Lipschitz Homotopy Convergence of Alexandrov Spaces

Abstract

We introduce the notion of good coverings of metric spaces, and prove that if a metric space admits a good covering, then it has the same locally Lipschitz homotopy type as the nerve complex of the covering. As an application, we obtain a Lipschitz homotopy stability result for a moduli space of compact Alexandrov spaces without collapsing.

1 Introduction

For given n and $D, v_0>0$, let ${\mathcal {A}}(n,D,v_0)$ denote the set of isometry classes of compact n-dimensional Alexandrov spaces with curvature $\ge -1$, diameter $\le D$, and volume $\ge v_0$. Perelman’s stability theorem has played important roles in the geometry of Alexandrov spaces with curvature bounded below. This theorem implies that the set of homeomorphism classes of spaces in ${\mathcal {A}}(n,D,v_0)$ is finite. Although he also claimed the Lipschitz version of the stability theorem is true, it has not yet been appeared.

We formulate our results for general metric spaces having good coverings. We say that a locally finite open covering of a metric space is good if any non-empty intersection in the covering has a Lipschitz strong deformation retraction to a point (see Definition 2.6 for the detail).

We use a symbol $\tau (\epsilon _1, \epsilon _2,\dots , \epsilon _k)$ to denote a positive continuous function satisfying $\lim _{\epsilon _1, \epsilon _2, \dots , \epsilon _k \rightarrow 0} \tau (\epsilon _1, \epsilon _2, \dots , \epsilon _k) = 0$.

The main theorems of the present paper are stated as follows.

Theorem 1.1

Let M be a $\sigma$-compact metric space having a good covering $\mathcal U$. Then M has the same locally Lipschitz homotopy type as the nerve of $\mathcal U$.

We remark that in Theorem 1.1 if M is compact, it has the same Lipschitz homotopy type as the nerve of $\mathcal U$.

Theorem 1.2

There exists a positive number $\epsilon =\epsilon _n(D,v_0)$ such that if $M,M'\in {\mathcal {A}}(n,D,v_0)$ have the Gromov–Hausdorff distance $d_{GH}(M,M') <\epsilon$, then M has the same Lipschitz homotopy type as $M'$. More precisely if $\theta :M\rightarrow M'$ is an $\epsilon$-approximation, then there is a Lipschitz homotopy equivalence $f:M\rightarrow M'$ satisfying that $|f(x),\theta (x)|<\tau (\epsilon )$ for all $x\in M$.

As a direct consequence of Theorem 1.2, we have.

Corollary 1.3

The set of Lipschitz homotopy types of Alexandrov spaces in ${\mathcal {A}}(n,D,v_0)$ is finite.

This provides a weaker version of “the finiteness of bi-Lipschitz homeomorphism classes” mentioned above.

In Corollary 1.3 we prove that every M and $M'$ in ${\mathcal {A}}(n,D,v_0)$ with small Gromov–Hausdorff distance have the same Lipschitz homotopy type through isomorphic nerves of some good coverings on them. However it was shown in [1] and [14] that there is an almost isometric map from a closed domain of an almost regular part of M to a closed domain of an almost regular part of $M'$. John Lott asked us if one can extend such an almost isometric map to a Lipschitz homotopy equivalence $M\rightarrow M'$. The answer is yes:

Theorem 1.4

Let $\delta$ be a sufficiently small positive number with respect to n. For given compact n-dimensional Alexandrov space M with curvature $\ge -1$ and a closed domain D in the $\delta$-regular part of M, there exists an $\epsilon =\epsilon _{M,D}>0$ satisfying the following: Let $M'$ be a compact n-dimensional Alexandrov space with curvature $\ge -1$ and with $d_{GH}(M,M')<\epsilon$, and let $\theta :M\rightarrow M'$ be an $\epsilon$-approximation. Then there is a Lipschitz homotopy equivalence $f:M\rightarrow M'$ such that

• (1) the restriction of f to D is $\tau (\epsilon )$-almost isometric;

• (2) $|f(x),\theta (x)|<\tau (\epsilon )$ for all $x\in M$.

Theorem 1.1 has an application to the set of homotopies of mapping between two metric spaces. Let [X, Y] denote the set of all homotopy classes of continuous maps from X to Y, and $[X,Y]_{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}$ the set of all locally Lipschitz homotopy classes of locally Lipschitz maps from X to Y. In Corollary 1.3 of [7], we proved that if K is a simplicial complex and Y is a locally Lipschitz contractible metric space, then the natural map $[K,Y]_{{\mathrm {loc}}\text {-}{\mathrm {Lip}}} \rightarrow [K, Y]$ is bijective.

Using Theorem 1.1 and Corollary 1.3 of [7], we obtain the following.

Corollary 1.5

Let X be a $\sigma$-compact metric space admitting a good covering, and Y a locally Lipschitz contractible metric space. Then, the natural map $[X,Y]_{{\mathrm {loc}}\text {-}{\mathrm {Lip}}} \rightarrow [X,Y]$ is bijective.

In particular, every continuous map from X to Y is homotopic to a locally Lipschitz one.

As an immediate consequence of Corollary 1.5, we have the following for instance.

Corollary 1.6

Let X be a finite-dimensional compact Alexandrov space with curvature bounded below, and Y a locally Lipschitz contractible metric space. Then every continuous map from X to Y is homotopic to a Lipschitz map.

Organization The rest of the present paper consists of Sects. 2–6. In Sect. 2, we recall the notions of Lipschitz homotopies, Alexandrov spaces and good coverings needed in this paper. Sections 3 and 4 are devoted to prove Theorem 1.1, where we employ a basic strategy in the proof of Theorem 9.4.15 of [13]. Since the argument in [13] is only topological, we need to proceed in the category of (locally) Lipschitz maps. In Sect. 3, we consider the case when metric spaces are compact, and deal with the non-compact case in Sect. 4. Using Theorem 1.1 and a stability result of nerves of good coverings in [9], we prove Theorem 1.2 and Corollaries 1.3 and 1.5 in Sect. 5. In Sect. 6, we prove Theorem 1.4 by developing a gluing method in [1].

2 Preliminaries

In this paper, the distance between two points x, y in a metric space is denoted by |xy| or |x, y|. The open metric ball around x of radius r is denoted by B(x, r). To prove the main result, we prepare several terminologies.

2.1 Homotopies in the Category of (Locally) Lipschitz Maps

Let X and Y be metric spaces.

Definition 2.1

We say that a subset A of X is a locally Lipschitz strong deformation retract of X if there is a Lipschitz map $F : X \times [0,1] \rightarrow X$ such that $F(x,0)=x$, $F(x,1) \in A$ and $F(a,t)=a$ for any $x \in X$, $a \in A$ and $t \in [0,1]$. Then, the map F is called a locally Lipschitz strong deformation retraction of X to A.

Definition 2.2

Two maps $h_0, h_1:X\rightarrow Y$ are said to be locally Lipschitz homotopic if there exists a locally Lipschitz map $h : X \times [0,1] \rightarrow Y$ such that $h_i = h(\cdot , i)$   $(i=0,1)$.

We say that X and Y are locally Lipschitz homotopy equivalent if there are locally Lipschitz maps $f : X \rightarrow Y$ and $g : Y \rightarrow X$ such that $g \circ f$ and $f \circ g$ are locally Lipschitz homotopic to $1_X$ and $1_Y$, respectively. In this case, f and g are called locally Lipschitz homotopy equivalences.

In the above definition, if a locally Lipschitz homotopy can be chosen to be a Lipschitz one, then it is called a Lipschitz homotopy. For other notions appeared in Definitions 2.1 and 2.2, we use similar terminologies. From definition, if Y is a (locally) Lipschitz strong deformation retract of X, then X and Y are (locally) Lipschitz homotopy equivalent.

Let X be an unbounded metric space, and $f:X\rightarrow X$ a Lipschitz map whose image is a bounded subset. Then it follows from definition that f is not Lipschitz homotopic to $1_X$. In particular if a metric space X is Lipschitz homotopy equivalent to a bounded metric space, then X is also bounded.

2.2 The Gromov–Hausdorff Distance

A map $f : X \rightarrow Y$ between metric spaces is called an $\epsilon$-approximation if it satisfies

• $||f(x),f(y)| - |x,y|| < \epsilon$ for all $x,y \in X$;

• for any $y \in Y$, there is an $x \in X$ such that $|f(x),y|< \epsilon$.

The Gromov–Hausdorff distance$d_{\mathrm {GH}}(X,Y)$ between X and Y is defined as

$$\begin{aligned} d_{\mathrm {GH}}(X,Y) := \inf \left\{ \epsilon > 0 \,\left| \begin{aligned}&\text {there exist }\epsilon \text {-approximations } \\&X \rightarrow Y \text {and } Y \rightarrow X \end{aligned} \right. \right\} . \end{aligned}$$

A bijective map $f : X \rightarrow Y$ is called an $\epsilon$-almost isometry if both f and $f^{-1}$ are Lipschitz with Lipschitz constants at most $1+\epsilon$.

2.3 Alexandrov Spaces and Good Coverings

We briefly recall the definition of Alexandrov spaces and their properties. For details, we refer to [1]. A complete metric space X is called an Alexandrov space if it is a length space and for any $p \in X$, there exist $\kappa \in {\mathbb {R}}$ and a neighborhood U of p such that for any $x,y,z \in U \setminus \{p\}$, we have

$$\begin{aligned} \tilde{\angle }_\kappa xpy + \tilde{\angle }_\kappa ypz + \tilde{\angle }_\kappa zpx \le 2 \pi , \end{aligned}$$

where $\tilde{\angle }_\kappa xpy$ is defined as the angle of a comparison triangle $\tilde{\Delta }xpy=\Delta \tilde{x}\tilde{p}\tilde{y}$ at $\tilde{p}$ in the complete simply connected surface $M_{\kappa }$ of constant curvature $\kappa$. It is known that the Hausdorff dimension of X coincides with its Lebesgue covering dimension [1, 12], which is called the dimension of X. When $\kappa$ is chosen to be independent of the choice of points $p \in X$, we say that X is of curvature $\ge \kappa$. When X is of dimension n, its volume is measured by the n-dimensional Hausdorff measure.

