Diffeomorphism Stability and Codimension Three

Abstract

Given $k\in \mathbb {R}$, v, $D>0$, and $n\in \mathbb {N}$, let $\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ be a Gromov–Hausdorff convergent sequence of Riemannian n–manifolds with sectional curvature $\ge k$, volume $>v$, and diameter $\le D$. Perelman’s Stability Theorem implies that all but finitely many of the $M_{\alpha }$s are homeomorphic. The Diffeomorphism Stability Question asks whether all but finitely many of the $M_{\alpha }$s are diffeomorphic. We answer this question affirmatively in the special case when all of the singularities of the limit space occur along Riemannian manifolds of codimension $\le 3$. We then describe several applications. For instance, if the limit space is an orbit space whose singular strata are of codimension $\le 3$, then all but finitely many of the $M_{\alpha }$s are diffeomorphic.

Appendices

Appendix A: How to Glue $C^{1}$–Close Submersions

In this section we prove Theorem 5.3 and Corollary 5.6. Before doing so we establish several inductive gluing tools in Sect. 7.1, and we prove a result about stability of intersection patterns in Sect. 7.6.

1.1 Tools to Glue $C^{1}$–Close Submersions

In this subsection we prove Key Lemma 7.5, the main inductive gluing lemma that will allow us to prove Theorem 5.3. First we establish several preliminary results.

Lemma 7.2

(Submersion Isotopy Lemma) Let $G\subset M$ be an open subset of a Riemannian n–manifold M. Let $\pi :G\rightarrow \mathbb {R}^{l}$ be an $\eta$–almost Riemannian submersion, and let $p:G\rightarrow \mathbb {R}^{l}$ be any submersion with

$$\begin{aligned} \left| p-\pi \right| _{C^{1}}<\varepsilon . \end{aligned}$$

There are positive numbers $\eta _{1}$ and $\varepsilon _{1}$ that only depend on l so that if $\eta \in \left( 0,\eta _{1}\right)$ and $\varepsilon \in \left( 0,\varepsilon _{1}\right)$, then the homotopy $H:G\times [0,1]\rightarrow \mathbb {R}^{l}$,

$$\begin{aligned} H_{t}\equiv \pi +t(p-\pi ), \end{aligned}$$

from p to $\pi$ has the following properties.

• 1. $H_{t}$ is a submersion.

• 2. $\left| H_{t}-\pi \right| _{C^{1}}<\varepsilon$ and $\left| H_{t}-p\right| _{C^{1}}<\varepsilon$.

• 3. $|H_{t}-\pi |_{C^{0}}\le |p-\pi |_{C^{0}}$ and $|H_{t}-p|_{C^{0}}\le |p-\pi |_{C^{0}}$.

• 4. If $Z\subset G$ is open and $q:Z\rightarrow \mathbb {R}^{l}$ is a submersion with $\left| q-\pi \right| _{C^{1}}<\varepsilon$ and $\left| q-p\right| _{C^{1}}<\varepsilon$, then $\left| H_{t}-q\right| _{C^{1}}<\varepsilon$.

• 5. If $Z\subset G$ is open and $q:Z\rightarrow \mathbb {R}^{l}$ is a submersion with $\left| q-\pi \right| _{C^{0}}<\xi$ and $\left| q-p\right| _{C^{0}}<\xi$, then $\left| H_{t}-q\right| _{C^{0}}<\xi$.

• 6. $|DH_{\left( x,t\right) }\left( 0,\frac{\partial }{\partial t}\right) |\le |p-\pi |_{C^{0}}$.

• 7. If F is a subset of G with $\pi _{k}\circ p|_{F}=\pi _{k}\circ \pi |_{F}$, then $\pi _{k}\circ H_{t}|_{F}=\pi _{k}\circ p|_{F}=\pi _{k}\circ \pi |_{F}$ for all t. Here $\pi _{k}:\mathbb {R}^{l}\rightarrow \mathbb {R} ^{k}$ is projection to the first k–factors.

Proof

There is an $\varepsilon _{\mathrm {Riem}}>0$ so that for any Riemannian submersion $\pi _{\mathrm {Riem}}:G\rightarrow \mathbb {R}^{l}$, any map $h:G\rightarrow \mathbb {R}^{l}$ is a submersion, provided

$$\begin{aligned} \left| h-\pi _{\mathrm {Riem}}\right| _{C^{1}}<\varepsilon _{\mathrm { Riem}}. \end{aligned}$$

Take $\eta _{1}=\varepsilon _{1}=\frac{\varepsilon _{\mathrm {Riem}}}{2}$. Then any map $h:G\rightarrow \mathbb {R}^{l}$ is a submersion provided

$$\begin{aligned} \left| h-\pi \right| _{C^{1}}<\varepsilon _{1}. \end{aligned}$$

Since

$$\begin{aligned} H_{t}\equiv \pi +t(p-\pi ), \end{aligned}$$

Conclusions 2, 3, 4, and 5 follow from convexity of balls in Euclidean space, and Conclusion 7 follows from the definition of H. Conclusion 1 follows from Conclusion 2 and our choice of $\varepsilon$. Conclusion 6 follows from

$$\begin{aligned} DH_{\left( x,t\right) }\left( 0,\frac{\partial }{\partial t}\right) =p\left( x\right) -\pi \left( x\right) . \end{aligned}$$

$\square$

Lemma 7.3

For $\zeta >0$, let $W\Subset V\Subset G\subset M$ be three nonempty, open, pre-compact sets that satisfy

$$\begin{aligned} \mathrm {dist(}\overline{W},\overline{G\setminus V}\mathrm {)}>\zeta . \end{aligned}$$

There is a $C^{\infty }$ function $\omega :G\longrightarrow \left[ 0,1\right]$ that satisfies

• 1.
  $$\begin{aligned} \omega \left( x\right) =\left\{ \begin{array}{cc} 0 &{} \text {for }x\in W \\ 1 &{} \text {for }x\in G\setminus V \end{array} \right. \end{aligned}$$
• 2.
  $$\begin{aligned} \left| \nabla \omega \right| \le \frac{2}{\zeta }. \end{aligned}$$

