数理解析研究所講究録
第 1922 巻 2014 年 65-77

65

Local and end deformation properties of uniform embeddings in -cones$\kappa$

Tatsuhiko Yagasaki

Kyoto Institute of Technology

1. INTRODUCTION
This article is a continuation of systematic study of topological properties of spaces of
uniform embeddings and groups of uniform homeomorphisms in metric manifolds [2, 4, 5,
8, 9]. Since the notion of uniform continuity and uniform topology depends on the choiceof metrics, it is essential to select reasonable classes of metric manifolds $(M, d)$ .

using Alexander’sJ. M. Kister [4] treated the case of the standard Euclidean spaceof bounded uniform homeomorphisms
trick. This argument implies that the groupendowed with the uniform topology is contractible in a weak sense of [7, Section 5.6].
ofconsidered the case where $M$ is the interior of a compact manifold $N$
A.V.
and the metric is a restriction of some metric on $N$ . Recently, in [8] we considered theclass of metric covering spaces over compact manifolds and obtained a local deformation$\mathbb{R}^{n}$$\mathcal{H}_{b}^{u}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\check{C}ernavski_{\dot{1}}[2]$$d$theorem for uniform embeddings in those spaces. From this local deformation theorem wealso deduced a global deformation result on groups of uniform homeomorphisms of metricmanifolds with finitely many Euclidean ends. As a corollary, it follows thatiscontractible as a topological space.
$\mathcal{H}_{b}^{u}(\mathbb{R}^{n})$This expository article is mainly based on [9], where we study deformation property of
uniform embeddings in the -cone ends
over compact Lipschitz metric manifolds.
of bounded uniform
First we interpret the (local) contractibility of the grouphomeomorphisms of a metric space (X, d) endowed with the uniform topology in termof continuous selection of uniform isotopies (Section 3). This clarifies the meaning of
the weaker notion of (local) contractibility of
due to [7, Section 5.6], which isdefined by continuous selection of level-wise bounded uniform isotopies. We also list up
basic properties of Alexander isotopies on -cones as level-wise bounded isotopies. Local
deformation property (LD) of uniform embeddings was introduced in the earlier versionof [9]. We extend the results to the -cone ends over compact Lipschitz metric manifolds(Section 4). We also introduce the notion of end deformation property (ED) of uniform
embeddings in proper product ends and extend the result on Euclidean ends in [8] to thecase of -cone ends of any compact Lipschitz metric manifolds (Section 5). The preliminary
section 2 is devoted to the basics on metric manifolds and spaces of uniform embeddings.
$(\kappa\leq 0)$$\kappa$$\mathcal{H}_{b}^{u}(X, d)$$\mathcal{H}_{b}^{u}(X, d)$$\kappa$$\kappa$$0$66
2. METRIC MANIFOLDS AND SPACES OF UNIFORM EMBEDDINGSThis section includes basics on metric spaces and uniform embeddings needed in thispaper.

2.1. Metric manifolds.
In this article, maps between topological spaces are always assumed to be continuous.
induced by the metric
As usual, any metric space (X, d) is given the topology
metric space is said to be proper if any bounded closed subset is compact.
let
Suppose (X, d) is a metric space. For $A\subset X$ anddenote the open$X$in and call a uniform-neighborhood of in $(X, d)$ . For $A,$ $B\subset X$ , we write
for someneighborhood of in $X$ if
An -manifold means a topological -manifold $M$ possibly with boundary. The symbols
as a manifold.
and Int $M$ denote the boundary and interior of
A metric -manifold means an -manifold with a fixed compatible metric. A basic(the simply connected complete Riemannian -manifold withmodel is the space-form
for $\kappa=1,$ $-1$ respecconstant sectional curvature ). For example,
is theis the Euclidean -space with the standard Euclidean metrictively. Here,is the standard unit -sphere in
hyperbolic -space anddefined by $\cos d_{1}(x, y)=\langle x,$ , whereis thewith the standard spherical metricfor any
is homothetic toO.
standard Euclidean inner product. Note that$d.$$\tau_{d}$$\epsilon\in[0, \infty]$$A$$\epsilon$$O_{\epsilon}(A)$$B$$A\subset_{u}B$$A$$\epsilon>0.$$O_{\epsilon}(A)\subset B$

$n$

$n$

$\partial M$$\mathbb{J}f$

$n$

$n$

$n$

$M_{\kappa}^{n}$$M_{\kappa}^{n}=\mathbb{S}^{n},$$\kappa$$\mathbb{R}^{n}$$\mathbb{R}^{n},$$0,$$\mathbb{H}^{n}$$d_{0},$$n$$\mathbb{S}^{n}=\{x\in \mathbb{R}^{n+1}|\Vert x\Vert=1\}$$n$$\mathbb{H}^{n}$$\mathbb{R}^{n+1}$$n$$d_{1}$$\langle,$$y\rangle$$\rangle$$\kappa>$$M_{\pm 1}^{n}$$M_{\pm\kappa}^{n}$

Let

$A$

$\mathbb{R}_{\geq 0}^{n}=\{x\in \mathbb{R}^{n}|x_{n}\geq 0\}.$Definition 2.1. A Lipschitz metric -manifold is a metric -manifold $(M, d)$ which satisfies
the following condition:
$n$

$n$

$(\natural$$)$

For each point $x\in M$ there exists an open neighborhood of in $M$ and an open.
such that $(U, d|_{U})$ is bi-Lipschitz homeomorphic tosubset of
$U$

$V$

$x$

$(V, d_{0}|_{V})$$\mathbb{R}_{\geq 0}^{n}$

Any Riemannian manifold with the induced path-length metric is a Lipschitz metricmanifold.