Complete Riemannian manifolds and orbifolds, the quotient spaces of complete Riemannian manifolds by isometric actions, and the Gromov–Hausdorff limits of sequences of complete Riemannian manifolds with a uniform lower sectional curvature bound are typical examples of Alexandrov spaces.

For $m \in {\mathbb {N}}$ and $\delta > 0$, a point p in an Alexandrov space X of curvature $\ge \kappa$ is called $(m,\delta )$-strained if there exist pairs of points $\{(a_i, b_i)\}_{i=1}^m$ such that

$$\begin{aligned}&\tilde{\angle }_{\kappa } a_i p b_i> \pi - \delta ,&\tilde{\angle }_{\kappa } a_i p b_j> \pi /2-\delta , \\&\tilde{\angle }_{\kappa } a_i p a_j> \pi /2 - \delta ,&\tilde{\angle }_{\kappa } b_i p b_j > \pi /2- \delta \end{aligned}$$

for all $1 \le i \ne j \le m$. The set $\{(a_i, b_i)\}$ is called an $(m, \delta )$-strainer atp. The length$\ell$ of the strainer $\{(a_i, b_i)\}$ at p is defined as

$$\begin{aligned} \ell := \min \{|p, a_i|, |p, b_i| \mid 1 \le i \le m \}. \end{aligned}$$

From now on, we shall use the convention

$$\begin{aligned} \tilde{\angle } xyz :=\tilde{\angle }_{\kappa } xyz, \end{aligned}$$

when the curvature lower bound is understood.

In an n-dimensional Alexandrov space X, a point $p \in X$ is called $\delta$-regular if it is $(n, \delta )$-strained and $\delta \ll 1/n$. The set of all $\delta$-regular points is called a $\delta$-regular part, and is denoted by $\mathcal R_X(\delta )$.

In the present paper, we are concerned with a moduli space of Alexandrov spaces with curvature bounded from below by a uniform constant, say $\kappa$. Rescaling the metric, we assume $\kappa =-1$ without loss of generality. Thus we deal with the moduli space $\mathcal A(n,D,v_0)$ as explained in the introduction.

Theorem 2.3

[1] Suppose that X is n-dimensional and $\delta$ is sufficiently small with $\delta \ll 1/n$. If p is $(n,\delta )$-strained by an $(n,\delta )$-strainer $\{(a_i, b_i)\}_{i=1}^n$ with length $\ell$, then the map $\varphi$ defined by

$$\begin{aligned} \varphi (x) := (|a_1, x|, \ldots , |a_n, x|) \end{aligned}$$

is a $\tau (\delta , \sigma /\ell )$-almost isometry from $B(p, \sigma )$ to an open subset of ${\mathbb {R}}^n$.

We will use Theorem 2.3 in Sect. 6.

Perelman proved the following theorem, called the topological stability theorem.

Theorem 2.4

([10], see also [5]) Let $D > 0$ and $n \in {\mathbb {N}}$ be fixed. Let $M_j$ be a sequence of n-dimensional compact Alexandrov spaces of diameter $\le D$ and curvature $\ge -1$ which converges to an n-dimensional compact Alexandrov space M as $j \rightarrow \infty$. Then, there is $j_0$ such that $M_j$ and M are homeomorphic for all $j \ge j_0$.

In particular, the set of homeomorphism types of spaces in the moduli space ${\mathcal {A}}(n,D,v_0)$ is finite.

The last statement follows from the fact that ${\mathcal {A}}(n,D,v_0)$ is compact with respect to the Gromov–Hausdorff distance.

We shall define a new notion of good coverings for metric spaces. In [9], we have proved that any Alexandrov space has a covering with geometrically and topologically good properties:

Theorem 2.5

[9] For any open covering of a finite-dimensional Alexandrov space X, there is an open covering ${\mathcal {U}}$ of X which is a refinement of the original covering, satisfying the following: Let $V=\bigcap _{i=0}^k U_{j_i}$ be any non-empty intersection of finitely many elements of $\mathcal U$. Then

• (1) V is convex in the sense that every minimal geodesic joining any two points in V is contained in V;

• (2) there exists a point $p \in V$ such that (V, p) is homeomorphic to a cone (C, v), where v is the apex of C;

• (3) there exists a Lipschitz strong deformation retraction $h : V \times [0,1] \rightarrow V$ of V to the point p as above (2) such that $|h_t(x), p|$ is non-increasing in $t\in [0,1]$ for each $x \in V$.

By extracting some fundamental properties that the covering ${\mathcal {U}}$ in Theorem 2.5 posses, we define good coverings for general metric spaces as follows.

Definition 2.6

A locally finite open covering ${\mathcal {U}}=\{U_j\}_{j \in J}$ of a metric space X is good if it satisfies the following:

• (1) the closure of each element of ${\mathcal {U}}$ is compact;

• (2) every non-empty intersection $\bigcap _{i=0}^k U_{j_i}$ of finitely many elements of $\mathcal U$ has a Lipschitz strong deformation retraction to a point p.

Any such a point p as in (2) is called a center of $\bigcap _{i=0}^k U_{j_i}$. We also say that ${\mathcal {U}}$ is a good r-cover if $\mathrm {diam}\,(U_j)<r$ for any $j\in J$.

3 Proof of Theorem 1.1 (Compact Case)

In this section, we prove Theorem 1.1 in the case when M is compact. We deal with the non-compact case in the next section. For the proof of Theorem 1.1, we employ a basic strategy in the proof of Theorem 9.4.15 of [13], where it is proved that if a topological space has a locally finite covering all of whose non-empty intersection are contractible, then it has the same homotopy type as the nerve of the covering. Since the argument there is only topological, we have to proceed in the category of (locally) Lipschitz maps.

3.1 Setting and Strategy

Let M be a compact metric space having a good cover ${\mathcal {U}}=\{U_j\}_{j \in J}$. Note that J is a finite set since $\mathcal U$ is locally finite and M is compact. Let $J=\{1, 2, \ldots , N\}$. Let $\mathcal N_{\mathcal U}$ denote the nerve of ${\mathcal {U}}$, which is a simplicial complex with the set of vertexes $\{U \in {\mathcal {U}} \mid U \ne \emptyset \}$, and whose k-simplices are unordered $(k+1)$-tuples of elements in ${\mathcal {U}}$ so that $\bigcap _{i=0}^k U_{j_i} \ne \emptyset$. We denote by $|{\mathcal {N}}_{\mathcal U}|\subset {\mathbb {R}}^N$ its geometric realization, where we assume that jth vertex $v_j :=\left< U_{j}\right>$ of ${\mathcal {N}}_{\mathcal U}$ is given by

$$\begin{aligned} v_j= (0, \ldots , {\overset{j}{1}}, \ldots , 0)\in {\mathbb {R}}^N. \end{aligned}$$

Let $\theta : V(\mathcal N_{\mathcal U}) \rightarrow [0,1]$ be a function defined on the set of vertices of $\mathcal N_{\mathcal U}$ satisfying

• (1) $\sum _{j\in J} \theta (v_j)=1;$

• (2) ${\mathrm {supp}}(\theta )$ defines a simplex $\sigma _{\theta }$ of $\mathcal N_{\mathcal U}$.

Since $\theta$ defines the point $\sum _{j \in J} \theta (v_j) v_j$ of $\sigma _{\theta }$, it can be considered as an element of $|\mathcal N_{\mathcal U}|$. From now on, we identify a function $\theta$ satisfying (1), (2) with an element $\sum _{j \in J} \theta (v_j) v_j \in |\mathcal N_{\mathcal U}|$. That is,

$$\begin{aligned} |\mathcal N_{\mathcal U}| = \left\{ \theta \equiv \sum _{j\in J} \theta (v_j) v_j \,\Big |\, \theta \,\, {\mathrm{satisfies \, above}}\,\, (1), (2) \right\} . \end{aligned}$$

This will be useful later on (see the proof of Lemma 3.1, for instance). For any subset $A \subset J$, we put

$$\begin{aligned} U_A:=\bigcap _{j\in A} U_j. \end{aligned}$$

Each simplex $\sigma \in \mathcal N_{\mathcal U}$ defines a subset $A(\sigma )\subset J$, and we also use the symbol

$$\begin{aligned} U_{\sigma }=U_{A(\sigma )}. \end{aligned}$$

By definition, there is a Lipschitz contraction $\varphi :U_{\sigma }\times [0,1]\rightarrow U_{\sigma }$ to a point $p_{\sigma }$ of $U_{\sigma }$.

We define a function $f_j$ on M by

$$\begin{aligned} f_j(x) = \frac{|x,U_j^c|}{|x,U_j^c| + |x,p_j|}, \end{aligned}$$

where $U_j^c$ denotes the complement of $U_j$ and $p_j$ is a center of $U_j$ to which $U_j$ has a Lipschitz strong deformation retraction. Since $|x,U_j^c| + |x,p_j|\ge |p_j, U_j^c|/2>0$, it is straightforward to check that $f_j$ is Lipschitz. Set

$$\begin{aligned} \xi _j(x)=\frac{f_j(x)}{\sum _i f_i(x)}. \end{aligned}$$

Then $\{\xi _j\}_{j\in J}$ defines a partition of unity dominated with $\mathcal U$ satisfying

• (1) ${\mathrm {supp}}(\xi _j) =\bar{U}_j;$

• (2) each $\xi _j$ is Lipschitz;

• (3) $\sum _j \xi _j = 1$.