Proof

Approximate $\mathrm {dist}(\overline{W},\cdot )$ and $\mathrm {dist}( \overline{G\setminus V},\cdot )$ by smooth functions in the $C^{0}$ -topology. Choose sublevels $C_{1}$ and $C_{2}$ of these approximations so that $W\Subset C_{1}$, $G\setminus V\Subset C_{2}$, and $\mathrm {dist} (C_{1},C_{2})>\zeta$. Using the techniques of [5, 6, 9], approximate $\mathrm {dist}(C_{i},\cdot )$ by smooth functions $f_{C_{i}}$ that satisfy $f_{C_{i}}\ge 0$, $|\nabla f_{C_{i}}|\le 2$, and $f_{C_{i}}|_{C_{i}}\equiv 0$. Since

$$\begin{aligned} \mathrm {dist}(C_{1},x)+\mathrm {dist}(C_{2},x)\ge \mathrm {dist} (C_{1},C_{2})>\zeta , \end{aligned}$$

and the technique of [5, 6, 9] allows the approximation to be as close as we please in the $C^{0}$–topology, we can choose the $f_{C_{i}}$s so that they also satisfy

$$\begin{aligned} f_{C_{1}}+f_{C_{2}}>\zeta . \end{aligned}$$

Then the function

$$\begin{aligned} \omega \equiv \frac{f_{C_{1}}}{f_{C_{1}}+f_{C_{2}}} \end{aligned}$$

satisfies Property 1. Moreover,

$$\begin{aligned} \left| \nabla \omega \right|= & {} \left| \frac{\left( f_{C_{1}}+f_{C_{2}}\right) \nabla f_{C_{1}}-f_{C_{1}}\nabla \left( f_{C_{1}}+f_{C_{2}}\right) }{\left( f_{C_{1}}+f_{C_{2}}\right) ^{2}} \right| \\= & {} \left| \frac{f_{C_{2}}\nabla f_{C_{1}}-f_{C_{1}}\nabla f_{C_{2}}}{ \left( f_{C_{1}}+f_{C_{2}}\right) ^{2}}\right| \\\le & {} 2\frac{f_{C_{2}}+f_{C_{1}}}{\left( f_{C_{1}}+f_{C_{2}}\right) ^{2}} \\\le & {} \frac{2}{\zeta }, \end{aligned}$$

as claimed. $\square$

Lemma 7.4

(Submersion Deformation Lemma) Let $W\Subset V\Subset G\subset M$ satisfy the hypotheses of Lemma 7.3, and let $\omega :G\longrightarrow \left[ 0,1\right]$ be as in the conclusion of Lemma 7.3. Let $\pi :G\rightarrow \mathbb {R}^{l}$ be an $\eta$–almost Riemannian submersion, where $\eta$ is as in Lemma 7.2 . Let $p:G\rightarrow \mathbb {R}^{l}$ be a submersion satisfying

$$\begin{aligned} \left| p-\pi \right| _{C^{1}}<\varepsilon \text { and }\left| p-\pi \right| _{C^{0}}<\xi <\varepsilon , \end{aligned}$$

and let $\varepsilon _{1}$ be as in Lemma 7.2.

If $0<\varepsilon +\frac{2\left| p-\pi \right| _{C^{0}}}{\zeta } <\varepsilon _{1}$, then the map $\psi :G\rightarrow \mathbb {R}^{l}$

$$\begin{aligned} \psi \left( x\right) =\pi \left( x\right) +\omega (x)\cdot (p-\pi )\left( x\right) \end{aligned}$$

is a submersion with the following properties.

• 1.
  $$\begin{aligned} \psi =\left\{ \begin{array}{cc} \pi &{} \text {on }W \\ p &{} \text {on }G\setminus V \end{array} \right. \end{aligned}$$
• 2.
  $$\begin{aligned} \left| \psi -\pi \right| _{C^{1}}<\varepsilon +\frac{2\left| p-\pi \right| _{C^{0}}}{\zeta }\text { and }\left| \psi -p\right| _{C^{1}}<\varepsilon +\frac{2\left| p-\pi \right| _{C^{0}}}{\zeta } \end{aligned}$$

• 3. If $U\subset G$ is open and $q:U\rightarrow \mathbb {R}^{l}$ is a submersion with $\left| q-\pi \right| _{C^{1}}<\varepsilon$ and $\left| q-p\right| _{C^{1}}<\varepsilon$, then $\left| \psi -q\right| _{C^{1}}<\varepsilon +\frac{2\left| p-\pi \right| _{C^{0}}}{\zeta }$.

• 4. If $U\subset G$ is open and $q:U\rightarrow \mathbb {R}^{l}$ is a submersion with $\left| q-\pi \right| _{C^{0}}<\xi$ and $\left| q-p\right| _{C^{0}}<\xi$, then $\left| \psi -q\right| _{C^{0}}<\xi$.

• 5. If F is a subset of G with $\pi _{k}\circ p|_{F}=\pi _{k}\circ \varphi |_{F}$, then $\pi _{k}\circ \psi |_{F}=\pi _{k}\circ p|_{F}=\pi _{k}\circ \varphi |_{F}$, where $\pi _{k}:\mathbb {R}^{l}\rightarrow \mathbb {R }^{k}$ is projection to the first k–factors.

Proof

Part 1 is a consequence of the definitions of $\psi$ and $\omega$.

Let $H_{t}:G\rightarrow \mathbb {R}^{l}$ be the isotopy from Lemma 7.2. Since $\psi (x)=H_{\omega (x)}(x)$, Parts 4 and 5 follow from Parts 5 and 7 of Lemma 7.2.