2.2. Spaces of uniform embeddings.
Suppose (X, d) andare metirc spaces. A map : $(X, d)arrow(Y, \rho)$ is said to be
there is a $\delta>0$ such that if $x’\in X$ and(i) uniformly continuous if for each$d(x, x’)<\delta$ then $\rho(f(x), f(x’))<\epsilon$ , (ii) coarsely uniform if for each $R>0$ there is a $S>0$
such that if $x’\in X$ and $d(x, x’)<R$ then $\rho(f(x), f(x’))<S$ , (iii) Lipschitz if there
exists $K>0$ such that $\rho(f(x), f(x’))\leq Kd(x, x’)$ for any $x’\in X.$
A map : $(X, d)arrow(Y, \rho)$ is called a uniform (coarsely uniform, Lipschitz) homeomorare uniformly continuous (coarsely uniform,
phism if is bijective and both and$(Y, \rho)$$f$$\epsilon>0$

$x,$

$x,$

$x,$

$f$

$f$

$f$

$f^{-1}$

67
Lipschitz, respectively). A uniform embedding is a uniform homeomorphism onto its im-age.
$\rho$

Let $C((X, d), (Y, \rho))$ denote the space of continuous maps
on induces the $\sup$ -metric on $C(X, Y)$ defined by
$Y$

$f$

: $(X, d)arrow(Y, \rho)$ . The metric$\tilde{\rho}$$\tilde{\rho}(f, g)=\sup\{\rho(f(x), g(x))|x\in X\}\in[0, \infty].$

The topology on $C((X, d), (Y, \rho))$ induced by the $\sup$ -metric is called the uniform topology. Below the space $C((X, d), (Y, \rho))$ and its subspaces are endowed with the $\sup$ -metricand the uniform topology, unless otherwise specified. For notational simplicity, the metricis denoted by the same symbol . Let $C^{u}((X, d), (Y, \rho))$ denote the subspace consisting$\tilde{\rho}$$\tilde{\rho}$$\tilde{\rho}$$\rho$of uniformly continuous maps : $(X, d)arrow(Y, \rho)$ .
Letdenote the group of homeomorphisms of (X, d) onto itself. This group andits subgroups are endowed with the $\sup$ -metric and the uniform topology. Let
$f$

$h$

$\mathcal{H}(X, d)$$\mathcal{H}_{b}(X, d)$$\{h\in \mathcal{H}(X, d)|d(h, id_{X})<\infty\}$$:=$(the subgroup of bounded homeomorphisms).

Letdenote the subgroup of uniform homeomorphisms of (X, d) onto itself. It
is not necessarily a topological groupis a topological group, while the whole group
under the uniform topology. We also use the following notations:$\mathcal{H}^{u}(X, d)$$\mathcal{H}(X, d)$$\mathcal{H}_{b}^{u}(X, d):=\mathcal{H}^{u}(X, d)\cap \mathcal{H}_{b}(X, d)$,$\mathcal{H}^{u}(X, d;C):=\{h\in \mathcal{H}^{u}(X, d)|h=id on C\}$

denote the open

-neighborhood of, so itis an open (and closed) subgroup ofalso includes. The groupinincludes the connected component of;
the subgroups
for $C\subset X$ , and for $\alpha>0$ let
in. The group$\mathcal{H}^{u}(X, d)$$\mathcal{H}^{u}(id_{X}, \alpha;X, d)$$id_{X}$$\alpha$$\mathcal{H}^{u}(X, d)$$\mathcal{H}_{b}^{u}(X, d)$$id_{X}$$\mathcal{H}^{u}(X, d)$$\mathcal{H}_{b}^{u}(X, d)$$\mathcal{H}_{c}(X, d)\subset \mathcal{H}_{0}(X, d)$supp h : compact{ $h\in \mathcal{H}(X, d)|h$ : asymptotic to id}.

(i)$\mathcal{H}_{c}(X, d)=\{h\in \mathcal{H}(X, d)|$(ii)$\mathcal{H}_{0}(X, d)=$$\},$Here (i) supp h $=cl_{X}\{x\in X|h(x)\neq x\}$ and (ii) is asymptotic to id if for any
there exists a compact subset $K$ of (X, d) such that $d(h(x), x)<\epsilon(x\in X-K)$ .
For subsets $F,$ $C\subset X$ , let $\mathcal{E}^{u}(F, (X, d);C)$ denote the space of uniform embeddings$\epsilon>0$$h$$f:(F, d|_{F})arrow(X, d)$with $f=id$ on $F\cap C$

with the $\sup$ -metric and the uniform topology.
Suppose $M=(M, d)$ is a metric -manifold and $C\subset M$ . An embedding : $Farrow M$
denote the subspace ofis said to be proper if $f^{-1}(\partial M)=F\cap\partial M$ . Let
consisting of proper uniform embeddings and for $\alpha>0$ let.
denote the open -neighborhood of the inclusion map : $F\subset M$ in the space
$n$

$F,$

$f$

$\mathcal{E}_{*}^{u}(F, M;C)$$\mathcal{E}_{*}^{u}(i_{F}, \alpha;F, M;C)$$\mathcal{E}^{u}(F, M;C)$$\alpha$$i_{F}$

Let $\mathcal{E}_{*}^{u}(F, M;C)_{b}=\{f\in \mathcal{E}_{*}^{u}(F, M;C)|d(f, i_{F})<\infty\}.$$\mathcal{E}_{*}^{u}(F, M;C)$68
By deformations of uniform embeddings we mean the following type of homotopies inspaces of uniform embeddings:
is a neighborhood ofDefinition 2.2. Suppose
over $H$ in
An admissible deformation of$\mathcal{W}$$i_{F}$

in$\mathcal{E}_{*}^{u}(F, M;D)$$\mathcal{W}$and $H\subset F,$ $D\subset C.$
means a homotopy$\mathcal{E}_{*}^{u}(F, M;C)$$\varphi:\mathcal{W}\cross[0, 1]arrow \mathcal{E}_{*}^{u}(F, M;D)$which satisfies the following conditions:(i) $\varphi_{0}(f)=f$ , (ii) $\varphi_{1}(f)=id$ on $H,$
, then $\varphi_{t}(f)=id$ on $F\cap\partial M(t\in[0,1$(iii) if $f=id$ on(1) For each$f\in \mathcal{W}$$F\cap\partial M$(2) $\varphi_{t}(i_{F})=i_{F}(t\in[0,1$2.3.

$\kappa$-cones.

