
(D3b)′ y(sj,n−1 dj )y −1 = sj,n−1 dj ,
−1
= sj,n−1 d−1
(D3c)′ b(sj,n−1 d−1
j ,
j )b
−1
(D3d)′ am;l (sj,n−1 dj )a−1
=
m;l

−1 −1
−1 −1

[(sl,n−1 dl ) , (am;n−1 a−1
] (sj,n−1 d−1

m )
j )


−1 −1
−1 −1
[(sl,n−1 dl ) , (am;n−1 am ) ] for l > j,
−1
−1
−1 −1

 (am;n−1 am ) (sj,n−1 dj )(am;n−1 am ) for l = j,

 s
−1
for l = j,
j,n−1 dj

24

R. KOBAYASHI AND G. OMORI

−1
(D3e)′ rm;l (sj,n−1 d−1
j )rm;l =

−1
−1
−1
[(sl,n−1 d−1
)−1 , rm;n−1
] (sj,n−1 dj−1 )[(sl,n−1 dl−1 )−1 , rm;n−1
]

l


for l > j,

r−1 (s
d−1 )rm;n−1 for l = j,

 m;n−1 −1j,n−1 j
sj,n−1 dj
for l < j,
−1 −1
′
(D3f) sl,t (sj,n−1 dj )sl,t =

−1 −1
−1
−1
(sj,n−1 d−1
 {(st,n−1 d−1
t )(sj,n−1 dj )}
j ){(st,n−1 dt )(sj,n−1 dj )}




for l = j,



−1
−1 −1
[(st,n−1 d−1
, (sl,n−1 d−1
] (sj,n−1 dj−1 )
t )
l )
−1 −1
−1 −1

[(st,n−1 dt ) , (sl,n−1 dl ) ] for l < j < t,


−1

−1

(sl,n−1 d−1
(sj,n−1 d−1
for t = j,

j )(sl,n−1 dl )
l )

 s
−1
for the other cases,
j,n−1 dj
−1
(D3g)′ s̄l,t (sj,n−1 d−1
j )s̄l,t =

−1 −1
−1 −1 −1
{[s̄l,n−1 d−1
][sl,n−1 d−1
r1;n−1 ]}−1

l , (st,n−1 dt )
l , r1;n−1 (s̄t,n−1 dt )


−1

(sj,n−1 dj )



−1 −1
−1 −1 −1

{[s̄l,n−1 d−1
][sl,n−1 d−1
r1;n−1 ]}

l , (st,n−1 dt )
l , r1;n−1 (s̄t,n−1 dt )



for
l
>
j,



−1 −1
−1 −1 −1


{[s̄j,n−1 d−1
](sj,n−1 d−1
r1;n−1 }−1

j , (st,n−1 dt )
j )r1;n−1 (s̄t,n−1 dt )

−1


 (sj,n−1 dj )
−1 −1
−1 −1
{[s̄j,n−1 d−1
](sj,n−1 dj−1 )r1;n−1 (s̄t,n−1 d−1
r1;n−1 }
t )
j , (st,n−1 dt )


for
l
=
j,



−1

−1
−1

[(s̄l,n−1 d−1
, st,n−1 d−1
(sj,n−1 dj−1 )[(s̄l,n−1 d−1
, st,n−1 d−1

t ]
t ]
l )
l )



 for l < j < t,


−1 −1 −1
−1
−1 −1

 {(s̄l,n−1 d−1
} (sj,n−1 d−1
}
j ){(s̄l,n−1 dl )(sj,n−1 dj )

l )(sj,n−1 dj )



for t = j,


sj,n−1 d−1
for t < j,
j
−1
′
(D3h) dl (sj,n−1 d−1
)d
= sj,n−1 d−1
j
j ,
l
−1 −1
′
(D4a) am (s̄j,n−1 dj )am =
 −1
−1
−1
−1
−1
−1
−1
 {r1;n−1 r2;n−1 (a1;n−1 a1 )}−1 (s̄j,n−1 dj ){r1;n−1 r2;n−1 (a1;n−1 a1 )}
for m = 1,

s̄j,n−1 d−1
for m ≥ 2,
j
−1
−1
−2
(D4b)′ y(s̄j,n−1 d−1
)y
= {r1;n−1
(a1;n−1 a−1
r2;n−1 r1;n−1 }−1 (s̄j,n−1 d−1
1 )
j
j )
−1
−1 −2
{r1;n−1 (a1;n−1 a1 ) r2;n−1 r1;n−1 },
−1
−1
−1
−1 −1
(D4c)′ b(s̄j,n−1 d−1
= {r1;n−1
(a2;n−1 a−1
r4;n−1 (a3;n−1 a−1
2 )
3 )r3;n−1
j )b
−1 −1
−1 −1
−1
−1
−1
(a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )} (s̄j,n−1 dj ){r1;n−1 (a2;n−1 a−1
2 )
−1
−1
−1 −1
−1
r4;n−1
(a3;n−1 a−1
3 )r3;n−1 (a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )},

PRESENTATION FOR MAPPING CLASS GROUP
−1
(D4d)′ am;l (s̄j,n−1 d−1
j )am;l =
 −1
−1
−1 −1 −1
{r1;n−1 r2;n−1 (a1;n−1 a−1
} (s̄j,n−1 d−1

1 )(s̄l,n−1 dl )
j )

 {r−1 r−1 (a
−1
−1 −1

a
)(s̄
d
)
}
for
m
=
1, l < j,
1;n−1
l,n−1

1;n−1
2;n−1
1
l


−1 −1 −1
−1
−1 −1
−1

{(s̄
d
)
r
r
(a
a
)}
(s̄
d

l,n−1
1;n−1
j,n−1
1;n−1 2;n−1
1
j )
l


−1 −1 −1
−1
−1


{(s̄
d
)
r
r
(a
a
)}
for
m
=
1, l > j,
l,n−1 l
1;n−1 2;n−1 1;n−1 1


−1
−1 −1
−1 −1


{(a
a
)r
·
·
·
(a
a
)r
(a
m−1;n−1 m−1 m−1;n−1
2;n−1 2

2;n−1 1;n−1 a1 )}


−1

(s̄j,n−1;m+1 dj )



−1 −1
−1
{(am−1;n−1 a−1
m−1 )rm−1;n−1 · · · (a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )}

for m ≥ 2, l = j,


−1
−1 −1
−1 −1


{(a
m−1;n−1 am−1 )rm−1;n−1 · · · (a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )}



−1
−1
−1
−1
−1
−1

{(s̄l,n−1;m dl ) rm;n−1 (s̄l,n−1;m+1 dl )} (s̄j,n−1;m dj )



