Geometry and Foliations 2013
Komaba, Tokyo, Japan

Groups of uniform homeomorphisms ofcovering spaces
Tatsuhiko YAGASAKI

1.

Introduction

The uniform topology is one of basic topologies on function spaces. In thisnote we report some results on local and global deformation properties ofspaces of uniform embeddings and groups of uniform homeomorphisms inmetric manifolds endowed with the uniform topology.
Our main goal is to understand local or global topological properties ofgroups of uniform homeomorphisms of metric manifolds endowed with theuniform topology (for example, local contractibility, homotopy type, local
or global topological type as infinite-dimensional manifolds, etc). Since thenotions of uniform continuity and uniform topology depend on the choice ofmetrics, we are also interested in dependence of those topological properties
on the behavior of metrics in neighborhoods of ends of manifolds.
In [6] we studied the formal behaviour of local deformation property
in the space of uniform embeddings and showed that this property is preserved by the restriction and union of domains of uniform embeddings.
This observation reduces our problem to the study of simpler pieces. In [2]A.V. Černavskiı̆ considered the case where the manifold M is the interiorof a compact manifold N and the metric d is a restriction of some metric on
N . Recently, in [5] we treated the class of metric covering spaces over compact manifolds. In this case we can deduce a local deformation theorem foruniform embeddings from the Edwards-Kirby local deformation theorem
for embeddings of compact spaces and the classical Arzela-Ascoli theorem
for equi-continuous families of maps ([5, Theorem 1.1]). The additivity oflocal deformation property implies that any metric manifold with a locallygeometric group action also has the same local deformation property ([6,Theorem 4.1]).
The local deformation property for uniform embeddings implies the local contractibility of the group of uniform homeomorphisms. Our next aim
is to study its global deformation property. The most standard example
is the Euclidean space Rn with the standard Euclidean metric. Its relevant feature is the existence of similarity transformations. This enables us
to deduce a global deformation for uniform embeddings in the Euclidean
end from the local one. Since this property is preserved by bi-Lipschitzc 2013 Tatsuhiko Yagasaki
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homeomorphisms, we obtain a global deformation theorem for the groupof uniform homeomorphisms of any metric manifold with finitely many biLipschitz Euclidean ends ([5, Theorem 1.2]). This implies, for instance, thecontractibility of the identity components of the groups of uniform homeomorphisms of Rn and any non-compact 2-manifold with finitely manybi-Lipschitz Euclidean ends ([5, Example 1.1]).
In the succeeding sections we explain some details of the statements
described in this introduction. Section 2 contains local deformation results
for uniform embeddings. Section 3 includes global deformation results foruniform homeomorphisms.

2.

Local deformation property for uniform embeddings

2.1. Suppose (X, d) is a metric space. For subsets A, B of X we writeA ⊂u B and call B a uniform neighborhood of A in X if B contains theε-neighborhood Oε (A) of A in X for some ε > 0.
A map f : (X, d) → (Y, ρ) between metric spaces is said to be uniformly
continuous if for each ε > 0 there is a δ > 0 such that if x, x� ∈ X andd(x, x� ) < δ then ρ(f (x), f (x� )) < ε. The map f is called a uniform homeomorphism if f is bijective and both f and f −1 are uniformly continuous.
A uniform embedding is a uniform homeomorphism onto its image.
A metric manifold means a separable topological manifold possibly withboundary assigned a fixed metric. Suppose (M, d) is a metric n-manifold.
For subsets X and C of M , let E∗u (X, M ; C) denote the space of proper
uniform embeddings f : (X, d|X ) → (M, d) such that f = id on X ∩ C.
This space is endowed with the uniform topology induced from the supmetric��d(f, g) = sup d(f (x), g(x)) | x ∈ X ∈ [0, ∞] (f, g ∈ E∗u (X, M ; C)).
Definition 2.1. For a subset A of M we say that A has the local deformation property for uniform embeddings in (M, d) and write A : (LD)M if
the following holds:(∗) for any subset X of A, any uniform neighborhoods W � ⊂ W of X in(M, d) and any subsets Z ⊂u Y of M there exists a neighborhood Wof the inclusion map iW : W ⊂ M in E∗u (W, M ; Y ) and a homotopy
φ : W × [0, 1] −→ E∗u (W, M ; Z) which satisfy the following conditions:(1) For each h ∈ W(i) φ0 (h) = h,(ii) φ1 (h) = id on X,(iii) φt (h) = h on W − W � and φt (h)(W ) = h(W ) (t ∈ [0, 1]),