First we recall the definition of -cones over metric spaces (cf. [1, Chapter I.5]). The) by collapsing the subset
cone $C(Y)$ over a set is obtained from the set
to one point. The equivalence class of the point $(x, s)\in Y\cross[O, \infty$ ) is denoted by $sx$ . For$sx\in C(Y)$ and $t\in[0, \infty$ ) the scalor multiplication is defined by $t(sx)=(ts)x$ . Let$\kappa$$Y$$Y\cross[O, \infty$$Y\cross\{O\}$$C(Y)_{r} :=\{ty\in C(Y)|t\geq r\} (r\geq 0)$ .

: $Yarrow Y’$ induces the associated function$C(f)$ : $C(Y)arrow C(Y’)$ : $C(f)(ty)=tf(y)$ .
on $Y$ is define by
Suppose $(Y, d)$ is a metric space. The metric

Any function$f$$d_{\pi}$$d_{\pi}(x, y) := \min\{d(x, y), \pi\}(x, y\in Y)$.

over $(Y, d)$ is the metric space, the -cone
Definition 2.3. For
on $C(Y)$ is defined by the following formula:where the metric$\kappa\leq 0$$\kappa$$C_{\kappa}(Y, d)$$(C(Y),\tilde{d}_{\kappa})$,$\tilde{d}_{\kappa}$(i)$for\kappa=0$(ii) for $\kappa<0$ ::$\tilde{d}_{\kappa}(sx, ty)^{2}=s^{2}+t^{2}-2st\cos d_{\pi}(x, y)$,$\cosh(\sqrt{-\kappa}\tilde{d}_{\kappa}(sx, ty))$$=\cosh(\sqrt{-\kappa}s)\cosh(\sqrt{-\kappa}t)-\sinh(\sqrt{-\kappa}s)\sinh(\sqrt{-\kappa}t)\cos d_{\pi}(x, y)$

The -cone ends over $(Y, d)$ mean the subspaces $C_{\kappa}(Y, d)_{r}=(C(Y)_{r},\tilde{d}_{\kappa})(r>0)$ of the.
-cone$\kappa$$C_{\kappa}(Y, d)$$\kappa$

The scalor multiplication$C_{\kappa}(Y, d)\cross[O, \infty$) $\ni(u, t)\mapsto tu\in C_{\kappa}(Y, d)$ is continuous.

Example 2.1. (cf. [1, Proposition 5.8]) Consider the standard unit $(n-1)$ -sphereis isometric towith the standard spherical metric . Then, the -cone$\mathbb{S}^{n-1}$$\mathbb{R}^{n}$for$d_{1}$$\kappa\leq 0$. An isometry$\varphi$:$C_{\kappa}(\mathbb{S}^{n-1})\approx M_{\kappa}^{n}$$\kappa$$C_{\kappa}(\mathbb{S}^{n-1})$

is obtained by$\varphi(ty)=\exp_{0}(ty) (ty\in C_{\kappa}(\mathbb{S}^{n-1}))$,in$M_{\kappa}^{n}$.

69
where is any fixed point ofidentified to the unit sphere$0$$M_{\kappa}^{n},$$\exp_{0}$$S(T_{o}M_{\kappa}^{n})$:is the exponential map andin the tangent space$\mathbb{S}^{n-1}$$T_{o}M_{\kappa}^{n}arrow M_{\kappa}^{n}$is$T_{o}M_{\kappa}^{n}.$The next remark is useful for the unified treatment of the cases $\kappa=0$ and $\kappa<0.$

Remark 2.1. (1) Consider the functions$\lambda_{\kappa}$:$\mathbb{R}arrow \mathbb{R}(\kappa\leq 0)$

defined by$\lambda_{\kappa}(u)=\{\begin{array}{ll}\frac{u}{2} (\kappa=0)\sinh(\frac{\sqrt{-\kappa}}{2}u) (\kappa<0) .\end{array}$

It is seen that$\lambda_{\kappa}$

is a diﬀeomorphism which is a monotonically increasing odd

function. We use the abbreviation:(2) We have the following equalities for the metric$\lambda_{\kappa}^{2}(u)=(\lambda_{\kappa}(u))^{2}.$(i)$\tilde{d}_{\kappa}(\kappa\leq 0)$:$(sx, ty\in C(Y))$$\lambda_{\kappa}^{2}(\tilde{d}_{\kappa}(sx, ty))=\lambda_{\kappa}^{2}(s-t)+\lambda_{\kappa}(2s)\lambda_{\kappa}(2t)\sin^{2}\frac{1}{2}d_{\pi}(x, y)$(ii) (a)$\tilde{d}_{\kappa}(sy, ty)=|s-t|\leq\tilde{d}_{\kappa}(sx, ty)$

, (b)

,

$\lambda_{\kappa}(\tilde{d}_{\kappa}(tx, ty))=\lambda_{\kappa}(2t)\sin\frac{1}{2}d_{\pi}(x, y)$.

: $(Y, d)arrow(Y’, d’)$ is a map between metric spaces. Then, for any
is continuous. If $K\geq 1$ andthe -cone extension $C_{\kappa}(f)\equiv C(f)$ :: $(Y, d_{\pi})arrow(Y’, d_{\pi}’)$ .
: $(Y, d)arrow(Y’, d’)$ is $K$ -Lipschitz, then so is the map
Suppose$\kappa\leq 0$$f$$C_{\kappa}(Y, d)arrow C_{\kappa}(Y’, d’)$$\kappa$$f_{\pi}\equiv f$$f$Lemma 2.1. The following conditions are equivalent.

(1)$f_{\pi}$

is Lipschitz.

(2)$C_{\kappa}(f)$(3)

is Lipschitz.

$C_{\kappa}(f)$is uniformly continuous.