−1 −1

{(s̄l,n−1;m d−1
rm;n−1 (s̄l,n−1;m+1 d−1

l )
l )}


−1
−1
−1


{(am−1;n−1 am−1 )rm−1;n−1 · · · (a2;n−1 a−1
2 )r2;n−1 (a1;n−1 a1 )}




for m ≥ 2, l > j,


s̄j,n−1 d−1
for the other cases,
j
−1
(D4e)′ rm;l (s̄j,n−1 d−1
)r
j
m;l =
 −1
−1
−1
{r1;n−1 (s̄l,n−1 d−1
(sl,n−1 d−1
(s̄j,n−1 dj−1 )

l )
l )r1;n−1 }


−1
−1
−1

−1

 {r1;n−1 (s̄l,n−1 dl ) (sl,n−1 dl )r1;n−1 } for m = 1, l < j,


−1
−1
−1
−1

 {(sj,n−1 dj )r1;n−1 } (s̄j,n−1 dj ){(sj,n−1 dj )r1;n−1 }



for m = 1, l = j,



−1
−1

{(s̄l,n−1 dl−1 )−1 r1;n−1
(sl,n−1 d−1
(s̄j,n−1 dj )−1

l )r1;n−1 }


−1


{(s̄l,n−1 dl−1 )−1 r1;n−1
(sl,n−1 d−1
for m = 1, l > j,

l )r1;n−1 }

−1
−1 −1
−1 −1


{(a
a
)r
·
·
·
(a
a
)r
m−1;n−1
m−1;n−1
2;n−1

m−1
2
2;n−1 (a1;n−1 a1 )}


−1

 (s̄j,n−1;m dj )
−1 −1
−1
{(am−1;n−1 a−1
m−1 )rm−1;n−1 · · · (a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )}



for m ≥ 2, l = j,


−1
−1 −1
−1 −1


{(a
m−1;n−1 am−1 )rm−1;n−1 · · · (a2;n−1 a2 )r2;n−1 (a1;n−1 a1 )}



−1
−1
−1
−1

rm;n−1 (s̄l,n−1;m dl )}−1

l )
 {(s̄l,n−1;m d−1



 (s̄j,n−1;m dj −1)

−1


{(s̄l,n−1;m dl )−1 rm;n−1
(s̄l,n−1;m d−1

l )}

−1
−1 −1
−1


{(a
a
)r
·
·
·
(a
m−1;n−1 m−1 m−1;n−1
2;n−1 a2 )r2;n−1 (a1;n−1 a1 )}




for m ≥ 2, l > j,


 s̄
−1
for m ≥ 2, l < j,
j,n−1 dj
−1 −1
′
(D4f) sl,t (s̄j,n−1 dj )sl,t =

−1
−1
(s̄l,n−1 d−1
(s̄j,n−1 d−1
for t = j,

j )(s̄l,n−1 dl )
l )


−1

−1 −1
−1 −1

[(s̄t,n−1 dt ) , (s̄l,n−1 dl ) ] (s̄j,n−1 d−1

j )


−1 −1
−1
[(s̄t,n−1 d−1
)
,
(s̄
d
)
]
for
l
<
j < t,
l,n−1 l
t
−1
−1 −1
−1
−1
−1

{(s̄
d
)(s̄
d
)}
(s̄
d
){(s̄
t,n−1 t
j,n−1 j
j,n−1 j
t,n−1 dt )(s̄j,n−1 dj )}




for l = j,


 s̄
−1
for the other cases,
j,n−1 dj

25

26

R. KOBAYASHI AND G. OMORI

−1
(D4g)′ s̄l,t (s̄j,n−1 d−1
j )s̄l,t =

−1
−1

[(s̄t,n−1 d−1
), r1;n−1
(sl,n−1 d−1
)−1 r1;n−1 ] (s̄j,n−1 d−1
t
j )

l

−1
−1
−1 −1


[(s̄
d
),
r
(s
d
)
r
]
for
t
<
j,
t,n−1 t
1;n−1

1;n−1 l,n−1 l
 −1

−1 −1
−1
−1
−1 −1

{r
(s
d
)
r
}(s̄
d
){r
(s
r1;n−1 }−1

l,n−1 l
1;n−1
j,n−1 j
1;n−1 l,n−1 dl )

 1;n−1
for t = j,
−1 −1
−1
for l = j,
 (st,n−1 d−1
t )(s̄j,n−1 dj )(st,n−1 dt )


−1

−1
−1
 [(s̄l,n−1 d )−1 , (st,n−1 d )] (s̄j,n−1 d−1 )[(s̄l,n−1 d−1 )−1 , (st,n−1 d−1 )]

t
t
j
l
l


 for l > j,



s̄j,n−1 d−1
for l < j < t,
j
−1
′
(D4h) dl (s̄j,n−1 d−1
)d
= s̄j,n−1 dj−1 .
j
l

Since jx0 (f˜(x)) = f˜jx0 (x)f˜−1 for x ∈ π1 (Ng,n−1 , x0 ) and f ∈ M(Ng,n−1 ), it is
′
useful for obtaining wx,f to find a product wx,f
of generators for π1 (Ng,n−1 )+ in
′
Lemma 5.5 such that f˜(x) = wx,f . For example, in Relation (D1e)′ for m = i,
ri;l (xi+1 xi ) is represented by the loop as on the right-hand side of Figure 14. Thus
we have
ri;l (xi+1 xi )

−1 2
−1
−2 −1 2 −1
∈ π1 (Ng,n−1 )+
= {x−2
i yl xi }yl (xi+1 xi )ȳi;l {xi yl xi }

and also have
−1
ri;l (ai;n−1 a−1
i )ri;l

−1
−1
−1
= {ri;n−1
(sl,n−1 d−1
ri;n−1 }(sl,n−1 d−1
l )
l )(ai;n−1 ai )
−1 −1
−1
(s̄l,n−1;i d−1
{ri;n−1 (sl,n−1 d−1
ri;n−1 }−1 .
l )
l )

Therefor M+ (Ng,n−1 , x0 ) has the presentation which is obtained from the finite
presentation for M(Ng,n−1 ) by adding generators ai;n−1 for 1 ≤ i ≤ g − 1, ri;n−1
for 1 ≤ i ≤ g, si,n−1 and s̄i,n−1 for 1 ≤ i ≤ n − 2, and Relations (D1a)′ -(D4h)′ .
Next, we apply Lemma 5.1 to the exact sequence 5.2. Let X2′ be the set of added
generators ai;n−1 , ri;n−1 , si;n−1 and s̄i;n−1 to the presentation for M+ (Ng,n−1 , x0 ).
Note that dn−1 commutes with every element of M(Ng,n ). M(Ng,n ) admits the
presentation which has the generating set X1 ∪ X2′ and relations as follows:
(1) For each relator r of the finite presentation for M+ (Ng,n−1 , x0 ) and the lift
r̃ of r with respect to ι∗ , there exists εr ∈ Z such that
r
r̃ = dεn−1
.