339(iv) if h = id on W ∩ ∂M , then φt (h) = id on W ∩ ∂M (t ∈ [0, 1]).
(2) φt (iW ) = iW (t ∈ [0, 1]).
In the case where A = M we omit the subscript M in the symbol(LD)M .
The celebrated Edwards-Kirby local deformation theorem [3] can berestated in the next form.
Theorem 2.2. (Edwards-Kirby [3]) Any relatively compact subset K of M
satisfies the condition (LD)M .
The condition (LD)M has the following formal properties:
Proposition 2.3. ([6, Proposition 3.1, Corollary 3.1, Remark 3.2])(1) The property (LD) is preserved by any uniform homeomorphism (i.e.,if (M, d) is uniformly homeomorphic to (N, ρ), then (M, d) : (LD)⇐⇒ (N, ρ) : (LD).)(2) (Restriction) (i) Suppose A ⊂ B ⊂ M . Then, B : (LD)M =⇒A : (LD)M .
(ii) Suppose A ⊂u N ⊂ M and N is an n-manifold. Then, A : (LD)N⇐⇒ A : (LD)M .
(3) (Additivity) (i) Suppose A ⊂u U ⊂ M and B ⊂ M . Then, U ,B : (LD)M =⇒ A ∪ B : (LD)M .
(ii) Suppose M = A ∪ B, A, B are n-manifolds and A − B ⊂u A.
Then, A, B : (LD) =⇒ M : (LD).
(4) Suppose K is a relatively compact subset of M and A ⊂ M . Then,A : (LD)M ⇐⇒ A ∪ K : (LD)M .

(5) (Neighborhoods of ends) Suppose M = K ∪ ∪m
i=1 Li , K is relativelycompact, each Li is an n-manifold and closed in M , and d(Li , Lj ) > 0for any i �= j. Then, M : (LD) ⇐⇒ Li : (LD) (i = 1, · · · , m).
2.2. Metric covering projections and Geometric group actions
Our next aim is to seek concrete examples which have the local deformation property for uniform embeddings. The following notion is a natural
metric version of Riemannian covering projections.
Definition 2.4. A map π : (X, d) → (Y, ρ) between metric spaces iscalled a metric covering projection if it satisfies the following conditions:

340(�)1 There exists an open cover U of Y such that for each U ∈ U theinverse π −1 (U ) is the disjoint union of open subsets of X each ofwhich is mapped isometrically onto U by π.
(�)2 For each y ∈ Y the fiber π −1 (y) is uniformly discrete in X.
(�)3 ρ(π(x), π(x� )) ≤ d(x, x� ) for any x, x� ∈ X.

Here, a subset A of X is said to be uniformly discrete if there exists anε > 0 such that d(x, y) ≥ ε for any distinct points x, y ∈ A. Note that if
Y is an n-manifold, then so is X and ∂X = π −1 (∂Y ). From the EdwardsKirby local deformation theorem [3] and the Arzela-Ascoli theorem we candeduce the local deformation theorem for uniform embeddings [5, Theorem
1.1].
Theorem 2.5. If π : (M, d) → (N, ρ) is a metric covering projection andN is a compact metric manifold, then (M, d) satisfies the condition (LD).
In term of covering transformations, this theorem corresponds to thecase of free group actions. For the non-free case, we have the followinggeneralization. Recall that an action Φ of a discrete group G on a metricspace X is called geometric if it is proper, cocompact and isometric. (cf.
[1, Chapter I.8]). More generally we say that the action Φ is (i) locallyisometric if for every x ∈ X there exists ε > 0 such that each g ∈ G maps
Oε (x) isometrically onto Oε (gx), and (ii) locally geometric if it is proper,cocompact and locally isometric.
Corollary 2.6. ([6, Theorem 4.1]) A metric manifold (M, d) satisfies thecondition (LD) if it admits a locally geometric group action.
Example 2.7. The Euclidean space Rn with the standard Euclidean metric admits the canonical geometric action of Zn and the associated Riemannian covering projection π : Rn → Rn /Zn onto the flat torus. Therefore,
Rn has the property (LD). From Proposition 2.3 (4) and (3) it follows
that the Euclidean ends Rnr = Rn − Or (0) (r > 0) and the half spaceRn≥0 = {x ∈ Rn | xn ≥ 0} also have the property (LD).

3.