The polyhedron $|\mathcal N_{\mathcal U}|$ has the distance induced from the metric of ${\mathbb {R}}^{N}$ defined as

$$\begin{aligned} d(x, y) = \max _{1\le i\le N} \, |x_i-y_i|. \end{aligned}$$

In the rest of this section, we are going to construct metric spaces $\mathcal D(\mathcal U)$ and $\mathcal M(p)$ together with natural bi-Lipschitz embeddings

(3.1)

and prove that their images are Lipschitz strong deformation retracts of the target spaces. This strategy comes from [13]. The most complicated part is a construction of a Lipschitz strong deformation retraction from $\mathcal M(p)$ to $\iota (\mathcal D(\mathcal U))$, which will be done simplex-wisely by means of provided Lipschitz strong deformation retractions of $U_\sigma$’s to their centers.

We divide the proof into three steps.

Step 1 We consider the following subspace of the product metric space $|\mathcal N_{\mathcal U}|\times M$ defined as

$$\begin{aligned} \mathcal D(\mathcal U):=\{(\theta , x)\in |\mathcal N_{\mathcal U}|\times M \,|\, x\in U_{{\mathrm {supp}}\,\theta } \} . \end{aligned}$$

Let $p:\mathcal D(\mathcal U)\rightarrow |\mathcal N_{\mathcal U}|$ and $q:\mathcal D(\mathcal U)\rightarrow M$ be the projections:

$$\begin{aligned} p(\theta , x)=\theta ,\,\,\, q(\theta , x)=x. \end{aligned}$$

Lemma 3.1

There exists a Lipschitz map $\tau :M\rightarrow \mathcal D(\mathcal U)$ such that

• (1) $\tau$ is a section of the map q (i.e., $q \circ \tau =1_M$);

• (2) $\tau (M)$ is a Lipschitz strong deformation retract of $\mathcal D(\mathcal U)$.

Proof

Define $\tau :M\rightarrow \mathcal D(\mathcal U)$ by

$$\begin{aligned} \tau (x) = (\Theta (x), x), \end{aligned}$$

where $\Theta (x)\in |\mathcal N_{\mathcal U}|$ is defined by $\Theta (x)(v_j) = \xi _j(x)$, $j\in J$. Obviously $\tau$ and $\Theta :M\rightarrow |{\mathcal {N}}_{\mathcal U}|$ are Lipschitz. For any $x\in M$, let $s(x):=\{j\in J\,|\, x\in U_j\}$, which forms a simplex of $\mathcal N_{\mathcal U}$. For any $(\theta , x)\in \mathcal D(\mathcal U)$, ${\mathrm {supp}}(\theta )$ defines a face of s(x). Thus we can define the Lipschitz map $H:\mathcal D(\mathcal U)\times [0,1] \rightarrow \mathcal D(\mathcal U)$ by

$$\begin{aligned} H( \theta ,x,s) =(s\Theta (x) + (1-s)\theta , x) \end{aligned}$$

(3.2)

satisfying $H(\theta ,x, 0)=1_{\mathcal D(\mathcal U)}(\theta , x)$, $H(\theta , x, 1)=(\Theta (x),x)=\tau (x)$ and $H(\tau (x),s)=\tau (x)$ for every $s\in [0,1]$. Obviously H is Lipschitz. This completes the proof. $\square$

Corollary 3.2

M has the same Lipschitz homotopy type as $\mathcal D(\mathcal U)$.

Proof

Let $q':\tau (M)\rightarrow M$ be the restriction of q to $\tau (M)$. Since $\tau$ is Lipschitz and $q'$ is 1-Lipschitz with $q' \circ \tau =1_M$ and $\tau \circ q'=1_{\tau (M)}$, M and $\tau (M)$ are bi-Lipschitz homeomorphic to each other. The conclusion follows from Lemma 3.1. $\square$

Step 2 For $L>0$ (see (3.3) for the proper choice of L), consider the mapping cylinder of p:

$$\begin{aligned} \mathcal M(p):= \mathcal D(\mathcal U)\times [0,L]\amalg |\mathcal N_{\mathcal U}|/(\theta , x,L)\sim \theta . \end{aligned}$$

Recall that

$$\begin{aligned} \mathcal D(\mathcal U)=\bigcup _{\sigma \in \mathcal N_{\mathcal U}} \sigma \times U_{{\sigma }} \subset |\mathcal N_{\mathcal U}|\times M. \end{aligned}$$

The canonical correspondence $[(\theta , x,t)]\rightarrow (\theta , [(x,t)])$ gives rise to the identification

$$\begin{aligned} \mathcal M(p)=\bigcup _{\sigma \in \mathcal N_{\mathcal U}} \sigma \times K(U_{{\sigma }}) \subset |\mathcal N_{\mathcal U}|\times K(M), \end{aligned}$$

where $K(V)=V\times [0,L]/V\times L$ denotes the Euclidean cone. From now on, we consider the metric of $\mathcal M(p)$ induced from that of the product metric $|\mathcal N_{\mathcal U}|\times K(M)$, where the metric of the Euclidean cone $K(M)=M\times [0,L]/M\times L$ is defined as

$$\begin{aligned} |[x,t],&[x',t']|^2 \\&= (L- t)^2 + (L-t')^2 -2(L-t)(L-t')\cos (\min \{\pi , |x,x'|\}), \end{aligned}$$

for $[x,t],[x',t']\in K(M)$.

Note that there is a natural isometric embedding ${\Psi }: |\mathcal N_{\mathcal U}| \rightarrow \mathcal M(p)$ defined by

$$\begin{aligned} {\Psi }(\theta )=(\theta , [x,L]) =[\theta ]{=(\theta , v_M)}, \end{aligned}$$

where $v_M$ denotes the vertex of K(M).

Lemma 3.3

$|\mathcal N_{\mathcal U}|$ is a Lipschitz strong deformation retract of $\mathcal M(p)$.

Proof

Define ${\Psi '}:\mathcal M(p)\rightarrow |\mathcal N_{\mathcal U}|$ by

$$\begin{aligned} {\Psi '}(\theta , [x,t]) = \theta . \end{aligned}$$

Since

$$\begin{aligned} | {\Psi '}(\theta , [x,t]) ,&{\Psi '}(\theta ', [x',t']) | \\&= |\theta ,\theta '| \\&\le \sqrt{|\theta ,\theta '|^2 + |[x,t], [x',t']|^2} \\&=|(\theta , [x,t]), (\theta ', [x',t'])|, \end{aligned}$$

and since $|{\Psi }(\theta _1), {\Psi }(\theta _2)| =|\theta _1,\theta _2|$, both ${\Psi }$ and ${\Psi '}$ are 1-Lipschitz.

Note that ${\Psi \circ \Psi '}(\theta , [x,t])=[\theta ]=(\theta , [x, L])$ and ${\Psi '\circ \Psi }=1_{|\mathcal N_{\mathcal U}|}$. Define $F:\mathcal M(p)\times [0,1]\rightarrow \mathcal M(p)$ by

$$\begin{aligned} F(\theta ,[x,t],s) = (\theta , [x, (1-s)t+sL]). \end{aligned}$$

Then $F_0=1_{\mathcal M(p)}$ and $F_1={\Psi \circ \Psi '}$. We show that F is Lipschitz. Since it suffices to prove that it is locally Lipschitz, let us assume that $(\theta ,[x,t],s)$ and $(\theta ',[x',t'],s')$ are close to each other. We then have

$$\begin{aligned} |F(\theta ,[x,t],s), F(\theta ',[x',t'],s')|^2 = |\theta ,\theta '|^2 + |[x,u], [x',u']|^2, \end{aligned}$$

where we set $u=(1-s)t+sL$, $u'=(1-s')t'+s'L$, and

$$\begin{aligned} |[x,u],&[x',u']|^2 \\&= (L-u)^2 +(L-u')^2 -2(L-u)(L-u')\cos |x,x'| \\&\le (u-u')^2 + (L-u)(L-u')|x,x'|^2. \end{aligned}$$

Since $|u-u'|\le (1-s')|t-t'|+(L-t)|s-s'|$, we have

$$\begin{aligned} |F(\theta ,[x,t],&s), F(\theta ',[x',t'],s')|^2 \\&\le |\theta ,\theta '|^2 + 2|t-t'|^2 +2|s-s'|^2+ (L-u)(L-u')|x, x'|^2. \end{aligned}$$

Similarly we have

$$\begin{aligned} |(\theta ,[x,t],s),&(\theta ',[x',t'],s')|^2\\&\ge |\theta ,\theta '|^2 + |t-t'|^2+ \frac{1}{2}(L-t)(L-t')|x, x'|^2 {+} |s-s'|^2. \end{aligned}$$

Combining those inequalities, we conclude that F is Lipschitz. $\square$

Step 3 Let us define $\iota :\mathcal D(\mathcal U)\rightarrow \mathcal D(\mathcal U)\times 0\subset \mathcal M(p)$ by $\iota (\theta ,x)=(\theta ,x,0)$. In this last step, we prove

Proposition 3.4

There exists a Lipschitz strong deformation retraction $\Phi :\mathcal M(p)\times [0,1] \rightarrow \mathcal M(p)$ of $\mathcal M(p)$ to $\mathcal D(\mathcal U)\times 0$.

The compact case of Theorem 1.1 now follows from Corollary 3.2, Lemma 3.3 and Proposition 3.4.

Let

$$\begin{aligned} L > 6. \end{aligned}$$

(3.3)

Let $k_0$ denote the dimension of $\mathcal N$. For each $0\le k\le k_0$, let $\mathcal N^{(k)}$ denote the k-skeleton of $\mathcal N_{\mathcal U}$, and $\mathcal D^k := p^{-1}(|\mathcal N^{(k)}|)$ and $p^k:=p|_{\mathcal D^k}:\mathcal D^k \rightarrow \mathcal N^{(k)}$. Let $\mathcal M(p^k)$ denote the mapping cone of $p^k$:

$$\begin{aligned} \mathcal M(p^k):= \mathcal D^k\times [0,L]\amalg |\mathcal N^{(k)}|/(\theta , x,L)\sim \theta . \end{aligned}$$

As before, we have

$$\begin{aligned} \mathcal D^k= & {} \bigcup _{\sigma \in \mathcal N^{(k)}} \sigma \times U_\sigma ,\\ \mathcal M(p^k)= & {} \bigcup _{\sigma \in \mathcal N^{(k)}} \sigma \times K(U_{\sigma }) \subset |\mathcal N_{\mathcal U}|\times K(M). \end{aligned}$$

Lemma 3.5

For each k, There exists a Lipschitz strong deformation retraction $\Phi ^k:\mathcal M(p^k)\times [0,1] \rightarrow \mathcal M(p^k)$ of $\mathcal M(p^k)$ to $\mathcal D^k\times 0\bigcup \mathcal M(p^{k-1})$.

