Homeomorphism Groups of Non-Compact Surfaces
with the Whitney Topology

Taras Banakh(Ivan Franko National University of Lviv, Ukraine)

Kotaro Mine

(Institute of Mathematics, University of Tsukuba)

Katsuro Sakai(Institute of Mathematics, University of Tsukuba)

Tatsuhiko Yagasaki (Kyoto Institute of Technology)

General Topology Symposium 2010
Institute of Mathematics, University of Tsukuba, Tsukuba, Ibaraki
December 20, 2010

1
Main Problem
Local / Global Topological Properties ofGroups of Homeomorphismsof Non-Compact n-Manifolds
with Whitney Topology
Other Typical Topologies(1) Compact-Open Topology(2) Uniform Topology

Different
Properties !!

(3) Direct Limit Topology
Compact-Open Topology ←→ Tychonoff Products · Weak Products
Whitney Topology ←→ Box Products · Small Box Products

2References. — Whitney topology
BMSY — T. Banakh, K. Mine, K. Sakai, T. Yagasaki
Local Topological Type :BMSY, Homeomorphism and diffeomorphism groups ofnon-compact manifolds with the Whitney topology
Topology Proceedings, 37 (2011) 61 - 93 (e-published).
Global Topological Type :BMS, Classifying homeomorphism groups of infinite graphs
Topology Appl. 157 (2009) 108–122.
BMSY, On homeomorphism groups of non-compact surfaces
endowed with the Whitney topology — arXiv:1004.3015

3§1. Box products · Small box products(Xn)n :

(Xn, ∗n)n :Q
Box product : nXn = Productn XnQBox Top :(Un ⊂ Xn : Open subset)n Un
Small box product :n Xn⊂ nXn(x0, x1, . . . , xk , ∗k+1, ∗k+2, . . . )(, )nXn = (nXn,n Xn )◦ Xn : metrizable =⇒◦ω

l2 ≈ l2 × R∞(, )ω X = (, )n∈ω Xn Xn : paracompact

R∞ = lim Rn−→ nTop. Classification of LF spaces by P. Mankiewicz◦ ω l2 : not locally connected, not normal

4§2. Homeo Groups with Whitney topology
M : Non-compact, Connected n-manifold
possibly with boundaryH(M ) : Homeo Group of M (Whitney topology)◦ h ∈ H(M ) has the basic nbd system :U(h) = g ∈ H(M ) : (h, g) ≺ U(U ∈ cov(M ))◦ H(M ) : Top group
H(M )0 : Identity connected component of H(M )Hc(M ) :Compact support

H(M, A) :Identity on A

(A ⊂ M )5
General Topological Properties of (H(M ), Hc(M ))H(M ) : Not GoodHc(M ) :(1) Hc(M ) : Paracompact, Locally contractible(2) Any compact subset of Hc(M ) has common compact support(3) H(M )0 ⊂ Hc(M ) : Open normal subgroupht

H(M )0 = {h ∈ H(M ) | h ' idM with Compact Support }(4) Mc(M ) = Hc(M )/H(M )0 : Mapping Class Group(Discrete top)
Hc(M ) ≈ H(M )0 × Mc(M ) (as Top Spaces)6(5) (Mi)i∈N : Sequence of compact subsets of M
s.t. Mi ⊂ IntM Mi+1, M = ∪iMiS◦ Hc(M ) = i H(M, M \ Mi)
Hc(M ) = g-lim−→i H(M, M \ Mi)(Direct limit in Category of Top Groups)◦ Hc(M ) with Direct Limit Top :not Top Group

Local Top Type of (H(M ), Hc(M )) :Thm.

(H(R), Hc(R)) ≈ (, )ω l2(BMS)

Conjecture. (H(M ), Hc(M )) ≈` (, )ω l2
Thm. n = 1, 2 : OK(n ≥ 3 : Open)7
Global Top Type of Hc(M )ωl2 ≈ l2 × R∞

n=1:

BMS Hc(R) ≈n=2:

BMSY (arXiv:1004.3015)

M : Non-Compact Connected 2-manifold possibly with boundaryThm. (1) H(M )0 ≈ l2 × R∞(2) Hc(M ) ≈ H(M )0 × Mc(M )Countable DiscreteMc(M ) ≈1 point
Exceptional Case :(M : Generic)(M : Exceptional)

M ≈X −K◦ X = Annulus, Disk or Möbius band,◦ K = Compact subset of one boundary circle of X, K 6= ∅

8
M : Non-compact PL 2-manifold possibly with boundary
Prop. HcPL(M ) ⊂ Hc(M ) : Homotopy Dense◦ A ⊂ X : Homotopy Dense⇐⇒ ∃ ϕt : X → X : ϕ0 = idX , ϕt(X) ⊂ A (t ∈ (0, 1])9§3. Ideas of Proof.
M : Non-Compact Connected 2-manifold possibly with boundaryThm 1.