3. UNIFORM ISOTOPIES AND LEVEL-WISE UNIFORM ISOTOPIES
3.1. Uniform isotopies.
Suppose (X, d) is a metric space. An isotopy on $X$ is a homeomorphism
By
which preserves , 1 -factor $(i.e., H(x, t)=(H_{t}(x), t)((x, t)\in X\cross[O,$consisting of all isotopies on $X$ . If $H$ is an
we denote the subgroup of
isotopy on $X$ , thenfor each $t\in[0$ , 1 and we obtain the associated function$[0, 1]arrow \mathcal{H}(X)$$\hat{H}(t)=H_{t}.$::The product space $X\cross[0$ , 1 is given the metric defined by$H\in \mathcal{H}(X\cross[O, 1])$$[0$$\mathcal{H}(X\cross[O, 1])^{I}$

$1$

$]$

$\mathcal{H}(X\cross[0,1])$$H_{t}\in \mathcal{H}(X)$$]$$\hat{H}$$\tilde{d}$$]$$\tilde{d}((x, t), (y, s))=d(x, y)+|t-s| ((x, t), (y, s)\in X\cross[O, 1$

Definition 3.1. An isotopy $H$ on $X$ is said to be(i)(bounded) uniform isotopy on (X, d) if(ii) a level-wise bounded (uniform) isotopy on (X, d) if$H\in \mathcal{H}_{(b)}^{u}(X\cross[O, 1],\tilde{d})$$a$,$H_{t}\in \mathcal{H}_{b}^{(u)}(X, d)$for each $t\in$$[0, 1 ].$denote the subgroups ofconsisting of (bounded) uniform isotopies and level-wise bounded (uniform) isotopies on$(X, d)$ , endowed with the $\sup$ -metric and the uniform topology defined byLet$\mathcal{H}_{(b)}^{u}(X\cross[O, 1],\tilde{d})^{I}$

and$\mathcal{H}^{lb(u)}(X\cross[O, 1],\tilde{d})^{I}$$\mathcal{H}(X\cross[O, 1])^{I}$$\tilde{d}.$70Uniform isotopies are reinterpreted correctly as continuous paths in$\mathcal{H}^{u}(X, d)$.

Lemma 3.1.
(1) If $H$ is (bounded) uniform isotopy on $(X, d)$ , then $H$ is a level-wise (bounded), $\hat{H}(t)=H_{t}$ is continuous.
uniform isotopy and the function :induces (bounded) uniform isotopy on (X, d) by(2) Any map$a$$\hat{H}$$[0, 1]arrow \mathcal{H}_{(b)}^{u}(X, d)$$\tilde{h}$$h:[0, 1]arrow \mathcal{H}_{(b)}^{u}(X, d)$$a$$\tilde{h}(x, t)=(h(t)(x), t) ((x, t)\in X\cross[O, 1$(3) There exists a natural isometry$\eta$: $(\mathcal{H}_{(b)}^{u}(X\cross[0,1],\tilde{d})^{I}, d)arrow(C([0,1], (\mathcal{H}_{(b)}^{u}(X, d), d))_{u}, d):\eta(H)=\hat{H},$$\eta^{-1}(h)=\tilde{h}.$From this lemma we obtain the following exponential law.
Proposition 3.1. For any topological space$(C(Z\cross[0,1], (\mathcal{H}_{(b)}^{u}(X, d), d d)$$\vec{\cong}$$Z$

there exist natural isometries$(C(Z, (C([0,1], (\mathcal{H}_{(b)}^{u}(X, d), d))_{u}, d d)$$\frac{ノ\eta_{\#}}{\backslash \cong} (C(Z, (\mathcal{H}_{(b)}^{u}(X\cross[0,1],\tilde{d})^{I}, d d)$.

.
3.2. Local contractibility ofIn the literature some authors introduced weaker notions of local contractibility of homeomorphism groups of noncompact manifolds (cf. [7, Section 5.6]). (One should note thatthese weaker notions are also denoted by the same terminology “local contractibility” andthis ambiguity might lead to some misunderstanding.) In this subsection, for the group$\mathcal{H}^{u}(X, d)$we clarify the relationship between the weaker notion and the standard notionof local contractibility.
First note that a topological group is locally contractible if and only if a neighborhoodof the unit element of is contractible in . Thus, Proposition 3.1 induces the followinginterpretation in term of continuous selection of uniform isotopies.
$\mathcal{H}_{b}^{u}(X, d)$

$G$

$G$

$G$

is locally contractible if and only if there exists a neighborin
and a map :such that for eachis a uniform isotopy from toin (1).
is contractible if and only if we can take

Corollary 3.1. (1)hood ofthe image(2)$\mathcal{U}$$id_{X}$$\mathcal{H}_{b}^{u}(X, d)$$\mathcal{H}_{b}^{u}(X, d)$$\Phi$$h$$\Phi(h)$$\mathcal{H}_{b}^{u}(X, d)$$\mathcal{U}arrow \mathcal{H}_{b}^{u}(X\cross[0,1],\tilde{d})^{I}$$h\in \mathcal{U}$$id_{X}.$$\mathcal{U}=\mathcal{H}_{b}^{u}(X, d)$

The general formulation in [7, Section 5.6] leads to the following definition for$\mathcal{H}_{b}^{u}(X, d)$:is locally contractible if there exist a neighborDefinition 3.2. We say thatinand a map
hood of$\mathcal{H}_{b}^{(u)}(X, d)$$\mathcal{U}$$id_{X}$$\mathcal{H}_{b}^{(u)}(X, d)$$\Phi:\mathcal{U}arrow \mathcal{H}^{lb(u)}(X\cross[0,1],\tilde{d})^{I}$

$*$

71
is a level-wise bounded (uniform) isotopy from tothe imagesuch that for each,
. If we can take
. The map is called a local contraction of.
is contractible and call the map a contraction ofthen we say that$h\in \mathcal{U}$$h$$\Phi(h)$$*$$\Phi$$id_{X}$$*$$*$$\mathcal{H}_{b}^{(u)}(X, d)$$\mathcal{U}=\mathcal{H}_{b}^{(u)}(X, d)$$\mathcal{H}_{b}^{(u)}(X, d)$$\Phi$$\mathcal{H}_{b}^{(u)}(X, d)$is (locally) contractible*.
Obviously, if
is (locally) contractible, thenIn the next subsection we see that the Alexander isotopies in a -cone yield a contraction*,but they do not induce a continuous contraction in general.
$\mathcal{H}_{b}^{u}(X, d)$$\mathcal{H}_{b}^{u}(X, d)$$\kappa$

