Realization of the mapping class group of handlebody by diffeomorphisms

0.08 MB·English

by  Susumu Hirose

REALIZATION OF THE MAPPING CLASS GROUP OF
HANDLEBODY BY DIFFEOMORPHISMS
9 SUSUMU HIROSE
0
0
2 Abstract. For the oriented 3-dimensional handlebody constructed from a 3-ball
by attaching g 1-handles, it is shown that the natural surjection from the group of
n
orientationpreservingdiffeomorphismsof itto the mapping class groupofit has no
a
J section when g is at least 6.

5
1

] LetM beann-dimensionalcompactorientedmanifoldandS beasubsetof∂M. We
T
G denotethegroupoforientationpreserving diffeomorphisms ofM whoserestrictions on
.
h S are identity by Diff(M,S), the subgroups of them consisting of elements that are
t
a isotopic to identity by Diff (M,S), and the quotient group Diff(M,S)/Diff (M,S)
m 0 0
by M(M,S). For an element f of Diff(M,S), let [f] be the element of M(M,S)
[

1 represented by f. The homomorphism πM,S from Diff(M,S) to M(M,S) defined
v
8 by πM,S(h) = [h] is a surjection. Let Γ be a subgroup of M(M,S). We call a
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homomorphism s from Γ to Diff(M,S) which satisfies π ◦s = id a section for π
1 M,S Γ M,S
2 over Γ. Morita [7] showed that the natural surjection from Diff2(Σ ) to the mapping
. g
1
0 class group M(Σ ) of Σ has no section over M(Σ ) when g ≥ 5. Markovic [5] (when
g g g
9
0 g ≥ 6) and Markovic and Saric [6] (when g ≥ 2) showed that the natural surjection
:
v fromHomeo(Σ ) to M(Σ ) hasno section over M(Σ ). Byusing the different method
g g g
i
X
from them, Franks and Handel [2] showed that the natural surjection from Diff(Σ )
g
r
a to M(Σ ) has no section over M(Σ ) when g ≥ 3.
g g
Let H be an oriented 3-dimensional handlebody of genus g, which is an oriented
g
3-manifold constructed from a 3-ball by attaching g 1-handles. Let Σ be an oriented
g
closed surface of genus g, then ∂H = Σ . The restriction to the boundary defines
g g
a homomorphism ρ : Diff(H ) → Diff(Σ ), and ρ induces a injection M(H ) ֒→
∂ g g ∂ g
M(Σ ) since H is an irreducible 3-manifold. We will show:
g g
1991 Mathematics Subject Classification. 57M60, 57N10.
This research was supported by Grant-in-Aid for Scientific Research (C) (No. 20540083), Japan
Society for the Promotion of Science.
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Theorem 1. If g ≥ 6, there is no section for π : Diff(H ) → M(H ) over M(H ).
Hg g g g
For contradiction, we assume that there is a section s : M(H ) → Diff(H ). Let
g g
Γ be a subgroup of M(H ), and i be the inclusion from Γ to M(H ). Then Γ is a
g Γ g
subgroup of M(Σ ), and the composition ρ ◦s◦i is a section for π : Diff(Σ ) →
g ∂ Γ Σg g
M(Σ ) over Γ. Therefore, if we can find a subgroup Γ of M(H ), over which there is
g g
no section for π , then Theorem 1 follows.
Σg
Let D be a 2-disk in Σ , and Σ be Σ \intD. Let c be an essential simple closed
g g,1 g
curve on Σ such that Σ \c is not connected, then the closure of one component of
g g
Σ \ c is diffeomorphic to Σ and the closure of the other component of Σ \ c is
g g1,1 g
diffeomorphic to Σ . We remark that g = g +g and g ,g ≥ 1. These diffeomor-
g2,1 1 2 1 2
phisms induce injections M(Σ ,∂Σ ) → M(Σ ) and M(Σ ,∂Σ ) → M(Σ )
g1,1 g1,1 g g2,1 g2,1 g
(see [8]). By these injections, we consider M(Σ ,∂Σ ) and M(Σ ,∂Σ ) as
g1,1 g1,1 g2,1 g1,1
subgroups of M(Σ ). From Theorem 1.6 in [2] proved by Franks and Handel, we see:
g
Theorem 2. [2] Let Γ be a nontrivial finitely generated subgroup of M(Σ ,∂Σ )
1 g1,1 g1,1
such that H1(Γ ,R) = 0, and µ be an element of M(Σ ,∂Σ ) which is represented
1 g2,1 g2,1
by a pseudo-Anosov homeomorphism on intΣ . Then there is no section for π :
