Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results.

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Tatsuhiko Yagasaki
Kyoto Institute of Technology
October 17, 2011
Joint work with

Tatsuhiko Yagasaki

Taras Banakh (Ukraine)

Mapping class groups of non-compact surfaces

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..

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Remarks on mapping class groups
of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary
H(S) : Group of homeomorphisms of S

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary
H(S) : Group of homeomorphisms of S!
"
Hc (S) = h ∈ H(S) | Supp h : compact

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary
H(S) : Group of homeomorphisms of S!
"
Hc (S) = h ∈ H(S) | Supp h : compactht!
"
Hc (S)0 = h ∈ Hc (S) | h " idM with compact support

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary
H(S) : Group of homeomorphisms of S!
"
Hc (S) = h ∈ H(S) | Supp h : compactht!
"
Hc (S)0 = h ∈ Hc (S) | h " idM with compact support.
Definition 1.
Mapping Class Group..
Mc (S) = Hc (S)/Hc (S)0(Discrete top)

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

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. (Isotopy classes of homeomorphisms with cpt supp)..

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Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Mapping class groups of surfaces
S : a connected 2-manifold possibly with boundary
H(S) : Group of homeomorphisms of S!
"
Hc (S) = h ∈ H(S) | Supp h : compactht!
"
Hc (S)0 = h ∈ Hc (S) | h " idM with compact support.
Definition 1.
Mapping Class Group..
Mc (S) = Hc (S)/Hc (S)0(Discrete top)
Caution on Punctures !
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

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. (Isotopy classes of homeomorphisms with cpt supp)..

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Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — I
H(S)w : Homeo Group with Whitney topology

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — I
H(S)w : Homeo Group with Whitney topology
S : Compact :Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — I
H(S)w : Homeo Group with Whitney topology
S : Compact :Whitney topology = Compact - Open topology
H(S)w : !2 -manifold

H(S)w0 : Contractible (≈ !2 )Tatsuhiko Yagasaki

except few cases

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — I
H(S)w : Homeo Group with Whitney topology
S : Compact :Whitney topology = Compact - Open topology
H(S)w : !2 -manifold

H(S)w0 : Contractible (≈ !2 )except few cases

S : Non-compact :Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — I
H(S)w : Homeo Group with Whitney topology
S : Compact :Whitney topology = Compact - Open topology
H(S)w : !2 -manifold

H(S)w0 : Contractible (≈ !2 )except few cases

S : Non-compact :Joint Work on Whitney topology
by T. Banakh, K. Sakai, K. Mine, T. Yagasaki(Univ. of Tsukuba)
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — II

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — II.
Proposition 1.
..
H(S)w : Top group.
(Bad space !)◦ Hc (S)0 = Identity connected component of H(S)www. ◦ Hc (S) ≈ Hc (S)0 × Mc (S)..

Tatsuhiko Yagasaki(as Top Spaces)

Mapping class groups of non-compact surfaces.

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Hc (S)w ⊃ Hc (S)0 : Open normal subgroup

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Homeomorphism groups of surfaces — II.
Proposition 1.
..
H(S)w : Top group.
(Bad space !)◦ Hc (S)0 = Identity connected component of H(S)w. Hc (S)0 ≈ l2 × R..
w∞(as Top Spaces).
(R∞ = dirlimn Rn )(Contractible)

Tatsuhiko Yagasaki.

Mapping class groups of non-compact surfaces

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.

w
w
. ◦ Hc (S) ≈ Hc (S)0 × Mc (S)..
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Proposition 2.
..
Hc (S)w : !2 × R∞ -manifold.

Hc (S)w ⊃ Hc (S)0 : Open normal subgroup

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Problem
S : Non-compact connected 2-manifold possibly with boundary

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Problem
S : Non-compact connected 2-manifold possibly with boundary
Mc (S) = Hc (S)/Hc (S)0

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Problem
S : Non-compact connected 2-manifold possibly with boundary
Mc (S) = Hc (S)/Hc (S)0

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

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Problem.
..
. (Geometric) Group Structure of Mc (S)..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Problem
S : Non-compact connected 2-manifold possibly with boundary
Mc (S) = Hc (S)/Hc (S)0.
Questions.
(1st step)..
(1) When is Mc (S) trivial ?
.(2) When is Mc (S) finitely generated ?
..