The construction of the Lipschitz strong deformation retraction $\Phi ^k$ in Lemma 3.5 will be done simplex-wisely. This is based on the following sublemma.

Sublemma 3.6

For each k-simplex $\sigma \in \mathcal N_{\mathcal U}$, there exists a Lipschitz strong deformation retraction of $\sigma \times K(U_{\sigma })$ to $(\sigma \times U_{\sigma }\times 0)\bigcup \partial \sigma \times K(U_{\sigma })$.

Since $\partial \sigma \times K(U_{\sigma })\subset \mathcal M(p^{k-1})$, applying Sublemma 3.6 to each k-simplex of $\mathcal N_{\mathcal U}$, we obtain Lemma 3.5.

By using Lemma 3.5 repeatedly, we have a finite sequence of Lipschitz retractions:

$$\begin{aligned} \begin{aligned} \mathcal M(p)= \mathcal M(p^{k_0}) \longrightarrow&\mathcal D(\mathcal U)\times 0\bigcup \mathcal M(p^{k_0-1}) \longrightarrow \cdots \\&\longrightarrow D(\mathcal U)\times 0\bigcup \mathcal M(p^0) \longrightarrow \mathcal D(\mathcal U)\times 0. \end{aligned} \end{aligned}$$

(3.4)

From (3.4), we conclude that $\mathcal D(\mathcal U)\times 0$ is a Lipschitz strong deformation retract of $\mathcal M(p)$. Thus all we have to do is to prove Sublemma 3.6.

Remark 3.7

From (3.4), one might think that $k_0=\dim \mathcal N_{\mathcal U}<\infty$ is essential in the argument below. However, we can generalize the argument of this section to the general case of $\dim \mathcal N_{\mathcal U}=\infty$. This will be verified in Sect. 4.

The following is the important first step in the proof of Sublemma 3.6, which is the case of $k=0$.

Claim 3.8

Let U be an element of $\mathcal U$. Then there exists a Lipschitz strong deformation retraction $K(U) \times [0,1] \rightarrow K(U)$ of K(U) to $U\times 0$.

Proof

Let $\varphi :U \times [0,L] \rightarrow U$ be a Lipschitz strong deformation retraction to $p \in U$. We may assume that $\varphi (x,t) = p$ for all $t \ge L/2$ and $x \in U$. Define the retraction $r:K(U)\rightarrow U\times 0\subset K(U)$ by

$$\begin{aligned} r([x,t]) := [\varphi (x,t), 0]. \end{aligned}$$

First we show that r is Lipschitz. Again we way assume that [x, t] and $[x',t']$ are sufficiently close. Note that

$$\begin{aligned} |r([x,t]), r([x',t'])|&\le |[\varphi (x,t),0],[\varphi (x',t),0]| + |[\varphi (x',t),0],[\varphi (x',t'),0]| \\&\le L|\varphi (x,t), \varphi (x',t)| + L|\varphi (x',t), \varphi (x',t')| \\&\le CL|x,x'| + CL|t-t'|. \end{aligned}$$

From here on, we use the symbols $C, C_1,C_2,\ldots$ to denote some uniform positive constants.

If both t and $t'$ are greater than L / 2, then $\varphi (x,t)=\varphi (x',t')=p$. Therefore we may assume that $t,t'\le L/2$. Then we have

$$\begin{aligned} |[x,t], [x',t']|^2&\ge (t-t')^2 + (L-t)(L-t')|x,x'|^2/2 \\&\ge (t-t')^2 + (L^2/8)|x,x'|^2. \end{aligned}$$

Combining the two inequalities, we have

$$\begin{aligned} |r([x,t]), r([x',t'])|&\le CL|x,x'| + C_1L|t-t'| \\&\le C (1+L)|[x,t], [x',t']|. \end{aligned}$$

Now let $g:[0,L]\times [0,1]\rightarrow [0,1]$ be a Lipschitz function such that

• $g(t,s) =1$ on $[0,L]\times [0,1/3];$

• $g(t,1)=0$ for all $0\le t\le L$,

and define $\Phi :K(U)\times [0,1]\rightarrow K(U)$ by

$$\begin{aligned} \Phi ([x,t],s) = [\varphi (x,st), g(t,s)t]. \end{aligned}$$

Note that $\Phi ([x,t],0)=[x,t]$, $\Phi ([x,t],1)=r([x,t])$.

To show that $\Phi$ is Lipschitz, let ([x, t], s) and $([x',t'],s')$ be elements of $K(U)\times [0,1]$ sufficiently close to each other. By triangle inequalities, it suffices to show the following:

• (1) $|\Phi ([x,t],s), \Phi ([x',t],s)| \le C_1|[x,t],[x',t]|;$

• (2) $|\Phi ([x,t],s), \Phi ([x,t'],s)| \le C_2L|[x,t],[x,t']|;$

• (3) $|\Phi ([x,t],s), \Phi ([x,t],s')| \le C_3L(1+L)|s-s'|$.

We show (1)

$$\begin{aligned} |\Phi ([x,t],s), \Phi ([x',t],s)|&=|[\varphi (x,st), g(t,s)t],[\varphi (x',st), g(t,s)t]| \\&\le |L-g(t,s)t||\varphi (x,st), \varphi (x',st)| \\&\le |L-g(t,s)t| C |x,x'| \\&\le |L-g(t,s)t| |x, x'|. \end{aligned}$$

If $s\le 1/3$, then $|L-g(t,s)t| |x,x'| = (L-t)|x,x'| \le 2|[x,t],[x',t]|$. If $s\ge 1/3$ and $t\ge L/2$, then $ts\ge L_0$, and therefore $|\Phi ([x,t],s), \Phi ([x',t],s)|=0$. If $s\ge 1/3$ and $t\le L/2$, then $|L-g(t,s)t| |x,x'|\le L|x,x'|\le 2(L-t)|x,x'| \le 3|[x,t],[x',t]|$.

We show (2)

$$\begin{aligned} |\Phi ([x,t],s), \Phi ([x,t'],s)|^2&=|[\varphi (x,st), g(t,s)t],[\varphi (x,st'), g(t',s)t']|^2 \\&\le |g(t,s)t-g(t',s)t'|^2 + L^2 |\varphi (x,st), \varphi (x,st')|^2, \\&\le |g(t,s)t-g(t',s)t'|^2 + L^2 C_1|st-st'|^2, \end{aligned}$$

where obviously

$$\begin{aligned} |g(t,s)t-g(t',s)t'|&\le |g(t,s)t-g(t',s)t| + |g(t',s)t-g(t',s)t'| \\&\le C(1+L)|t-t'|. \end{aligned}$$

Thus we have $|\Phi ([x,t],s), \Phi ([x,t'],s)|\le C_2(1+L)|[x,t],[x,t']|$.

We show (3)

$$\begin{aligned} \begin{aligned} |\Phi ([x,t],s), \Phi ([x,t],s')|^2&=|[\varphi (x,st), g(t,s)t],[\varphi (x,s't), g(t,s')t]|^2 \\&\le |g(t,s)t-g(t,s')t|^2 + C_1 L^2 |st-s't|^2 \\&\le C_2L^2|s-s'|^2 + C_3L^4|s-s'|^2\\&\le C_4L^2(1+L^2)|s-s'|^2. \end{aligned} \end{aligned}$$

(3.5)

This shows that $\Phi$ is Lipschitz, and together with (3.5) this completes the proof of Claim 3.8. $\square$

Next we consider the general case.

Proof of Sublemma 3.6

Let $\sigma$ be any simplex of $\mathcal N$. Note that $\sigma \times 0\cup \partial \sigma \times [0,L]$ is a Lipschitz strong deformation retract of $\sigma \times [0,L]$. Let $r:\sigma \times [0,L]\rightarrow \sigma \times 0\cup \partial \sigma \times [0,L]$ be a Lipschitz strong deformation retraction defined by the radial projection from the point $(x^*,2L)\in \sigma \times {\mathbb {R}}$, where $x^*$ is the barycenter of $\sigma$. Let us represent r as

$$\begin{aligned} r(x,t) =(\psi _0(x,t), u(x,t))\in \sigma \times 0\cup \partial \sigma \times [0,L]\subset \sigma \times [0,L]. \end{aligned}$$

Define the retraction $f:\sigma \times K(U) \rightarrow \sigma \times U\times 0\cup \partial \sigma \times K(U)$ by

$$\begin{aligned} f(x,[y,t])=(\psi _0(x,t), [\varphi (y,t-u(x,t)), w(x,t)]), \end{aligned}$$

where $w:\sigma \times [0,L]\rightarrow [0,L]$ is defined as follows: Let us consider the following closed subsets of ${\mathbb {R}}^{N+1}$:

$$\begin{aligned}&\Omega _0=\{(x,t)\in \sigma \times [0,L]\,|\,u(x,t)\le L/10\}, \\&\Omega _1=\{(x,t)\in \sigma \times [0,L]\,|\,u(x,t)\ge L/2\}. \end{aligned}$$

Note that $|\Omega _0,\Omega _1|\ge c>0$ for some constant $c>0$. Let $s_i(x,t)=|(x,t), \Omega _i|$, $i=1,,2$, and define w by

$$\begin{aligned} w(x,t) := \frac{s_1(x,t)}{s_0(x,t)+s_1(x,t)} u(x,t) + \frac{s_0(x,t)}{s_0(x,t)+s_1(x,t)} t. \end{aligned}$$

