Persistence, Metric Invariants, and Simplification

Home , Injective metric space, Intrinsic metric, Metric space, Size function

Persistence, Metric Invariants, and Simpliﬁcation

Dissertation

Presented in Partial Fulﬁllment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University

Osman Berat Okutan, M.S.

Graduate Program in Mathematics

The Ohio State University

2019

Dissertation Committee:

Facundo M´emoli,Advisor Matthew Kahle Jean-Franc¸oisLafont c Copyright by

Osman Berat Okutan

2019 Abstract

Given a metric space, a natural question to ask is how to obtain a simpler and faithful approximation of it. In such a situation two main concerns arise: How to construct the approximating space and how to measure and control the faithfulness of the approximation.

In this dissertation, we consider the following simplification problems: Finite approximations of compact metric spaces, lower cardinality approximations of filtered simplicial complexes, tree metric approximations of metric spaces and finite metric graph approximations of compact geodesic spaces. In each case, we give a simplification construction, and measure the faithfulness of the process by using the metric invariants of the original space, including the Vietoris-Rips persistence barcodes.

ii For Esra and Elif Beste

iii Acknowledgments

First and foremost I’d like to thank my advisor, Facundo M´emoli,for our discussions and for his continual support, care and understanding since I started working with him. I’d like to thank Mike Davis for his support especially on Summer 2016. I’d like to thank Dan

Burghelea for all the courses and feedback I took from him. Finally, I would like to thank my wife Esra and my daughter Elif Beste for their patience and support. This research was supported by NSF grants AF 1526513, DMS 1723003, CCF 1740761, IIS-1422400 and

CCF-1526513 .

iv Vita

May 2010 ...... B.S. in Mathematics Bilkent University May 2012 ...... M.S. in Mathematics Bilkent University September 2012 - present ...... Graduate Teaching Associate Department of Mathematics The Ohio State University.

Publications

Research Publications

O. Okutan, E. Yalcin “Free actions on products of spheres at high dimensions”. Algebraic & Geometric Topology 13, pp: 2087-2099, 2013.

Fields of Study

Major Field: Mathematics Specialization: Topological Data Analysis, Metric Geometry

v Table of Contents

Page

Abstract ...... ii

Dedication ...... iii

Acknowledgments ...... iv

Vita...... v

List of Figures ...... ix

1. Introduction ...... 1

1.1 Content ...... 3

2. Background ...... 9

2.1 Metric Geometry ...... 9 2.1.1 Gromov-Hausdorﬀ Distance ...... 10 2.1.2 Hyperbolicity ...... 15 2.1.3 Injective/Hyperconvex Spaces ...... 17 2.2 Persistence ...... 19 2.2.1 Persistence Sequence ...... 22 2.3 Interleaving Distance ...... 27 2.4 Metric Graphs ...... 28 2.5 Reeb Graphs ...... 35 2.5.1 Stability of Reeb Metric Graphs ...... 39 2.5.2 Smoothings ...... 40 2.6 Diﬀerential Topology ...... 42

vi 3. A Geometric Characterization of Vietoris-Rips Filtration ...... 46

3.1 Introduction ...... 46 3.2 Persistence via Metric Pairs ...... 48 3.3 Isomorphism ...... 52 3.3.1 Stability of Metric Homotopy Pairings ...... 54 3.4 Application to the Vietoris-Rips Filtration ...... 56 3.4.1 Products and Wedge Sums ...... 56 3.5 Applications to the Filling Radius ...... 60 3.5.1 Bounding Barcode Length via Spread ...... 60 3.5.2 Bounding the Filling Radius ...... 62 3.5.3 Stability ...... 64

4. Finite Approximations of Compact Metric Spaces ...... 67

4.1 Introduction ...... 67 4.2 Discrete Length Structures ...... 68 4.3 Discrete Length structures on Metric Spaces ...... 69 4.4 Proof of Theorem 4.1.1 ...... 73

5. Metric Graph Approximations of Geodesic Spaces ...... 75

5.1 Introduction ...... 75 5.2 Graph Approximations ...... 79 5.3 Tree Approximations ...... 82

6. The Distortion of the Reeb Quotient Map on Riemannian Manifolds ...... 92

6.1 Introduction ...... 92 6.2 Distortion of the Reeb Quotient Map ...... 95 6.3 Thickness ...... 97 6.3.1 A Calculation: Thickened Filtered Graphs ...... 99 6.4 The Diameter of a Fiber of the Reeb Quotient Map ...... 103 6.5 The Bound of Theorem 6.1.1 for Thickened Graphs ...... 105