(2) For each generator x of the finite presentation for M+ (Ng,n−1 , x0 ) and the
lift x̃ of x with respect to ι∗ ,
[dn−1 , x̃] = 1.
Thus we obtain Relations (D0) from the relations in (2) above. Note that Relations (D0) are obtained from Relation (I).
We compute εr for each r. We have the homomorphism ι′∗ : M(N ′ ) ֒→ M(Ng,n )
induced by the inclusion ι′ : N ′ ֒→ Ng,n . Thus for each relator r of the finite
presentation for M+ (Ng,n−1 , x0 ) which obtained from a relator of the finite presentation for M(Ng,n−1 ), we have εr = 0. Hence we obtain Relations (A1)-(B8)
and (D0)-(D4g) for 1 ≤ k ≤ n − 2 in M(Ng,n ). By the inductive argument on n
and the argument below, we can show that Relations (A1)-(B8) and (D0)-(D4g)
for 1 ≤ k ≤ n − 2 are obtained from Relations (I) and (III).

PRESENTATION FOR MAPPING CLASS GROUP

27

We remark that the lift of Relations (D1h)′ , (D2h)′ , (D3h)′ and (D4h)′ are
obtained from Relations (D0). Hence Relations (D1h), (D2h), (D3h) and (D4h)
are obtained from Relation (I).
Relations (D1a)′ for m 6= i − 1, i + 1, (D1b)′ for i ≥ 3, (D1c)′ for i 6= 4, (D1d)′
for m ≤ i − 2, m = i and m ≥ i + 2, (D1e)′ for m ≤ i − 1 and m ≥ i − 2, (D1f)′ ,
(D1g)′ for i ≥ 2, (D1h)′ , (D2a)′ for m 6= i − 1, (D2b)′ for i 6= 2, (D2c)′ for i ≥ 5,
(D2d)′ for m ≤ i − 2 and m ≥ i + 1, (D2e)′ , (D2f)′ , (D2g)′ for i ≥ 2, (D2h)′ and
(D3a)′ -(D4h)′ are obtained from the braid relations. Hence for their relators r’s, we
have εr = 0. Thus we obtain Relations (D1a) for m 6= i − 1, i + 1, (D1b) for i ≥ 3,
(D1c) for i 6= 4, (D1d) for m ≤ i − 2, m = i and m ≥ i + 2, (D1e) for m ≤ i − 1
and m ≥ i − 2, (D1f), (D1g) for i ≥ 2, (D2a) for m 6= i − 1, (D2b) for i 6= 2, (D2c)
for i ≥ 5, (D2d) for m ≤ i − 2 and m ≥ i + 1, (D2e), (D2f), (D2g) for i ≥ 2 and
(D3a)-(D4g) when k = n − 1 by Relation (I).
For the other relators, i.e. Relations (D1a)′ for m = i − 1, i + 1, (D1b)′ for
i = 1, 2, (D1c)′ for i = 4, (D1d)′ for m = i − 1, i + 1, (D1e)′ for m = i, i + 1, (D1g)′
for i = 1, (D2a)′ for m = i − 1, (D2b)′ for i = 2, (D2c)′ for i = 1, 2, 3, 4, (D2d)′ for
m = i − 1, i and (D2g)′ for i = 1 when k = n− 1, we compute their εr in Section 5.5.
By the computations in Section 5.5, we can show that the remaining relations of
presentation in Proposition 3.2 are obtained from Relations (I) and (III) and we
have completed the proof of Proposition 3.2 except for computations of their εr ’s.


Figure 13. The subsurface N ′ of Ng,n which is diffeomorphic to Ng,n−1 .

Figure 14. Loop ri;l (xi+1 xi ) on Ng,n−1 based at x0 for 1 ≤ i ≤
n − 2.

28

R. KOBAYASHI AND G. OMORI

5.5. Computing εr . In this section, we compute εr for Relators (D1a)′ for m =
i − 1, i + 1, (D1b)′ for i = 1, 2, (D1c)′ for i = 4, (D1d)′ for m = i − 1, i + 1, (D1e)′
for m = i, i + 1, (D1g)′ for i = 1, (D2a)′ for m = i − 1, (D2b)′ for i = 2, (D2c)′ for
i = 1, 2, 3, 4, (D2d)′ for m = i − 1, i and (D2g)′ for i = 1 when k = n − 1. We define
ȳi
ȳi;j

:= x−1
1 yi x1

for i = 1, . . . , n − 2,

:= {(x2 x1 )−1 x22 · · · (xi xi−1 )−1 x2i }−1 ȳi {(x2 x1 )−1 x22 · · · (xi xi−1 )−1 x2i }
for 1 ≤ i ≤ n − 2, 2 ≤ j ≤ g.

Remark that ȳi;j = x−1
By using Relations (L+), (L-) and (L0) (see
j yi xj .
Lemma 5.2 and 5.3), when k = n − 1, we can compute εr as follows: εr = 2
for Relation (D1e)′ when m = i + 1, εr = 1 for Relation (D1d)′ when m = i + 1,
(D1h)′ when m = i − 1 and (D2b)′ when i = 2, εr = −1 for Relation (D1b)′
when i = 2, (D1c)′ when i = 4 and (D1d)′ , εr = −2 for Relation (D1b)′ when
i = 1, (D1e)′ when m = i and εr = 0 for the other relations. In particular, Relations (D1a) when m = i − 1, i + 1 and (D1b) when i = 2 are obtained from a single
braid relation and one of Relations (L+), (L-) and (L0), and for Relations (D1b)
when i = 1, (D1c) when i = 4, (D1d) when m = i + 1, (D2a) when m = i − 1,
(D2b) when i = 2, (D2d) when m = i, we can compute εr easily. As examples, we
compute εr for Relation (D1e)′ when m = i and Relation (D2c)′ when i = 2 by
using figures. For the other cases, we give computations of εr by only deformations
of the expressions.
For Relation (D1e)′ when m = i and k = n − 1, we have the following relation
in M(Ng,n ) by Figure 15:
∆(yl )∆((xi+1 xi ))∆(ȳi;l )−1

(L+)

−1
∆((yl (xi+1 xi ))∆(ȳi;l
)dn−1

(L+)

−1 2
∆(yl (xi+1 xi )ȳi;l
)dn−1 .