3.3. Alexander isotopies in -cones.
In this section we discuss basic properties of Alexander isotopies in the -cone $C_{\kappa}(X, d)=$onover any compact metric space (X, d) (cf. [4]). The metric,1
onand the metricinduces the $\sup$-metricon$\kappa$$\kappa$$(C(X),\tilde{d}_{\kappa})(\kappa\leq 0)$$\tilde{d}_{\kappa}$$d_{\kappa}\approx$$C_{\kappa}(X, d)$$\tilde{d}_{\kappa}$

The -cone$\kappa$$C_{\kappa}(X, d)$$].$admits the radial transformations$\theta_{t}\in \mathcal{H}(C_{\kappa}(X, d))$For each$C_{\kappa}(X, d)\cross[O$$\mathcal{H}(C_{\kappa}(X, d))$:$\theta_{t}(u)=tu$$(t\in(O,$$\infty$

we obtain$h\in \mathcal{H}_{b}(C_{\kappa}(X, d))$$h_{t}\in \mathcal{H}_{b}(C_{\kappa}(X, d))(t\in[O, 1])$:$h_{t}=\{\begin{array}{ll}\theta_{t}h(\theta_{1/t}) (t\in(0,1id (t=0) ,\end{array}$and the Alexander isotopy for ;$h$$\Phi(h):C_{\kappa}(X, d)\cross[0, 1]arrow C_{\kappa}(X, d)\cross[0, 1 ]$:$\Phi(h)(u, t)=(h_{t}(u), t)=\{\begin{array}{ll}(th(\frac{1}{t}u), t) (t\in(O, 1(u, 0) (t=0) ,\end{array}$which is a level-wise bounded isotopy on:
The function$\eta_{h}$$C_{\kappa}(X, d)$$[0, 1]arrow \mathcal{H}_{b}(C_{\kappa}(X, d))$from id to$h.$: $\eta_{h}(t)=h_{t}$ is always continuous at $t=0.$

Lemma 3.2. The isotopy
is a uniform isotopy on
continuous and $h_{t}\in \mathcal{H}_{b}^{u}(C_{\kappa}(X, d))(t\in[0,1 In$ this case,$\Phi(h)$$C_{\kappa}(X, d)$$\Phi(h)$if and only if$\eta_{h}$isis bounded.

We obtain an isometric embedding$\Phi:\mathcal{H}_{b}(C_{\kappa}(X, d))\ni h\mapsto\Phi(h)\in \mathcal{H}^{lb}(C_{\kappa}(X, d)\cross[O, 1], d_{\kappa})^{I}\approx,$

which forms a contraction of$*$$\Phi$:$\mathcal{H}_{b}(C_{\kappa}(X,$$d$(For $\kappa=0$ , it restricts to a contraction*
However, the associate functions$\mathcal{H}_{b}^{u}(C_{0}(X, d))arrow \mathcal{H}^{lbu}(C_{0}(X, d)\cross[0,1],d_{0})^{I}.)\approx$$\varphi$: $\mathcal{H}_{b}(C_{\kappa}(X, d))\cross[O, 1]arrow \mathcal{H}_{b}(C_{\kappa}(X, d \varphi(h, t)=h_{t}$and :are not continuous for any compact Lipschitz
metric manifold (X, d) as shown in the examples below.
$\varphi$$\mathcal{H}_{b}^{u}(C_{0}(X, d))\cross[O, 1]arrow \mathcal{H}_{b}^{u}(C_{0}(X, d))$Example 3.1. For any non-zero vector(i)$v\in \mathbb{R}^{n}$$h((2k+1)v)=(2k+1)v(k\in \mathbb{N})$

we can findand$h\in \mathcal{H}_{b}^{u}(\mathbb{R}^{n})$such that

72(ii) there exists $c>0$ such that $d(h(2kv), 2kv)>c(k\in \mathbb{N})$ .

For any such , the function$h$is not continuous at $t=1.$$\eta_{h}$is continuous at $t=1$ . Then, there exists $t_{0}\in[0$ , 1)
, there exists
such
such that $d(h_{t}, h)<c$ for any $t\in(t_{0},1].$ Since. It follows that
that

Proof Assume the contrary, that$\eta_{h}$$k\in \mathbb{N}$$\frac{2k}{2k+1}arrow 1(karrow\infty)$$t:= \frac{2k}{2k+1}>t_{0}$$h_{t}(2kv)=th( \frac{1}{t}2kv)=th((2k+1)v)=t(2k+1)v=2kv$

so that$d(h_{t}, h)\geq d(h_{t}(2kv), h(2kv))=d(2kv, h(2kv))>c.$

This contradicts $d(h_{t}, h)<c.$$\square$This argument extends to the -cone case.
$\kappa$

Example 3.2. For any compact Lipschitz metric manifold (X, d) there exists
is not continuous at $t=1.$
for which the function$h\in \mathcal{H}_{b}^{u}(C_{\kappa}(X, d))$$\eta_{h}$One can expect that the function restricts to continuous contractions of some subgroups ofConsider the following subgroup of$\varphi$$\mathcal{H}_{b}^{u}(C_{\kappa}(X,$$d$$\mathcal{H}_{b}^{u}(C_{\kappa}(X,$is a uniform isotopy$:=\{h\in \mathcal{H}_{b}^{u}(C_{\kappa}(X, d))|\Phi(h)$$\mathcal{G}=\mathcal{G}(C_{\kappa}(X, d))$$d$Proposition 3.2.