Note that w is Lipschitz and has the property

$$\begin{aligned} w(x,t) = {\left\{ \begin{array}{ll} u(x,t)\,\, &{}{\mathrm{if}} \,\,u(x,t)\le L/10 \\ t \,\, &{}{\mathrm{if}} \,\, u(x,t)\ge L/2. \end{array}\right. } \end{aligned}$$

Note also that f is the identity on $\sigma \times U\times 0\cup \partial \sigma \times K(U)$, and therefore it defines a retraction of $\sigma \times U\times 0\cup \partial \sigma \times K(U)$. We show that f is Lipschitz. It suffices to show that the second component

$$\begin{aligned} f_2(x,[y,t])=([\varphi (y,t-u(x,t)), w(x,t)]) \end{aligned}$$

of f is Lipschitz. As before, we may assume that (x, [y, t]) and $(x',[y',t'])$ are sufficiently close to each other. Letting $u=u(x,t)$, $u'=u(x',t)$, $w=w(x,t)$, $w'=w(x',t)$ we have

$$\begin{aligned} |f_2(x,&[y,t]), f_2(x',[y,t])|^2 \\&=|[\varphi (y,t-u), w], [\varphi (y,t-u'),w']|^2 \\&\le (L-w)^2 + (L-w')^2 -2(L-w)(L-w')\cos |\varphi (y,t-u), \varphi (y,t-u')| \\&\le (w-w')^2 +(L-w)(L-w')|\varphi (y,t-u), \varphi (y,t-u')|^2 \\&\le (w-w')^2 + C_1L^2(u-u')^2\\&\le C_2(1+L^2)|x,x'|^2, \end{aligned}$$

and

$$\begin{aligned} |f_2(x,[y,t]), f_2(x,[y',t])|&=|[\varphi (y,t-u), w], [\varphi (y',t-u),w]| \\&\le (L-w)|\varphi (y,t-u), \varphi (y',t-u)|, \end{aligned}$$

where since $|[y,t],[y',t]|\ge (1/2)(L-t)|y,y'|$, we may assume that $t\ge 9L/10$. If $t\ge 9L/10$ and $u(x,t)\le L/2$, then $\varphi (\cdot , t-u)=p$. If $t\ge 9L/10$ and $u(x,t) > L/2$, then $w(x,t)=t$, and we have

$$\begin{aligned} |f_2(x,[y,t]), f_2(x',[y,t])|&\le (L-t) C|y,y'| \\&\le C|[y,t], [y',t]|. \end{aligned}$$

Finally letting $u=u(x,t)$, $u'=u(x,t')$, $w=w(x,t)$, $w'=w(x,t')$ we have

$$\begin{aligned} |f_2(x,[y,t]), f_2(x,[y,t'])|^2&=|[\varphi (y,t-u), w], [\varphi (y,t-u'),w']|^2 \\&\le (w-w')^2 + L^2|\varphi (y,t-u), \varphi (y,t-u')|^2 \\&\le (w-w')^2 + C_1L^2(u-u')^2 \\&\le C_2(1+L^2)|t-t'|^2. \end{aligned}$$

Thus f is Lipschitz.

Now define the homotopy $\Phi :\sigma \times K(U)\times [0,1]\rightarrow \sigma \times K(U)$ by

$$\begin{aligned}&\Phi (x, [y,t],s) \\&\quad = ((1-s)x+s\psi _0(x,t), [\varphi (y,\mu (s)(t-u(x,t)),(1-\nu (s))t+\nu (s)w(x,t)]), \end{aligned}$$

where $\mu$ and $\nu$ are Lipschitz functions on [0, 1] satisfying

$$\begin{aligned} \mu (s) = {\left\{ \begin{array}{ll} 1 \,\, &{} {\mathrm{if}} \,\,s \ge 2/3 \\ 0 \,\, &{} {\mathrm{if}} \,\, s\le 1/2, \end{array}\right. } \,\, \nu (s) = {\left\{ \begin{array}{ll} 1 \,\, &{}{\mathrm{if}} \,\,s \ge 3/4 \\ 0 \,\, &{}{\mathrm{if}} \,\, s\le 2/3. \end{array}\right. } \end{aligned}$$

Obviously, $\Phi (\cdot , 0)=1_{\sigma \times K(U)}$, $\Phi (\cdot , 1)=f$ and $\Phi (\cdot ,s)$ fixes each point of $\sigma \times U\times 0 \cup \partial \sigma \times K(U)$. We show that $\Phi$ is Lipschitz. It suffices to show that the second component

$$\begin{aligned} \Phi _2(x,[y,t],s)=([\varphi (y,\mu (s)(t-u(x,t)),(1-\nu (s))t+\nu (s)u(x,t)]) \end{aligned}$$

of $\Phi$ is Lipschitz. As before, we may assume that (x, [y, t], s) and $(x',[y',t'],s')$ are sufficiently close to each other. Letting $u=u(x,t)$, $u'=u(x',t)$, $w=w(x,t)$, $w'=w(x',t)$ and $\mu =\mu (s)$, $\nu =\nu (s)$, we have

$$\begin{aligned} |\Phi _2(x,&[y,t],s), \Phi _2(x',[y,t],s)|^2 \\&=|[\varphi (y,\mu (t-u)), (1-\nu )t +\nu w], [\varphi (y,\mu (t-u')), (1-\nu )t +\nu w']|^2 \\&\le \nu ^2(w-w')^2 +L^2|\varphi (y,\mu (t-u)), \varphi (y,\mu (t-u')|^2 \\&\le \nu ^2(w-w')^2 + C_1L^2\mu ^2(u-u')^2\\&\le C_2(1+L^2)|x,x'|^2, \end{aligned}$$

and

$$\begin{aligned} |\Phi _2(x,&[y,t],s), \Phi _2(x,[y',t],s)| \\&=|[\varphi (y,\mu (t-u)), (1-\nu )t +\nu w], [\varphi (y',\mu (t-u)), (1-\nu )t +\nu w]| \\&\le (L-(1-\nu )t -\nu w)|\varphi (y,\mu (t-u), \varphi (y',\mu (t-u)| \\&\le (L-(1-\nu )t )C|y,y'|, \end{aligned}$$

where if $t<9L/10$, then $L|y,y'|\le CL|[y,t], [y',t]|$. Hence we may assume that $t\ge 9L/10$. If $u(x,t)>L/2$ then $w(x,t)=t$. If $u(x,t)\le L/2$ and $s\ge 2/3$, then $\mu (s)=1$ and $\varphi (\cdot , \mu (t-u))=p$. If $u(x,t)\le L/2$ and $s\le 2/3$, then $\nu =0$. Thus we conclude that

$$\begin{aligned} |\Phi _2(x,[y,t],s), \Phi _2(x,[y',t],s)|&\le (L-t) C|y,y'| \\&\le C|[y,t], [y',t]|. \end{aligned}$$

Next letting $u=u(x,t)$, $u'=u(x,t')$, $w=w(x,t)$, $w'=w(x,t')$, we have

$$\begin{aligned} |\Phi _2(x,&[y,t],s),\Phi _2(x,[y,t'],s)|^2 \\&=|[\varphi (y,\mu (t-u)), (1-\nu )t +\nu w], [\varphi (y,\mu (t'-u')), (1-\nu )t' +\nu w']|^2 \\&\le ((1-\nu )(t-t') + \nu (w-w'))^2 + L^2 |\varphi (y,\mu (t-u), \varphi (y, \mu (t'-u'))| \\&\le C(1+L^2)(t-t')^2. \end{aligned}$$

Finally letting $\mu '=\mu (s')$, $\nu '=\nu (s')$, we have

$$\begin{aligned} |\Phi _2(x,&[y,t],s), \Phi _2(x,[y,t],s')|^2 \\&=|[\varphi (y,\mu (t-u)), (1-\nu )t +\nu w], [\varphi (y,\mu '(t-u)), (1-\nu ')t +\nu ' w]|^2 \\&\le (t(\nu '-\nu )+w(\nu -\nu '))^2 + L^2 |\varphi (y,\mu (t-u), \varphi (y, \mu '(t-u))|^2 \\&\le L^2(\nu -\nu ')^2 + C_1L^2(\mu -\mu ')^2 \\&\le C_2L^2(s-s')^2. \end{aligned}$$

Thus $\Phi$ is Lipschitz. This completes the proof of Sublemma 3.6. $\square$

This completes the proof of Proposition 3.4. We have just proved the compact case of Theorem 1.1.

By the above discussion, we have the following commutative diagram:

From Lemmas 3.1, 3.3, and Proposition 3.4 together with (3.1), we have the following.

Corollary 3.9

Let $M, {\mathcal {U}}, \{\xi _j\}_{j \in J}$ be the same as in this section. Then the natural map

$$\begin{aligned} \Theta : M \ni x \mapsto (\xi _j(x))_{j \in J} \in |\mathcal N_{\mathcal U}| \end{aligned}$$

is a Lipschitz homotopy equivalence.

