Diffeomorphism Groups of Non-Compact Manifolds
with the Compact-Open C ∞ Topology

Tatsuhiko Yagasaki(Kyoto Institute of Technology, Japan)

The 25th Summer Conference onTopology and its Applications.
Kielce, Poland, July 25-30, 2010.

1
Main Problem
Study of Local and Global Topological Properties ofDiffeomorphism Groups of Non-Compact C ∞ n-Manifolds(1) The Whitney Topology has been studied in
Joint Works by T. Banakh, K. Mine, K. Sakai and T. Yagasaki.
(2) In this talk, we discuss the Compact-Open C ∞ Topology.
This talk is based upon
the paper :[∗] T. Yagasaki, Groups of volume-preserving diffeomorphisms of noncompact
manifolds and mass flow toward ends, Trans. AMS, 362 (2010), 5745-5770.
two preprints:[1] T. Yagasaki, Homotopy types of diffeomorphism groups of noncompact
2-manifolds, arXiv:0109183v3.
QP[2] T. Yagasaki, Topological ( ω `2, ω `2)-factors of diffeomorphism groupsof non-compact manifolds, arXiv:1003.2833.

2§.1. Groups of Diffeomorphisms andSubgroups of Volume-Preserving Diffeomorphisms
M : Connected C ∞ n-Manifold
D(M ) = Group of Diffeomorphisms of M
If M is Orientable,(Compact-Open C ∞ Topology)

for any Volume Form ω on M ,D(M ; ω) = Subgroup of ω-Preserving Diffeomorphisms of M
(1) When M is Compact, Moser’s Thm =⇒SDR
D (M ) ⊃ D(M ; ω)+(2) When M is Non-Compact,
D+(M ) ⊃ D(M ; ω)∪∪D(M )0 ⊃ D(M ; ω)0 ⊃ ker cω0∪∪
Dc(M )∗1⊃
Dc(M ; ω)∗1
D (M )∗1 =cc

H

End-Charge Homo.
cω0 : D(M ; ω)0 −→ S(M ; ω)h7−→ cω0 (h)h ∈ D (M ) | ∃ Isotopy h ' idM with Compact Support

2-1
Properties of Dc(M )∗1(1) By Definition, each h ∈ Dc(M )∗1 is Isotopic to idM with Compact Support.
(2) However, Path in Dc(M )∗1 need not have Common Compact Support.
(3) hi → idM in Dc(M )∗1 if Supp hi → ∞.
Sharp Contrast to Whitney C ∞-Topology
D(M )w = D(M ) with Whitney C ∞-Topology(Joint Work by T. Banakh, K. Mine, K. Sakai, T. Yagasaki)
Fact 1.1.
(1) (D(M )w )0 = Dc(M )∗1 as Sets
(2) Any compact subset of Dc(M )w has a common compact support.
On Ends of M
Compact-Open C ∞-Topology
Too WeakvsWhitney C ∞ Topology
Too Strong3In the paper[∗] T. Yagasaki, Groups of volume-preserving diffeomorphisms of noncompact
manifolds and mass flow toward ends, Trans. AMS, 362 (2010), 5745-5770.
we have shown that
D(M ) ⊃ D(M ; ω)
∪∪
SDR
SDR
D(M )0 ⊃ D(M ; ω)0 ⊃ ker cω0∪Dc(M )∗1∪⊃

Dc(M ; ω)∗1

Remaining Problems
(1) Homotopy Type Topological Type of D(M )0 and D(M ; ω)0(2) Relations between D(M )0 ⊃ Dc(M )∗1 and ker cω0 ⊃ Dc(M ; ω)∗1In n = 2 we can answer these questions.

4§2. 2-dim case
M : Non-Compact Connected C ∞ 2-Manifold without Boundary
Exceptional Case — Plane, Open Möbius Band and Open Annulus
Generic Case — All Other cases§2.1. Groups of Diffeomorphisms of Non-Compact 2-manifolds
D(M )0 ⊃ Dc(M )0 ⊃ Dc(M )1 ⊃ Dc(M )∗1Theorem 2.1.
(1) D(M )0 : Topological `2-Manifold(2) (i) Exceptional Case :D(M )0 ' S1

D(M )0 ≈ `2 × S1(ii) Generic Case :D(M )0 ' ∗D(M )0 ≈ `2

Theorem 2.2.
(1) D(M )0 ⊃ Dc(M )∗1 : Homotopy Dense (HD)∃ ϕt : D(M )0 → D(M )0 s.t. ϕ0 = id, ϕt(D(M )0) ⊂ Dc(M )∗1 (0 < t ≤ 1)
(2) D(M )0 ⊃ Dc(M )0 = Dc(M )1 ⊃ Dc(M )∗1 : Homotopy Equi.