7. Reeb Posets and Metric Tree Approximations ...... 109

7.1 Introduction ...... 109 7.2 Posets ...... 111 7.3 Reeb Constructions ...... 115 7.3.1 Poset Paths and Length Structures ...... 115

vii 7.3.2 Reeb Posets ...... 117 7.3.3 Reeb Tree Posets ...... 119 7.4 Hyperbolicity for Reeb Posets ...... 122 7.5 Approximation ...... 124 7.6 An Application to Metric Graphs and Finite Metric Spaces ...... 126 7.7 Example where Φ ∼ Υ...... 129

8. Quantitative Simpliﬁcation of Filtered Simplicial Complexes ...... 132

8.1 Introduction ...... 132 8.2 Gromov-Hausdorff and Interleaving Type Distances between Filtered Sim- plicial Complexes ...... 139 8.2.1 Gromov-Hausdorff Distance between Filtered Simplicial complexes 139 F 8.2.2 The Interleaving Type Distance dI between Filtered Simplicial Com- plexes ...... 143 F 8.2.3 Remarks About the Definition of dI ...... 149 8.2.4 Stability and the Proof of Theorem 8.1.1 ...... 150 8.3 The Vertex Quasi-distance and Simplification ...... 152

8.3.1 Computing δX (v, w): The Procedure ComputeCodensityMatrix() . 155 8.3.2 Specializing δX (v, w) According to Homology Degree ...... 158 8.4 An Application to the Vietoris-Rips Filtration of Finite Metric Spaces and Graphs ...... 160 8.4.1 Finite Metric Spaces ...... 160 8.4.2 Application to Metric Graphs ...... 163 F 8.5 Classiﬁcation of Filtered Simplicial Complexes via dI ...... 168 F 8.6 An Example where dI dGH ...... 171 8.7 Chain Construction ...... 173

9. Future work ...... 176

Bibliography ...... 178

viii List of Figures

Figure Page

1.1 The space on the right is a simpliﬁcation of the space on the left...... 2

1.2 B(X, r) is the r-neighborhood of X in E. Notice how the small loop in X is ﬁlled...... 3

1.3 A four point approximation of a metric space with three components. . . . .4

2.1 A metric graph ...... 29

2.2 Reeb graph Xh of the height function h on X...... 35

3.1 A big sphere X with a small handle. In this case, as r > 0 increases, 2 Br(X, κ(X)) changes homotopy type from that of X to that of S as soon as r > r0 for some r0 < FillRad(X)...... 63

2 ab a a 6.1 TA = a2+b2 depends only on b and converges to 0 as b → 0...... 98

6.2 A 2-dimensional thickened ﬁltered graph...... 99

6.3 A 2-dimensional thickened 3-fork...... 100

6.4 An inverse 2-dimensional thickened fork...... 102

6.5 A vertical fork...... 106

7.1 A ﬁnite metric space embedded in a metric graph ...... 110

7.2 Let R ≥ r > 0 and consider the metric graph from the ﬁgure. Let Zn be the ﬁ- nite subset {p, x0, . . . , xn, y1, . . . , yn}. We show that Φ(Zn) ∼ 2 log(4n) hyp(Zn) and Υp(Zn) = 2 log(4n + 4) hyp(Zn)...... 130

ix 8.1 These two ﬁnite spaces have the same Vietoris-Rips PH≥1, see Example 8.4.3. 133

∗ ∗ 8.2 X := ∆3...... 148

8.3 A simple metric graph...... 166

F 8.4 These two ﬁltered simplicial complexes are at 0 dI -distance while they are at 1 Gromov-Hausdorﬀ distance at least 2 ...... 172

x Chapter 1: Introduction

In Data Analysis, a data set can generally be endowed with a metric structure. This enables the analysis of the data set not just from a statistical point of view, but also from a geometric point of view. Topological Data Analysis tries to combine and take advantage of both the quantitative (but albeit often noisy) nature of Data and the qualitative nature of

Topology [19].