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

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Problem.
..
. (Geometric) Group Structure of Mc (S)..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — I
G : a countable group

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — I
G : a countable group.
Definition.
..
Asymptotic dimension :.

asdim(G ) ≤ n ⇐⇒∀ F ⊂ G : a finite subset ∃ U : a cover of G
#
−1
s.t.
U : finite
U∈U U∀ x ∈ G xF meets at most n + 1 sets U ∈ U. asdim(G ) = ∞ ⇐⇒ asdim(G ) -≤ n for each n..
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

.

asdim(G ) = n ⇐⇒ asdim(G ) ≤ n, asdim(G ) -≤ n − 1

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II

Tatsuhiko Yagasaki.
(Z-rank)

Mapping class groups of non-compact surfaces.

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Definition.
..
n. rZ (G ) = sup{n ≥ 0 : Z "→ G }..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II.
(Z-rank)(1) asdim(Zn ) = n (n ≥ 0)

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

.

.
Definition.
..
n. rZ (G ) = sup{n ≥ 0 : Z "→ G }..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II.
(Z-rank)(1) asdim(Zn ) = n (n ≥ 0)(2) rZ (G ) ≤ asdim(G )Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

.

.
Definition.
..
n. rZ (G ) = sup{n ≥ 0 : Z "→ G }..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II.
(Z-rank)(1) asdim(Zn ) = n (n ≥ 0)(2) rZ (G ) ≤ asdim(G )(3) rZ (G ) = 0 ⇐⇒ G : a torsion group

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

.

.
Definition.
..
n. rZ (G ) = sup{n ≥ 0 : Z "→ G }..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Notions on Countable groups — II.
(Z-rank)(1) asdim(Zn ) = n (n ≥ 0)(2) rZ (G ) ≤ asdim(G )(3) rZ (G ) = 0 ⇐⇒ G : a torsion group(4) G : abelian =⇒ rZ (G ) = free rank of G = asdim(G )Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

.

.
Definition.
..
n. rZ (G ) = sup{n ≥ 0 : Z "→ G }..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. When is Mc (S) trivial ?

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. When is Mc (S) trivial ?

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

.

.

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Theorem 1.
..
The following conditions are equivalent:(1) Mc (S) = {1};(2) rZ (Mc (S)) = 0 (Torsion group);(3) asdim(Mc (S)) = 0;
(4) S ≈ N \ K
N : Disk, Annulus, or Möbius band
K : Non-empty compact subset
of one boundary circle of N.
..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — I

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — I

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

.

.

.

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Definition.
..
S : Finite type ⇐⇒ S ≈ N \ F ;N : Compact surface, F ⊂ N \ ∂N : Finite subset.
..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — I.
Definition.
..
S : Semi-finite type ⇐⇒ S ≈ N \ (F ∪ K ) ;N : Compact surface, F ⊂ N \ ∂N : Finite subset
K ⊂ ∂N : Compact subset.
..

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

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Definition.
..
S : Finite type ⇐⇒ S ≈ N \ F ;N : Compact surface, F ⊂ N \ ∂N : Finite subset.
..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — II

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — II

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

.

.

.

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Lemma.
..
The following conditions are equivalent:(1) π1 (S) : Finitely presented;(2) H1 (S; Z) : Finitely generated;.(3) S : Semi-finite type..

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — II.

.

.
Lemma.
.
..
The following conditions are equivalent:(1) π1 (S) : Finitely presented;(2) H1 (S; Z) : Finitely generated;.(3) S : Semi-finite type..
.
.
.
Theorem.
(M. Bestvina, K. Bromberg, K. Fujiwara)..
S : Compact=⇒ asdim Mc (S) < ∞(Semi-finite type).
..
.

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. When is Mc (S) finitely generated ?

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. When is Mc (S) finitely generated ?
.
Theorem 2.
..
The following conditions are equivalent:(1) Mc (S) : Finitely generated;.

(2) Mc (S) : Finitely presented;(3) rZ (Mc (S)) < ∞;.(5) S : Semi-finite type..

Tatsuhiko Yagasaki.

Mapping class groups of non-compact surfaces.

(4) asdim(Mc (S)) < ∞;

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. When is Mc (S) finitely generated ?
.
Theorem 2.
..
The following conditions are equivalent:(1) Mc (S) : Finitely generated;.

(2) Mc (S) : Finitely presented;(3) rZ (Mc (S)) < ∞;.(5) S : Semi-finite type..