=

=

−1
Note that ∆(ri;l (xi xi+1 )) = ri;l (ai;n−1 a−1
i )ri;l by the braid relation. Hence we
have
−1
−1
−1 −1
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }(sl,n−1 d−1
l )(ai;n−1 ai )(s̄l,n−1;i dl )
−1
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }−1

=

−1
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }∆(yl )∆((xi+1 xi ))∆(ȳi;l )−1
−1
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }−1

=

−1
−1 2
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }∆(yl (xi+1 xi )ȳi;l
)dn−1
−1
{ri;n−1
(sl,n−1 dl−1 )−1 ri;n−1 }−1

(I)

=

−1
−1
−1
∆({ri;n−1
(sl,n−1 d−1
ri;n−1 }(yl (xi+1 xi )ȳi;l
))d2n−1
l )

=

−1 2
−1
−2 −1 2 −1 2
∆({x−2
)dn−1
i yl xi }yl (xi+1 xi )ȳi;l {xi yl xi }

=

∆(ri;l (xi xi+1 ))d2n−1

(I)

−1 2
ri;l (ai;n−1 a−1
i )ri;l dn−1 .

=

Thus εr = −2 for Relation (D1e)′ when m = i and k = n − 1, and we obtain
Relation (D1e) for m = i and k = n − 1 by Relations (I) and (L+). Hence
Relation (D1e) for m = i and k = n − 1 is obtained from Relations (I) and (III).

PRESENTATION FOR MAPPING CLASS GROUP

29

Figure 15. On the upper side, we explain that the relation
∆(yl )∆((xi+1 xi )) = ∆((yl (xi+1 xi ))dn−1 is obtained from Relation (L+). As a similarity, the arrow from the upper right side
−1
to lower left side explain that the relation ∆((yl (xi+1 xi ))∆(ȳi;l
)=
−1
∆(yl (xi+1 xi )ȳi;l )dn−1 is obtained from Relation (L+). “≃” means
a deformation of the loop by a homotopy fixing x0 .
For Relation (D2c) when i = 2 and k = n − 1, we note that b(x22 ) is represented
by a loop as on the lower right side of Figure 21 and ∆(b(x22 )) = br2;n−1 b−1 by the
braid relation. By Figure 16, we have
−1
−1 −1
(a1;n−1 a−1
1 )(a3;n−1 a3 )r2;n−1 r1;n−1 (a1;n−1 a1 )

=

−1 −1
2
2
(a1;n−1 a−1
1 )∆(x4 x3 )∆(x2 )∆(x1 )(a1;n−1 a1 )

(L+)

−1 −1
2 2
dn−1
(a1;n−1 a−1
1 )∆(x4 x3 )∆(x2 x1 )(a1;n−1 a1 )

(L+)

=

−1 −1 2
2 2
(a1;n−1 a−1
dn−1
1 )∆((x4 x3 )x2 x1 )(a1;n−1 a1 )

(I)

2 2
2
∆((a1;n−1 a−1
1 )((x4 x3 )x2 x1 ))dn−1

=

=

in M(Ng,n ). By Figure 17 and Figure 18, we have the following relation in M(Ng,n ).
−1
−1
(a2;n−1 a−1
r3;n−1 (a3;n−1 a−1
r4;n−1 (a2;n−1 a−1
2 )
3 )
2 )

=

−1
−1
−1
(a2;n−1 a−1
r3;n−1 r4;n−1 r4;n−1
(a3;n−1 a−1
r4;n−1 (a2;n−1 a−1
2 )
3 )
2 )

=

−1
−1
(a2;n−1 a−1
∆(x23 )∆(x24 )r4;n−1
∆(x4 x3 )−1 r4;n−1 (a2;n−1 a−1
2 )
2 )

(L+),(I)

=

(L-)

=

−1
−1
(a2;n−1 a−1
∆(x23 x24 )∆(r4;n−1
((x4 x3 )−1 ))(a2;n−1 a−1
2 )dn−1
2 )
−1
−1
(a2;n−1 a−1
∆(x23 x24 r4;n−1
((x4 x3 )−1 ))(a2;n−1 a−1
2 )
2 )

30

R. KOBAYASHI AND G. OMORI
(I)

=

−1 2 2 −1
∆((a2;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 ))).
2 )

Let ζ1 and ζ2 be simple closed curves on Ng,n as in Figure 19. Since ι∗ (tζ1 ) ∈
−1
2 2
M+ (Ng,n−1 , x0 ) fixes x22 and ∆((a1;n−1 a−1
1 )((x4 x3 )x1 x1 )) = tζ1 tζ2 , we have
−1 2
2 2
2
∆((a1;n−1 a−1
1 )((x4 x3 )x1 x1 ))(x2 ) = tζ2 (x2 ). We remark that the loop as on the
upper right side of Figure 20 is homotopic to the loop as on the lower right side
of Figure 20 by a homotopy fixing x0 as in Figure 20. By the relations above and
Figure 21, we have
−1 −1
{(a3;n−1 a−1
(a1;n−1 a1−1 )(a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r2;n−1 r1;n−1
−1
−1
−1
(a1;n−1 a−1
r2;k (a2;n−1 a−1
r3;n−1 (a3;n−1 a−1
r4;n−1
1 )
2 )
3 )
−1
−1
(a2;n−1 a−1
2 ){(a3;n−1 a3 )(a1;n−1 a1 )}
(L+),(I)

=

−1 −1
2 2
∆((a1;n−1 a−1
{(a3;n−1 a−1
1 )((x4 x3 )x2 x1 ))r2;k
3 )(a1;n−1 a1 )}
−1
−1
(a2;n−1 a−1
r3;n−1 (a3;n−1 a−1
r4;n−1 (a2;n−1 a−1
2 )
3 )
2 )
−1
2
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}dn−1

(L+),(L-),(I)

=

−1 −1
2 2
{(a3;n−1 a−1
∆((a1;n−1 a−1
3 )(a1;n−1 a1 )}
1 )((x4 x3 )x2 x1 ))r2;k
−1 2 2 −1
∆((a2;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 ))){(a3;n−1 a3−1 )(a1;n−1 a−1
2 )
1 )}

d2n−1
=

−1 −1
2
2 2
{(a3;n−1 a−1
∆((a1;n−1 a−1
3 )(a1;n−1 a1 )}
1 )((x4 x3 )x2 x1 ))∆(x2 )
2 2 −1
2 2
∆((a1;n−1 a−1
∆((a1;n−1 a−1
1 )((x4 x3 )x2 x1 ))
1 )((x4 x3 )x2 x1 ))
−1
−1 2 2 −1
∆((a2;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 ))){(a3;n−1 a−1
2 )
3 )(a1;n−1 a1 )}

d2n−1
(I)

