Table Of ContentA NORMAL GENERATING SET FOR THE TORELLI GROUP OF A
COMPACT NON-ORIENTABLE SURFACE
RYOMA KOBAYASHI
6
1
0
2 Abstract. For a compact surface S, let I(S) denote the Torelli group of S. For a
compact orientable surface Σ, I(Σ) is generated by BSCC maps and BP maps (see [10]
n
a and [11]). For a non-orientable closedsurface N, I(N) is generatedby BSCC maps and
J BP maps (see [5]). In this paper, we give an explicit normal generating set for I(Nb),
g
3 where Nb is a genus-g compact non-orientable surface with b boundary components for
g
2
g ≥4 and b≥1.
]
T
G 1. Introduction
.
h For g ≥ 1 and b ≥ 0, let Nb denote a genus-g compact connected non-orientable surface
t g
a with b boundary components, and let N = N0. In this paper, we regard Nb as a surface
m g g g
obtained by attaching g M¨obius bands to a sphere with g +b boundary components, as
[
shown in Figure 1. We say each of these M¨obius bands attached to this sphere a cross cap.
1 The mapping class group M(Nb) of Nb is the group consisting of isotopy classes of all
v g g
7 diffeomorphisms over Nb which fix each point of the boundary. The Torelli group I(Nb)
g g
5 of Nb is the subgroup of M(Nb) consisting of elements acting trivially on the integral
2 g g
6 first homology group H (Nb;Z) of Nb. The Torelli group of a compact orientable surface
1 g g
0
is generated by BSCC maps and BP maps (see [10] and [11]). In particular, Johnson
.
1 [6] showed that the Torelli group of an orientable closed surface is finitely generated by
0
BP maps. Hirose and the author [5] showed that I(N ) is generated by BSCC maps and
6 g
1 BP maps for g ≥ 4. In this paper, we give an explicit normal generating set for I(Nb)
g
:
v consisting of BSCC maps and BP maps, for g ≥ 4 and b ≥ 1.
i
X
r
a
attach
{ {
g b
Figure 1. A genus-g compact connected non-orientable surface with b
boundary components.
Let N be a compact connected non-orientable surface. For a simple closed curve c on
N, we call c an A-circle (resp. an M-circle) if its regular neighborhood is an annulus
(resp. a M¨obius band), as shown in Figure 2. Foran A-circle c, we can define the mapping
class t , called the Dehn twist about c, and the direction of the twist is indicated by a
c
1
2 R.KOBAYASHI
(a) A-circles. (b) M-circles.
Figure 2.
c
t
c
Figure 3. The Dehn twist t about c.
c
small arrow written beside c as shown in Figure 3. We can notice that an A-circle (resp.
an M-circle) passes through cross caps even times (resp. odd times).
Let α, β, β′, γ, δ , ρ and σ be simple closed curves on Nb as shown in Figure 4. The
i i ij g
main result of this paper is as follows.
Theorem 1.1. Let g ≥ 5 and b ≥ 0. In M(Nb), I(Nb) is normally generated by t ,
g g α
t t−1, t , t and t for 1 ≤ i,j ≤ b−1 with i < j. In M(Nb), I(Nb) is normally
β β′ δi ρi σij 4 4
generated by t , t t−1, t , t , t and t for 1 ≤ i,j ≤ b−1 with i < j.
α β β′ δi ρi σij γ
β
δ
i
α
β' i j
(a) Loops α, β, β′ and δ .
i
ρ σ
i ij
γ
i j
(b) Loops γ, ρ and σ .
i ij
Figure 4.
In this paper, for f,g ∈ M(Nb), the composition gf means that we first apply f and
g
then g.
2. Basics on mapping class groups for non-orientable surfaces
2.1. On mapping class groups for non-orientable surfaces.
Mapping class groups for orientable surfaces are generated by only Dehn twists. Lick-
orish showed that M(N ) is generated by Dehn twists and Y-homeomorphisms and that
g
the subgroup of M(N ) generated by all Dehn twists is an index 2 subgroup of M(N )
g g
(see [7, 8]). Hence M(N ) is not generated by only Dehn twists. On the other hand,
g
A NORMAL GENERATING SET FOR I(Nb) 3
g
since a Y-homeomorphism acts trivially on H (N ;Z/2Z), M(N ) is not generated by
1 g g
only Y-homeomorphisms.