.
∞

◦ S : not Semi-finite type =⇒ ⊕ Z ⊂ Mc (S)∞(⊕ Z : Free abelian group of Infinite rank)
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

(4) asdim(Mc (S)) < ∞;

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Tatsuhiko YagasakiŇ := S \ IntS N

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Ň := S \ IntS N(i) C(N) : the set of connected components of N

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Ň := S \ IntS N(i) C(N) : the set of connected components of N$(ii) Mc (S, N) = Hc (S, N) Hc (S, N)0

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Ň := S \ IntS N(i) C(N) : the set of connected components of N$(ii) Mc (S, N) = Hc (S, N) Hc (S, N)0
Hc (S, N) := {h ∈ Hc (S) : h|N = idN }Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Ň := S \ IntS N(i) C(N) : the set of connected components of N$(ii) Mc (S, N) = Hc (S, N) Hc (S, N)0

Hc (S, N) := {h ∈ Hc (S) : h|N = idN }!
"Hc (S, N)0 := h ∈ Hc (S, N) | ∃ ht : h " idS rel N(cpt supp)

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — I
S : a surface (connected)(1) N ⊂ S : a 2-submanifold

Ň := S \ IntS N(i) C(N) : the set of connected components of N$(ii) Mc (S, N) = Hc (S, N) Hc (S, N)0

Hc (S, N) := {h ∈ Hc (S) : h|N = idN }!
"Hc (S, N)0 := h ∈ Hc (S, N) | ∃ ht : h " idS rel N(cpt supp)(2) Mc (S, Ň1 )

φN1 ,N2! Mc (S, Ň2 )
!!!
"
!!!
"""
"
!!!
"
"
(N1 ⊂ N2 ⊂ S)!"
φN1#""" φN2
Mc (S)
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

N ∈ N (S)Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

N ∈ N (S)(i) N : Retract of S

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

N ∈ N (S)(i) N : Retract of S

φN(ii) Mc (N, FrS N) ≈ Mc (S, Ň) −→ Mc (S) : injective

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

N ∈ N (S)(i) N : Retract of S

φN(ii) Mc (N, FrS N) ≈ Mc (S, Ň) −→ Mc (S) : injective
φN1 ,N2! Mc (S, Ň2 )Mc (S, Ň1 )
!!!
""
!!!
"""
!!!
"
"
(N1 ⊂ N2 in N (S))!"φN1#""" φN2
Mc (S)

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Relative Mapping Class Groups — II&'
& a compact connected 2-submanifold&(3) N (S) := N ⊂ S && ∀ L ∈ C(Ň) : non-compact%

N ∈ N (S)(i) N : Retract of S

φN(ii) Mc (N, FrS N) ≈ Mc (S, Ň) −→ Mc (S) : injective
φN1 ,N2! Mc (S, Ň2 )Mc (S, Ň1 )
!!!
""
!!!
"""
!!!
"
"
(N1 ⊂ N2 in N (S))!"φN1#""" φN2
Mc (S)◦ Direct Limit in Gr
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — III
S : a non-compact surface

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — III
S : a non-compact surface.
Proposition.
..
S : semi-finite type

.

.
..

.
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

⇐⇒ ∃ M ∈ N (S) such thateach L ∈ C(M̌) satisfies the next conditions :Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — III
S : a non-compact surface.
Proposition.
..
S : semi-finite type

.

.
..

(∗1 ) L includes no handle,(∗2 ) L does not include two boundary circles of S,(∗3 ) L is not separated by a circle C ⊂ Int L
into two non-compact connected 2-submanifolds.
.
Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

⇐⇒ ∃ M ∈ N (S) such thateach L ∈ C(M̌) satisfies the next conditions :Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — IV
D : Disk,

A : Annulus

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — IV
D : Disk,
A : Annulus.
Lemma.
..
S : a non-compact orientable surface, (∗1 ), (∗2 ), (∗3 ).

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces

.

.
..

.

Mapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — IV
D : Disk,
A : Annulus.
Lemma.
..
S : a non-compact orientable surface, (∗1 ), (∗2 ), (∗3 )

.

.
..

.

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

(1) S has no boundary circle=⇒ S ≈ D \ K for a non-empty compact subset K of ∂DMapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results. Surfaces of semi-finite type — IV
D : Disk,
A : Annulus.
Lemma.
..
S : a non-compact orientable surface, (∗1 ), (∗2 ), (∗3 ).

(2) S has a boundary circle=⇒ S ≈ A \ K for a non-empty compact subset K
of one boundary circle of A.
..
.

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces.

(1) S has no boundary circle=⇒ S ≈ D \ K for a non-empty compact subset K of ∂DMapping class groups of surfaces
Homeomorphism groups of surfaces
Some Results

End of Talk
Thank you very much !

Tatsuhiko Yagasaki

Mapping class groups of non-compact surfaces