For any $x\in G$ and any $v\in T_{x}M$,

$$\begin{aligned} D\psi _{x}\left( v\right) =D\pi _{x}\left( v\right) +\omega \left( x\right) D\left( p-\pi \right) _{x}\left( v\right) +\left\langle \nabla \omega ,v\right\rangle \left( p-\pi \right) \left( x\right) . \end{aligned}$$

(7.4.1)

Since $\left| p-\pi \right| _{C^{1}}<\varepsilon$, $\left| \omega \right| \le 1$, and $\left| \nabla \omega \right| \le \frac{2}{\zeta }$,

$$\begin{aligned} \left| D\psi _{x}-D\pi _{x}\right|\le & {} \varepsilon +\left| \nabla \omega \right| |p-\pi |_{C^{0}} \\\le & {} \varepsilon +\frac{2|p-\pi |_{C^{0}}}{\zeta }. \end{aligned}$$

By rewriting $\psi$ as $\psi =p+\left( 1-\omega \right) \cdot (\pi -p)$, a similar argument gives

$$\begin{aligned} \left| D\psi _{x}-Dp_{x}\right| \le \varepsilon +\frac{2|p-\pi |_{C^{0}}}{\zeta }. \end{aligned}$$

Combining the previous two displays gives us Part 2.

If q is as in Part 3, then by Part 4 of Lemma 7.2,

$$\begin{aligned} \left| \left( Dq\right) _{x}-\left( D\pi _{x}+\omega \left( x\right) D\left( p-\pi \right) _{x}\right) \right| <\varepsilon . \end{aligned}$$

Combined with Eq. (7.4.1) this gives us Part 3.

Combining Part 2 with our hypothesis that $\varepsilon +\frac{2\left| p-\pi \right| _{C^{0}}}{\zeta }<\varepsilon _{1}$, we see that $\psi$ is a submersion. $\square$

Key Lemma 7.5

Let $\tilde{M}$ and S be compact Riemannian manifolds. Let

$$\begin{aligned} \tilde{W}\Subset \tilde{V}\Subset \tilde{G},\tilde{O}\subset & {} \tilde{M} \text { and} \\ G,O\subset & {} S \end{aligned}$$

be pre-compact open sets with

$$\begin{aligned} \mathrm {dist(closure}\left( \tilde{W}\right) ,\mathrm {closure}\left( \tilde{G }\setminus \tilde{V}\right) \mathrm {)}>\zeta \text {.} \end{aligned}$$

Let $p_{O}:\tilde{O}\longrightarrow O$, $p_{G}:\tilde{G}\longrightarrow G$ and $\mu :G\longrightarrow \mathbb {R}^{l}$ be $\eta$–almost Riemannian submersions with $\mu$ a coordinate chart.

Suppose $\tilde{W}\cap \tilde{O}\ne \emptyset$, $p_{G}\left( \tilde{W} \right) \cap p_{O}\left( \tilde{O}\right) \ne \emptyset$, and the restrictions of $p_{O}$ and $p_{G}$ to $\tilde{O}\cap \tilde{G}$ satisfy

$$\begin{aligned}&\left| \mu \circ p_{O}-\mu \circ p_{G}\right| _{C^{1}}<\varepsilon \text { and} \end{aligned}$$

(7.5.1)

$$\begin{aligned}&\left| \mu \circ p_{O}-\mu \circ p_{G}\right| _{C^{0}}<\xi <\varepsilon , \end{aligned}$$

(7.5.2)

where $\varepsilon +\frac{2\xi }{\zeta }<\varepsilon _{1}$, and $\varepsilon _{1}$ is as in Lemma 7.2.

Then there is a submersion

$$\begin{aligned} P:\tilde{W}\cup \tilde{O}\longrightarrow P\left( \tilde{W}\cup \tilde{O} \right) \subset S \end{aligned}$$

so that

$$\begin{aligned} P=\left\{ \begin{array}{cc} p_{G} &{} \mathrm {on}\,\,\,\tilde{W} \\ p_{O} &{} \mathrm {on}\,\,\,\tilde{O}\setminus \tilde{V}, \end{array} \right. \end{aligned}$$

(7.5.3)

and in addition, the following hold.

• 1. On $\tilde{G}\cap \tilde{O}$,

  $$\begin{aligned} \left| \mu \circ P-\mu \circ p_{G}\right| _{C^{1}}<\varepsilon + \frac{2\xi }{\zeta }\text { and }\left| \mu \circ P-\mu \circ p_{O}\right| _{C^{1}}<\varepsilon +\frac{2\xi }{\zeta }. \end{aligned}$$

• 2. If $\tilde{U}\subset \tilde{G}\cap \tilde{O}$ is open and $q: \tilde{U}\longrightarrow S$ is a submersion with $\left| \mu \circ q-\mu \circ p_{G}\right| _{C^{1}}<\varepsilon$ and $\left| \mu \circ q-\mu \circ p_{O}\right| _{C^{1}}<\varepsilon$, then $\left| \mu \circ P-\mu \circ q\right| _{C^{1}}<\varepsilon +\frac{2\xi }{\zeta }$.

• 3. If $\tilde{U}\subset \tilde{G}\cap \tilde{O}$ is open and $q: \tilde{U}\longrightarrow S$ is a submersion with $\left| \mu \circ q-\mu \circ p_{G}\right| _{C^{0}}<\xi$ and $\left| \mu \circ q-\mu \circ p_{O}\right| _{C^{0}}<\xi$, then $\left| \mu \circ P-\mu \circ q\right| _{C^{0}}<\xi$.

• 4. If F is a subset of $\tilde{O}\cap \tilde{G}$ with $\pi _{k}\circ \mu \circ p_{O}|_{F}=\pi _{k}\circ \mu \circ p_{G}|_{F}$, then $\pi _{k}\circ \mu \circ P|_{F}=\pi _{k}\circ \mu \circ p_{O}|_{F}=\pi _{k}\circ \mu \circ p_{G}|_{F}$. Here $\pi _{k}:\mathbb {R}^{l}\rightarrow \mathbb {R}^{k}$ is projection to the first k–factors.