Chillingworth [2] found a finite generating set for M(N ). M(N ) and M(N1) are
g 1 1
trivial(see[3]). FinitepresentationsforM(N ),M(N1),M(N )andM(N )areobtained
2 2 3 4
by [7], [12], [1] and [14], respectively. Paris-Szepietowski [9] obtained a finite presentation
of M(Nb) for b = 0,1 and g +b > 3.
g
Let N be a non-orientable surface, and let a and m be an oriented A-circle and an
M-circle on N respectively such that a and m mutually intersect transversely at only one
point. We now define a Y-homeomorphism Y . Let K be a regular neighborhood of a∪b
m,a
in N, and let M be a regular neighborhood of m in the interior of K. We can see that
K is homeomorphic to the Klein bottle with one boundary component. Y is defined as
m,a
the isotopy class of a diffeomorphism over N which is described by pushing M once along
a and which fixes each point of the boundary and the exterior of K (see Figure 5).
K a
m
Y
m,a
M
Figure 5. The Y-homoemorphism Y .
m,a
2.2. On Torelli groups for non-orientable surfaces.
Let c be an A-circle on a non-orientable surface N such that N \ c is not connected.
We call t a bounding simple closed curve map, for short a BSCC map. For example, in
c
Theorem 1.1, t , t , t , t and t are BSCC maps. Let c and c be A-circles on N such
α γ δi ρi σij 1 2
that N\c is connected, N\(c ∪c ) is not connected and one of its connected components
i 1 2
is an orientable surface with two boundary components. We call t t−1 a bounding pair
c1 c2
map, for short a BP map. For example, in Theorem 1.1, t t−1 is a BP map.
β β′
Hirose and the author [5] obtained the following theorem.
Theorem 2.1 ([5]). For g ≥ 5, I(N ) is normally generated by t and t t−1 in M(N ).
g α β β′ g
I(N ) is normally generated by t , t t−1 and t in M(N ).
4 α β β′ γ 4
Theorem 1.1 is a natural extension of Theorem 2.1.
We can check that all BSCC maps and BP maps are in I(Nb). In addition, by Theo-
g
rem 1.1, we have that I(Nb) is generated by BSCC maps and BP maps. We do not know
g
whether or not I(Nb) can be finitely generated.
g
2.3. Capping, Pushing and Forgetful homomorphisms.
Let N be a compact non-orientable surface. Take a point ∗ in the interior of N. Let
M(N,∗) denote the group consisting of isotopy classes of all diffeomorphisms over N
which fix ∗ and each point of the boundary, and let I(N,∗) denote the subgroup of
M(N,∗) consisting of elements acting trivially on H (N;Z)
1
Take a point ∗ in the interior of Nb−1. We regard Nb as a subsurface of Nb−1
g g g
not containing ∗. The natural embedding Nb ֒→ Nb−1 induces the homomorphism
g g
Cb : M(Nb) → M(Nb−1,∗), called the capping homomorphism. We have the following
g g g
lemma.
Lemma 2.2 (cf. [4, 13]). kerCb is generated by t .
g δb
4 R.KOBAYASHI
Since t is in I(Nb) we obtain the following.
δb g
Corollary 2.3. kerCb| is generated by t .
g I(Ngb) δb
We remark that Cb and Cb| are not surjective.
g g I(Ngb)
The pushing homomorphism Pb−1 : π (Nb−1,∗) → M(Nb−1,∗) is defined as follows.
g 1 g g
For x ∈ π (Nb−1,∗) take a representative oriented loop x˜ based at ∗. Pb−1(x) is described
1 g g
by pushing ∗ once along x˜ and fixes each point of exterior of neighborhood of x˜. Note
that Pb−1 is an anti-homomorphism, that is, for x,y ∈ π (Nb−1,∗) we have Pb−1(xy) =
g 1 g g
Pb−1(y)Pb−1(x). The forgetful homomorphism Fb−1 : M(Nb−1,∗) → M(Nb−1) is defined
g g g g g
naturally. Note that Fb−1 is surjective.
g
We have the natural exact sequence
Pb−1 Fb−1
π (Nb−1,∗) −g→ M(Nb−1,∗) −g→ M(Nb−1) −→ 1.