=

−1 −1
−1
2
2 2
{(a3;n−1 a−1
∆(t−1
3 )(a1;n−1 a1 )}
ζ2 (x2 ))∆((a1;n−1 a1 )((x4 x3 )x2 x1 ))
−1
−1 2 2 −1
∆((a2;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 ))){(a3;n−1 a−1
2 )
3 )(a1;n−1 a1 )}

d2n−1
(L-)

=

2
−1
{(a3;n−1 a3−1 )(a1;n−1 a−1
∆(t−1
1 )}
ζ2 (x2 ))
−1 −1 2 2 −1
2 2
∆((a1;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 )))
1 )((x4 x3 )x2 x1 ) · (a2;n−1 a2 )
−1
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}dn−1

(L-)

=

−1 −1
(x22 ) ·
{(a3;n−1 a−1
∆(tζ−1
3 )(a1;n−1 a1 )}
2
−1 −1 2 2 −1
2 2
(a1;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 )))
1 )((x4 x3 )x2 x1 ) · (a2;n−1 a2 )
−1
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}

(I)

=

−1 −1 −1 2
∆({(a3;n−1 a−1
(tζ2 (x2 ) ·
3 )(a1;n−1 a1 )}
−1 −1 2 2 −1
2 2
(a1;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 ))))
1 )((x4 x3 )x2 x1 ) · (a2;n−1 a2 )

=

∆(b(x22 ))

(I)

br2;n−1 b−1 .

=

PRESENTATION FOR MAPPING CLASS GROUP

31

Thus εr = 0 for Relation (D2c)′ when i = 2 and k = n − 1, and we obtain
Relation (D2c) for i = 2 and k = n − 1 by Relations (I) and (III).

Figure 16. Relations ∆(x22 )∆(x21 ) = ∆(x22 x21 )dn−1 and
∆(x4 x3 )∆(x21 x22 )
=
∆((x4 x3 )x22 x21 )dn−1
and
loop
−1
2 2
(a1;n−1 a1 )((x4 x3 )x2 x1 ).

Figure 17. Relation ∆(x23 )∆(x24 ) = ∆(x23 x24 )dn−1 .
For Relation (D1d) when m = i − 1 and k = n − 1, we have
−1
−1
−1
−1
[(ai−1;n−1 a−1
, (sl,n−1 d−1
](ai;n−1 a−1
i−1 )
i )(sl,n−1 dl )(ai−1;n−1 ai−1 )
l )

=

−1
−1
{(sl,n−1 d−1
∆(xi xi−1 )∆(yl )∆(xi+1 xi ){(sl,n−1 d−1
l )
l )(ai−1;n−1 ai−1 )}

(ai−1;n−1 a−1
i−1 )}
(L+)

=

−1
−1
{(sl,n−1 d−1
∆(xi xi−1 )∆(yl (xi+1 xi )){(sl,n−1 d−1
l )(ai−1;n−1 ai−1 )}
l )

(ai−1;n−1 a−1
i−1 )}dn−1

32

R. KOBAYASHI AND G. OMORI

−1
Figure
18. Relation
∆(x23 x24 )∆(r4;n−1
((x4 x3 )−1 ))
=
−1
−1
2 2 −1
−1
−1
∆(x3 x4 r4;n−1 ((x4 x3 ) ))dn−1 and loops r4;n−1 ((x4 x3 ) ) and
−1 2 2 −1
(a2;n−1 a−1
(x3 x4 r4;n−1 ((x4 x3 )−1 )).
2 )

Figure 19. Simple closed curves ζ1 and ζ2 on Ng,n .

(L0)

=

−1
−1
{(sl,n−1 d−1
∆((xi xi−1 )yl (xi+1 xi )){(sl,n−1 d−1
l )(ai−1;n−1 ai−1 )}
l )

(ai−1;n−1 a−1
i−1 )}dn−1
(I)

=

−1
−1
∆({(sl,n−1 d−1
((xi xi−1 )yl (xi+1 xi )))dn−1
l )(ai−1;n−1 ai−1 )}

=

∆(ai−1;l (xi+1 xi ))dn−1

(I)

−1
ai−1;l (ai;n−1 a−1
i )ai−1;l dn−1 .

=

PRESENTATION FOR MAPPING CLASS GROUP

33

2
Figure 20. Loop t−1
ζ2 (x2 ).

Thus εr = −1 for Relation (D1d)′ when m = i − 1 and k = n − 1, and we obtain
Relation (D1d) when m = i − 1 and k = n − 1 by Relations (I) and (III).
For Relation (D1e) when m = i + 1 and k = n − 1, we have
−1
−1
ri+1;n−1
(sl,n−1 d−1
ri+1;n−1 (s̄l,n−1;i+1 dl−1 )(ai;n−1 a−1
i )
l )

=
(I),(L-)

=

(L-)

−1
ri+1;n−1
∆(yl )ri+1;n−1 ∆(ȳl;i+1 )∆(xi+1 xi )
−1
∆(ri+1;n−1
(yl ))∆(ȳl;i+1 (xi+1 xi ))d−1
n−1

=

−1
∆(ri+1;n−1
(yl )ȳl;i+1 (xi+1 xi ))d−2
n−1

=

∆(ri+1;l (xi+1 xi ))d−2
n−1

(I)

−1
−2
ri+1;l (ai;n−1 a−1
i )ri+1;l dn−1 .

=

Thus εr = −2 for Relation (D1e)′ when m = i + 1 and k = n − 1, and we obtain
Relation (D1e) when m = i + 1 and k = n − 1 by Relations (I) and (III).
For Relation (D1g) when i = 1 and k = n − 1, we have
−1
−1 −1
−1 −1
−1
−1
[(s̄l,k d−1
, st,k d−1
(sl,k d−1
t ]
l )
l )(a1;k a1 )r1;k (s̄t,k dt )r1;k (sl,k dl )
−1 −1
−1
r1;k (s̄t,k d−1
r1;k [(s̄l,k d−1
, (st,k d−1
t )
t )]
l )

=

−1
−1
−1
∆(yl )−1
[(s̄l,k d−1
, st,k d−1
∆(yl )∆(x2 x1 )r1;k ∆(ȳt )r1;k
t ]
l )
−1
−1
[(s̄l,k d−1
, (st,k d−1
r1;k ∆(ȳt )−1 r1;k
t )]
l )

34

R. KOBAYASHI AND G. OMORI

−1 −1
2 2
Figure 21. Relations ∆((a1;n−1 a−1
1 )((x4 x3 )x2 x1 ))∆((a2;n−1 a2 )
−1
−1
(x23 x24 r4;n−1 ((x4 x3 )−1 )))
=
∆(a1;n−1 a1 )((x4 x3 )x22 x21 )·
−1 −1 2 2 −1
−1
(a2;n−1 a2 ) (x3 x4 r4;n−1 ((x4 x3 ) )))d−1
n−1
−1
2
2 2
·
and
∆(t−1
ζ2 (x2 ))∆(a1;n−1 a1 )((x4 x3 )x2 x1 )
−1 −1 2 2 −1
−1
(a2;n−1 a2 ) (x3 x4 r4;n−1 ((x4 x3 ) )))
=
−1
2
2 2
(x
)
·
a
a
∆(t−1
)((x
x
)x
x
)
·
1;n−1 1
4 3 2 1
2
ζ2
−1 −1 2 2 −1
−1
−1
2
(a2;n−1 a2 ) (x3 x4 r4;n−1 ((x4 x3 ) )))dn−1 and loop b(x2 ).