g2,1 Σg
Diff(Σ ) → M(Σ ) over hΓ ,µi.
g g 1
We assume g ≥ 6. The 3-manifold Σ × [0,1] is diffeomorphic to H . Let D
2,1 4 1
be a 2-disk in int∂Σ × [0,1] ⊂ ∂(Σ × [0,1]), D and D be disjoint 2-disks on
2,1 2,1 2 3
∂Hg−6, and D4 be 2-disk on ∂H2. Along these 2-disks, we glue Σ2,1 × [0,1], Hg−6
and H such that D = D , D = D , then the 3-manifold obtained as a result
2 1 2 3 4
is diffeomorphic to H . By the above construction, we get two natural inclusions
g
Σ ×[0,1] ֒→ H and H ֒→ H . These inclusions induce natural homomorphisms
2,1 g 2 g
i : M(Σ ×[0,1],∂Σ ×[0,1]) → M(H ) and i : M(H ,D ) → M(H ). If [h] is
1 2,1 2,1 g 2 2 4 g
in M(Σ × [0,1],∂Σ × [0,1]) (resp. M(H ,D )) represented by h ∈ Diff(Σ ×
2,1 2,1 2 4 2,1
[0,1],∂Σ ×[0,1]) (resp. Diff(H ,D )), then i ([h]) (resp. i ([h]) is represented by
2,1 2 4 1 2
extending h to H using the identity mapping on H \Σ ×[0,1] (resp. H \H ).
g g 2,1 g 2
We define homomorphisms Π : Diff(Σ ,∂Σ ) → Diff(Σ ×[0,1],∂Σ ×[0,1]) by
2,1 2,1 2,1 2,1
Π(h) = h×id , andI : Diff(Σ ×[0,1],∂Σ ×[0,1]) → Diff(H )bytheidentity on
[0,1] 1 2,1 2,1 g
H \Σ ×[0,1], then thecomposition I ◦Πinduces aninjectionP : M(Σ ,∂Σ ) →
g 2,1 1 2,1 2,1
2
M(H ). By applying Corollary 4.2 of [8] to the subsurface Σ ×{0,1} ⊂ ∂H , the
g 2,1 g
injectivity of P is shown. Korkmaz [4] showed that H (M(Σ ,∂Σ ),Z) = Z/10Z,
1 2,1 2,1
hence H1(M(Σ ,∂Σ ),R) = 0. Therefore, Γ = P(M(Σ ,∂Σ )) satisfies the
2,1 2,1 1 2,1 2,1
assumption of Theorem 2 when g = g −2, g = 2.
1 2
FathiandLaudenbach[3]constructedapseudo-Anosovhomeomorphismφon∂(H )
2
which is a restriction of a homeomorphism on H . Definition of pseudo-Anosov home-
2
omorphisms and terminologies (e.g., singular foliation) related to them can be found
in [1]. Any pseudo-Anosov homeomorphism preserves the set of singular points of the
singular foliation which is preserved by this homeomorphism. Since the number of
singular points of singular foliation is finite, a proper power of φ, say φn, fixes some
points. Letpbeapointfixed byφn, thenφn defines apseudo-Anosov homeomorphism
on ∂(H )\p = intΣ . Let µ be an element of M(Σ ,∂Σ ) ⊂ M(Σ ) represented
2 2,1 2,1 2,1 g
by this homomorphism, then µ is an element of M(H ) and satisfies the assumption
g
of Theorem 2 when g = g −2, g = 2.
1 2
Then hP(M(Σ ,∂Σ )),µi is a subgroup of M(H ) and, by Theorem 2, there
2,1 2,1 g
is no section hP(M(Σ ,∂Σ )),µi → M(Σ ). Therefore, there is no section for
2,1 2,1 g
π : Diff(H ) → M(H ) over M(H ).
Hg g g g
References
[1] A. J. Casson, S. A. Bleiler,Automorphisms of surfaces after Nielsen and Thurston, London
Mathematical Society Student Texts, 9. Cambridge University Press, Cambridge, 1988.
[2] J. Franks, M. Handel, Global fixed points for centralizers and Morita’s Theorem, Geometry
and Topology 13, (2009), 87–98.
[3] A. Fathi, F. Laudenbach, Diff´eomorphisms pseudo-Anosov et d´ecomposition de Heegaard,C.
R. Acad. Sc. Paris, t. 291, S´erie A, (1980), 423B5.
[4] M. Korkmaz, Low-dimensional homology groups of mapping class groups: a survey, Turkish
J. Math. 26, (2002), 101–114.
[5] V. Markovic, Realization of the mapping class group by homeomorphisms, Invent. Math. 168,
(2007), 523–566.
[6] V. Markovic, D. Saric, The mapping class group cannot be realized by homeomorphisms,
preprint (arXiv:0807.0182)
[7] S. Morita, Characteristic classes of surface bundles, Invent. Math. 90, (1987), 551–577.
[8] L. Paris, D. Rolfsen, Geometric subgroups of mapping class groups, J. Reine Angew. Math.
521, (2000), 47–83.
DepartmentofMathematics, FacultyofScienceandEngineering, SagaUniversity,
Saga, 840–8502 Japan
E-mail address: [email protected]
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