1 g g g
Since ImPb−1 is in I(Nb−1,∗) and Fb−1(I(Nb−1,∗)) is equal to I(Nb−1), we have the
g g g g g
exact sequence
π (Nb−1,∗) −→ I(Nb−1,∗) −→ I(Nb−1) −→ 1.
1 g g g
3. A normal generating set fot Cb(I(Nb)) in M(Nb−1,∗)
g g g
By the capping homomorphism Cb : M(Nb) → M(Nb−1,∗) we have that a normal
g g g
generating set for I(Nb) in M(Nb) consists of t and lifts by Cb of normal generators of
g g δb g
Cb(I(Nb)) in M(Nb−1,∗). Thus in this section we consider a normal generating set for
g g g
Cb(I(Nb)) in M(Nb−1,∗).
g g g
Let α and β be oriented loops on Nb−1 based at ∗ as shown in Figure 6, and let x
i j g i
and y be the elements of π (Nb−1,∗) corresponding to α and β respectively. Note that
j 1 g i j
π (Nb−1,∗) is generated by x and y for 1 ≤ i ≤ g and 1 ≤ j ≤ b−1.
1 g i j
*
α α β β
1 g 1 b-1
{ {
g b-1
Figure 6. The oriented loops α and β on Nb−1.
i j g
Let p : π (Nb−1,∗) → π (N ,∗) denote the natural surjection defined as p(x ) = x and
1 g 1 g i i
p(y ) = 1, for b ≥ 1. For x ∈ π (Nb−1,∗) we can denote p(x) = xε1xε2 ···xεt, where t is
j 1 g i1 i2 it
the word length of p(x) and ε = ±1. We define
k
O (x) = ♯{i | i = i},
i 2k−1 2k−1
E (x) = ♯{i | i = i},
i 2k 2k
Γb−1 = {x ∈ π (Nb−1,∗) | O (x) = E (x),1 ≤ i ≤ g}.
g 1 g i i
For example, for x = x y x x−1y y−2x x−1y3x−1 ∈ π (Nb−1,∗), since p(x) =
1 2 2 3 5 1 1 2 4 3 1 g
x x x−1x x−1x−1, we have that O (x) and E (x) are equal to 1 (resp. 0) for i = 1,2,3
1 2 3 1 2 3 i i
(resp. i ≥ 4), and hence x is in Γb−1.
g
A NORMAL GENERATING SET FOR I(Nb) 5
g
In this section we prove the following three propositions.
Proposition 3.1. Im Fb−1| is equal to I(Nb−1).
g Cgb(I(Ngb)) g
Proposition 3.2. kerFb−1| is equal to Pb−1(Γb−1).
g Cgb(I(Ngb)) g g
Proposition 3.3. Pb−1(Γb−1) is the normal closure of Pb−1(x2) and Pb−1(y ) for 1 ≤ j ≤
g g g g g j
b−1.
By Proposition 3.1 and Proposition 3.2, we have the short exact sequence
1 −→ Γb−1 −→ Cb(I(Nb)) −→ I(Nb−1) −→ 1.
g g g g
Hence Cb(I(Nb)) is the normal closure of Pb−1(Γb−1) and lifts by Fb−1| of normal
g g g g g Cgb(I(Ngb))
generators of I(Nb−1). In addition, by Proposition 3.3 we obtain the following.
g
Corollary 3.4. In M(Nb−1,∗), Cb(I(Nb)) is normally generated by Pb−1(x2), Pb−1(y )
g g g g g g j
and lifts by Fb−1| of normal generators of I(Nb−1), for 1 ≤ j ≤ b−1.
g Cgb(I(Ngb)) g
3.1. Proof of Proposition 3.1.
ToproveProposition3.1, itsufficestoshowthatforanyϕ ∈ I(Nb−1)thereisϕ ∈ I(Nb)
g g
such that (Fb−1 ◦Cb)(ϕ) = ϕ.
g g
e
Let γ and δ be oriented loops on Nb as shown in Figure 7, and let c and d be the
i j g i i
elements of H (Nb;Z)ecorresponding to γ and δ respectively. As a Z-module, H (Nb,Z)
1 g i j 1 g
has a presentation
H (Nb,Z) = hc ,...,c ,d ,...,d | 2(c +···+c )+(d +···+d ) = 0i,
1 g 1 g 1 b 1 g 1 b
and is isomorphic to Zg+b−1.