(L+),(I)

=

−1
−1
[(s̄l,k d−1
, st,k d−1
∆(yl (x2 x1 ))∆(r1;k (ȳt ))∆(yl )−1 ∆(r1;k (ȳt ))−1
t ]
l )
−1
[(s̄l,k d−1
, (st,k d−1
t )]dn−1
l )

(I)

=

−1
−1
[(s̄l,k d−1
, st,k d−1
∆(yl (x2 x1 ))∆(∆(r1;k (ȳt ))(yl ))−1
t ]
l )
−1
[(s̄l,k d−1
, (st,k d−1
t )]dn−1
l )

(L-)

=

−1
−1
[(s̄l,k d−1
, st,k d−1
∆(yl (x2 x1 )∆(r1;k (ȳt ))(yl )−1 )
t ]
l )
−1
[(s̄l,k d−1
, (st,k d−1
t )]
l )

(I)

=
=

−1
−1
∆([(s̄l,k d−1
, st,k d−1
(yl (x2 x1 )∆(r1;k (ȳt ))(yl )−1 ))
t ]
l )
∆(s̄l;t (xi+1 xi ))

(I)

−1
s̄l;t (ai;n−1 a−1
i )s̄l;t .

=

PRESENTATION FOR MAPPING CLASS GROUP

35

Thus εr = 0 for Relation (D1g)′ when i = 1 and k = n − 1, and we obtain
Relation (D1g) when i = 1 and k = n − 1 by Relations (I) and (III).
For Relation (D2c) when i = 1 and k = n − 1, we have
−1
−1
−1 −1
(a1;n−1 a−1
(a3;n−1 a−1
(a2;n−1 a−1
r4;n−1 (a3;n−1 a−1
1 )
3 )
2 )
3 )
−1
−1
−1
r3;n−1
(a2;n−1 a−1
2 )r2;n−1 (a1;n−1 a1 )

=

−1
−1
(a1;n−1 a−1
(a3;n−1 a−1
∆(x3 x2 )−1 ∆(x24 )−1 (a3;n−1 a−1
3 )
1 )
3 )
−1
r3;n−1
∆(x3 x2 )r3;n−1 ∆(x23 )−1 ∆(x22 )−1 (a1;n−1 a−1
1 )

(L-),(I)

=

−1
−1
−1
(a1;n−1 a−1
(a3;n−1 a−1
∆((x3 x2 )−1 x−2
1 )
3 )
4 )(a3;n−1 a3 )
−1
−2
−1 −2
∆(r3;n−1
(x3 x2 ))∆(x−2
3 x2 )(a1;n−1 a1 )dn−1

(I),(L+)

=

−1
−2 −2
−1
−1
(a1;n−1 a−1
∆((a3;n−1 a−1
((x3 x2 )−1 x−2
1 )
3 )
4 ))∆(r3;n−1 (x3 x2 )x3 x2 )
−1
(a1;n−1 a−1
1 )dn−1

(L+)

=

−1
−2 −2
−1
−1
(a1;n−1 a−1
∆((a3;n−1 a−1
((x3 x2 )−1 x−2
1 )
3 )
4 )r3;n−1 (x3 x2 )x3 x2 )

(a1;n−1 a−1
1 )
(I)

=

−1
−2 −2
−1
−1
∆((a1;n−1 a−1
((a3;n−1 a−1
((x3 x2 )−1 x−2
1 )
3 )
4 )r3;n−1 (x3 x2 )x3 x2 ))

=

∆(b(x21 ))

(I)

b(r1;n−1 )b−1 .

=

Thus εr = 0 for Relation (D2c)′ when i = 1 and k = n − 1, and we obtain
Relation (D2c) when i = 1 and k = n − 1 by Relations (I) and (III).
For Relation (D2c) when i = 3 and k = n − 1, we have
−1 −1 −1
−1
−1
{(a3;n−1 a−1
r4;n−1 (a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r3;n−1 (a2;n−1 a2 )
−1
−1
−1 −1
−1
r2;n−1
(a1;n−1 a−1
r3;n−1 (a3;n−1 a−1
1 )r1;n−1 (a2;n−1 a2 )
3 )
−1
−1
(a1;n−1 a−1
{(a3;n−1 a−1
1 )
3 )(a1;n−1 a1 )}

=

−1 −1 −1
−1
−1
{(a3;n−1 a−1
r4;n−1 (a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r3;n−1 (a2;n−1 a2 )
−1
−1
−1
r3;n−1 (a3;n−1 a−1
r2;n−1
∆(x2 x1 )r2;n−1 ∆(x22 )−1 ∆(x21 )−1 (a2;n−1 a−1
2 )
3 )
−1
∆(x2 x1 )−1 {(a3;n−1 a−1
3 )(a1;n−1 a1 )}

(I),(L-)

=

−1 −1 −1
−1
−1
{(a3;n−1 a−1
r4;n−1 (a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r3;n−1 (a2;n−1 a2 )
−1 −1
−1
−2
−1
r3;n−1 (a3;n−1 a−1
∆(r2;n−1
(x2 x1 ))∆(x−2
3 )
2 x1 )(a2;n−1 a2 )
−1
−1
∆(x2 x1 )−1 {(a3;n−1 a−1
3 )(a1;n−1 a1 )}dn−1

(L+)

=

−1 −1 −1
−1
−1
{(a3;n−1 a−1
r4;n−1 (a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r3;n−1 (a2;n−1 a2 )
−1
−2
−1 −1
−1
∆(r2;n−1
(x2 x1 )x−2
r3;n−1 (a3;n−1 a−1
2 x1 )(a2;n−1 a2 )
3 )
−1
∆(x2 x1 )−1 {(a3;n−1 a−1
3 )(a1;n−1 a1 )}