Corollary 3.10

Let M, ${\mathcal {U}}=\{U_j\}_{j=1}^N$ and $\mathcal N_{\mathcal U}$ be the same as in this section, and $\zeta :|\mathcal N_{\mathcal U}|\rightarrow M$ a Lipschitz homotopy inverse to $\Theta : M \rightarrow |\mathcal N_{\mathcal U}|$. For every $\theta \in |\mathcal N_{\mathcal U}|$, let $\sigma$ be the open simplex of $\mathcal N_{\mathcal U}$ containing $\theta$ with $\sigma =\langle U_{j_0}, \ldots , U_{j_k}\rangle$. Then we have

$$\begin{aligned} \zeta (\theta ) \in \bigcup _{i=0}^k \,U_{j_i}. \end{aligned}$$

(3.6)

Proof

Let $H:\mathcal D(\mathcal U)\times [0,1] \rightarrow \mathcal D(\mathcal U)$ be a Lipschitz strong deformation retraction of $\mathcal D(\mathcal U)$ to $\tau (M)$ given in (3.2), and set $H_1:=H(\,\cdot \,, 1)$. Let $\Phi :\mathcal M(p)\times [0,1] \rightarrow \mathcal M(p)$ be a Lipschitz strong deformation retraction of $\mathcal M(p)$ to $\mathcal D(\mathcal U)\times 0$ given in Proposition 3.4, and set $\Phi _1:=\Phi (\,\cdot \,, 1)$. From our argument in this section, we have the following commutative diagram:

Note that $\tau ^{-1}\circ H_1(\mu ,x)=x$ for every $(\mu ,x)\in \mathcal D(\mathcal U)$. Therefore, we can write

$$\begin{aligned} \Phi _1\circ \Psi (\theta )=(\eta (\theta ),\zeta (\theta ))\in \mathcal D(\mathcal U), \end{aligned}$$

where $\eta (\theta )\in |\mathcal N_{\mathcal U}|$ and $\zeta (\theta )\in M$. If $\sigma _1$ denotes the open simplex of $\mathcal N_{\mathcal U}$ containing $\eta (\theta )$, then it follows from the definition of $\mathcal D(\mathcal U)$ that $\zeta (\theta )\in U_{\sigma _1}$. From Sublemma 3.6 together with (3.4), we see that $\sigma _1$ is a face of $\sigma$, which yields (3.6). $\square$

4 Non-compact Case

We prove Theorem 1.1 for the general case. Let M be a $\sigma$-compact metric space admitting a good covering ${\mathcal {U}}=\{U_j\}_{j\in J}$. From the local finiteness of ${\mathcal {U}}$, J is countable, and therefore we may assume $J={\mathbb {N}}$. Since $\mathcal U$ is locally finite, the number of $U_j$’s meeting each $\bar{U}_i$ is finite. It follows that the nerve $\mathcal N_{\mathcal U}$ is locally finite. Note that $|\mathcal N_{\mathcal U}|\subset {\mathbb {R}}^{\infty }$ in this case. Note that the Lipschitz constant of the strong deformation retraction $U_j\times [0,1]\rightarrow U_j\times [0,1]$ of $U_j$ to a point of $U_j$ depends on j, and that $\dim \mathcal N_{\mathcal U}=\infty$ in general. From the local finiteness of $\mathcal N_{\mathcal U}$, basically we can do the same construction as in Sect. 3 to obtain the spaces $\mathcal D(\mathcal U)$, $\mathcal M(p)$ in the general case, too. We also have the natural embeddings in a similar manner:

Note that the map $\tau :M\rightarrow \mathcal D(\mathcal U)$ defined by

$$\begin{aligned} \tau (x) = (\Theta (x), x),\quad \Theta (x)(v_j) = \xi _j(x) \quad (j\in J) \end{aligned}$$

is locally bi-Lipschitz. In a way similar to Corollary 3.2, we see that $\tau (M)$ has the same locally Lipschitz homotopy type as $\mathcal D(\mathcal U)$. Note also that the natural embedding ${\Psi }: |\mathcal N_{\mathcal U}| \rightarrow \mathcal M(p)$ defined by

$$\begin{aligned} {\Psi (\theta )=(\theta , [x,L]) =[\theta ]} \end{aligned}$$

is isometric, and we see that $|\mathcal N_{\mathcal U}|$ is a locally Lipschitz strong deformation retract of $\mathcal M(p)$ in a way similar to Lemma 3.3. Therefore to complete the proof of Theorem 1.1 in the general case, we only have to check the following:

Lemma 4.1

There exists a locally Lipschitz strong deformation retraction of $\mathcal M(p)$ to $\mathcal D(\mathcal U)\times 0$.

Proof

Recall that in the compact case in Sect. 3, the strong deformation retraction $\Phi :\mathcal M(p)\times [0,1] \rightarrow \mathcal M(p)$ of $\mathcal M(p)$ to $\mathcal D(\mathcal U)\times 0$ is constructed simplex-wisely from higher dimensions to lower dimensions through Lemma 3.5. Therefore the Lipschitz constant of $\Phi$ depends on the dimension $k_0$ of $\mathcal N_{\mathcal U}$.

Since $\dim \mathcal N_{\mathcal U}$ could be infinite in the present case, first of all, we have to verify that the map $\Phi :\mathcal M(p)\times [0,1] \rightarrow \mathcal M(p)$ is well defined. For any point $(\theta , [x,t])\in \mathcal M(p)$, let $\sigma _\theta$ be the simplex whose interior contains $\theta$. Let L be the star of $\sigma _\theta$, which is a finite subcomplex because of the local finiteness of $\mathcal N_{\mathcal U}$. Then

$$\begin{aligned} \mathcal M_L(p) := \bigcup _{\sigma \in L} \, \sigma \times K(U_{\sigma }) \end{aligned}$$

provides an open subset of $\mathcal M(p)$ containing $(\theta , [x,t])$. Set

$$\begin{aligned} \mathcal D_L:= \bigcup _{\sigma \in L} \, \sigma \times U_{\sigma }. \end{aligned}$$

From the argument in Lemma 3.5, we have a Lipschitz strong deformation retraction $\Phi _L:\mathcal M_L(p) \times [0,1]\rightarrow \mathcal M_L(p)$ of $\mathcal M_L(p)$ to $\mathcal D_L\times 0$. Since $\mathcal M(p)$ is a locally finite union of such open subsets $\mathcal M_L(p)$, we can construct a strong deformation retraction $\Phi :\mathcal M(p) \times [0,1]\rightarrow \mathcal M(p)$ of $\mathcal M(p)$ to $\mathcal D\times 0$ simplex-wisely in a similar manner. Since $\Phi |_{\mathcal M_L(p)\times [0,1]}=\Phi _L$ and the construction is simplex-wise, $\Phi$ is locally Lipschitz. This completes the proof. $\square$

5 Proofs of Theorem 1.2 and Corollary 1.5

To prove Theorem 1.2, we need the following result, which follows from the proof of [9, Theorem 1.2].

Theorem 5.1

[9] For every $M\in \mathcal A(n,D,v_0)$, let $M_i$ be a sequence in $\mathcal A(n,D,v_0)$ converging to M as $i \rightarrow \infty$. Then for any $\mu >0$, there exists a good $\mu$-covering $\mathcal U = \{U_j\}_{j \in J}$ of M satisfying the following. For every $\epsilon _i$-approximations $\phi _i : M \rightarrow M_i$ with $\epsilon _i \rightarrow 0$, there exist a good $2\mu$-covering $\mathcal U_i = \{U_{ij}\}_{j \in J}$ of $M_i$ and $\nu _i$-approximations $\varphi _i : M \rightarrow M_i$ with $\nu _i\rightarrow 0$ such that for sufficiently large i

• (1) $\varphi _i(p_j)$ is a center of $U_{ij}$ in the sense of Definition 2.6 for every $j \in J$;

• (2) the corresponding $U_{j} \mapsto U_{ij}$ induces an isomorphism $\mathcal N_{\mathcal U} \rightarrow \mathcal N_{\mathcal U_i}$ between the nerves of $\mathcal U$ and $\mathcal U_i$;

• (3) $\lim _{i \rightarrow \infty } \sup _{x \in M} |\phi _i(x), \varphi _i(x)| = 0$.

Definition 5.2

We call such a $\mathcal U_i$ given in Theorem 5.1 a lift of $\mathcal U$ with respect to $\varphi _i$.

Proof of Theorem 1.2

Due to [9, Theorem 1.2], there exist $\epsilon >0$ and finitely many spaces $M_1, \dots , M_N \in {\mathcal A(n,D,v_0)}$ and finite simplicial complexes $K_1, \dots , K_N$ such that

• $\bigcup _{i=1}^N U_{\epsilon }^{\mathrm {GH}}(M_i) = \mathcal A(n,D,v_0)$;

• any $M \in U_\epsilon ^{\mathrm {GH}}(M_i)$ admit a good covering whose nerve complex is isomorphic to $K_i$.

Here, $U_\epsilon ^{\mathrm {GH}}(X)$ denotes the $\epsilon$-neighborhood of X in $\mathcal A(n,D,v_0)$ with respect to the Gromov–Hausdorff distance. From this and Theorem 1.1, we obtain the first conclusion of Theorem 1.2.

We prove the second conclusion by contradiction. Suppose it does not hold. Then we would have sequences $\{M_i\}$, $\{M_i'\}$ in $\mathcal A(n,D,v_0)$ with $d_{GH}(M_i,M_i')<{\delta _i}$, $\lim \delta _i=0$, together with $\delta _i$-approximation $\theta _i:M_i\rightarrow M_i'$ such that

$$\begin{aligned} \sup _{x\in M_i} {|\theta _i(x), {k_i}(x)|}>c>0, \end{aligned}$$

(5.7)

for any Lipschitz homotopy equivalence ${k_i}:M_i\rightarrow M_i'$, where c is a constant not depending on i. Passing to a subsequence, we may assume that both $M_i$ and $M_i'$ converge to an Alexandrov space $M \in \mathcal A(n,D,v_0)$. We introduce a new positive number $\mu \ll c$. By Theorem 5.1, one can take a good $\mu$-cover $\mathcal U$${= \{U_j\}_{j \in J}}$ of M and a good $2\mu$-cover ${\mathcal {U}}_i$$=\{U_{ij}\}_{j \in J}$ of $M_i$ such that $\mathcal U_i$ is a lift of $\mathcal U$ with respect to some $\nu _i$-approximation $\varphi _i:M\rightarrow M_i$, where $\lim _{i\rightarrow \infty }\nu _i=0$. Let $\psi _i :M_i \rightarrow M$ be an $\nu _i$-approximation, which is an almost inverse of $\varphi _i$, in the sense that

$$\begin{aligned} \sup _{x \in M_i} |\varphi _i \circ \psi _i(x), x| \le \nu _i, \quad \sup _{x \in M} |\psi _i \circ \varphi _i(x), x| \le \nu _i. \end{aligned}$$