γ γ δ δ
1 g 1 b
{ {
g b
Figure 7. The oriented loops γ and δ on Nb.
i j g
For f ∈ M(Nb) we denote by f the automorphism over H (Nb;Z) induced by f. Since
g ∗ 1 g
f fixes each point of the boundary of Nb, we have that f (d ) = d for 1 ≤ j ≤ b.
g ∗ j j
For any ϕ ∈ I(Nb−1) there exists ψ ∈ M(Nb) such that (Fb−1 ◦ Cb)(ψ) = ϕ. For
g g g g
1 ≤ i ≤ g there is an integer n such that ψ (c ) = c + n d . Let γ and γ be simple
i ∗ i i i b ij ij
closed curves on Ngb as shown in Figure 8, and let τij = tγijtγeij, for 1 ≤ i < j ≤ g. τij is
a mapping class which is described by pushing the b-th boundary componenet once along
between γ and γ . We can check
ij ij
e ci −db (t = i),
(τij)∗(ct) = cj +db (t = j),
c (t 6= i,j).
t
6 R.KOBAYASHI
γ
ij
~
γ
ij
i j
Figure 8. The loops γ and γ .
ij ij
e
Let τ = τng−1 ···τn2τn1 and ϕ = τψ. Since (Fb−1◦Cb)(τ) = 1 we have (Fb−1◦Cb)(ϕ) =
g−1g 2g 1g g g g g
ϕ. For 1 ≤ i ≤ g −1 we calculate
e e
ϕ (c ) = τ (ψ (c ))
∗ i ∗ ∗ i
= τ (c +n d )
∗ i i b
e
= τ (c )+n τ (d )
∗ i i ∗ b
= (τ )ni(c )+n d
ig ∗ i i b
= (c −n d )+n d
i i b i b
= c .
i
In addition, we calculate
ϕ (c ) = τ (ψ (c ))
∗ g ∗ ∗ g
= τ (c +n d )
∗ g g b
e
= τ (c )+n τ (d )
∗ g g ∗ b
g−1
= (c + n d )+n d
g k b g b
Xk=1
g
= c + n d .
g k b
Xk=1
Let c = 2(c +···+c )+(d +···+d )(= 0). We see
1 g 1 b
ϕ (c) = 2(ϕ (c )+···+ϕ (c ))+(ϕ (d )+···+ϕ (d ))
∗ ∗ 1 ∗ g ∗ 1 ∗ b
g
e = 2(ce +···+c e+c + en d )+(d +·e··+d )
1 g−1 g k b 1 b
X
k=1
g
= c+2 n d
k b
X
k=1
g
= 2 n d .
k b
X
k=1
g
Since ϕ (c) = 0 we have n = 0, and hence ϕ (c ) = c . Hence we have that ϕ is
∗ k ∗ g g ∗
Xk=1
the ideentity. Therefore we conclude that ϕ is in I(eNb) with (Fb−1◦Cb)(ϕ) = ϕ. Thues we
g g g
complete the proof of Proposition 3.1.
e e
3.2. Proof of Proposition 3.2.
A NORMAL GENERATING SET FOR I(Nb) 7
g
Note that Pb−1(x) can be lifted by Cb if and only if the word length of p(x) ∈ π (N ,∗)
g g 1 g
is even. Let π+(Nb−1,∗) denote the subgroup of π (Nb−1,∗) consisting of x such that the
1 g 1 g
word length of p(x) is even. For x ∈ π+(Nb−1,∗), let x denote the automorphism over
1 g ∗
H (Nb,Z) induced by the natural lift by Cb of Pb−1(x). For example, the natural lifts of
1 g g g
Pb−1(x2) and Pb−1(y ) are t and t t−1, respectively.
g g g j ρb σjb δj
Wecanseethat(x2) = 1and(y ) = 1. Hencewehavethat(x x ) , (x x−1) , (x−1x ) ,
i ∗ j ∗ i j ∗ i j ∗ i j ∗
(x−1x−1) and (x x )−1 are mutually equal. Let x = (x x ) for 1 ≤ i,j ≤ g. Note that
i j ∗ j i ∗ ij i j ∗
(γ ) (i < j),
ij ∗
x =
ij (cid:26) (γj−i1)∗ (j < i).