Proof

By Lemma 7.4 there is a submersion $\psi :\tilde{G}\cap \tilde{O}\longrightarrow \psi \left( \tilde{G}\cap \tilde{O}\right) \subset \mathbb {R}^{l}$ so that

$$\begin{aligned} \psi =\left\{ \begin{array}{cc} \mu \circ p_{G} &{} \text {on }\tilde{W}\cap \tilde{O} \\ \mu \circ p_{O} &{} \text {on }\left( \tilde{G}\setminus \tilde{V}\right) \cap \tilde{O} \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} \left| \psi -\mu \circ p_{G}\right| _{C^{1}}<\varepsilon +\frac{2\xi }{\zeta }\text { and }\left| \psi -\mu \circ p_{O}\right| _{C^{1}}<\varepsilon +\frac{2\xi }{\zeta }. \end{aligned}$$

Therefore, the map

$$\begin{aligned} P:\tilde{W}\cup \tilde{O}\longrightarrow S \end{aligned}$$

defined by

$$\begin{aligned} P:=\left\{ \begin{array}{cc} p_{G} &{} \text {on }\,\,\,\tilde{W} \\ \mu ^{-1}\circ \psi &{} \text {on }\,\,\,\tilde{G}\cap \tilde{O} \\ p_{O} &{} \text {on }\,\,\,\tilde{O}\setminus \tilde{V} \end{array} \right. \end{aligned}$$

is a well defined submersion satisfying Eq. (7.5.3). Combining the definition of P with Parts 2, 3, 4, and 5 of the Submersion Deformation Lemma gives us Parts 1, 2, 3, and 4. $\square$

1.2 Stability of Intersection Patterns

Proposition 7.7

Let $\mathcal {C}$ be an ordered collection of m open subsets of a compact metric space X. Suppose that $\mathcal {C}$ and $\text {cl}\left( \mathcal {C}\right)$ have the same intersection pattern. Let $\mathcal {X}$ be the collection of compact subsets of X equipped with the Hausdorff metric, and let $\mathcal {X}^{m}$ be the m–fold product of $\mathcal {X}$.

There is a neighborhood $\mathcal {N}$ of cl$\left( \mathcal {C} \right)$ in $\mathcal {X}^{m}$ with the following property: If $\mathcal {D}$ is a collection of m open subsets of X with cl$\left( \mathcal {D }\right) \in \mathcal {N}$, then $\mathcal {D}$ and $\mathcal {C}$ have the same intersection pattern.

Proof

Since $\mathcal {C}$ and cl$\left( \mathcal {C}\right)$ have the same intersection pattern, there is an $\varepsilon >0$ so that if $C_{i},C_{j}\in \mathcal {C}$ are disjoint, then $\mathrm {dist}\left( c_{i},c_{j}\right) >\varepsilon$ for all $c_{i}\in C_{i}$ and $c_{j}\in C_{j}$. It follows that if $D_{i},D_{j}\in \mathcal {D}$ are close enough to $C_{i}$ and $C_{j}$, then $D_{i}$ and $D_{j}$ are disjoint.

On the other hand, if $x\in C_{i}\cap C_{j}$, then there is an $\eta >0$ so that $B\left( x,\eta \right) \subset C_{i}\cap C_{j}$. It follows that $D_{i}\cap D_{j}\ne \emptyset$ if the Hausdorff distances satisfy

$$\begin{aligned} \mathrm {dist}_{Haus}\left( C_{i},D_{i}\right)<\frac{\eta }{10}\text { and } \mathrm {dist}_{Haus}\left( C_{j},D_{j}\right) <\frac{\eta }{10}. \end{aligned}$$

$\square$

Proposition 7.8

Adopt the hypotheses of Theorem 5.3, and let

$$\begin{aligned} P_{k}:\cup _{i=1}^{k}\tilde{B}_{i}\left( \rho \right) \longrightarrow P_{k}\left( \cup _{i=1}^{k}\tilde{B}_{i}\left( \rho \right) \right) \subset S \end{aligned}$$

be a submersion with

$$\begin{aligned} \left| \left. P_{k}-\mu _{i}^{-1}\circ \tilde{p}_{i}\right| _{\tilde{ B}_{i}\left( \rho \right) }\right| _{C^{0}}<\xi \end{aligned}$$

(7.8.1)

for all i. If $\xi$ is sufficiently small, then

$$\begin{aligned} P_{k}\left( \cup _{i=1}^{k}\tilde{B}_{i}\left( \rho \right) \right) \cap B_{k+1}\left( \rho \right) \ne \emptyset \end{aligned}$$

if and only if

$$\begin{aligned} \cup _{i=1}^{k}B_{i}\left( \rho \right) \cap B_{k+1}\left( \rho \right) \ne \emptyset . \end{aligned}$$

Proof

We have $P_{k}\left( \cup _{i=1}^{k}\tilde{B}_{i}\left( \rho \right) \right) =\cup _{i=1}^{k}P_{k}\left( \tilde{B}_{i}\left( \rho \right) \right)$, and Inequalities (7.8.1) and (5.3.1) give us that $\cup _{i=1}^{k}P_{k}\left( \tilde{B}_{i}\left( \rho \right) \right)$ is Hausdorff close to $\cup _{i=1}^{k}B_{i}\left( \rho \right)$. So by Proposition 7.7,

$$\begin{aligned} \cup _{i=1}^{k}B_{i}\left( \rho \right) {\displaystyle \bigcap }B_{k+1}\left( \rho \right) \ne \emptyset \end{aligned}$$

if and only if

$$\begin{aligned} P_{k}\left( \cup _{i=1}^{k}\tilde{B}_{i}\left( \rho \right) \right) {\displaystyle \bigcap }B_{k+1}\left( \rho \right) \ne \emptyset . \end{aligned}$$

$\square$

1.3 Proofs of Theorem 5.3, Corollary 5.5, and Corollary 5.6

Proof of Theorem 5.3

Choose $\varepsilon _{0}>0$ so that

$$\begin{aligned} \varepsilon _{0}<\frac{\varepsilon _{1}}{2}, \end{aligned}$$

where $\varepsilon _{1}$ is as in Lemma 7.2. Choose $\xi _{0},\eta >0$ so that the conclusion of Proposition 7.8 holds with $\xi =\xi _{0}$ and so that

$$\begin{aligned} \left( 1+\eta \right) ^{2\left( \mathfrak {o}-1\right) }\varepsilon _{0}+ \frac{2}{\rho }\xi _{0}\left( \mathfrak {o}-1\right) \left( 1+\eta \right) ^{2\left( \mathfrak {o}-1\right) }<\frac{\varepsilon _{1}}{2}. \end{aligned}$$