FσCorollary 2.1. D (M )0 ⊃ G ⊃ Dc(M )∗1
Qω Pω `2 ,`2 -manifold.
(1) D(M )0, G is
Qω Pω (2) (i) Exceptional Case :D(M )0, G ≈`2,`2 × S1
Qω Pω (ii) Generic Case :D(M )0, G ≈`2,`2cExample 2.1. When M = R2,(1) D(R2)0 ' S1This homotopy equivalence is induced fromLoop of θ rotations ϕ(θ) (θ ∈ [0, 2π])

(2) Dc(R2) = Dc(R2)∗1 ⊂ D(R2)0 : HD
∃ ϕt(θ) (t ∈ [0, 1]) : Homotopy of Loops such thatϕ1(θ) = ϕ(θ) and ϕt(θ) ∈ Dc(R2) for t < 1.
We can take ϕt(θ) as Loop of Truncated θ rotations
such that Level of Truncation r = rt(θ) satisfies Conditions :(i) for each t < 1, r → ∞ as θ → 2π(ii) r → ∞ uniformly as t → 15

7-1
TruncatedθRotationr = level of truncation

θr

6Comparison with Whitney C ∞-Topology(Joint Work by T. Banakh, K. Mine, K. Sakai, T. Yagasaki)
Theorem.

(D(M )w )0 = (Dc(M )w )0 ≈ R∞ × `2 ' ∗D(M )w = D(M ) with Whitney C ∞-Topology
R∞ : Direct Limit of the Tower

R1 ⊂ R2 ⊂ R 3 ⊂ · · ·§2.2 Groups of Volume-Preserving Diffeomorphisms

7of Non-Compact 2-manifolds
M : Non-Compact Connected Orientable C ∞ 2-Manifold without Boundaryω : Volume Form on M
H-typeD(M )0HD⊃SDR ∪Dc(M )0∪D(M ; ω)0 ⊃ ker cω0 ⊃ Dc(M ; ω)0 ⊃ Dc(M ; ω)1 ⊃ Dc(M ; ω)∗1SDR
Theorem 2.3. (1) D(M ; ω)0 : Topological `2-manifold(2) (i) Exceptional Case :(ii) Generic Case :D(M ; ω)0 ≈ `2 × S1
D(M ; ω)0 ≈ `2

Theorem 2.4. (1) ker cω0 ⊃ Dc(M ; ω)∗1 : Homotopy Dense(2) ker cω0 ⊃ Dc(M ; ω)0 = Dc(M ; ω)1 ⊃ Dc(M ; ω)∗1 : Homo. Equi.

8Stability Property of (D(M )0, Dc(M )0)(n : arbitrary)`2-Stability:(1) (`2)2 ≈ `2 =⇒ If X is an `2-manifold, then X × `2 ≈ X.
(2) When M is Compact,

D(M ) : a Frechet manifold∴ `2-stable`2-factor of D(M ) :If M is Orientable, Moser’s Thm =⇒D+(M ) ≈ D(M ; ω) × V(M ; ω)V(M ; ω) = µ ∈ V(M ) | µ ∼ ω, µ(M ) = ω(M ) : Convex ≈ `2
Even if M is Non-Orientable,
we can use Volume Density instead of Volume Form.
Qω

`2,Pω



`2 -Stability:
PωQ`2 = (xi)i ∈ ω `2 | xi = 0 except for finitely many i9When M is Non-Compact,
we can apply Moser’s Thm to a discrete sequence of n-disks in M
Dc(M )0 ⊃ G ⊃ Dc(M )∗1to obtain the following conclusion:QωPω`2 -stableQω Pω 
`2,`2 ≈ (D(M )0, G)
i.e., (D(M )0, G) ×

Theorem. (D(M )0, G) :Qω

`2,Pω



Characterization of`2,`2 -manifoldsQω Pω (X, A) :`2 ,`2 -manifold⇐⇒(1) X : Separable, Completely Mentrizable ANR,(2) A : Fσ , Homotopy Dense in X
Qω Pω (3) (X, A) :`2 ,`2 -stableA ⊂ X : Homotopy Dense (HD)⇐⇒ ∃ ϕt : X → X : Homotopy s.t. ϕ0 = idX , ϕt(X) ⊂ A (0 < t ≤ 1)
Q
P[2] T. Yagasaki, Topological ( ω `2, ω `2)-factors of diffeomorphism groups
of non-compact manifolds, arXiv:1003.2833.