Geometric intuition tells us that metric spaces have topological features and these features have quantitative properties like size. To be able to recognize these quantitative properties, one needs to go beyond the topological structure induced by the metric, since a given topological space can be metrized in many different ways. Furthermore, as an example, if we consider a finite metric subspace of a unit circle, it is supposed to have an inherited circularity property as a metric space, but the underlying topological space is just discrete. Similarly, if we only look at the underlying topology, finite metric spaces have discrete topology, which can not explain any expected topological feature of the metric space. The main insight of persistence ([69, 38, 19]) for metric spaces is the following: A metric space does not simply induce a topological space, but a family of topological spaces indexed by non-negative real numbers. This family generally arises as a filtration. Then one can observe how topological features change as we change the index.

1 Figure 1.1: The space on the right is a simpliﬁcation of the space on the left.

Given a metric space X, the most common way of obtaining a family of topological spaces

r r is via the Vietoris-Rips filtration (VR (X))r≥0 , where VR (X) is the simplicial complex with the vertex set X and simplices given by finite subsets of X with diameter less than or equal to r. There is also a more geometric method of obtaining a filtration, which is equivalent to the Vietoris-Rips filtration up to homotopy. Given a metric space X, there are several natural metric spaces into which X can be isometrically embedded; for example the Kuratowski space κ(X) and the tight span E(X) [34]. These spaces have many nice metric and topological properties, which mainly follow from being hyperconvex [53]. They also have very nice categorical properties in the category of metric spaces [53]. Now, let H be such a natural space associated to X. Then, we can look at open or closed r-neighborhoods

Br(X) of X in H, and investigate how the topology of this ﬁltration changes. We make this interpretation precise in Chapter 3. Note that for r = 0 we have the underlying topological space of X itself and as r increases it starts to look more like H.

One of the main concerns of Data Analysis is obtaining simpler and faithful representations of data. In this spirit, the main theme of this thesis is utilizing persistence and metric invariants for simpliﬁcation of metric objects. More speciﬁcally we try to answer questions like the following: Let X be a metric object (it can be a metric space, a geodesic

2 E E B(X,r) X X

Figure 1.2: B(X, r) is the r-neighborhood of X in E. Notice how the small loop in X is ﬁlled.

space, a filtered simplicial complex etc). How can we measure the complexity of X? Given a measurement of complexity, how can we construct new metric objects from X with less complexity and how much does that object differ from X? What type of quantification of difference should be used for such comparison? Given a family F of simple metric objects, how similar can X be to an object in F ?

1.1 Content

In Chapter 2, we give the main deﬁnitions and results in Metric Geometry, Topological

Data Analysis, and Topology which we are going to use in the rest of this thesis. This chapter includes many classical concepts and results as well as new ones which we introduce. In particular, the stability result of Reeb graphs (Theorem 2.5.8) and the eﬀect of -smoothings on the ﬁrst Betti number of a metric graph (Proposition 2.5.10) are novel results we proved in Chapter 2 and they are essential for the rest of our work.

3 z x y

Figure 1.3: A four point approximation of a metric space with three components.

In Chapter 3 we establish a precise relationship (i.e. a ﬁltered homotopy equivalence)

between the Vietoris-Rips simplicial ﬁltration of a metric space and a more geometric (or

extrinsic) way of assigning a persistence module to a metric space, which consists of ﬁrst

embedding it into a larger space and then taking the persistence homology of the ﬁltration

obtained by considering the increasing neighborhoods of the original space inside the ambient

space (see Figure 1.2). These neighborhoods are also metric spaces and we beneﬁt from this,

for example, in obtaining a short proof of the K¨unnethformula for persistent homology.

In Chapter 4, we consider ﬁnite approximations of compact metric spaces given an upper bound for the cardinality of the approximating space (see Figure 1.3). The main result (Theorem 4.1.1) of this chapter has a striking similarity with one of the main results

(Theorem 5.1.1) about graph approximations of compact geodesic spaces in Chapter 5, which hints a deeper and more general result about approximations in general.

In Chapter 5, we consider graph approximations of compact geodesic spaces (see Figure

1.1). A classical result in metric geometry is that any compact geodesic space can be approximated in the Gromov-Hausdorﬀ sense arbitrarily well by ﬁnite metric graphs. The classical construction for this approximation consists of taking an -net N from the compact geodesic

4 space and inducing a graph structure on N based on the proximity of its points. However, this construction does not provide any control on the first Betti number of the approximating graphs. If we interpret a graph approximation to a geodesic space as a simplification of the space, we encounter the following problem: As gets smaller, the approximating graphs mentioned above become quite complicated themselves, i.e. their first Betti numbers grow without bound. To get a better handle on this problem, we introduce the following invariant.