Next we partition $\left\{ \tilde{B}_{i}\left( 3\rho \right) \right\} _{i=1}^{m_{l}}$ into $\mathfrak {o}$ subcollections of pairwise disjoint balls, where $\mathfrak {o}$ is the first order of $\left\{ \tilde{B} _{i}\left( 3\rho \right) \right\} _{i=1}^{m_{l}}$. To begin, we take $\mathcal {\tilde{B}}_{1}\left( 3\rho \right)$ to be a maximal subcollection of $\left\{ \tilde{B}_{i}\left( 3\rho \right) \right\} _{i=1}^{m_{l}}$ that is pairwise disjoint, and in general, for $j\in \left\{ 2,\ldots ,\mathfrak {o }\right\}$, we take $\mathcal {\tilde{B}}_{j}\left( 3\rho \right)$ to be a maximal pairwise disjoint subcollection of $\left\{ \tilde{B}_{i}\left( 3\rho \right) \right\} _{i=1}^{m_{l}}\setminus \left\{ \mathcal {\tilde{B}} _{1}\left( 3\rho \right) \cup \cdots \cup \mathcal {\tilde{B}}_{j-1}\left( 3\rho \right) \right\}$. Then every element of $\mathcal {\tilde{B}} _{j}\left( 3\rho \right)$ intersects at least one element from each of $\mathcal {\tilde{B}}_{1}\left( 3\rho \right) ,\cdots ,\mathcal {\tilde{B}} _{j-1}\left( 3\rho \right)$, so the first order of the collection $\mathcal { \tilde{B}}_{1}\left( 3\rho \right) \cup \cdots \cup \mathcal {\tilde{B}} _{j}\left( 3\rho \right)$ is at least j. Therefore for $j\ge \mathfrak {o} +1$, $\mathcal {\tilde{B}}_{j}\left( 3\rho \right) =\emptyset$, and $\mathcal {\tilde{B}}_{1}\left( 3\rho \right) \cup \cdots \cup \mathcal {\tilde{ B}}_{\mathfrak {o}}\left( 3\rho \right) =\left\{ \tilde{B}_{i}\left( 3\rho \right) \right\} _{i=1}^{m_{l}}$.

We let $\mathcal {\tilde{B}}_{j}\left( \rho \right)$ be the $\rho$–balls that have the same centers as the $\mathcal {\tilde{B}}_{j}\left( 3\rho \right)$s, and we let $\mathcal {B}_{j}\left( 3\rho \right)$ and $\mathcal {B }_{j}\left( \rho \right)$ be the corresponding subcollections of $\left\{ B_{j}\left( 3\rho \right) \right\} _{j=1}^{m_{l}}$ and $\left\{ B_{j}\left( \rho \right) \right\} _{j=1}^{m_{l}}$. We use the superscript $^{u}$ to denote the union of one of these subcollections. Thus for example, $\mathcal { \tilde{B}}_{1}^{u}\left( 3\rho \right)$ is the subset of M obtained by taking the union of each ball in $\mathcal {\tilde{B}}_{1}\left( 3\rho \right)$.

For each $j\in \left\{ 1,2,\ldots ,\mathfrak {o}\right\}$ and each i with $\tilde{B}_{i}\left( 3\rho \right) \in \mathcal {\tilde{B}}_{j}\left( 3\rho \right)$, we let

$$\begin{aligned} \hat{p}_{j}:\mathcal {\tilde{B}}_{j}^{u}\left( 3\rho \right) \longrightarrow \mathbb {R}^{l}\text { } \end{aligned}$$

be given by

$$\begin{aligned} \hat{p}_{j}|_{B_{i}\left( 3\rho \right) }=\tilde{p}_{i}, \end{aligned}$$

and

$$\begin{aligned} \hat{\mu }_{j}:\mathcal {B}_{j}^{u}\left( 3\rho \right) \longrightarrow \mathbb {R}^{l} \end{aligned}$$

be given by

$$\begin{aligned} \hat{\mu }_{j}|_{B_{i}\left( 3\rho \right) }=\mu _{i}. \end{aligned}$$

The proof is by induction on the index j of the $\mathcal {\tilde{B}} _{j}\left( 3\rho \right)$s. To formulate our induction statement for $k\in \left\{ 1,\ldots ,\mathfrak {o}\right\}$, we set

$$\begin{aligned} \mathcal {E}_{k}=\left( 1+\eta \right) ^{2\left( k-1\right) }\varepsilon + \frac{2}{\rho }\xi \left( k-1\right) \left( 1+\eta \right) ^{2\left( k-1\right) }. \end{aligned}$$

(7.9.2)

Our $k^{th}$ statement asserts the existence of a submersion

$$\begin{aligned} P_{k}:\cup _{j=1}^{k}\mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \longrightarrow P_{k}\left( \cup _{j=1}^{k}\mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \right) \subset S \end{aligned}$$

so that for all $s\in \left\{ 1,2,\ldots ,\mathfrak {o}\right\}$ on $\cup _{j=1}^{k}\mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \cap \mathcal { \tilde{B}}_{s}^{u}\left( 3\rho \right)$,

$$\begin{aligned}&\left| \hat{\mu }_{k}\circ P_{k}-\hat{\mu }_{k}\circ \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right| _{C^{0}}<\left( 1+\eta \right) ^{2k}\xi \text { and} \end{aligned}$$

(7.9.3)

$$\begin{aligned}&\left| \hat{\mu }_{k}\circ P_{k}-\hat{\mu }_{k}\circ \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right| _{C^{1}}<\mathcal {E}_{k}. \end{aligned}$$

(7.9.4)

Setting $P_{1}=\hat{\mu }_{1}^{-1}\circ \hat{p}_{1}$ and appealing to Eqs. (5.3.2) and (5.3.3) anchors the induction.