Groups of uniform homeomorphismsuSuppose (X, d) is a metric space and A is a subset of X. Let HA(X, d)
denote the group of uniform homeomorphisms of (X, d) onto itself whichu
fix A pointwise, endowed with the uniform topology. Let HA(X, d)0 denoteu
the connected component of the identity map idX of X in HA(X, d). We

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also consider the subgroupuu
HA(X, d)b = {h ∈ HA(X, d) | d(h, idX ) < ∞}.
uu
It is easily seen that HA(X, d)0 ⊂ HA(X, d)b since the latter is both closed
uand open in HA (X, d).
The group Hu (M, d) is locally contractible if a metric manifold (M, d)satisfies the condition (LD) in Section 2. Hence, our main concern in thissection is in the study of its global deformation property.
The most standard model space Rn has the similarity transformations

kγ : Rn ≈ Rn : kγ (x) = γx(γ > 0).

Conjugation with these similarity transformations enables us to deduce aglobal deformation property for uniform embeddings in the Euclidean ends
Rnr = Rn − Or (0) (r > 0) from the local one. Since this global deformation
property is preserved by bi-Lipschitz equivalence, we can transfer to a moregeneral setting of metric spaces with finitely many bi-Lipschitz Euclideanends.
Recall that a map h : (X, d) → (Y, ρ) between metric spaces is said
to be Lipschitz if there exists a constant C > 0 such that ρ(h(x), h(x� )) ≤Cd(x, x� ) for any x, x� ∈ X. The map h is called a bi-Lipschitz homeomorphism if h is bijective and both h and h−1 are Lipschitz maps. Abi-Lipschitz n-dimensional Euclidean end of a metric space (X, d) means aclosed subset L of X which admits a bi-Lipschitz homeomorphism of pairs,θ : (Rn1 , ∂Rn1 ) ≈ ((L, FrX L), d|L ) and d(X − L, Lr ) → ∞ as r → ∞, where
FrX L is the topological frontier of L in X and Lr = θ(Rnr ) for r ≥ 1. Weset L� = θ(Rn2 ) and L�� = θ(Rn3 ).
Theorem 3.1. ([5, Theorem 1.2]) Suppose X is a metric space and L1 , · · · ,Lm are mutually disjoint bi-Lipschitz Euclidean ends of X. Let L� =L�1 ∪ · · · ∪ L�m and L�� = L��1 ∪ · · · ∪ L��m . Then there exists a strong deformation retraction φ of Hu (X)b onto HLu �� (X) such thatφt (h) = h on h−1 (X − L� ) − L� for any (h, t) ∈ Hu (X)b × [0, 1].
This theorem leads to the following conclusions.
Example 3.2. (1) Hu (Rn )b is contractible for every n ≥ 1. In fact, thereexists a strong deformation retraction of Hu (Rn )b onto HRu n3 (Rn ) and thelatter is contractible by Alexander’s trick.
(2) Suppose N is a compact connected 2-manifold with a nonempty
boundary and C = ∪mi=1 Ci is a nonempty union of some boundary circles ofN . If the noncompact 2-manifold M = N −C is assigned a metic d such thatfor each i = 1, · · · , m the end Li of M corresponding to the boundary circle

342Ci is a bi-Lipschitz Euclidean end of (M, d), then Hu (M, d)0 � HLu �� (M )0 ≈HC (N )0 � ∗.
We close the section with a question on the topological type of thegroup Hu (Rn )b . In [4] we studied the topological type of Hu (R)b as an
infinite-dimensional manifold and showed that it is homeomorphic to �∞ .
Example 3.2 (1) leads to the following conjecture.
Conjecture 3.3. Hu (Rn )b is homeomorphic to �∞ for any n ≥ 1.

References[1] M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, GMW 319,
Springer-Verlag, Berlin, 1999.
[2] A.V. Černavskiı̆, Local contractibility of the group of homeomorphisms of a manifold, (Russian) Mat. Sb. (N.S.), 79 (121) (1969) 307–356.
[3] R. D. Edwards and R. C. Kirby, Deformations of spaces of imbeddings, Ann. ofMath. (2) 93 (1971), 63–88.
[4] K. Mine, K. Sakai, T. Yagasaki and A. Yamashita, Topological type of the group ofuniform homeomorphisms of the real lines, Topology Appl., 158 (2011), 572–581.
[5] T. Yagasaki, Groups of uniform homeomorphisms of covering spaces, to appear in
J. Math. Soc. Japan (Article in Press JMSJ 6460), (arXiv:1203.4028).
[6] T. Yagasaki, On local deformation property for uniform embeddings, preprint,(arXiv:1301.3265).

Graduate School of Science and Technology, Kyoto Institute of Technology
Kyoto, 606-8585, Japan
E-mail: yagasaki@kit.ac.jp