Given a compact geodesic space X, deﬁne:

X δn := {dGH(X,G): G a ﬁnite metric graph β1(G) ≤ n},

where dGH denotes the Gromov-Hausdorﬀ distance. We study the rate of decay of this sequence, speciﬁc elements in the sequence like δX , δX and metric graph constructions for β1(X) 0

X obtaining upper and lower bounds. Note that the element δ0 corresponds to metric tree approximations.

In Chapter 6, we analyze approximations of compact Riemannian manifolds by Reeb graphs of Morse functions. We generalize the results of Gromov and Zinov’ev [43, 73], which give measure theoretic bounds on the distortion of the Reeb quotient map of a Morse function, to arbitrary n-dimensional closed Riemannian manifolds. In order to do this, we introduce a metric invariant Tf associated to any ﬁltered metric space f : X → R (see Section 6.3) which we refer to as the thickness of f : X → R. This invariant gives a quantitative measure of how the volume of level sets of f is distributed with respect to their diameters.

In Chapter 7, we consider tree approximations of metric spaces. From the standpoint of applications, datasets which can be associated a tree representation can be readily visualized because trees are planar graphs. When a dataset does not directly lend itself to being represented as by a tree, motivated by the desire to visualize it, the question arises of what

5 is the closest tree to the given dataset. In this sense, one would then want to have (1) ways

of quantifying the treeness of data and, (2) eﬃcient methods for actually computing a tree

that is (nearly) optimally close to the given dataset. There are three diﬀerent but related

ways in which trees can be mathematically described. The ﬁrst one is poset theoretic: a

tree is a partially ordered set such that any two elements less than a given element are

comparable, or in other words there is a unique way to go down the poset. The second is

graph theoretic: a tree is a graph without loops. Finally, there is the metric way: a tree

metric space is a metric space which can be embedded in a metric tree (graph). This last

description is the bridge between data analysis and combinatorics of trees. Through it, we

can ask and eventually answer the following questions: How tree-like is a given metric data set? How does this treeness aﬀect its geometric features? How can we obtain a tree which is close to a given dataset?

For a metric space X, there exists a metric invariant called hyperbolicity (see Section 2.1.2) denoted by hyp(X) such that hyp(X) ≥ 0 and it is equal to zero if and only if X is a tree

metric. A natural question that ensues is whether the relaxed condition that hyp(X) be

small (instead of hyp(X) = 0), guarantees the existence of a tree metric on X which is close

to dX . In this respect, in [42] Gromov shows that for each ﬁnite metric space (X, dX ), there

exists a tree metric tX on X such that

||dX − tX ||∞ ≤ Υ(X) := 2 hyp(X) log(2|X|), where |X| is the cardinality of X. Despite the seemingly unsatisfactory fact that Υ(X) blows up with the cardinality of X (unless hyp(X) = 0), it is known that this bound is asymptotically tight [27]. This suggests searching for alternative bounds which may perform better in more restricted scenarios. We reﬁne Gromov’s bound Υ(X) by identifying a quantity Φ(X) that is related to but often much smaller than Gromov’s Υ(X).

6 In Chapter 8, we consider simpliﬁcations of ﬁltered simplicial complexes. For a subset

t I of R, a ﬁltered simplicial complex indexed over I is a family (X )t∈I of simplicial complexes such that for each t ≤ t0 in I, Xt is contained in Xt0 . Filtered simplicial complexes arise in topological data analysis for example as Vietoris-Rips or Cechˇ complexes of metric spaces

[37]. Simplicial complexes have the advantage of admitting a discrete description, hence they are naturally better suited for computations when compared to arbitrary topological spaces.

A useful and computationally feasible way of analyzing the scale dependent features of a

ﬁltered simplicial complex is through persistent homology and persistence diagrams/barcodes

[19, 37]. Given a ﬁltered simplicial complex X∗, for a given k ∈ N, eﬃcient computation of

∗ its k-th dimensional persistent homology PHk(X ) is studied in many papers, for example

[38, 75, 31, 39].