Since the collection $\left\{ \mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \right\} _{j=1}^{\mathfrak {o}}$ has first order $\mathfrak {o,}\left( \cup _{j=1}^{k}\mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \right) \cap \mathcal {\tilde{B}}_{k+1}^{u}\left( \rho \right) \ne \emptyset$. Combining this with $\mathcal {E}_{k}<\mathcal {E}_{\mathfrak {o}}<\varepsilon _{1}$ allows us to apply Key Lemma 7.5 with $p_{O}=P_{k}$ and $p_{G}=\hat{\mu }_{k+1}^{-1}\circ \hat{p}_{k+1}$ to get a new submersion

$$\begin{aligned} P_{k+1}:\cup _{j=1}^{k+1}\mathcal {\tilde{B}}_{j}^{u}\left( \rho \right) \longrightarrow P_{k+1}\left( \cup _{j=1}^{k+1}\mathcal {\tilde{B}} _{j}^{u}\left( \rho \right) \right) \subset S. \end{aligned}$$

It remains to verify hypotheses (7.9.3)$_{k+1}$ and ( 7.9.4) $_{k+1}$. The induction hypothesis,(7.9.3)$_{k}$, combined with our hypothesis that the differentials of the $\hat{\mu }_{i}$s are $\left( 1+\eta \right)$–bi-lipshitz gives

$$\begin{aligned} \left| \hat{\mu }_{k{+}1}\circ P_{k}{-}\hat{\mu }_{k+1}\circ \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right| _{C^{0}}&{=}&\left| \left( \hat{\mu } _{k+1}\circ \hat{\mu }_{k}^{-1}\right) \circ \left( \hat{\mu }_{k}\circ P_{k}{-} \hat{\mu }_{k}\circ \hat{\mu }_{s}^{-1}\circ \hat{p}_{s}\right) \right| _{C^{0}} \\< & {} \left( 1+\eta \right) ^{2}\left( 1+\eta \right) ^{2k}\xi \\= & {} \left( 1+\eta \right) ^{2\left( k+1\right) }\xi . \end{aligned}$$

So by Part 3 of the Key Lemma 7.5,

$$\begin{aligned} \left| \hat{\mu }_{k+1}\circ P_{k+1}-\hat{\mu }_{k+1}\circ \left( \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right) \right| _{C^{0}}<\left( 1+\eta \right) ^{2\left( k+1\right) }\xi , \end{aligned}$$

and ( 7.9.3)$_{k+1}$ holds.

Combining ( 7.9.4)$_{k}$ with the fact that the differentials of the $\hat{\mu }_{i}$s are $\left( 1+\eta \right)$–bi-lipshitz gives

$$\begin{aligned} \left| \hat{\mu }_{k+1}\circ P_{k}-\hat{\mu }_{k+1}\circ \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right| _{C^{1}}= & {} \left| \hat{\mu } _{k+1}\circ \hat{\mu }_{k}^{-1}\circ \left( \hat{\mu }_{k}\circ P_{k}-\hat{\mu } _{k}\circ \hat{\mu }_{s}^{-1}\circ \hat{p}_{s}\right) \right| _{C^{1}} \\< & {} \left( 1+\eta \right) ^{2}\left( \mathcal {E}_{k}\right) . \end{aligned}$$

So by Part 2 of Key Lemma 7.5 and ( 7.9.3)$_{k}$,

$$\begin{aligned}&\left| \hat{\mu }_{k+1}\circ \hat{P}_{k+1}-\hat{\mu }_{k+1}\circ \hat{\mu } _{s}^{-1}\circ \hat{p}_{s}\right| _{C^{1}} \\&\quad <\left( 1+\eta \right) ^{2}\left( \mathcal {E}_{k}\right) +\frac{2}{\rho }\left( 1+\eta \right) ^{2k}\xi \\&\quad =\left( 1+\eta \right) ^{2}\left( \left( 1+\eta \right) ^{2\left( k-1\right) }\varepsilon +\frac{2}{\rho }\xi \left( k-1\right) \left( 1+\eta \right) ^{2\left( k-1\right) }\right) \\&\qquad +\frac{2}{\rho }\left( 1+\eta \right) ^{2k}\xi \\&\quad =\left( 1+\eta \right) ^{2k}\varepsilon +\frac{2}{\rho }\xi k\left( 1+\eta \right) ^{2k} \\&\quad =\mathcal {E}_{k+1}. \end{aligned}$$

To complete the proof, we need to establish Eq. (5.3.4). To do so, we re-index so that $\tilde{B}_{m_{l}}\left( \rho \right) \subset$ $\mathcal {\tilde{B}}_{\mathfrak {o}}^{u}\left( 3\rho \right)$ and notice that

$$\begin{aligned} P|_{\mathcal {\tilde{B}}_{\mathfrak {o}}^{u}\left( \rho \right) }=P_{\mathfrak { o}}|_{\tilde{B}_{m_{l}}\left( \rho \right) }=\hat{\mu }_{\mathfrak {o} }^{-1}\circ \hat{p}_{\mathfrak {o}} \end{aligned}$$

by Equation (7.5.3). $\square$

Proof of Corollary 5.5

This is a consequence of Part 4 of Key Lemma 7.5 and the observation that at the $k^{th}$–stage of the induction, we glue $p_{O}=P_{k}$ to $p_{G}=\hat{\mu }_{k+1}^{-1}\circ \hat{p}_{k+1}$. $\square$

Proof of Corollary 5.6

First apply Theorem 5.3 to construct a submersion $\tilde{P}:\cup _{i=1}^{m_{l}}\tilde{B}_{i}\left( \rho \right) \longrightarrow S$ that is close to the $\tilde{p}_{i}$s in the sense that Inequalities (5.3.5) and (5.3.6) hold.