(I)

=

−1 −1
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}
−1
−1
−1
−1
−2 −2
r4;n−1
∆({(a3;n−1 a−1
3 )r3;n−1 (a2;n−1 a2 )}(r2;n−1 (x2 x1 )x2 x1 ))r4;n−1

36

R. KOBAYASHI AND G. OMORI
−1
∆(x24 )−1 ∆(x2 x1 )−1 {(a3;n−1 a−1
3 )(a1;n−1 a1 )}
(I),(L-)

=

−1 −1
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}
−1
−1
−1
−1
−2 −2
∆({r4;n−1
(a3;n−1 a−1
3 )r3;n−1 (a2;n−1 a2 )}(r2;n−1 (x2 x1 )x2 x1 ))
−1
−1
∆(x−2
){(a3;n−1 a−1
4 (x2 x1 )
3 )(a1;n−1 a1 )}

(L+)

=

−1 −1
{(a3;n−1 a−1
3 )(a1;n−1 a1 )}
−1
−1
−1
−1
−2 −2 −2
∆({r4;n−1
(a3;n−1 a−1
3 )r3;n−1 (a2;n−1 a2 )}(r2;n−1 (x2 x1 )x2 x1 )x4
−1
(x2 x1 )−1 ){(a3;n−1 a−1
3 )(a1;n−1 a1 )}

(I)

=

−1 −1
−1
−1
−1
∆({(a3;n−1 a−1
({r4;n−1
(a3;n−1 a−1
3 )(a1;n−1 a1 )}
3 )r3;n−1 (a2;n−1 a2 )}
−1
−2 −2
−1
(r2;n−1
(x2 x1 )x−2
))
2 x1 )x4 (x2 x1 )

=

∆(b(x23 ))

(I)

b(r3;n−1 )b−1 .

=

Thus εr = 0 for Relation (D2c)′ when i = 3 and k = n − 1, and we obtain
Relation (D2c) when i = 3 and k = n − 1 by Relations (I) and (III).
For Relation (D2c) when i = 4 and k = n − 1, we have
−1 −1
r4;n−1 (a2;n−1 a−1
r2;n−1
2 )r1;n−1 (a1;n−1 a1 )
−1
−1
−1
(a2;n−1 a−1
r3;n−1 (a3;n−1 a−1
r4;n−1 (a3;n−1 a−1
2 )
3 )
3 )(a1;n−1 a1 )

=

−1
−1
∆(x24 )∆(x3 x2 )∆(x21 )(a1;n−1 a−1
∆(x22 )(a1;n−1 a−1
1 )
1 )∆(x2 x1 )
−1
(a2;n−1 a2−1 )−1 ∆(x23 )(a2;n−1 a−1
∆(x4 x3 )−1
2 ) ∆(x3 x2 )

r4;n−1 ∆(x4 x3 )∆(x2 x1 )
(I),(L0)

=

−1 2
(x2 ))∆(x2 x1 )−1
∆(x24 )∆(x3 x2 )∆(x21 )∆((a1;n−1 a−1
1 )
−1 2
∆((a2;n−1 a−1
(x3 ))∆((x3 x2 )−1 (x4 x3 )−1 )r4;n−1 ∆(x4 x3 )∆(x2 x1 )
2 )

(L+)

=

−1 2
∆(x24 (x3 x2 )x21 )∆((a1;n−1 a−1
(x2 ))∆(x2 x1 )−1
1 )
−1 2
∆((a2;n−1 a−1
(x3 ))∆((x3 x2 )−1 (x4 x3 )−1 )r4;n−1 ∆(x4 x3 x2 x1 )d3n−1
2 )

(L-)

=

−1 2
∆(x24 (x3 x2 )x21 )∆((a1;n−1 a−1
(x2 ))∆(x2 x1 )−1
1 )
−1 2
∆((a2;n−1 a−1
(x3 )(x3 x2 )−1 (x4 x3 )−1 )r4;n−1 ∆(x4 x3 x2 x1 )d2n−1
2 )

(L0)

=

−1 2
∆(x24 (x3 x2 )x21 )∆((a1;n−1 a−1
(x2 ))∆((x2 x1 )−1
1 )
−1 2
(a2;n−1 a−1
(x3 )(x3 x2 )−1 (x4 x3 )−1 )r4;n−1 ∆(x4 x3 x2 x1 )d2n−1
2 )

(L-)

=

−1 2
∆(x24 (x3 x2 )x21 )∆((a1;n−1 a−1
(x2 )(x2 x1 )−1
1 )
−1 2
(a2;n−1 a−1
(x3 )(x3 x2 )−1 (x4 x3 )−1 )r4;n−1 ∆(x4 x3 x2 x1 )dn−1
2 )

(L-)

=

−1 2
−1 2
∆(x24 (x3 x2 )x21 (a1;n−1 a−1
(x2 )(x2 x1 )−1 (a2;n−1 a−1
(x3 )(x3 x2 )−1
1 )
2 )

(x4 x3 )−1 )∆(x4 x3 x2 x1 ) ∆(x4 x3 x2 x1 )−1 ∆(x24 )∆(x4 x3 x2 x1 )
(L+),(I)

=

−1 2
−1 2
∆(x24 (x3 x2 )x21 (a1;n−1 a−1
(x2 )(x2 x1 )−1 (a2;n−1 a−1
(x3 )(x3 x2 )−1
1 )
2 )

(x4 x3 )−1 x4 x3 x2 x1 )∆(∆(x4 x3 x2 x1 )−1 (x24 ))dn−1

PRESENTATION FOR MAPPING CLASS GROUP
(L-)

=

37

−1 2
−1 2
∆(x24 (x3 x2 )x21 (a1;n−1 a−1
(x2 )(x2 x1 )−1 (a2;n−1 a−1
(x3 )(x3 x2 )−1
1 )
2 )

(x4 x3 )−1 x4 x3 x2 x1 ∆(x4 x3 x2 x1 )−1 (x24 ))
=

∆(b(x24 ))

(I)

b(r4;n−1 )b−1 .

