Abelianization and
Nielsen realization problem of
the mapping class group of handlebody
.
Susumu Hirose
Saga University
Susumu Hirose (Saga University)
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
1 / 30
Contents
What is the mapping class group of handlebody M(H g )?
Abelianization of the mapping class group of handlebody
Nielsen realization problem of the mapping class group of
handlebody
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
2 / 30
What is M(H g )?
Σ g : a closed oriented surface of genus g．
3-dimensional handlebody H g = an orientable 3-manifold
constructed from 3-ball with attaching g 1-handles．
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
Σ g : a closed oriented surface of genus g．
3-dimensional handlebody H g = an orientable 3-manifold
constructed from 3-ball with attaching g 1-handles．
M(Σ g ) :=
Susumu Hirose (Saga University)
Diff + (Σ g )
isotopy
M(H g ) :=
Diff + (H g )
isotopy
Abelianization and Nielsen realization problem of the mapping class group of handlebody
3 / 30
What is M(H g )?
Σ g : a closed oriented surface of genus g．
3-dimensional handlebody H g = an orientable 3-manifold
constructed from 3-ball with attaching g 1-handles．
M(Σ g ) :=
Diff + (Σ g )
isotopy
M(H g ) :=
Diff + (H g )
isotopy
∂H g = Σ g ⇒
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
3 / 30
What is M(H g )?
Σ g : a closed oriented surface of genus g．
3-dimensional handlebody H g = an orientable 3-manifold
constructed from 3-ball with attaching g 1-handles．
M(Σ g ) :=
Diff + (Σ g )
isotopy
M(H g ) :=
Diff + (H g )
isotopy
∂H g = Σ g ⇒ restriction to the boundary Diff + (H g ) → Diff + (Σ g )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
3 / 30
What is M(H g )?
Σ g : a closed oriented surface of genus g．
3-dimensional handlebody H g = an orientable 3-manifold
constructed from 3-ball with attaching g 1-handles．
M(Σ g ) :=
Diff + (Σ g )
isotopy
M(H g ) :=
Diff + (H g )
isotopy
∂H g = Σ g ⇒ restriction to the boundary Diff + (H g ) → Diff + (Σ g )
H g is irreducible ⇒ M(H g ) → M(Σ g ) is injective.
i.e. M(H g ) is a subgroup of M(Σ g ).
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
3 / 30
What is M(H g )?
b bound a disk in H g , but T l (b) does not bound disk in H g .
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
4 / 30
What is M(H g )?
b bound a disk in H g , but T l (b) does not bound disk in H g .
⇒ T l ∈ M(Σ g ) but T l < M(H g )
⇒ M(H g ) is a proper subgroup of M(Σ g ).
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
M(Σ g ) v.s. M(H g )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn [1938], Lickorish [1964])
M(H g ): ○ (Shin’ichi Suzuki [1977])
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
5 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn [1938], Lickorish [1964])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
5 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn [1938], Lickorish [1964])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
5 / 30
What is M(H g )?
Disk twist δ d1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
6 / 30
What is M(H g )?
Disk twist δ d1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
6 / 30
What is M(H g )?
Disk twist δ d1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
6 / 30
What is M(H g )?
Disk twist δ d1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
6 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn, Lickorish [1964], Humphries [1977])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
7 / 30
What is M(H g )?
Cyclic translation ρ
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
Cyclic translation ρ
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
8 / 30
What is M(H g )?
Cyclic translation ρ
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
8 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn, Lickorish [1964], Humphries [1977])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
9 / 30
What is M(H g )?
Twisting a knob ω1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
10 / 30
What is M(H g )?
Twisting a knob ω1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
10 / 30
What is M(H g )?
Twisting a knob ω1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
10 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn, Lickorish [1964], Humphries [1977])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
11 / 30
What is M(H g )?
Interchanging two knobs ρ12
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
Interchanging two knobs ρ12
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
Interchanging two knobs ρ12
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
Interchanging two knobs ρ12
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
Interchanging two knobs ρ12
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
Interchanging two knobs ρ12
Δ
ρ212 = δ−1
δ δ
∆ k1 k2
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
12 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finitely generated?