Since the first order of $\left\{ B_{i}\left( 3\rho _{R}\right) \right\} _{i\in I_{R}}$ is $\mathfrak {o}$, as in the proof of Theorem 5.3, for each $j\in \left\{ 1,2,\ldots ,\mathfrak {o}\right\}$, we construct a subcollection $\mathcal {B}_{j}\left( 3\rho _{R}\right)$ of $\left\{ B_{i}\left( 3\rho _{R}\right) \right\} _{i\in I_{R}}$ so that the balls of $\mathcal {B}_{j}\left( 3\rho _{R}\right)$ are pairwise disjoint, and the collection $\mathcal {B}_{1}\left( 3\rho _{R}\right) \cup \cdots \cup \mathcal {B}_{j}\left( 3\rho _{R}\right)$ has first order at least j.

For each $j\in \left\{ 1,2,\ldots ,\mathfrak {o}\right\}$, we set

$$\begin{aligned} p_{j}\equiv \mu _{j}\circ Q\circ R:\mathcal {B}_{j}^{u}\left( 3\rho _{R}\right) \longrightarrow \mathbb {R}^{l}, \end{aligned}$$

and note that since the $p_{j}$s are all coordinate representations of the same submersion, $Q\circ R$,

$$\begin{aligned} \text {Inequalities }(5.3.2)\text { and }(5.3.3)\text { hold with }\xi =\varepsilon =0 \end{aligned}$$

(7.9.5)

and the $p_{i}$s playing the role of the $\tilde{p}_{i}$s. Using this, for each $j\in \left\{ 1,2,\ldots ,\mathfrak {o}\right\}$, we successively apply the proof of Theorem 5.3 to deform $\tilde{P}$ on each $\mathcal {B}_{j}\left( 3\rho _{R}\right)$ so that it ultimately equals $Q\circ R$ on $\cup _{j=1}^{\mathfrak {o}}\mathcal {B}_{i}^{u}\left( \rho _{R}\right)$. For the first deformation, this is possible because Inequalities (5.6.1), (5.6.2), and (7.9.5) tell us that the $p_{j}$s are close to the $\tilde{p}_{j}$s. Via (5.3.5) and (5.3.6) it follows that the $p_{j}$s are close to local representations of $\tilde{P}$. In other words, we have that Inequalities (7.5.1) and (7.5.2) hold with $p_{O}=\tilde{P}$ and $p_{G}=Q\circ R$. This continues to be possible for subsequent deformations because Parts 2 and 3 of Key Lemma 7.5 tell us our deformations preserve Inequalities (7.5.1) and (7.5.2), provided $\xi$ and $\varepsilon$ are sufficiently small.

To explain why $P=Q\circ R$ on $\cup _{j=1}^{\mathfrak {o}}\mathcal {B} _{j}^{u}\left( \rho _{R}\right)$, we let $\tilde{P}_{0}$, $\tilde{P} _{1},\ldots ,\tilde{P}_{\mathfrak {o}}$ be the deformations of $\tilde{P}= \tilde{P}_{0}$. By combining Eq. (7.5.3) with the fact that $p_{1}=\mu _{1}\circ Q\circ R$, it follows that

$$\begin{aligned} \tilde{P}_{1}\equiv Q\circ R \end{aligned}$$

on $\mathcal {B}_{1}^{u}\left( \rho _{R}\right)$. By the same reasoning, we have

$$\begin{aligned} \tilde{P}_{k}\equiv Q\circ R \end{aligned}$$

on $\mathcal {B}_{k}^{u}\left( \rho _{R}\right)$, and Part 4 of Lemma 7.5 gives, via induction, that after the $k^{th}$ deformation, we have

$$\begin{aligned} \tilde{P}_{k}\equiv Q\circ R \end{aligned}$$

on $\cup _{j=1}^{k}\mathcal {B}_{j}^{u}\left( \rho _{R}\right)$. So setting $P\equiv \tilde{P}_{\mathfrak {o}}$, we see that $P=Q\circ R$ on $\cup _{j=1}^{ \mathfrak {o}}\mathcal {B}_{j}^{u}\left( \rho _{R}\right)$. $\square$

Appendix B: Conventions and Notations

We assume throughout that all metric spaces are complete, and the reader has a basic familiarity with Alexandrov spaces, including but not limited to the seminal paper by Burago, Gromov, and Perelman ([2]). Let X, $\mathcal {S}=\left\{ S_{i}\right\} _{i\in I}$, $\mathcal {N}$, and $\mathcal {K}$ be as in Theorem C, and let p, x, and y be points of X.

• 1. We call minimal geodesics in X segments.

• 2. We denote comparison angles with $\tilde{\sphericalangle }$.

• 3. We let $\Sigma _{p}X$ and $T_{p}X$ denote the space of directions and tangent cone at p, respectively, and we let $*$ denote the cone point.

• 4. For a geodesic direction $v\in T_{p}X$, we let $\gamma _{v}$ be the segment whose initial direction is v.

• 5. Following [24], given a subset $A\subset X$, $\Uparrow _{x}^{A}\subset \Sigma _{x}$ denotes the set of directions of segments from x to A, and $\uparrow _{x}^{A}\in$ $\Uparrow _{x}^{A}$ denotes the direction of a single segment from x to A. For $x\in S_{i}\subset X$ and $A\subset S_{i}$, we write $\left( \uparrow _{x}^{A}\right) _{S_{i}}$ or $\left( \Uparrow _{x}^{A}\right) _{S_{i}}$ if we are referring to intrinsic segments of S and $\left( \uparrow _{x}^{A}\right) _{X}$ or $\left( \Uparrow _{x}^{A}\right) _{X}$ if we are referring to extrinsic segments of X.

• 6. For a differentiable map $\Phi$ we write $D\Phi$ for the differential of $\Phi$. If $\Phi$ is real valued, we write $D_{v}\left( \Phi \right)$ for the derivative of $\Phi$ in the v direction.