=

Thus εr = 0 for Relation (D2c)′ when i = 4 and k = n − 1, and we obtain
Relation (D2c) when i = 4 and k = n − 1 by Relations (I) and (III).
For Relation (D2d) when m = i − 1 and k = n − 1, we have
−1
−1
−1
{(sl,n−1 d−1
(ai−1;n−1 a−1
i−1 )(sl,n−1 dl )ri;n−1 ri−1;n−1
l )(ai−1;n−1 ai−1 )}
−1
−1
−1
(ai−1;n−1 a−1
ri;n−1 (s̄l,n−1;i d−1
i−1 )
l ){(sl,n−1 dl )(ai−1;n−1 ai−1 )}

=

−1
−1
−1
{(sl,n−1 d−1
(ai−1;n−1 ai−1
)∆(yl )∆(x2i )∆(x2i−1 )
l )(ai−1;n−1 ai−1 )}
−1
−1
−1
(ai−1;n−1 a−1
∆(x2i )(s̄l,n−1;i d−1
i−1 )
l ){(sl,n−1 dl )(ai−1;n−1 ai−1 )}

(L+)

=

−1
−1 −1
−1
2 2
{(sl,n−1 d−1
(ai−1;n−1 a−1
i−1 )∆(yl xi xi−1 )(ai−1;n−1 ai−1 )
l )(ai−1;n−1 ai−1 )}

(I)

−1
−1
2 2
{(sl,n−1 d−1
∆((ai−1;n−1 a−1
i−1 )(yl xi xi−1 ))∆(ȳl;i )
l )(ai−1;n−1 ai−1 )}

=

−1
−1
2
(s̄l,n−1;i d−1
∆(x2i )(s̄l,n−1;i dl−1 ){(sl,n−1 d−1
l )
l )(ai−1;n−1 ai−1 )}dn−1
(I)

=

−1
−1
2 2
{(sl,n−1 d−1
∆((ai−1;n−1 a−1
i−1 )(yl xi xi−1 ))∆(ȳl;i )
l )(ai−1;n−1 ai−1 )}
−1 2
2
∆((s̄l,n−1;i d−1
(xi )){(sl,n−1 dl−1 )(ai−1;n−1 a−1
i−1 )}dn−1
l )

(L-)

=

−1
−1
2 2
{(sl,n−1 d−1
∆((ai−1;n−1 a−1
i−1 )(yl xi xi−1 )ȳl;i )
l )(ai−1;n−1 ai−1 )}
−1
−1 2
∆((s̄l,n−1;i d−1
(xi )){(sl,n−1 d−1
l )
l )(ai−1;n−1 ai−1 )}dn−1

(L-)

=

−1
−1
2 2
{(sl,n−1 d−1
∆((ai−1;n−1 a−1
i−1 )(yl xi xi−1 )ȳl;i
l )(ai−1;n−1 ai−1 )}
−1
−1 2
(xi )){(sl,n−1 d−1
(s̄l,n−1;i d−1
l )
l )(ai−1;n−1 ai−1 )}

(I)

=

−1
−1
2 2
∆({(sl,n−1 d−1
((ai−1;n−1 a−1
i−1 )(yl xi xi−1 )ȳl;i
l )(ai−1;n−1 ai−1 )}
−1 2
(s̄l,n−1;i d−1
(xi )))
l )

=

∆(ai−1;n−1 (x2i ))

(I)

ai−1;n−1 (ri;n−1 )a−1
i−1;n−1 .

=

Thus εr = 0 for Relation (D2d)′ when m = i − 1 and k = n − 1, and we obtain
Relation (D2d) when m = i − 1 and k = n − 1 by Relations (I) and (III).
For Relation (D2g) when i = 1 and k = n − 1, we have
−1 −1
−1
−1 −1
[st,n−1 d−1
][r1;n−1 (s̄t,n−1 d−1
]r1;n−1
t , (s̄l,n−1 dl )
t )r1;n−1 , (sl,n−1 dl )

=

−1
−1
∆(ȳl )∆(x21 )(s̄t,n−1 d−1
(st,n−1 d−1
(st,n−1 d−1
t )
t )∆(ȳl )
t )
−1
−1
−1
r1;n−1
(sl,n−1 d−1
r1;n−1 ∆(ȳt )−1 r1;n−1
(sl,n−1 d−1
l )
l )r1;n−1

(I)

=

−1
−1
∆((st,n−1 d−1
∆(ȳl )∆(y¯t )(s̄t,n−1 d−1
∆(x21 )(s̄t,n−1 d−1
t )(ȳl ))
t )
t )
−1
−1
∆(r1;n−1
(sl,n−1 d−1
r1;n−1 (ȳt ))−1
l )

(I)

=

−1
−1 2
∆((st,n−1 d−1
∆(ȳl )∆(y¯t )∆((s̄t,n−1 d−1
(x1 ))
t )(ȳl ))
t )

38

R. KOBAYASHI AND G. OMORI
−1
−1
∆(r1;n−1
(sl,n−1 d−1
r1;n−1 (ȳt ))−1
l )
(L-)

=

−1
−1 2
∆((st,n−1 d−1
∆(ȳl y¯t )∆((s̄t,n−1 d−1
(x1 ))
t )(ȳl ))
t )
−1
−1
∆(r1;n−1
(sl,n−1 d−1
r1;n−1 (ȳt ))−1 d−1
n−1
l )

(L-)

=

−1
−1 2
∆((st,n−1 d−1
∆(ȳl y¯t (s̄t,n−1 d−1
(x1 ))
t )(ȳl ))
t )
−1
−1
∆(r1;n−1
(sl,n−1 d−1
r1;n−1 (ȳt ))−1 d−2
n−1
l )

(L+)

=

−1
−1 2
∆((st,n−1 d−1
ȳl y¯t (s̄t,n−1 d−1
(x1 ))
t )(ȳl )
t )
−1
−1
∆(r1;n−1
(sl,n−1 d−1
r1;n−1 (ȳt ))−1 d−1
n−1
l )

(L+)

=

−1
−1 2 −1
−1
∆((st,n−1 d−1
ȳl y¯t (s̄t,n−1 d−1
(x1 )r1;n−1 (sl,n−1 d−1
r1;n−1 (ȳt )−1 )
t )(ȳl )
t )
l )

=

∆(s̄l,t (x21 ))

(I)

s̄l,t (r1;n−1 )s̄−1
l,t .

=

Thus εr = 0 for Relation (D2g)′ when i = 1 and k = n − 1, and we obtain
Relation (D2g) when i = 1 and k = n − 1 by Relations (I) and (III).
Acknowledgement: The authors would like to express his gratitude to Hisaaki
Endo, for his encouragement and helpful advices. The second author was supported
by JSPS KAKENHI Grant number 15J10066.
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39

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(Ryoma Kobayashi) Department of General Education, Ishikawa National College
of Technology, Tsubata, Ishikawa, 929-0392, Japan
E-mail address: kobayashi ryoma@ishikawa-nct.ac.jp
(Genki Omori) Department of Mathematics, Tokyo Institute of Technology, Ohokayama, Meguro, Tokyo 152-8551, Japan
E-mail address: omori.g.aa@m.titech.ac.jp