M(Σ g ): ○ (Dehn, Lickorish [1964], Humphries [1977])
M(H g ): ○ (Shin’ichi Suzuki [1977])
Generators of M(H g ) :
Disk twist δ d1
Cyclic translation ρ
Twisting a knob ω1
Interchanging two knobs ρ12
Slidings θ12 and ξ12
Susumu Hirose (Saga University)
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
13 / 30
What is M(H g )?
Slidings θ12 and ξ12
D : a disk on ∂H g−1 ,
l: an oriented loop on ∂H g−1 whose base is the center of D.
S(l) : Spin map about l
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
14 / 30
What is M(H g )?
Slidings θ12 and ξ12
D : a disk on ∂H g−1 ,
l: an oriented loop on ∂H g−1 whose base is the center of D.
S(l) : Spin map about l
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
14 / 30
What is M(H g )?
Slidings θ12 and ξ12
D : a disk on ∂H g−1 ,
l: an oriented loop on ∂H g−1 whose base is the center of D.
S(l) : Spin map about l
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
14 / 30
What is M(H g )?
Slidings θ12 and ξ12
D : a disk on ∂H g−1 ,
l: an oriented loop on ∂H g−1 whose base is the center of D.
S(l) : Spin map about l
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
14 / 30
What is M(H g )?
Slidings θ12 and ξ12
θ1,2 = S(a1,2 ), ξ1,2 = S(b1,2 )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
M(Σ g ) v.s. M(H g )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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What is M(H g )?
M(Σ g ) v.s. M(H g )
finite presentation
M(Σ g ): Hatcher-Thurston [1980], Harer [1983], Wajnryb [1983]
M(H g ): Wajnryb [1998]
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
16 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finite presentation
M(Σ g ): Hatcher-Thurston [1980], Harer [1983], Wajnryb [1983]
M(H g ): Wajnryb [1998]
.
virtual cohomological dimension
M(Σ g ): 4g-5 (Harer [1986]) M(H g ): 4g-5 (H. [2003])
.
Susumu Hirose (Saga University)
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
16 / 30
What is M(H g )?
M(Σ g ) v.s. M(H g )
finite presentation
M(Σ g ): Hatcher-Thurston [1980], Harer [1983], Wajnryb [1983]
M(H g ): Wajnryb [1998]
.
virtual cohomological dimension
M(Σ g ): 4g-5 (Harer [1986]) M(H g ): 4g-5 (H. [2003])
rational Euler number
M(Σ g ): −
B2g
2g
.
(Harer-Zagier [1986]) M(H g ): 0 (H. [2003])
Susumu Hirose (Saga University)
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Abelianization of M(H g )
Abelianization of M(Σ g )
If g ≥ 3, M(Σ g )
ab
.
= 0 (Powell [1978])
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
17 / 30
Abelianization of M(H g )
Abelianization of M(Σ g )
If g ≥ 3, M(Σ g )
ab
.
= 0 (Powell [1978])
Abelianization of M(H g )
.
If g ≥ 3, M(H g ) ab is a finite Abelian group, therefore
H1 (M(H g ) ab ; R) = 0．(Wajnryb [1998] , reconsidered by H
[2009])
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
17 / 30
Abelianization of M(H g )
Abelianization of M(Σ g )
If g ≥ 3, M(Σ g )
ab
.
= 0 (Powell [1978])
Abelianization of M(H g )
.
If g ≥ 3, M(H g ) ab is a finite Abelian group, therefore
H1 (M(H g ) ab ; R) = 0．(Wajnryb [1998] , reconsidered by H
[2009])
If g = 2,
Wajnryb [1998] : M(H2 ) ab is a finite Abelian group.
H. [1997] : obtained a presentation for M(H2 ).
From this presentation, M(H2 ) ab = Z + Z2 + Z2 !?
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
17 / 30
Abelianization of M(H g )
Abelianization of M(Σ g )
If g ≥ 3, M(Σ g )
ab
.
= 0 (Powell [1978])
Abelianization of M(H g )
.