To reduce computational complexity, in the interest of being able to process large datasets, an important task is that of simplifying filtered simplicial complexes (that is, reducing the total number of simplices) in a way such that it is possible to precisely quantify the trade-off between degree of simplification and loss/distortion of homological features [38, 52, 71, 21,

31, 33, 14]. In this chapter we consider the effect on persistent homology of removing a vertex and all cells containing it. In this respect, our study is related to [71, Section 7] and [21, Section 6]. A standard measure of the change in persistent homology is called the interleaving distance[10], which is, by the Isometry Theorem [56, Theorem 3.4], isometric to the bottleneck distance between persistent barcodes. To quantify the distortion at the persistent homology level incurred by operations carried out at the simplicial level, we introduce an interleaving type distance for filtered simplicial complexes which is compatible with the distance between their persistent homology signatures. More precisely, persistent homology is stable with respect to this new metric. We bound the effect of removing a vertex

7 with respect to this new metric in terms of a new invariant that we call the codensity of the vertex, which in turn gives a bound on the change in persistent homology. Finally, we introduce a construction which we call the chain construction which takes an arbitrary family of simplicial complexes and produces a ﬁltered simplicial complex with the same persistent homology.

Finally, in Chapter 9, we discuss some open problems and future directions arising from our research.

8 Chapter 2: Background

In this chapter, we give the necessary deﬁnitions and results we are going to use in the

rest of the paper.

We start Section 2.1 by stating a result about connected metric spaces and Coarea formula for Riemannian manifolds. Then we review Gromov-Hausdorﬀ distance and introduce a novel

version of it for geodesic spaces. Then we review hyperbolicity and injective/hyperconvex

metric spaces. In Section 2.2, we review persistence modules and Vietoris-Rips ﬁltration.

Then we introduce persistence sequences.In Section 2.4, we review metric graphs and do

an analysis of paths in ﬁnite metric graphs. In Section 2.5, we review Reeb graphs and -

smoothings. Here we also prove novel results about the stability of Reeb graph construction

and the eﬀect of smoothings on the ﬁrst Betti number. Finally, in Section 2.6, we prove a

few results about Morse functions that we need later.

2.1 Metric Geometry

The following is a result about covers of connected metric spaces.

Proposition 2.1.1. Let X be a connected metric space and A be a ﬁnite cover of X. Then

X diam(X) ≤ diam(A). A∈A

9 For a proof of the Proposition 2.1.1, see the proof of [18, p. 53, Lemma 2.6.1].

For an integer k ≥ 0, we denote the kth Hausdorﬀ measure [18] on a metric space by µk.

We have the following coarea formula (see [40, Theorem 3.2.12, p. 249]).

Proposition 2.1.2 (Coarea Formula). If f : X → R is a smooth L-Lipschitz function deﬁned on an n-dimensional Riemannian manifold X, then for each t0 ≤ t1 ∈ R we have

Z t1 n−1 −1 n −1 µ (f (t)) µ (f [t0, t1]) ≥ dt. t0 L

2.1.1 Gromov-Hausdorﬀ Distance

Gromov-Hausdorﬀ distance is a way of measuring how similar two metric spaces are.

There are several equivalent ways of deﬁning the Gromov-Hausdorﬀ distance (see [18, Section

7.3]). We are going to deﬁne it using correspondences.

Deﬁnition 2.1.1 (Correspondences). • A correspondence R between two given sets X

and Y , is a relation between them such that for all x in X, there exists a y0 in Y such

that x R y0 and for each y in Y , there exists an x0 in X such that x0 R y.

• A correspondence between pointed sets (X, p) and (Y, q) is a correspondence R between

X and Y such that p R q.

• A correspondence R0 between X and Y is called a subcorrespondence of R if x R0 y

implies that x R y.

• If R is a correspondence between X and Y and S is a correspondence between Y and

Z, then we deﬁne the relation S ◦R between X and Z as follows: x S ◦R z if there exists

y in Y such that x R y and y S z. Note that S ◦ R is a correspondence between X,Z.

10 Note that the composition of pointed correspondences is a pointed correspondence and

the composition of subcorrespondences is a subcorrespondence of the composition.

Deﬁnition 2.1.2 (Distortion of a correspondence). Let (X, dX ) and (Y, dY ) be metric spaces and R be a correspondence between X and Y . The metric distortion dis(R) of the correspondence R is deﬁned as

0 0 dis(R) := sup |dX (x, x ) − dY (y, y )|. (x,y),(x0,y0)∈R

Remark 2.1.3. • If R0 is a subcorrespondence of R, then dis(R0) ≤ dis(R).

• dis(S ◦ R) ≤ dis(R) + dis(S).

Deﬁnition 2.1.3 (Gromov-Hausdorﬀ distance). Let X and Y be metric spaces.

