Groups of homeomorphisms and diffeomorphismsof non-compact manifolds with the Whitney topology

Taras Banakh
Kotaro Mine
Katsuro Sakai
Tatsuhiko Yagasaki(Ivan Franko National University of Lviv, Ukraine)(University of Tokyo, Japan)(Kanagawa University, Japan)(Kyoto Institute of Technology, Japan)

ICM Seoul, 2014
Short Communications in Topology Session
Aug. 16, 2014

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§1. Model spaces for Whitney Topology
Compact-Open Topology←→

Tychonoff Products / Weak Products of l2

Whitney Topology←→

Box Products / Small Box Products of l2

Definition.

ω = {0, 1, 2, · · · }(1) The (Countable) Box product of Top space X :ω X
Q— the usual countable productas a Setn∈ω X— is given Box Topology
This topology is generated byQn Un(Un ⊂ X : Open subset)(2) The Small box product of Pointed Space (X, ∗) :ω

X— Subspace of ω X consisting of finite seq’s (x0, x1, . . . , xi, ∗, ∗, . . . )(1) ω l2 : a bad space (not normal, not locally connected, etc.)(2) Top. Classification of LF spaces(P. Mankiewicz, 1974)

ωR ≈ R∞ ≡ dir lim {R1 ⊂ R2 ⊂ R3 ⊂ · · · }ωl2 ≈ l2 × R∞§2. Homeomorphism Groups with Whitney topology
M : Non-compact n-manifold (possibly with boundary)H(M ) : Homeo Group of M (Whitney topology)∗ O(h, U) := g ∈ H(M ) | g : U-close to h— Top group(h ∈ H(M ), U ∈ cov(M ))H0(M ) : the identity connected component of H(M )Hc(M ) ⊂ H(M ) : Subgroup of Homeo’s with compact support
Basic Property.

(BMSY, 2011)(1) Hc(M ) : Paracompact, Locally contractible(2) H0(M ) ⊂ Hc(M ) : Open normal subgroup(3) Mc(M ) = Hc(M )/H0(M ) : Mapping Class Group
Hc(M )≈H0(M ) × Mc(M )(Discrete Top)(as Top spaces)

homeo(4) (Mi)i∈N : Compact subsets of M s.t. Mi ⊂ IntM Mi+1, M = ∪iMi
G(Mi) := {h ∈ Hc(M ) | supp h ⊂ Mi}
Hc(M ) = g-lim−→i G(Mi) (Direct limit in Category of Top Groups)23

Topological Type1-dim case

(H(R), Hc(R)) ≈ (ω l2,

ωl2)(BMS, 2009)

2-dim case
M : Non-compact Connected 2-manifold (possibly with boundary)Thm 1. (H(M ), Hc(M )) ≈` (ω l2,Thm 2. (1) H0(M ) ≈ωωl2 )l2 ≈ l2 × R∞

(BMSY, 2011)(BMSY, 2014)(2) Hc(M ) ≈ H0(M ) × Mc(M )(Mc(M ) : discrete)homeo
1pt M = X − K : (∗)
Mc(M ) ≈Nin all other cases
homeo(∗) : X = Annulus, Disk or Möbius band,
K = Non-empty compact subset of one boundary circle of X
Remark. H(M )co with Compact-Open Topology(T. Yagasaki, 2000)
 1S × l2 if M = R2, S1 × R, S1 × [0, ∞) or M − ∂M
(H(M )co)0 ≈l2in all other cases.

4∞§3. Diffeomorphism Groups with Whitney C -topology
M : Non-compact C ∞ n-manifold (without boundary)D(M ) : Diffeo Group of M (Whitney C ∞-topology)— Top group

D0(M ) : the identity connected component of D(M )Dc(M ) ⊂ D(M ) : Subgroup of Diffeo’s with compact support
Basic Property.

(BMSY, 2011)(1) Dc(M ) : Paracompact(2) D0(M ) ⊂ Dc(M ) : Open normal subgroup(3) M∞c (M ) = Dc (M )/D0 (M ) : Mapping Class Group
Dc(M )≈D0(M ) × M∞c (M )(Discrete Top)(as Top spaces)

homeo(4) (Mi)i∈N : Compact subsets of M s.t. Mi ⊂ IntM Mi+1, M = ∪iMi
G(Mi) := {h ∈ Dc(M ) | supp h ⊂ Mi}
Dc(M ) = g-lim−→i G(Mi) (Direct limit in Category of Top Groups)

5Topological Type1-dim case

(D(R), Dc(R)) ≈ (ω l2,ωl2)(BY, 2010)

n-dim case
M : Non-compact Connected C ∞ n-manifold (without boundary)Thm 1. (D(M ), Dc(M )) ≈` (ω l2,ωl2)

Thm 2. Dc(M ) ≈ Open subset of l2 × R∞
D0(M ) ≈ L × R∞ for some top. l2-manifold L(1) D0(M ) ≈ l2 × R∞

(BMSY, 2011)(BY)( L ' D0(M ) )if n = 1, 2 orn = 3 and M : orientable, irreducible(2) D0(M ) ≈ D0(N ; ∂N ) × R∞
if N : Compact Connected C ∞ n-manifold with Boundary
M = Int N

∞§5. Criterion of Top Group ≈ l2 × R .
T. Banakh - D. Repovš (2009 - ) — Series of papers
Study of Direct limit of Uniform spaces and Top. LF-manifolds
Quotients of Hilbert manifolds groups, etc.
Thm.

(BMRSY, 2013)

G : Top Group, Non-metrizable, G = ∪nGn(∗1) Gn ⊂ G : Closed subgroup, Gn ⊂ Gn+1, Gn ≈ l2(∗2) p :n Gn → G : Open(∗3) Gn+1 → Gn+1/Gn admits a local section(∗4) each Z-point of Gn+1/Gn is a strong Z-point.
=⇒ G ≈ l2 × R∞

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7References.
BMRSY = T. Banakh, K. Mine, D. Repovš, K. Sakai, T. Yagasaki[0] BMSY, arXiv:0802.0337v1 (2008)−→ [2], [3], [4][1] BMS, Classifying homeomorphism groups of infinite graphs, Topology Appl., 157(2009) 108–122.
[2] BMSY, Homeomorphism and diffeomorphism groups of non-compact manifolds withthe Whitney topology, Topology Proceedings, 37 (2011) 61 – 93.
[3] BMSY, On homeomorphism groups of non-compact surfaces, endowed with theWhitney topology, Topology Appl., 164 (2014) 170 – 181.
[4] BY, Diffeomorphism groups of non-compact manifolds endowed with the Whitney
C ∞ -topology, to appear.
[5] MS, Open subsets of LF-spaces, Bull. Pol. Acad. Sci. Math., 56 (2008) 25–37.
[6] BR, A topological characterization of LF-spaces, Topology Appl., 159 (2012) 1475 –1488.
[7] BMRSY, Detecting topological groups which are (locally) homeomorphic to LFspaces, Topology Appl., 160 (2013) 2272 – 2284

8Thank you very muchfor your attention !