• 7. Given a subset $A\subset X$, we say that $\mathrm {dist}_{A}\left( \cdot \right)$ is $\left( 1-\varepsilon \right)$–regular at x if there is a $v\in \Sigma _{x}$ so that the derivative of $\mathrm {dist}_{A}\left( \cdot \right)$ in the direction v satisfies

  $$\begin{aligned} D_{v}\mathrm {dist}_{A}>1-\varepsilon . \end{aligned}$$

• 8. We let px denote a segment from p to x.

• 9. We let $\sphericalangle (x,p,y)$ denote the angle of a hinge formed by segments px and py and $\tilde{\sphericalangle }(x,p,y)$ denote the corresponding comparison angle.

• 10. Following [21], we let $\tau :\mathbb {R}^{k}\rightarrow \mathbb {R }_{+}$ be any function that satisfies

  $$\begin{aligned} \lim _{x_{1},\ldots ,x_{k}\rightarrow 0}\tau \left( x_{1},\ldots ,x_{k}\right) =0, \end{aligned}$$

  and, abusing notation, we let $\tau :\mathbb {R}^{k}\times \mathbb {R} ^{n}\rightarrow \mathbb {R}$ be any function that satisfies

  $$\begin{aligned} \lim _{x_{1},\ldots ,x_{k}\rightarrow 0}\tau \left( x_{1},\ldots ,x_{k}|y_{1},\ldots ,y_{n}\right) =0, \end{aligned}$$

  provided $y_{1},\ldots ,y_{n}$ remain fixed. When making an estimate with a function $\tau$, we implicitly assert the existence of such a function for which the estimate holds. $\tau$ often depends on the limit space X and/or its dimension, but we make no other mention of this.

• 11. We identify $\mathbb {R}^{l}$ with $\mathbb {R}^{l}\times \left\{ 0\right\}$, and we let $\pi _{l}:\mathbb {R}^{l}\times \mathbb {R} ^{n-l}\longrightarrow \mathbb {R}^{l}$ be orthogonal projection to the first l factors of $\mathbb {R}^{n}$.

• 12. For $\lambda \in \mathbb {R}$, we call a function $f:\mathbb {R} \longrightarrow \mathbb {R}$ (strictly) $\lambda$–concave if and only if the function $g(t)=f(t)-\lambda t^{2}/2$ is (strictly) concave.

• 13. If U is an open subset of an Alexandrov space X, we call $f:U\longrightarrow \mathbb {R}$, (strictly) $\lambda$–concave if and only if its restriction to every geodesic is (strictly) $\lambda$–concave.

• 14. We abbreviate the statement “$\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }$ converges to X in the Gromov–Hausdorff topology” with the symbols, $M_{\alpha }\overset{GH}{ \longrightarrow }X$. Similarly, if $f_{\alpha }:M\longrightarrow \mathbb {R}$ and $f:X\longrightarrow \mathbb {R}$, we abbreviate “$\left\{ f_{\alpha }\right\} _{\alpha =1}^{\infty }$ converges to f in the Gromov–Hausdorff topology” with the symbols,write $f_{\alpha }\overset{GH}{\longrightarrow }f$.

• 15. Let V and W be normed vector spaces. For a linear map $L:V\longrightarrow W$, we set $\left| L\right| =\max \left\{ \left. \left| L\left( \frac{v}{\left| v\right| }\right) \right| \text { }\right| \text { }v\in V\setminus \left\{ 0\right\} \right\}$.

• 16. Let $U\subset M$ be open and $\Phi :U\longrightarrow \mathbb {R}^{n}$ be $C^{1}$. We write

  $$\begin{aligned} \left| \Phi \right| _{C^{0}}\equiv & {} \sup _{x\in U}\left\{ \left| \Phi (x)\right| \right\} \text { and } \\ \left| \Phi \right| _{C^{1}}\equiv & {} \max \left\{ \left| \Phi \right| _{C^{0}},\sup _{x\in U}\left\{ \left| D\Phi _{x}\right| \right\} \right\} \end{aligned}$$

• 17. We call a submersion, $\pi$, $\eta$–almost Riemannian if and only if for all unit horizontal vectors,

  $$\begin{aligned} \left| D\pi \left( v\right) -1\right| <\eta . \end{aligned}$$

• 18. An $\eta$–embedding ($\eta$–homeomorphism) is an embedding (homeomorphism) that is also an $\eta$–Gromov–Hausdorff approximation.

• 19. Volume of subsets of Alexandrov spaces means rough volume as defined in [2].

• 20. For $\lambda >0$, we write

  $$\begin{aligned} \lambda X \end{aligned}$$

  for the metric spaces obtained from X by rescaling all distances by $\lambda$.

• 21. We write N or $N_{i}$ for an element of $\mathcal {N}$; K or $K_{i}$ for an element of $\mathcal {K};$ and S or $S_{i}$ for an element of $\mathcal {S}$. Thus we redundantly write

  $$\begin{aligned} \mathcal {S}= & {} \left\{ S_{i}\right\} _{i} \\= & {} \left\{ \mathcal {K}_{k}\right\} _{k}\cup \left\{ N_{n}\right\} _{n}. \end{aligned}$$

• 22. We set

  $$\begin{aligned} \mathcal {S}^{\mathrm {ext}}\equiv \mathcal {S}\cup \left( X\setminus \cup _{S\in \mathcal {S}}S\right) . \end{aligned}$$

• 23. We use superscripts to denote components of vectors in subspaces. So, for example, if V is a subspace of W, then $U^{V}$ is the component of U in V.

• 24. We write $\mathbb {S}^{n}$ for the unit sphere in $\mathbb {R}^{n+1}$.

• 25. We set

  $$\begin{aligned} B\left( p,r\right) \equiv \left\{ \left. x\in X\text { }\right| { \text {dist}}\left( x,p\right) <r\right\} . \end{aligned}$$

• 26. We use $A\Subset B$ to mean that the closure of A is contained in the interior of B.

• 27. We say that a collection of sets $\mathcal {C}$ has first order $\le \mathfrak {o}$ if and only if each $C\in \mathcal {C}$ intersects no more than $\mathfrak {o}-1$ other members of $\mathcal {C}$.