(i) The function restricts to a contraction :in the following sense:(ii) The subset is maximal in:is continuous,
and
If
then
are normal subgroups of and the contracand(2) Bothtion of restricts to the contractions of these subgroups.
(1)$\mathcal{H}_{b}^{u}(C_{\kappa}(X, d))$$\mathcal{G}$$(\natural$$S\subset \mathcal{H}_{b}^{u}(C_{\kappa}(X, d))$$)$$\mathcal{G}\cross[0, 1]arrow \mathcal{G}.$$\varphi$$\varphi$$\varphi$$S\cross[O, 1]arrow \mathcal{H}^{u}(C_{\kappa}(X, d))$$\mathcal{S}\subset \mathcal{G}.$$\mathcal{H}_{c}(C_{\kappa}(X, d))$$\mathcal{H}_{0}(C_{\kappa}(X, d))$$\mathcal{G}$$\mathcal{G}$$\varphi$

In particular, in the case.
traction of$\mathbb{R}^{n}=C_{0}(\mathbb{S}^{n-1})$, the function$\varphi$

restricts to a continuous con-$\mathcal{H}_{c}(\mathbb{R}^{n})$denote the group
Remark 3.1. Let
topology. For $n\geq 2$ , the function :$\mathcal{H}_{c}(\mathbb{R}^{n})_{co}$$\varphi$$(id_{R^{n}}, 0)$$\mathcal{H}_{c}(\mathbb{R}^{n})$endowed with the compact-open
is not continuous at$\mathcal{H}_{c}(\mathbb{R}^{n})_{co}\cross[0, 1]arrow \mathcal{H}_{c}(\mathbb{R}^{n})_{co}$.

is contractible in the usual sense. There we adopted theIn [8] we showed that
admits a metric covering projection onto the flat torus,
following strategy: (1) Sinceof a small open ballhas the property (LD) and there exists a contraction(see Section 4). (2) Given $\alpha>0$ , using and a similarity transformation , weinin
. (3) Thecan find $\beta>0$ and a contraction of the open ballinofiteration of (2) yields a sequence of contractions$\mathcal{H}_{b}^{u}(\mathbb{R}^{n})$$\mathbb{R}^{n}$$\mathbb{R}^{n}$$\mathcal{H}^{u}(id, \epsilon;\mathbb{R}^{n})$$\chi_{t}$$\mathcal{H}_{b}^{u}(\mathbb{R}^{n})$$\theta_{\gamma}$$\chi_{t}$$\psi_{t}$$\mathcal{H}^{u}(id, \alpha;\mathbb{R}^{n})$$\psi_{t}^{i}$$\mathcal{H}^{u}(id, \beta;\mathbb{R}^{n})$$\mathcal{H}^{u}(id, \alpha_{i};\mathbb{R}^{n})$$\mathcal{H}^{u}(id, \alpha_{i+1};\mathbb{R}^{n})$73
for some increasing sequence. A contraction ofis obtained bycomposing these contractions . In this argument, we need not change the scale factorof the similarity transformation continuously.
$\alpha_{i}\in \mathbb{R}(i\in \mathbb{N})$$\mathcal{H}_{b}^{u}(\mathbb{R}^{n})$$\psi_{t}^{i}$$\gamma$$\theta_{\gamma}$

4. LOCAL DEFORMATION PROPERTY OF UNIFORM EMBEDDINGSIn this section we summarize basic facts on local deformation property of uniform em-

beddings (LD) and discuss the case of -cone ends
over compact Lipschitz metricmanifolds. Throughout this section $M=(M, d)$ denotes a metric manifold.
$(\kappa\leq 0)$$\kappa$

4.1. Definition and basic facts.
Definition 4.1.
(1) An admissible tuple in $M$ is a tuple $(X, W’, W, Z, Y)$ such that $X\subset_{u}W’\subset W\subset M$
and $Z\subset_{u}Y\subset M$ . Let
denote the collection of all admissible tuples in $M.$
(2) An admissible deformation for $\mathcal{X}=(X, W’, W, Z, Y)\in S(M)$ is an admissible
deformation$\mathcal{S}(M)$:$\varphi$of a neighborhood
such that for each$\mathcal{W}$of$\mathcal{W}\cross[0, 1]arrow \mathcal{E}_{*}^{u}(W, lI4i;Z)$$i_{W}$

in$\mathcal{E}_{*}^{u}(W, M;Y)$$(f, t)\in \mathcal{W}\cross[0$(i) $\varphi_{t}(f)=f$ on $W-W’$,1over $X$ (in the sense of Definition 2.2)

$]$

and

(ii) $\varphi_{t}(f)(W)=f(W)$ .

Definition 4.2. (LD)(1) For(2) For $A\subset M$$\mathcal{X}\in S(M)$,$(LD)_{M}(\mathcal{X})\Leftrightarrow\exists$

an admissible deformation for$\mathcal{X}.$$A:(LD)_{M}$for any $\mathcal{X}=(X, W’, W, Z, Y)\in S(M)$ with $X\subset A$: $(LD)_{M}\Leftrightarrow(LD)_{M}(\mathcal{X})$ for any$\Leftrightarrow(LD)_{M}(\mathcal{X})$(3) $M$ : (LD)$\Leftrightarrow M$$\mathcal{X}\in S(M)$Basic Example 4.1. The local deformation theorem by A.V.
R. D. Edwards-R. C. Kirby ([3, Theorem 5.1]) is restated as follows:$\check{C}ernavski_{\dot{1}}$$(*)$

Any relatively compact subset $K$ of $M$ has the property $(LD)_{M}.$The condition (LD) has the following basic properties.

Proposition 4.1.
(1) Suppose $h:(\mathbb{J}l, d)\approx(lII’, d’)$ is a uniform homeomorphism. Then(i) for any $A\subset M,$(ii) $M$ : (LD)$\Leftrightarrow$(2)$A:(LD)_{M}$$M’$$\Leftrightarrow$$h(A):(LD)_{M}$ ,and: (LD)(i) Suppose $A\subset B\subset M$ . Then $B:(LD)_{M}$$\Rightarrow$$A:(LD)_{M}.$([2]) and74(ii) Suppose$A$$A\subset_{u}N\subset\Lambda 4$

and $N$ is an -manifold. Then,: $(LD)_{(N,d|_{N})}$$\Leftrightarrow$

$n$

$A$

: $(LD)_{M}.$(3) Suppose $A\subset_{u}U\subset M$ and $B\subset M$ . Then$U,$

$B:(LD)_{M}$$\Rightarrow$$A\cup B:(LD)_{M}.$(4) Suppose $K$ is a relatively compact subset of $M$ . Then, for any $A\subset M$$A\cup K$ : $(LD)_{M}.$: $(LD)_{M}$$A$$\Leftrightarrow$(5) Suppose $M=K \cup\bigcup_{i=1}^{m}L_{i},$ $K$ is relatively compact, eachclosed in $M$ , and $d(L_{i}, L_{j})>0$ for any $i\neq j$ . Then,$M$: (LD)(6) If $M$ : (LD), then$\Leftrightarrow$$(L_{i}, d|_{L_{i}})$$\mathcal{H}^{u}(M)$and$L_{i}$

is an -manifold and$n$: (LD) $(i=1, \cdots, m)$ .