If g ≥ 3, M(H g ) ab is a finite Abelian group, therefore
H1 (M(H g ) ab ; R) = 0．(Wajnryb [1998] , reconsidered by H
[2009])
If g = 2,
.
Wajnryb [1998] : M(H2 ) ab is a finite Abelian group.
H. [1997] : obtained a presentation for M(H2 ).
From this presentation, M(H2 ) ab = Z + Z2 + Z2 !?
Clement Radu Popescu [2000] : Wajnryb’s presentation of M(H2 )
and H.’s presentation of M(H2 ) are equivalent !!?? .
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
17 / 30
Abelianization of M(H g )
Observation 1
.
Conjugate in M(H g ) =⇒ equal in M(H g ) .
ab
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Abelianization of M(H g )
Observation 1
.
Conjugate in M(H g ) =⇒ equal in M(H g ) .
ab
.
.
ϕ, ϕ′ ∈ M(H g ) are conjugate =⇒ ∃ψ ∈ M(H g ) s.t. ϕ′ = ψϕψ−1
=⇒ in M(H g ) ab , ϕ′ = ψ + ϕ − ψ = ϕ.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
18 / 30
Abelianization of M(H g )
Observation 1
.
Conjugate in M(H g ) =⇒ equal in M(H g ) .
ab
.
ϕ, ϕ′ ∈ M(H g ) are conjugate =⇒ ∃ψ ∈ M(H g ) s.t. ϕ′ = ψϕψ−1
=⇒ in M(H g ) ab , ϕ′ = ψ + ϕ − ψ = ϕ.
.
Observation 2
Disk twists about the following disks are trivial in M(H g ) ab .
Especially, δ d1 = 0.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
18 / 30
Abelianization of M(H g )
Lantern relation
t a1 t a2 t a3 = t b0 t b1 t b2 t b3
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
19 / 30
Abelianization of M(H g )
Lantern relation
t a1 t a2 t a3 = t b0 t b1 t b2 t b3
.
D0 :
δ a1 δ a2 δ a3 = δ b0 δ D0 δ b2 δ b3
=⇒
Observation 1
3δ D0 = 4δ D0.
⇒ δ D0 = 0 ⇒ δ d1
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
19 / 30
Abelianization of M(H g )
Lantern relation
t a1 t a2 t a3 = t b0 t b1 t b2 t b3
.
D1 :
δ a1 δ a2 δ a3 = δ b0 δ D1 δ b2 δ b3
=⇒
Observation 1
3δ D0 = 3δ D0 + δ D1
.
⇒ δ D1 = 0
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
19 / 30
Abelianization of M(H g )
Lantern relation
t a1 t a2 t a3 = t b0 t b1 t b2 t b3
.
D2 :
δ a1 δ a2 δ a3 = δ D2 δ b1 δ b2 δ b3
=⇒
Observation 1
3δ D0 = 3δ D0 + δ D2
.
⇒ δ D2 = 0
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
19 / 30
Abelianization of M(H g )
Observation 3
.
In M(H g ) , θ12 = 0.
ab
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Abelianization of M(H g )
Observation 3
.
In M(H g ) , θ12 = 0.
ab
.
.
θ1,2 = S(a1,2 ), θ1,3 := S(a1,3 ), θ1,2,3 := S(a1,2,3 )
θ1,2 , θ1,3 , θ1,2,3 are conjugate =⇒
Observation 1
in M(H g ) ab , θ1,2 = θ1,3 = θ1,2,3
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
20 / 30
Abelianization of M(H g )
Observation 3
.
In M(H g ) , θ12 = 0.
ab
.
.
θ1,2 = S(a1,2 ), θ1,3 := S(a1,3 ), θ1,2,3 := S(a1,2,3 )
θ1,2 , θ1,3 , θ1,2,3 are conjugate =⇒
Observation 1
in M(H g ) ab , θ1,2 = θ1,3 = θ1,2,3
a1,2,3 = a1,3 · a1,2 (read from right to left)
⇒ θ1,2,3 ≡ θ1,3 · θ1,2 mod δ d1 (= δ d2 )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
20 / 30
Abelianization of M(H g )
Observation 3
.