Lemma 3.5. For 1 ≤ i,j,k,l ≤ g, we have x x = x x .
ij kl kl ij
Proof. If (i,j,k,l) = (i,i,i,i), (i,i,k,i), (i,i,i,l), (i,i,k,k), (i,j,i,j), (i,j,j,i) and
(i,i,k,l), it is clear that x x = x x . Hence we check the cases where (i,j,k,l) =
ij kl kl ij
(i,j,i,l), (i,j,k,j), (i,j,k,i) and (i,j,k,l). For any 1 ≤ i,j ≤ g since x acts trivially on
ij
d ,d ,...,d , we check the actions on c ,c ,...,c .
1 2 b 1 2 g
For any mutually different indices 1 ≤ i,j,l ≤ g we see
x (c −d ) = c −2d (t = i),
ij i b i b
x (c ) = c +d (t = j),
x x (c ) = ij j j b
ij il t x (c +d ) = c +d (t = l),
ij l b l b
x (c ) = c (t 6= i,j,l),
ij t t
x (c −d ) = c −2d (t = i),
il i b i b
x (c +d ) = c +d (t = j),
x x (c ) = il j b j b
il ij t x (c ) = c +d (t = l),
il l l b
x (c ) = c (t 6= i,j,l).
il t t
Hence we have x x = x x .
ij il il ij
For any mutually different indices 1 ≤ i,j,k ≤ g we see
x (c ) = c −d (t = i),
ij i i b
x (c +d ) = c +2d (t = j),
x x (c ) = ij j b j b
ij kj t x (c −d ) = c −d (t = k),
ij k b k b
x (c ) = c (t 6= i,j,k),
ij t t
x (c −d ) = c −d (t = i),
kj i b i b
x (c +d ) = c +2d (t = j),
x x (c ) = kj j b j b
kj ij t x (c ) = c −d (t = k),
kj k k b
x (c ) = c (t 6= i,j,k).
kj t t
Hence we have x x = x x .
ij kj kj ij
For any mutually different indices 1 ≤ i,j,k ≤ g we see
x (c +d ) = c (t = i),
ij i b i
x (c ) = c +d (t = j),
x x (c ) = ij j j b
ij ki t x (c −d ) = c −d (t = k),
ij k b k b
x (c ) = c (t 6= i,j,k),
ij t t
x (c −d ) = c (t = i),
ki i b i
x (c +d ) = c +d (t = j),
x x (c ) = ki j b j b
ki ij t x (c ) = c −d (t = k),
ki k k b
x (c ) = c (t 6= i,j,k).
ki t t
8 R.KOBAYASHI
Hence we have x x = x x .
ij ki ki ij
For any mutually different indices 1 ≤ i,j,k,l ≤ g we see
x (c ) = c −d (t = i),
ij i i b
x (c ) = c +d (t = j),
ij j j b
x x (c ) = x (c −d ) = c −d (t = k),
ij kl t ij k b k b
x (c +d ) = c +d (t = l),
ij l b l b
x (c ) = c (t 6= i,j,k,l),
ij t t
x (c −d ) = c −d (t = i),
kl i b i b
x (c +d ) = c +d (t = j),
kl j b j b
x x (c ) = x (c ) = c −d (t = k),
kl ij t kl k k b
x (c ) = c +d (t = l),
kl l l b
x (c ) = c (t 6= i,j,k,l).
kl t t
Hence we have x x = x x .
ij kl kl ij
(cid:3)
Thus we obtain the claim.
For x ∈ π+(Nb−1,∗) denote p(x) = xε1xε2 ···xε2l. Since (y ) = 1, we have x =
1 g i1 i2 i2l j ∗ ∗
x x ···x . For 1 ≤ i ≤ g, let s = ♯{k | i = i = i}, t = O (x) − s and
i1i2 i3i4 i2l−1i2l 2k−1 2k i
u = E (x)−s. Since x = 1 and x (c ) = c , by Lemma 3.5 we have
i ii kl i i
x (c ) = x ···x ·x ···x ·xs(c )
∗ i k(u)i k(1)i ij(t) ij(1) ii i
= x ···x ·x ···x (c )
k(u)i k(1)i ij(t) ij(1) i
= x ···x (c −td )
k(u)i k(1)i i b
= (c +ud )−td
i b b
= c +(u−t)d ,
i b
using the suitable indices j(1),...,j(t) and k(1),...,k(u). Therefore we have that x ∈
Γb−1 if and only if x = 1.