$\mathcal{H}^{u}(M, \partial M)$

are locally contractible.

4.2. Examples.
In this subsection we list some basic examples of metric manifolds with the property(LD). This enables us to deduce deformation results in more complicated metric manifolds
from those in simpler pieces.
[1] Metric covering spaces and geometric group actions
The notion of metric covering spaces was introduced in [8] as a natural metric versionof Riemannian coverings in the smooth category.

Definition 4.3. A map : $(X, d)arrow(Y, \rho)$ is called a metric covering projection if itsatisfies the following conditions:$\pi$

the inverse $\pi^{-1}(U)$ is
There exists an open cover of such that for eachthe disjoint union of open subsets of $X$ each of which is mapped isometrically ontobyis uniformly discrete in $X.$
For each $y\in Y$ the fiber$x’\in X.$$(*)_{3}\rho(\pi(x), \pi(x’))\leq d(x, x’)$ for any$\mathcal{U}$$(*)_{1}$

$U$

$Y$

$U\in \mathcal{U}$$\pi.$$\pi^{-1}(y)$$(*)_{2}$$x,$The next theorem follows from Basic Example 4.1 and the Arzela-Ascoli theorem.

Theorem 4.1. If : $(M, d)arrow(N, \rho)$ is a metric covering projection and $(N, \rho)$ is acompact metric manifold, then $(M, d)$ has the property (LD).
$\pi$This result extends to the case of metric manifolds with geometric group actions. An
action of (discrete) group $G$ on alocally compact metric space (X, d) is called geometric$\Phi$$a$if it is proper, cocompact and isometric (cf. [1, Chapter I.8]). We work in a slightly moregeneral setting.

Definition 4.4. The action$\Phi$

of

$G$

on

$X$

is said to be(1) locally isometric if for every $x\in X$ there exists
isometrically onto
each $g\in G$ maps$\epsilon\in(0, \infty$$(\natural)_{x}$$O_{\epsilon}(x)$] such that

$O_{\epsilon}(gx)$

, and

75
(2) locally geometric if it is proper, cocompact and locally isometric.

Theorem 4.2. A metric manifold has the property (LD) if it admits a locally geometricgroup action.
[2] The -cone ends
over compact Lipschitz metric manifolds
The property (LD) for a metric manifold depends only on the property (LD) for its ends
(Proposition $4.1(5)$ ). This observation leads us to study of the property (LD) for typical$(\kappa\leq 0)$$\kappa$

metric ends.
The Euclidean space, the hyperbolic space
and the cylinder over any compact
metric manifold have the property (LD), since they admit free geometric group actions(Theorem 4.2). Hence their ends also have the property (LD). This implies that thecone ends
have the property (LD). By Proposition 4.1 (1), (3) thisgeneralizes to the following form.
$\mathbb{R}^{n}$$\mathbb{H}^{n}$$\kappa-$$C_{\kappa}(\mathbb{S}^{n-1})_{1}(\kappa\leq 0)$

Theorem 4.3. Suppose $(N, d)$ is a compact Lipschitz metric manifold. Then for any

the -cone end$\kappa$$C_{\kappa}(N, d)_{1}$$\kappa\leq 0$

has the property (LD).

5. END DEFORMATION PROPERTY OF UNIFORM EMBEDDINGSIn this section we introduce the notion of end deformation property of uniform embeddings (ED) in proper product ends and study its basic feature.

5.1. Definition and basic facts.
Definition 5.1. An -dimensional proper product end is a metric -manifold $(L, d)$ such)that (i) the metric is proper and (ii) there exists a homeomorphism :for some compact $(n-1)$ -manifold $S.$
$n$

$n$

$d$

$\theta$

Suppose $(L, d)$ is a proper product end. A subset $F$ of$L$$S\cross[1, \infty$$\approx L$

is said to be cofinal if $L-F$

we denote the collection of cofinal closed subsets of$:=\theta(S\cross I)$ for. We fix a homeomorphism)):
and letandfor $r\geq 1$ . Note that (i) $F\subset L$ is cofinal if and only if
for some$r\in[1, \infty)$ and (ii) $d(L_{\{1\}}, L_{r})arrow\infty(rarrow\infty)$ since the metricis proper.
is relatively compact in

$L$

. By

$L$

$C\mathcal{F}(L)$$\theta$$L_{r}$$S\cross[1, \infty$$\approx L$$I\subset[1, \infty$$L_{I}$$L_{r}\subset F$$:=L_{[r,\infty)}$$d$Definition 5.2. Suppose $(L, d)$ is a proper product end. We say that $(L, d)$ has theproperty (ED) and write $(L, d)$ : (ED) if the following condition is satisfied:$(*)$

For any
with $H\subset F$ andand any $\alpha>0$ there exist $\beta>0,$
an admissible deformation over $H$ (in the sense of Definition 2.2)
$F\in C\mathcal{F}(L)$$\varphi$such that:$H\in C\mathcal{F}(L)$$\mathcal{E}_{*}^{u}(i_{F}, \alpha, F, (L, d))\cross[0_{\}}1]arrow \mathcal{E}_{*}^{u}(i_{F}, \beta;F, (L, d))$76$(*)_{1}\varphi_{t}(f)=f$on

for each$Fr_{L}F$

Remark 5.1. The deformation$\varphi$

,1

$].$

in Definition 5.2 also satisfies the following condition:for each$(*)_{2}\varphi_{t}(f)(F)=f(F)$$(f, t)\in \mathcal{E}_{*}^{u}(i_{F}, \alpha;F, (L, d))\cross[0$$(f, t)\in \mathcal{E}_{*}^{u}(i_{F}, \alpha;F, (L, d))\cross[0$