In M(H g ) , θ12 = 0.
ab
.
.
θ1,2 = S(a1,2 ), θ1,3 := S(a1,3 ), θ1,2,3 := S(a1,2,3 )
θ1,2 , θ1,3 , θ1,2,3 are conjugate =⇒
Observation 1
in M(H g ) ab , θ1,2 = θ1,3 = θ1,2,3
a1,2,3 = a1,3 · a1,2 (read from right to left)
⇒ θ1,2,3 ≡ θ1,3 · θ1,2 mod δ d1 (= δ d2 ) ⇒ θ1,2,3 = θ1,3 · θ1,2 · δ d1
⇒ θ1,2 = 0 in M(H g ) ab
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
20 / 30
Abelianization of M(H g )
Observation 4
In M(H g ) ab , ρ, ω1 , ρ12 ξ12 are elements of finite order.
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
21 / 30
Abelianization of M(H g )
Observation 4
In M(H g ) ab , ρ, ω1 , ρ12 ξ12 are elements of finite order.
In M(H g ) ab ,
.
ρ = id H g = 0,
= δ D1 = 0,
ξ1,2 = δ D0 − δ D0 = 0.
g
ω21
Susumu Hirose (Saga University)
ρ21,2
=
δ−1
D2
+ 2δ D1 = 0,
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
21 / 30
Abelianization of M(H g )
Observation 4
In M(H g ) ab , ρ, ω1 , ρ12 ξ12 are elements of finite order.
In M(H g ) ab ,
.
ρ = id H g = 0,
= δ D1 = 0,
ξ1,2 = δ D0 − δ D0 = 0.
g
ω21
ρ21,2
=
δ−1
D2
+ 2δ D1 = 0,
.
If g = 3, elements of M(H g ) ab are finite order.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Nielsen realization problem of M(H g )
X : an oriented manifold
Diff + (X)
π X : Diff + (X) → isotopy
= M(X) : a natural surjection
Γ ⊂ M(X)
subgroup
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Nielsen realization problem of M(H g )
X : an oriented manifold
Diff + (X)
π X : Diff + (X) → isotopy
= M(X) : a natural surjection
Γ ⊂ M(X)
subgroup
(Generalized) Nielsen realization problem
Is there a homomorphism s : Γ → Diff + (X) (a section of π X over
Γ) ?
Diff
(X)
; +
Γ
Susumu Hirose (Saga University)
v

v
v
vs
v
πX
/ M(X)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Nielsen realization problem of M(H g )
X = Σg
[Kerkhoff, 1983] Over any finite subgroup group Γ ⊂ M(Σ g ), there
is a section s : Γ → Diff + (Σ g ).
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
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Nielsen realization problem of M(H g )
X = Σg
[Kerkhoff, 1983] Over any finite subgroup group Γ ⊂ M(Σ g ), there
is a section s : Γ → Diff + (Σ g ).
[Morita, 1987] If g ≥ 5, there is no section s : M(Σ g ) → Diff 2+ (Σ g )
over Γ = M(Σ g ).
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
23 / 30
Nielsen realization problem of M(H g )
X = Σg
[Kerkhoff, 1983] Over any finite subgroup group Γ ⊂ M(Σ g ), there
is a section s : Γ → Diff + (Σ g ).
[Morita, 1987] If g ≥ 5, there is no section s : M(Σ g ) → Diff 2+ (Σ g )
over Γ = M(Σ g ).
[Markovic-Saric, 2008] If g ≥ 2, there is no section
.
s : M(Σ g ) → Homeo+ (Σ g ) over Γ = M(Σ g ).
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
23 / 30
Nielsen realization problem of M(H g )
X = Σg
[Kerkhoff, 1983] Over any finite subgroup group Γ ⊂ M(Σ g ), there
is a section s : Γ → Diff + (Σ g ).
[Morita, 1987] If g ≥ 5, there is no section s : M(Σ g ) → Diff 2+ (Σ g )
over Γ = M(Σ g ).
[Markovic-Saric, 2008] If g ≥ 2, there is no section
.
s : M(Σ g ) → Homeo+ (Σ g ) over Γ = M(Σ g ).