Note that $\theta _i \circ \varphi _i:M\rightarrow M_i'$ is a $2( \delta _i + \nu _i)$-approximation. Applying Theorem 5.1 to $\theta _i \circ \varphi _i$, we also obtain a $\nu _i'$-approximation $\varphi _i' : M \rightarrow M_i'$ with $\lim _{i\rightarrow \infty }\nu _i'=0$ and a lift $\mathcal U_i'$${= \{U_{ij}' \}_{j \in J}}$ of $\mathcal U$ with respect to $\varphi _i'$ such that

$$\begin{aligned} \lim _{i\rightarrow \infty } \sup _{x\in M} |\theta _i \circ \varphi _i(x), \varphi _i'(x)| = 0. \end{aligned}$$

(5.8)

Set $p_{ij}:= \varphi _i(p_j)$ and $p_{ij}':= \varphi _i'(p_j)$, which are centers of $U_{ij}$ and $U_{ij}'$, respectively. It follows that

$$\begin{aligned} \sup _{x \in M_i}|\theta _i(x), \varphi _i' \circ \psi _i(x)| \le \mu \end{aligned}$$

(5.9)

for large i. Let $\alpha _i : \mathcal N_{\mathcal U} \rightarrow \mathcal N_{\mathcal U_i}$ and $\alpha _i' : \mathcal N_{\mathcal U} \rightarrow \mathcal N_{\mathcal U_i'}$ be the isomorphisms given by the correspondence $U_j \mapsto U_{ij}$ and $U_j \mapsto U_{ij}'$, respectively. We now consider the following diagram:

Here, $\Theta , \zeta , \Theta _i, \zeta _i'$ are maps given by Corollaries 3.9 and 3.10, for $(M, \mathcal U)$, $(M_i, \mathcal U_i)$, and $(M_i', \mathcal U_i')$, respectively. For instance, $\Theta (x) = (\xi _j(x))_{j \in J}$ for $x \in M$, where $(\xi _j)_{j \in J}$ is a partition of unity by Lipschitz functions subordinate to $\left\{ \bar{U}_j \right\} _{j \in J}$, and $\zeta$ is a Lipschitz homotopy inverse of $\Theta$ given by Corollary 3.10. Now, we consider the compositions

$$\begin{aligned} {h_i:= \zeta \circ \alpha _i^{-1}\circ \Theta _i, \quad g_i:= \zeta _i'\circ \alpha '_i\circ \Theta ,} \end{aligned}$$

which are Lipschitz homotopy equivalences satisfying

$$\begin{aligned} \sup _{x \in M_i} |\psi _i(x), h_i(x)| \le 10 \mu ,\qquad \quad \sup _{x \in M} |\varphi _i'(x), g_i(x)| \le 10 \mu . \end{aligned}$$

(5.10)

Indeed, for $x \in M_i$, $\Theta _i(x)$ is contained in a unique open simplex . Then, $\alpha _i^{-1} \circ \Theta _i(x)$ is contained in . By the property of $\zeta$ stated in Corollary 3.10, we have $h_i(x) \in {U_{j_{\ell }}}$ for some $0\le \ell \le k$. On the other hands, since $x \in U_{i j_{\ell }}$, we have $|x, p_{i {j_\ell }}| \le 2\mu$ and $|\psi _i(x), p_{j_\ell }| \le 3 \mu$. Therefore, we obtain

$$\begin{aligned} |h_i(x), \psi _i(x)|&\le |h_i(x), p_{j_\ell }| + |p_{j_\ell }, \psi _i(x)| \le 4 \mu . \end{aligned}$$

Thus we obtain (5.10) for $h_i$. Similarly we obtain (5.10) for $g_i$. It follows from (5.10) and (5.9) that $\sup _{x\in M_i} |\theta _i(x), g_i \circ h_i(x)| \le 100 \mu$, which is a contradiction to (5.7). $\square$

For two metric spaces A and B, let us denote by

$$\begin{aligned}{}[A,B]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}} \end{aligned}$$

the set of all locally Lipschitz homotopy classes of locally Lipschitz maps from A to B. Let us denote by

$$\begin{aligned}{}[A,B] \end{aligned}$$

the set of all homotopy classes of continuous maps from A to B. For another metric space C and a locally Lipschitz map $f : A \rightarrow B$, we define a map $f^*: [B,C]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}} \rightarrow [A,C]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}}$ (and $f^*: [B,C] \rightarrow [A,C]$) by $f^*(g) := g \circ f$ up to locally Lipschitz homotopy (and up to homotopy, respectively). From the definition, for a locally Lipschitz map $g : B \rightarrow C$, we have

$$\begin{aligned} (g \circ f)^*= f^*\circ g^*. \end{aligned}$$

(5.11)

Proof of Corollary 1.5

Let us fix a good cover ${\mathcal {U}}$ of a $\sigma$-compact metric space X, and let K be the geometric realization of the nerve of ${\mathcal {U}}$. From [7, Corollary 1.3], the induced map

$$\begin{aligned}{}[{K}, Y]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}} \rightarrow [{K}, Y] \end{aligned}$$

is bijective. By Theorem 1.1, K is locally Lipschitz homotopy equivalent to X. Let $f : X \rightarrow {K}$ and $g : {K} \rightarrow X$ be locally Lipschitz homotopy equivalences such that $g \circ f$ and $f \circ g$ are locally Lipschitz homotopy equivalent to ${\mathrm {id}}_X$ and ${\mathrm {id}}_K$, respectively. By the contravariant property (5.11), the induced maps $g^*: [X,Y]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}} \rightarrow [{K}, Y]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}}$ and $f^*: [{K}, Y]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}} \rightarrow [X, Y]_{{{\mathrm {loc}}\text {-}{\mathrm {Lip}}}}$ are mutually inverse. So are $f^*: [{K}, Y] \rightarrow [X, Y]$ and $g^*: [X, Y] \rightarrow [{K}, Y]$. These imply the conclusion. $\square$

Remark that in the statement of Corollary 1.5, if X is compact, we obtain a natural bijection between the set of all Lipschitz homotopy classes of Lipschitz maps from X to Y and [X, Y].

A refinement of Corollary 1.5 is the following:

Corollary 5.3

Let X and Y be as in Corollary 1.5. For any continuous function $\epsilon : Y \rightarrow (0,\infty )$ and any continuous map $f : X \rightarrow Y$, there is a locally Lipschitz map $g : X \rightarrow Y$ which is homotopic to f and satisfies

$$\begin{aligned} |f(x), g(x)| < \epsilon (f(x)) \end{aligned}$$

for every $x \in X$.

Proof

This follows from [7, Corollary 4.4] and a discussion similar to the proof of Corollary 5.3. $\square$

6 Gluing with an Almost Isometry

For a small $1/n\gg \delta >0$, let ${\mathcal {R}}_M(\delta )$ the open set of M consisting of all $(n,\delta )$-strained points, which is called the $\delta$-regular part of M. In this section we prove Theorem 1.4 by making use of the notion of center of mass developed in [1] (see [2] for the original idea).

Proof of Theorem 1.4

Let $\theta :M\rightarrow M'$ be an $\epsilon$-approximation. Take $\mu >0$ such that the closed $3\mu$-neighborhood of D is contained in ${\mathcal {R}}_M(\delta )$. Let $D_1$ be the closed $2\mu$-neighborhood of D. We also denote by $D_0$ the closed $\mu$-neighborhood of D. By [1] and [14], for small enough $\epsilon >0$ with $\epsilon \ll \mu$, we have a $\tau (\delta )$-almost isometric map

$$\begin{aligned} g:D_1\rightarrow g(D_1)\subset {\mathcal {R}}_{M'}(2\delta ) \end{aligned}$$

such that $d(g(x),\theta (x))<\tau (\epsilon )$ for all $x\in D_1$. On the other hand from Theorem 1.2, we have a Lipschitz homotopy equivalence

$$\begin{aligned} f:M\rightarrow M' \end{aligned}$$

such that $d(f(x),\theta (x))<\tau (\epsilon )$ for all $x\in M$.

We shall construct a Lipschitz homotopy equivalence $h:M\rightarrow M'$ such that $h=g$ on D and $h=f$ on $M\setminus D_0$. Denote by E the closure of $D_1\setminus D$. Take $R>0$ such that each point $x\in E$ has an $(n,\delta )$-strainer of length $> R$. Let $\{x_i\}_{i=1}^N\ \subset E$ be a maximal family with $|x_i,x_j|\ge \delta R/2$ for each $i\ne j$. Then $\{B_i\}_{i=1}^N$ with $B_i:= B(x_i,\delta R/2)$ gives a covering of E. By Theorem 2.3, for each $1\le i\le N$ there are $\tau (\delta )$-almost isometric maps

$$\begin{aligned} f_i:B(x_i, 2\delta R) \rightarrow {\mathbb {R}}^n, \quad f_i':B(g(x_i), 2\delta R) \rightarrow {\mathbb {R}}^n. \end{aligned}$$

Let

$$\begin{aligned} d(x) =\min \{|D,x|, \mu \}. \end{aligned}$$

Note that the multiplicity of the covering $B(x_i, 2\delta R)$ is uniformly bounded by a constant $C_n$.

For every $x\in B_i$, let

$$\begin{aligned} h_i^0(x) := (f_i')^{-1}\left( \frac{d(x)}{\mu } f_i'(f(x)) +\left( 1-\frac{d(x)}{\mu }\right) f_i'(g(x))\right) . \end{aligned}$$

This extends to a Lipschitz map $h_i: \overline{M\setminus E}\cup B_i \rightarrow M'$ satisfying

$$\begin{aligned} h_i(x) = {\left\{ \begin{array}{ll} g(x), \,\,\, &{}x\in D \\ f(x), \,\,\, &{}x\in \overline{M\setminus D_0}, \end{array}\right. } \end{aligned}$$

and $|\theta (x), h_i(x)|<\tau (\epsilon )$ for all $x\in \overline{M\setminus E}\cup B_i$.