,1

$].$

Proposition 5.1. Suppose $(L, d)$ , $(L’, d’)$ are proper product ends and $h:(L, d)\approx(L’, d’)$is a uniform, coarsely uniform homeomorphism. If $(L, d)$ : (ED), then so is $(L’,$$d$Example 5.1. Suppose $(N, d)$ is a compact metric manifold. If $C_{0}(N, d)_{1}$ : (LD), then: (ED). In particular, the -cone end $C_{0}(N, d)_{1}$ over any compact Lipschitz
metric manifold $(N, d)$ has the property (ED).
$0$$C_{0}(N, d)_{1}$5.2. End deformation theorem for uniform embeddings.
Suppose $(M, d)$ is a metric -manifold. A proper product end of $M$ is a closed subset of$M$ such that $Fr_{M}L$ is compact and $(L, d|_{L})$ is a proper product end. Suppose
is a proper. Let
product end of $M$ . We say that the end is isolated if $d(M-L, L_{r})arrow\infty$ as.
denote the collection of closed subsets of $M$ such that
If a metric manifold $M$ has finitely many isolated proper product ends with the property(ED), then the spaces of uniform embeddings in $M$ admit strong deformation retractionsof the following form:
$L$

$n$

$L$

$L$

$rarrow\infty$$F$$C\mathcal{F}(\Lambda l, L)$$F\cap L\in C\mathcal{F}(L)$$L(m)\subset M$ are pairwise
Theorem 5.1. Suppose $(M, d)$ is a metric manifold and $L(1)$ ,$m,$disjoint isolated proper product ends of Al. If $(L(i), d|_{L(i)})$ : (ED) for each $i=1,$
then for any $F\subset M$ with $F\in C\mathcal{F}(M, L(i))(i=1, \cdots , m)$ and any $s_{i}>r_{i}>1$ with
$L(i)_{r_{i}}\subset Int_{M}(F\cap L(i))(i=1, \cdots, m)$ there exists a strong deformation retraction$\cdots,$$\cdots,$$\varphi$of$\mathcal{E}_{*}^{u}(F, M)_{b}$onto$\mathcal{E}_{*}^{u}(F, M;\bigcup_{i=1}^{m}L(i)_{s_{i}})_{b}$such that(1) for each$(f, t)\in \mathcal{E}_{*}^{u}(F, M)_{b}\cross[0$

,1

$]$

on $f^{-1}(M$ –Int, then $\varphi_{t}(f)(F)=f(F)$ ,(ii) ifon, then(iii) if $f=id$ on(i)$M( \bigcup_{i=1}^{m}L(i)_{r_{i}}))-Int_{M}(\bigcup_{i=1}^{m}L(i)_{r_{t}})$$\varphi_{t}(f)=f$,$\bigcup_{i=1}^{m}L(i)_{r_{i}}\subset f(F)$$F\cap\partial M$$\varphi_{t}(f)=id$$F\cap\partial M.$(2) for each $\alpha>0$ there exists $\beta>0$ such that$\varphi(\mathcal{E}_{*}^{u}(i_{F}, \alpha;F, M)\cross[0,1])\subset \mathcal{E}_{*}^{u}(i_{F}, \beta;F, M)$.

As a corollary we have the following deformation theorem for uniform homeomorphisms.
$L(m)\subset M$ are pairwise
Corollary 5.1. Suppose $(M, d)$ is a metric manifold and $L(1)$ ,disjoint isolated proper product ends of $M$ such that $(L(i), d|_{L(i)})$ has the property (ED)$m$ . Let $L_{r}=L(1)_{r}\cup\cdots\cup L(m)_{r}(r\geq 1)$ . Then there
and $L(i)_{2}\subset Int_{M}L(i)$ for $i=1,$$\cdots,$$\cdots,$

exists a strong deformation retraction$\varphi$of$\mathcal{H}_{b}^{u}(M)$onto$\mathcal{H}_{b}^{u}(\Lambda I;L_{3})$such that

77(1) for each$(h, t)\in \mathcal{H}_{b}^{u}(M)\cross[0$

,1

$]$

(i)

$\varphi_{t}(h)=h$on(ii) if $h=id$ on$h^{-1}$$\partial M$( $M$ –Int${}_{M}L_{2}$) –Int, then $\varphi_{t}(h)=id$ on${}_{M}L_{2},$$\partial M,$(2) for each $\alpha>0$ there exists $\beta>0$ such that$\varphi(\mathcal{H}^{u}(id_{M}, \alpha;M)\cross[O, 1])\subset \mathcal{H}^{u}(id_{M}, \beta;lII)$.

We close this article with a problem which requires further investigation.
Problem 5.1. Examine the property (ED) for the -cone ends $C_{\kappa}(M)_{1}(\kappa<0)$ overtypical compact Riemannian manifolds $M.$$\kappa$

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[7] T.B. Rushing, Topological embeddings, Academic Press, New York, 1973.
[8] T. Yagasaki, Groups of uniform homeomorphisms of covering spaces, to appear in J. Math. Soc. Japan(arXiv: 1203. 4028v2).
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GRADUATE SCHOOL OF SCIENCE AND TECHNOLOGY,KYOTO INSTITUTE OF TECHNOLOGY,
KyoTo, 606-8585, JAPAN-mail address: yagasaki@kit.ac.jp$E$F$\yen\beta$エt$\equiv\theta$$|\Phi$#/$\grave{}\not\in*\not\cong$エJ$\grave{}\grave{}$$\ovalbox{\tt\small REJECT}$$\equiv$$\mp^{arrow^{\backslash \backslash }}\Re_{J\iota}^{\pi}N^{\backslash }\backslash$$*\nearrow_{7^{\wedge}}\# 5_{p}\grave{J}\ovalbox{\tt\small REJECT} P//$