g ∗
Note that kerFb−1| is equal to the intersection of kerFb−1 and Cb(I(Nb)). For
g Cgb(I(Ngb)) g g g
any x ∈ Γb−1, since x = 1, we have that Pb−1(x) is in Cb(I(Nb)). In addition, since
g ∗ g g g
Pb−1(x) is in kerFb−1, we have that Pb−1(x) is in kerFb−1| . Hence we conclude
g g g g Cgb(I(Ngb))
Pb−1(Γb−1) ⊂ kerFb−1| .
g g g Cgb(I(Ngb))
For any ϕ ∈ kerFb−1| , since ϕ is in kerFb−1 there exists x ∈ π (Nb−1,∗) such
g Cgb(I(Ngb)) g 1 g
thatϕ = Pb−1(x). Inaddition, sinceϕisinCb(I(Nb))wehavex = 1,andhencex ∈ Γb−1.
g g g ∗ g
Hence ϕ is in Pb−1(Γb−1). Therefore we conclude kerFb−1| ⊂ Pb−1(Γb−1).
g g g Cgb(I(Ngb)) g g
Thus we complete the proof of Proposition 3.2.
3.3. Proof of Proposition 3.3.
Let πb−1 denote the finitely presented group with the generators x and y , and with the
g i j
relators x2, y and [x x ,x x ], where 1 ≤ i,i(1),i(2),i(3),i(4) ≤ g, 1 ≤ j ≤ b−1
i j i(1) i(2) i(3) i(4)
and [x,y] means xyx−1y−1. There is the natural surjection ψ : π (Nb−1,∗) → πb−1. We
1 g g
first show the following lemma.
Lemma 3.6. Γb−1 is equal to kerψ.
g
Proof. It is clear that Γb−1 ⊃ kerψ. We show Γb−1 ⊂ kerψ. Namely, it suffices to show
g g
that x ≡ 1 modulo x2, y and [x x ,x x ] for any x ∈ Γb−1.
i j i(1) i(2) i(3) i(4) g
For any x ∈ Γb−1 we can denote x ≡ x x ···x modulo x2 and y . Since
g i1 i2 i2l i j
O (x) = E (x), there exists 1 ≤ t ≤ l such that i = i . Hence, modulo x2, y
i1 i1 1 2t i j
A NORMAL GENERATING SET FOR I(Nb) 9
g
and [x x ,x x ], we calculate
i(1) i(2) i(3) i(4)
x ≡ x x ···x ·x ···x
i1 i2 i2t i2t+1 i2l
= [x x ,x x ]x x x x ·x x ···x ·x ···x
i1 i2 i3 i4 i3 i4 i1 i2 i5 i6 i2t i2t+1 i2l
≡ x x x x ·x x ···x ·x ···x
i3 i4 i1 i2 i5 i6 i2t i2t+1 i2l
= x x [x x ,x x ]x x x x ·x x ···x ·x ···x
i3 i4 i1 i2 i5 i6 i5 i6 i1 i2 i7 i8 i2t i2t+1 i2l
≡ x x x x x x ·x x ···x ·x ···x
i3 i4 i5 i6 i1 i2 i7 i8 i2t i2t+1 i2l
.
.
.
= x ···x [x x ,x x ]x x x x ·x ···x
i3 i2t−2 i1 i2 i2t−1 i2t i2t−1 i2t i1 i2 i2t+1 i2l
≡ x ···x ·x x ·x ·x ···x
i3 i2t−1 i2t i1 i2 i2t+1 i2l
≡ x ···x ·x ·x ···x .
i3 i2t−1 i2 i2t+1 i2l
Let x′ = x ···x · x · x ···x . It immediately follows that the word length of
i3 i2t−1 i2 i2t+1 i2l
x′ is 2l − 2 and x′ is in Γb−1. Since O (x′) = E (x′), there exists 2 ≤ t′ ≤ l such that
g i3 i3
i3 = i2t′. Then, similarly there exists x′′ such that the word length of x′′ is 2l−4, x′′ is in
Γb−1 and x′ ≡ x′′ modulo x2, y and [x x ,x x ]. Repeating the same operation,
g i j i(1) i(2) i(3) i(4)
we have that x ≡ x′ ≡ x′′ ≡ ··· ≡ 1 modulo x2, y and [x x ,x x ]. Therefore we
i j i(1) i(2) i(3) i(4)
have x ∈ kerψ.