• The Gromov-Hausdorﬀ distance dGH(X,Y ) is deﬁned as

1 d (X,Y ) := inf{dis(R): R is a correspondence between X and Y }. GH 2

• Let p and q be points in X and Y respectively. The pointed Gromov-Hausdorﬀ distance

is deﬁned as

1 d (X, p), (Y, q) := inf{dis(R): R is a correspondence between (X, p) and (Y, q)}. GH 2

The following remark is straightforward.

Remark 2.1.4. Let X and Y be metric spaces and p be a point in X. Then

dGH(X,Y ) = inf dGH (X, p), (Y, q) . q∈Y

Here, we are going to introduce a version of the Gromov-Hausdorﬀ distance for geodesic

spaces. We are going to achieve this by considering the following special type of correspon-

dences between geodesic spaces:

11 Deﬁnition 2.1.4 (Path correspondence). Let R be a correspondence between X and Y .

• A subcorrespondence R0 of R is called a path subcorrespondence if for each x R0 y and

0 0 0 0 x R0 y and for each continuous path α from x to x and β from y to y , there exists

continuous pathsα ˜ from x to x0 and β˜ from y to y0 so that α(t) R β˜(t) andα ˜(t) R β(t)

for each t. Note that we are assuming that the domain of β˜ is same as the domain of

α and the domain ofα ˜ is same as the domain of β.

• A correspondence R between X and Y is called a path correspondence if it has a path

subcorrespondence.

• Let p and q be points in X and Y respectively. A path correspondence between (X, p)

and (Y, q) is a correspondence R between X and Y with a path subcorrespondence R0

such that p R0 q.

The following remark is obvious.

Remark 2.1.5. If X,Y and Z are geodesic spaces and R and S are path correspondences between X and Y and Y and Z respectively, then S ◦ R is a path correspondence between X and Z.

Definition 2.1.5 (Length Gromov-Hausdorff distance). The length Gromov-Hausdorff dis-

` tance dGH(X,Y ) between geodesic spaces X and Y is deﬁned by

` dGH(X,Y ) := inf{dis(R): R is a path correspondence between X and Y }.

If p and q are points in X and Y respectively, we deﬁne the pointed length Gromov-Hausdorﬀ

` distance dGH((X, p), (Y, q)) by

` dGH((X, p), (Y, q)) := inf{dis(R): R is a path correspondence between (X, p) and (Y, q)}.

12 The following proposition follows from Remark 2.1.5.

Proposition 2.1.6. The length Gromov-Hausdorﬀ distance (resp. the pointed length Gro- mov -Hausdorﬀ distance) is a (pseudo)metric between geodesic spaces (resp. pointed geodesic spaces).

Proposition 2.1.7. • Let X and Y be geodesic spaces. Then,

` dGH(X,Y ) ≤ dGH(X,Y ) ≤ 2 dGH(X,Y ).

• If (X, p) and (Y, q) are pointed geodesic spaces, then

` dGH((X, p), (Y, q)) ≤ dGH((X, p), (Y, q)) ≤ 2 dGH((X, p), (Y, q)).

In order to prove Proposition 2.1.7, given a correspondence between X and Y , we need

to ﬁnd a path correspondence between X and Y whose distortion is at most controllably

larger than that of the original correspondence. To do this, we introduce the concept of

r-extension of a correspondence.

Deﬁnition 2.1.6 (r-extension of a correspondence). Let R be a correspondence between

metric spaces (X, dX ) and (Y, dY ) and r ≥ 0. We deﬁne the r-extension Rr of R as the

following correspondence between X,Y : x Rr y if there exists x0, y0 in X,Y such that x0 R y0

and dX (x, x0) + dY (y, y0) ≤ r.

Lemma 2.1.8. dis(Rr) ≤ dis(R) + 2r.

13 0 0 0 Proof. Let x Rry and x Rr y. There exists x0, x0 in X and y0, y0 in Y such that x0 R y0,

0 0 0 0 0 0 x0 R y0, dX (x, x0) + dY (y, y0) ≤ r and dX (x , x0) + dY (y , y0) ≤ r. Now, we have

0 0 0 0 0 0 0 0 |dX (x, x ) − dY (y, y )| ≤ |dX (x, x ) − dX (x0, x0)| + |dX (x0, x0) − dY (y0, y0)| + |dY (y0, y0) − dY (y, y )|

0 0 0 0 ≤ dX (x, x0) + dX (x , x0) + dis(R) + dY (y0, y) + dY (y0, y )

0 0 0 0 = dis(R) + dX (x, x0) + dY (y, y0) + dX (x , x0) + dY (y , y0)

≤ dis(R) + 2r.