X = Hg
[H. 2009] If g ≥ 5, there is no section s : M(H g ) → Diff + (H g ) over
Γ = M(H g ).
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
23 / 30
Nielsen realization problem of M(H g )
Assume : ∃ Γ
⊂
subgroup
M(H g ) s.t. there is no section of
πΣ g : Diff + (Σ g ) → M(Σ g ) over Γ.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
24 / 30
Nielsen realization problem of M(H g )
Assume : ∃ Γ
⊂
subgroup
M(H g ) s.t. there is no section of
πΣ g : Diff + (Σ g ) → M(Σ g ) over Γ. ⇒ there is no section of π H g over
M(H g ).

Γ
Susumu Hirose (Saga University)
@s b \ W
R N
pi
y
Diff + (H g )
O
s
πH g
/ M(H g ) 
J
G#
/ Diff + (Σ g )
πΣ g
/ M(Σ g )
Abelianization and Nielsen realization problem of the mapping class group of handlebody
24 / 30
Nielsen realization problem of M(H g )
Assume : ∃ Γ
⊂
subgroup
M(H g ) s.t. there is no section of
πΣ g : Diff + (Σ g ) → M(Σ g ) over Γ. ⇒ there is no section of π H g over
M(H g ).

Γ
@s b \ W
R N
pi
y
Diff + (H g )
O
s
πH g
/ M(H g ) 
J
G#
/ Diff + (Σ g )
πΣ g
/ M(Σ g )
Find Γ.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
24 / 30
Nielsen realization problem of M(H g )
g′ 5 g, D: a 2-disk on Σ g′ , Σ g′ ,1 = Σ g′ \ int D,
Diff + (Σ g′ ,1 , fix ∂ D)
M(Σ g′ ,1 ) =
.
isotopy
Σ g′ ,1 ⊂ Σ g , M(Σ g′ ,1 ) ⊂ M(Σ g ) · · · extension by identity.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
25 / 30
Nielsen realization problem of M(H g )
Theorem[Franks-Handel, 2009]
Γ1 : a nontrivial finitely generated subgroup of M(Σ g−2,1 ) s.t.
H1 (Γ1 , R) = 0,
µ : an element of M(Σ2,1 ) represented by a pseudo-Anosov
homeomorphism in Σ2,1 ,
⟨Γ1 , µ⟩ = a subgroup of M(Σ g ) generated by Γ1 ∪ {µ},
⇒ there is no section of πΣ g : Diff + (Σ g ) → M(Σ g ) over ⟨Γ1 , µ⟩.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
25 / 30
Nielsen realization problem of M(H g )
g′ 5 g,
D: a 2-disk on ∂H g′ , M(H g′ ,1 ) =
H g′ ⊂ H g , D ⊂
proper
Diff + (H g′ , fix D)
isotopy
.
Hg
M(H g′ ,1 ) ⊂ M(H g ) · · · extension by identity.
[Fathi-Laudenbach, 1980]
If g ≥ 2, ∃ ϕ g ∈ M(H g ) s.t. ϕ g |∂H g is a pseudo-Anosov homeo.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
26 / 30
Nielsen realization problem of M(H g )
g′ 5 g,
D: a 2-disk on ∂H g′ , M(H g′ ,1 ) =
H g′ ⊂ H g , D ⊂
proper
Diff + (H g′ , fix D)
isotopy
.
Hg
M(H g′ ,1 ) ⊂ M(H g ) · · · extension by identity.
[Fathi-Laudenbach, 1980]
If g ≥ 2, ∃ ϕ g ∈ M(H g ) s.t. ϕ g |∂H g is a pseudo-Anosov homeo.
The number of singular points of a singular foliation preserved by a
pseudo-Anosov homeo. is finite
⇒ ∃n > 0 s.t. (ϕ g ) n fixes a point p.
µ := [(ϕ2 ) n| H2 \{ p} ] ∈ M(H2,1 )
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
26 / 30
Nielsen realization problem of M(H g )
Proposition
If g′ ≥ 2, M(H g′ ) ab ≃ M(H g′ ,1 ) ab .