(cid:3)
Thus we obtain the claim.
We next show the following lemma.
Lemma 3.7. πb−1 has a presentation with
g
πb−1 = hx ,...,x | x2,...,x2,(x x x )2,1 ≤ i < j < k ≤ gi.
g 1 g 1 g i j k
Proof. It suffices to show that (x x x )2 is a product of conjugations of some
i j k
[x x ,x x ] and that [x x ,x x ] is a product of conjugations of some
i(1) i(2) i(3) i(4) i(1) i(2) i(3) i(4)
(x x x )2, modulo x2,x2,...,x2.
i j k 1 2 g
For any 1 ≤ i < j < k ≤ g, modulo x2,x2,...,x2 we see
1 2 g
(x x x )2 ≡ (x x x )x x (x x x )
i j k i j k j j i j k
= x x ·x x ·x x ·x x
i j k j j i j k
≡ x x ·x x ·(x x )−1(x x )−1
i j k j i j k j
= [x x ,x x ].
i j k j
If (i(1),i(2),i(3),i(4)) = (i,i,i,i), (i,i,i,j), (i,i,j,i), (i,j,i,i), (j,i,i,i)
(i,i,j,j), (i,j,i,j), (i,j,j,i), (i,i,j,k) and (i,j,k,k), we have immediately that
[x x ,x x ] ≡ 1 modulo x2,x2,...,x2. Hence we check the other cases.
i(1) i(2) i(3) i(4) 1 2 g
10 R.KOBAYASHI
For mutually different indices 1 ≤ i,j,k,l ≤ g, modulo x2,x2,...,x2 we see
1 2 g
[x x ,x x ] ≡ x x x x x x x x
i j i k i j i k j i k i
≡ x (x x x )2x−1,
i j i k i
[x x ,x x ] ≡ x x x x x x x x
i j k j i j k j j i j k
≡ (x x x )2,
i j k
[x x ,x x ] ≡ x x x x x x x x
i j k i i j k i j i i k
≡ (x x x )2,
i j k
[x x ,x x ] ≡ x x x x x x x x
i j j k i j j k j i k j
≡ (x x x )2,
i k j
[x x ,x x ] ≡ x x x x x x x x
i j k l i j k l j i l k
≡ (x x x )2x x x ·x x x (x x x )2x x
i j k k j i i j l l j i l k
≡ (x x x )2x x (x x x )2(x x )−1
i j k k l l j i k l
For the relator (x x x )2, applying conjugations and taking their inverses, it suffices to
i j k
consider the case i < j < k. Thus we obtain the claim. (cid:3)
By Lemma 3.6 and Lemma 3.7 we have the short exact sequence
1 −→ Γb−1 −→ π (Nb−1,∗) −→ πb−1 −→ 1,
g 1 g g
and hence Γb−1 is normally generated by x2,...,x2, y ,...,y and (x x x )2 for
g 1 g 1 b−1 i j k
1 ≤ i < j < k ≤ g in π (Nb−1,∗). Hence Pb−1(Γb−1) is normally generated by
1 g g g
Pb−1(x2),...,Pb−1(x2), Pb−1(y ),...,Pb−1(y ) and Pb−1((x x x )2) for 1 ≤ i < j < k ≤
g 1 g g g 1 g b−1 g i j k
g in M(Nb−1,∗). Representative loops of x2,...,x2 and (x x x )2 bound a M¨obius band
g 1 g i j k
(see Figure 9). Therefore Pb−1(x2),...,Pb−1(x2 ) and Pb−1((x x x )2) are conjugate to
g 1 g g−1 g i j k
Pb−1(x2). Thus we complete the proof of Proposition 3.3.
g g
*
x 2 x 2
1 g
(a) Representative loops of x2,x2,...,x2 bound
1 2 g
a Mo¨bius band.
*
(x x x )2
1 2 3
(b) Arepresentativeloopof(x1x2x3)2boundsaMo¨biusband. Sim-
ilarly a representativeloop of (x x x )2 bounds a Mo¨bius band for
i j k
1≤i<j <k ≤g.
Figure 9.