Now we are going to glue these Lipschitz maps $\{h_i\}$ to get a Lipschitz map $h:M\rightarrow M'$. Define a Lipschitz cut-off function $\varphi _i:M\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} \varphi _i(x) := {\left\{ \begin{array}{ll} 1- \frac{|x,x_i|}{\delta R}, &{} \,\,\, x\in B(x_i, \delta R) \\ 0, &{}\,\,\, {\mathrm{otherwise}}. \end{array}\right. } \end{aligned}$$

Let $F_i := \overline{M\setminus E}\cup B_1\cup \cdots \cup B_i$, and set

$$\begin{aligned} \psi _i(x) :=\sum _{j=1}^{i}\varphi _j(x),\,\, x\in M. \end{aligned}$$

Assuming that $h_{1\cdots i}: F_i\rightarrow M'$ is already defined in such a way that

$$\begin{aligned} {\left\{ \begin{array}{ll} |\theta (x), h_{1\cdots i}(x)|<\tau (\epsilon ), \, x\in F_i\\ h_{1\cdots i}(x) = {\left\{ \begin{array}{ll} g(x), \,\,\, &{}x\in D \\ f(x), \,\,\, &{}x\in \overline{M\setminus D_0}, \end{array}\right. } \end{array}\right. } \end{aligned}$$

(6.12)

define $h_{1\cdots i+1}: F_{i+1}\rightarrow M'$ by

Note that $h_{1\cdots i+1}$ also satisfies (6.12).

Finally we set $h:=h_{1\cdots N}:M\rightarrow M'$. Note that $|h(x), \theta (x)|<\tau (\epsilon )$ for all $x\in M$, and

$$\begin{aligned} h(x) = {\left\{ \begin{array}{ll} g(x), \,\,\, &{}x\in D \\ f(x), \,\,\, &{}x\in \overline{M\setminus D_0}. \end{array}\right. } \end{aligned}$$

Similarly we define $h':M'\rightarrow M$ by using the Lipschitz homotopy inverse $f'$ of f, $g':=g^{-1}$, $D':=g(D)$, $D_0':=g(D_0)$, $D_1':=g(D_1)$ and $d'=d\circ g^{-1}$ in place of f, g, D, $D_1$, and d. Note that every $y\in D_1'$ has $(n,2\delta )$-strainer of length $>R/2$. Obviously, $|h'\circ h(x), x|<\tau (\epsilon )$ and

$$\begin{aligned} h'\circ h(x) = {\left\{ \begin{array}{ll} x, \,\,\, &{}x\in D \\ f'\circ f (x), \,\,\, &{}x\in \overline{M\setminus D_0}. \end{array}\right. } \end{aligned}$$

To construct a Lipschitz homotopy between $1_M$ and $h'\circ h$, we use a method developed in [3]. We consider the product space $M\times M$ and denote by $\Delta \subset M\times M$ the diagonal. Introduce a positive constant $\sigma$ with $\epsilon \ll \sigma \ll \mu$ and take a sequence $0<\sigma _i <\sigma$ with $\lim \sigma _i = 0$. For every $\mathbf{x}:=(x_1,x_2)\in D_1\times D_1\cap A(\Delta ;\sigma _i,\sigma )$, let y denote the midpoint of a minimal geodesic joining $x_1$ and $x_2$, where $A(\Delta ;\sigma _i,\sigma ):=\overline{B(\Delta ,\sigma )\setminus B(\Delta ,\sigma _i)}$ is the annulus. Note that $\mathbf{y}:=(y,y)$ is the foot of a minimal geodesic from $\mathbf{x}$ to $\Delta$. It is possible to take points $z_1$ and $z_2$ of M such that

$$\begin{aligned}&\tilde{\angle } yx_i z_i > \pi -\tau (\delta ), i=1,2, \\&|y,z_1| =|y, z_2|, \,\, |\Delta , \mathbf{z}|=\sigma , \end{aligned}$$

where $\mathbf{z}:=(z_1,z_2)$. Then a direct computation shows that

$$\begin{aligned} |z_i, y| > |z_i, x_i| + (1-\tau (\delta )) |x_i, y|, \,\, i=1,2, \end{aligned}$$

and

$$\begin{aligned} | \mathbf{z}, \mathbf{y}| > |\mathbf{z}, \mathbf{x}| + (1-\tau (\delta )) |\mathbf{x}, \mathbf{y}|, \end{aligned}$$

which yields that

$$\begin{aligned} \tilde{\angle } \mathbf{z x y} > \pi -\tau (\delta ). \end{aligned}$$

The above argument shows that the distance function $d_{\Delta }$ from $\Delta$ is $(1-\tau (\delta ))$-regular on $D_1\times D_1\cap A(\Delta ;\sigma _i,\sigma )$. Now we consider a smooth approximation of a neighborhood $U_i$ of $D_1\times D_1\cap A(\Delta ;\sigma _i,\sigma )$. By [6] and [8], there are a smooth manifold $N_i$ and a bi-Lipschitz homeomorphism $\Phi _i:U_i\rightarrow N_i$ together with a gradient like unit vector field $X_i$ for $d_{\Delta }\circ \Phi _i^{-1}$ defined on $N_i$ such that if $\phi _i(\Phi _i(\mathbf{x}), t)$ denotes the integral curves of $-X_i$ starting at $\Phi _i(\mathbf{x})$, then for each $\mathbf{x}\in D_1\times D_1\cap A(\Delta ;\sigma _i,\sigma )$ with $|\mathbf{x},\Delta |=\sigma$,

$$\begin{aligned} |\Phi _i^{-1}\circ \phi _i(\Phi _i(\mathbf{x}),t_0),\Delta |=\sigma _i, \end{aligned}$$

for some $t_0<2(\sigma -\sigma _i)$. By combining the flow curves $\{\Phi _i^{-1}\circ \phi _i(\Phi _i(\mathbf{x}), t)\}_i$, we obtain a Lipschitz flow $\phi$ on $D_1\times D_1\cap \overline{B}(\Delta , \sigma )$ such that for each $\mathbf{x}\in D_1\times D_1\cap \overline{B}(\Delta , \sigma )$ with $|\mathbf{x},\Delta |=\sigma$, $\phi (\mathbf{x},s_0)\in \Delta$ for some $s_0<2\sigma$. For $\mathbf{x}=(x, h'\circ h(x))$, if we denote $\phi (\mathbf{x}, t) =(\phi ^1(\mathbf{x}, t), \phi ^2(\mathbf{x}, t))$, the union of $\phi ^1(\mathbf{x}, t)$ and $\phi ^2(\mathbf{x}, 1-t)$ provides the desired Lipschitz homotopy between $1_M$ and $h'\circ h$ on $D_1$.

We have just constructed a Lipschitz homotopy H(x, t) between $1_M$ and $h'\circ h$ on $D_1$. Recall that we have a Lipschitz homotopy F(x, t) between $1_M$ and $f'\circ f$. We have to glue F and H to get a Lipschitz homotopy G(x, t) between $1_M$ and $h'\circ h$ defined on M. Let $\rho :M\times [0,1]\rightarrow [0,1]$ be a Lipschitz function such that

$$\begin{aligned} \rho (x,t) = {\left\{ \begin{array}{ll} 0, \,\,\, &{} {\mathrm{on}} \,\, D\times [0,1]\cup D_0\times [1/2,1], \\ 1, \,\,\, &{} {\mathrm{on}} \,\, \overline{M\setminus D_1}\times [0,1]. \end{array}\right. } \end{aligned}$$

For every $(x,t)\in B_i\times [0,1]$, let

$$\begin{aligned} G_i^0(x,t) := f_i^{-1}\left( \rho (x,t) f_i( F(x,t)) + (1-\rho (x,t))f_i(H(x,t))\right) . \end{aligned}$$

This extends to a Lipschitz map $G_i:\overline{M\setminus E}\cup B_i\times [0,1]\rightarrow M$ satisfying

$$\begin{aligned} G_i(x,t) = {\left\{ \begin{array}{ll} x, \,\,\, &{} {\mathrm{on}}\,\,\,\overline{M\setminus E}\cup B_i\times 0,\\ f'\circ f(x), \,\,\, &{} {\mathrm{on}}\,\,\,\overline{M\setminus E}\cup B_i\times 1, \\ H(x,t), &{} {\mathrm{on}}\,\,\, D\times [0,1], \\ F(x,t), \,\,\, &{} {\mathrm{on}}\,\,\, \overline{M\setminus D_1}\times [0,1]. \\ \end{array}\right. } \end{aligned}$$

Assuming that $G_{1\cdots i}: F_i\times [0,1]\rightarrow M$ is already defined in such a way that

$$\begin{aligned} G_{1\cdots i}(x,t) = {\left\{ \begin{array}{ll} x, \,\,\, &{} {\mathrm{on}}\,\,\, F_i\times 0, \\ f'\circ f(x), \,\,\, &{} {\mathrm{on}}\,\,\,F_{i}\times 1, \\ H(x,t), \,\,\, &{} {\mathrm{on}}\,\,\, D\times [0,1], \\ F(x,t), \,\,\, &{} {\mathrm{on}} \,\,\, \overline{M\setminus D_1}\times [0,1], \end{array}\right. } \end{aligned}$$

(6.13)

define $G_{1\cdots i+1}: F_{i+1}\times [0,1]\rightarrow M$ by

Finally set $G:=G_{1\cdots N}$. Obviously G is Lipschitz, and $G=H$ on $D\times [0,1]$ and $G=F$ on $\overline{M\setminus D_1}\times [0,1]$, and thus G is a required Lipschitz homotopy between $1_M$ and $h'\circ h$. Similarly we obtain a Lipschitz homotopy between $1_{M'}$ and $h\circ h'$. This completes the proof of Theorem 1.4. $\square$

7 Further Problems

It is quite natural to expect that there should exist uniform Lipschitz constants of the Lipschitz homotopies in Theorems 1.2 and 1.4.