The next lemma shows that large enough extensions of a correspondence are path correspondences themselves.

Lemma 2.1.9. Let R be a correspondence between (X, dX ) and (Y, dY ) and r > dis(R)/2.

Then R is a path subcorrespondence of Rr.

Proof. Let x and x0 be points in X and y and y0 be points in Y such that x R y, x0 R y0. Let

α be a continuous path between x and x0. Take δ = r − dis(R)/2 > 0 . Take a partition t0 < t1 < ··· < tn of the domain of α such that diam(α([ti−1, ti])) ≤ δ/2. Let xi = α(ti) and

0 ˜ for each i choose yi in Y such that y0 = y, yn = y and xiRyi. Deﬁne a continuous path β ˜ from the domain of α to Y such that β|[ti−1,ti] is a length minimizing geodesic from yi−1 to

˜ 0 ˜ yi. Note that β is a path from y to y . Let us show that α(t) Rr β(t) for each t. Assume ˜ t is in [ti−1, ti]. Since β reduces to a length minimizing geodesic on [ti−1, ti] without loss of ˜ generality we can assume that dY (β(t), yi) ≤ dY (yi−1, yi)/2. Hence we have

˜ dX (xi, α(t)) + dY (yi, β(t)) ≤ δ/2 + dY (yi−1, yi)/2

≤ δ/2 + (δ + dis(R))/2

= δ + dis(R)/2 = r.

14 ˜ 0 Therefore, α(t) Rr β(t). Similarly, given a continuous path β(t) from y to y , we can construct

0 a continuous pathα ˜ from x to x such thatα ˜(t) Rr β(t) for all t. Hence, R is a path subcorrespondence of Rr.

Now, we can give a proof of Proposition 2.1.7.

Proof of Proposition 2.1.7. The ﬁrst inequalities in both items are obvious, since to deﬁne

` dGH we take inﬁmum of distortions over a subset of all correspondences. Let R be a correspondence between X and Y and r > dis(R)/2. Then, by Lemma 2.1.8

and Lemma 2.1.9 1 1 d` (X,Y ) ≤ dis(R ) ≤ dis(R) + r. GH 2 r 2

Taking the limit as r → dis(R)/2, we get

` dGH(X,Y ) ≤ dis(R).

Inﬁmizing over R, we get

` dGH(X,Y ) ≤ 2 dGH(X,Y ).

Similarly, we have

` dGH((X, p), (Y, q)) ≤ 2 dGH((X, p), (Y, q)).

2.1.2 Hyperbolicity

Hyperbolicty is a metric invariant measuring the treeness of a metric space. It is deﬁned

through the Gromov product, which is deﬁned as follows:

15 Deﬁnition 2.1.7 (Gromov product (See [18], Chapter 8)). Let (X, dX ) be a metric space and p, x and y be points in X. The Gromov product gp(x, y) is deﬁned by

1 g (x, y) := (d (p, x) + d (p, y) − d (x, y)). p 2 X X X

Definition 2.1.8 (Hyperbolicity (see [18], Definition 8.4.6) ). The p-hyperbolicity and hyperbolicity of X can be respectively defined as follows:

hypp(X) := inf{δ ≥ 0 : gp(x, z) ≥ min(gp(x, y), gp(y, z)) − δ for each x, y, z ∈ X},

hyp(X) := sup hypp(X). p∈X

Definition 2.1.9 (Tree metrics). A finite tree metric is a finite metric space T which can be isometrically embedded into a metric tree (i.e. a tree graph with a length structure, see

Deﬁnition 2.4.1).

The following two propositions shows explicitly what we mean by saying hyperbolicty is a measure of treeness.

Proposition 2.1.10. A ﬁnite metric space is a tree metric if and only if its hyperbolicity is

A proof of this can be found in [70, Theorem 5.14].

Proposition 2.1.11. Let (X, dX ) and (Y, dY ) be metric spaces. Then

|hyp(X) − hyp(Y )| ≤ 4 dGH(X,Y ).