(H(M ), Hc(M )) ≈` (, )ω l2

Thm 2.

H(M )0 ≈ l2 × R∞(Mi)i∈N : Sequence of Compact 2-submanifolds of M
s.t. Mi ⊂ IntM Mi+1, M = ∪iMi
Proof of Thm 1.
Li := Mi − IntM Mi−1
E(Li, M )∗ = {f |Li : f ∈ H(M )} (Compact - Open Top)
(H(M ), Hc(M )) ≈` (, )i E(L2i, M )∗ × (, )i H(M, M \ L2i−1)≈` (, )ω l2

10
Proof of Thm 2.

(H(M )0 ≈ l2 × R∞ ≈ωl2)[1] (2007 - 2008) BMSY (arXiv:0802.0337v1)
Lemma.

G : Top groupG = ∪iGi :Gi ⊂ G : Closed subgroup, Gi ⊂ Gi+1i Gi → G : Open(∗1) p :p(x1, . . . , xm, e, e, · · · ) = xm · · · x1(∗2)/πi : Gi k

Gi/Gi−1πisi = id

∃ si

=⇒

s=oo7
i ooioooooooosi (Gi /Gi−1 )H(M )0 =

Si Gi

ps
≈

i H(M, M \ Mi )0 ≈DDDDDDDDD!

p/

Gi (Gi /Gi−1 ) ≈i `2

11[2] (2009 - 2010) BMSY (arXiv:1004.3015)
BR : T. Banakh - D. Repovš,A topological characterization of LF-spaces (arXiv:0911.0609)
Topological Characterization of l2 × R∞
X : Top Space, Non-metrizable
X ≈ l2 × R∞ ⇐⇒ X ≈ u-lim−→ Xn (the uniform direct limit)(Xn)n∈N : a tower of metrizable uniform spaces such thateach Xn (i) uniformly locally equiconnected(ii) a uniform neighborhood retract in Xn+1
(iii) has a uniform frill in Xn+1
(iv) contractible in Xn+1
(v) an l2-manifold

12BMRSY(arXiv:1004.0305)

On topological groups (locally) homeomorphic to LF-spacesCor.

G : Top Group, Non-metrizableG = ∪iGi :(∗1) p :Gi ⊂ G : Closed subgroup, Gi ⊂ Gi+1i Gi → G : Openp(x1, . . . , xm, e, e, · · · ) = xm · · · x1(∗2) GLi : a uniform neighborhood retract of GLi+1(∗3) Gi ≈ l2=⇒ G ≈ l2 × R∞

13
§4. Homeo Groups with Compact-Open Top.
M : Non-Compact Connected 2-manifold possibly with boundary
Exceptional = Plane, Open Möbius Band,or Open / Half Open Annulus
H(M )co : Compact-Open TopologyhtHc(M )∗1 = {h ∈ H(M ) | h ' idM(Compact Support) }= H(M )0 in Whitney Topology
Thm. (T. Yagasaki, 2000, 2003) ((1) H(M )co
0 : Top l2 -manifold≈l2(Generic Case)l2 × S1 (Exceptional Case)∗(2) H(M )co0 ⊃ Hc (M )1 : Homotopy Dense

14Cor. (T. Yagasaki, arXiv:1003.2833)
Qω Pω co∗H(M )0 , Hc(M )1 : Topl2,l2 -manifold( Qω Pω l2,l2(Generic Case)≈Qω Pω l2,l2 × S1 (Exceptional Case)Example.

15M = R2

1H(R2)co'S: Homotopy Equivalence is induced from0

Loop of θ rotations

ϕ(θ) (θ ∈ [0, 2π])

Hc(R2)co ⊂ H(R2)co0 : Homotopy Dense∴ ∃ ϕt(θ) (t ∈ [0, 1]) : Homotopy of Loops such thatϕ1(θ) = ϕ(θ), ϕt(θ) ∈ Dc(R2) (t < 1)
We can take ϕt(θ) as Loop of Truncated θ rotations
Level of Truncation r = rt(θ)
need to satisfy Conditions :(i) ∀ t < 1, r → ∞ as θ → 2π(ii) r → ∞ uniformly as t → 1

θr

16
M = Open Annulus
Exercise.

h : Dehn Twist on M along Center circle of M

(1) h ∈ H(M )0 \ Hc(M )∗1(2) ∃ ht ∈ H(M )0 (0 ≤ t ≤ 1) such thath0 = h, h1 = idM and ht ∈ Hc(M )∗1 (0 < t ≤ 1)
(Hint: Introduce Reverse Dehn Twist from ∞)End of Talk
Thank you very much !