.
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
27 / 30
Nielsen realization problem of M(H g )
Proposition
If g′ ≥ 2, M(H g′ ) ab ≃ M(H g′ ,1 ) ab .
.
By the result in the previous section, if g′ ≥ 3,
.
M(H g′ ,1 ) ab ≃ M(H g′ ) ab is a finite Abelian group. Especially,
H1 (M(H g′ ,1 ); R) = 0.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
27 / 30
Nielsen realization problem of M(H g )
Proposition
If g′ ≥ 2, M(H g′ ) ab ≃ M(H g′ ,1 ) ab .
.
By the result in the previous section, if g′ ≥ 3,
.
M(H g′ ,1 ) ab ≃ M(H g′ ) ab is a finite Abelian group. Especially,
H1 (M(H g′ ,1 ); R) = 0.
If g = 5, let Γ1 = M(H g−2,1 ).
Γ1 , µ satisfy a condition in Theorem [Franks-Handel, 2009].
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
27 / 30
Nielsen realization problem of M(H g )
Proposition
If g′ ≥ 2, M(H g′ ) ab ≃ M(H g′ ,1 ) ab .
.
By the result in the previous section, if g′ ≥ 3,
.
M(H g′ ,1 ) ab ≃ M(H g′ ) ab is a finite Abelian group. Especially,
H1 (M(H g′ ,1 ); R) = 0.
If g = 5, let Γ1 = M(H g−2,1 ).
Γ1 , µ satisfy a condition in Theorem [Franks-Handel, 2009].
Γ = ⟨Γ1 , µ⟩ is a subgroup of M(H g ) which we need.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
27 / 30
Nielsen realization problem of M(H g )
Proposition
′
If g = 2, M(H g′ )
.
ab
≃ M(H g′ ,1 ) .
ab
.
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
28 / 30
Nielsen realization problem of M(H g )
Proposition
′
If g = 2, M(H g′ )
.
ab
≃ M(H g′ ,1 ) .
p ∈ int D, M(H g′ 1 ) =
Susumu Hirose (Saga University)
ab
Diff + (H g′ , fix p)
isotopy
.
.
Abelianization and Nielsen realization problem of the mapping class group of handlebody
28 / 30
Nielsen realization problem of M(H g )
Proposition
′
.
If g = 2, M(H g′ )
ab
≃ M(H g′ ,1 ) .
p ∈ int D, M(H g′ 1 ) =
ab
Diff + (H g′ , fix p)
isotopy
.
.
0
⟨δ D ⟩
M(H g′ ,1 )
0
/π1 (∂H g′ , p)
/M(H
g′
1
)
/M(H g′ )
/0
0
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
28 / 30
Nielsen realization problem of M(H g )
Proposition
′
.
If g = 2, M(H g′ )
ab
≃ M(H g′ ,1 ) .
p ∈ int D, M(H g′ 1 ) =
ab
Diff + (H g′ , fix p)
isotopy
.
.
0
⟨δ D ⟩
M(H g′ ,1 )
0
/π1 (∂H g′ , p)
/M(H
g′
1
)
/M(H g′ )
/0
0
δ D and elements of π1 (∂H g′ , p) are trivial in M(H g′ ,1 ) ab .
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
28 / 30
Nielsen realization problem of M(H g )
δ D:
δ a1 δ a2 δ a3 = δ b0 δ D δ b2 δ b3
=⇒
Observation 1
3δ D0 = 3δ D0 + δ D
⇒ δD = 0
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
29 / 30
Nielsen realization problem of M(H g )
δ D:
δ a1 δ a2 δ a3 = δ b0 δ D δ b2 δ b3
=⇒
Observation 1
3δ D0 = 3δ D0 + δ D
⇒ δD = 0
π1 (∂H g′ , p):
S(b) = δ D0 − δ D0 = 0
S(a) = 0 ( By the same
method as in Observation
3)
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
29 / 30
Thank you very much !!
Susumu Hirose (Saga University)
Abelianization and Nielsen realization problem of the mapping class group of handlebody
30 / 30