16 Proof. Let x, y, z and w be points in X. Let R be a correspondence between X and Y and p, q, r and s be points in Y related to x, y, w and z under R respectively. Then we have

dX (x, y) + dX (z, w) ≤ dY (p, q) + dY (r, s) + 2 dis(R)

≤ max(dY (p, r) + dY (q, s), dY (p, s) + dY (q, r)) + 2 dis(R) + 2 hyp(Y )

≤ max(dX (x, w) + dX (y, z), dX (x, z) + dX (y, w)) + 4 dis(R) + 2 hyp(Y ).

Hence,

hyp(X) − hyp(Y ) ≤ 2 dis(R).

Similarly, we have

hyp(Y ) − hyp(X) ≤ 2 dis(R).

Inﬁmizing over R, we get

|hyp(X) − hyp(Y )| ≤ 4 dGH(X,Y ).

2.1.3 Injective/Hyperconvex Spaces

Injective/hyperconvex metric spaces are metric spaces with nice intersection and extension properties. Another very nice property of these spaces is that every metric space can be isometrically and naturally embedded in an injective metric space and the neighborhoods in these embeddings carries information about the embedded metric space itself. We are going to see some examples below.

Deﬁnition 2.1.10 (Hyperconvex space). A metric space (X, dX ) is called hyperconvex if for every family (xi, ri)i∈I of xi in X and ri ≥ 0 such that dX (xi, xj) ≤ ri + rj for each i, j in I, there exists a point x such that dX (xi, x) ≤ ri for each i in I.

17 Deﬁnition 2.1.11. A metric space E is called injective if for each 1-Lipschitz map f :

X → E and isometric embedding of X into X˜, there exists a 1-Lipschitz map f˜ : X˜ → E

extending f.

For a proof of the following proposition, see [53, Proposition 2.2].

Proposition 2.1.12. A metric space is hyperconvex if and only if it is injective.

Example 2.1.13. For any set S, the space `∞(S) of real valued functions on S with `∞

norm is a hyperconvex space.

Deﬁnition 2.1.12. (X, dX ) is a metric space, the space `∞(X) is called the Kuratowski space and is denoted by κ(X). The map X → κ(X), p 7→ dX (p, ·) is an isometric embedding

and it is called the Kuratowski embedding. Hence every metric space can be isometrically

embedded in a hyperconvex space.

The following lemma is going to be useful when we consider neighborhoods of subspaces

of an injective metric space.

Lemma 2.1.14. In a hyperconvex space, any non-empty intersection of open balls is con-

tractible.

Proof. A geodesic bicombing of a geodesic space S is a map γ : S × S × [0, 1] → S such that

for each x and y in S, γ(x, y, ·) : [0, 1] → S is a constant speed length minimizing geodesic from x to x0. Note given a point p in S, restricting γ to S × {p} × [0, 1] gives a deformation retraction of S onto p. By [53, Proposition 3.8], a hyperconvex space (E, dE) has a geodesic bicombing γ such that for each x, y, x0 and y0 in E and t in [0, 1],

0 0 0 0 dE(γ(x, y, t), γ(x , y , t)) ≤ (1 − t)dE(x, x ) + tdE(y, y ).

18 In particular, by letting x0 = y0 = z, we get

dE(γ(x, y, t), z) ≤ max(dE(x, z), dE(y, z)).

Hence, if x and x0 is contained in an open ball of z, then γ(x, y, t) is contained in the same ball for each t in [0, 1]. Therefore, if U is a non-empty intersection of open balls in E, then

γ restricts to U × U × [0, 1] → U, which implies that U is contractible.

2.2 Persistence

The fundamental deﬁnitions and results about persistence can be found in [10]. We include some of them here for completeness. After reviewing the basics, we are going to show some relations of persistence with injective metric spaces and then introduce persistence sequences.

Deﬁnition 2.2.1 (Persistence module). A persistence module over R>0 is a family of vector spaces Vr>0 with morphisms fr,s : Vr → Vs for each r ≤ s such that

• fr,r = IdVr ,

• fs,t ◦ fr,s = fr,t for each r ≤ s ≤ t.

Deﬁnition 2.2.2 (Irreducible persistence modules). Given an interval I in R>0, the persistence module IR is deﬁned as follows. The vector space at r is R if r is in I and zero otherwise. Given r ≤ s, the morphism corresponding to (r, s) is identity if r, s are in I and zero otherwise.

For the following theorem, see [29].

19 Theorem 2.2.1. If V∗ is a persistence module such that each Vr is ﬁnite dimensional, then there is a family (Iλ)λ∈Λ of intervals, unique up to reordering, such that V∗ is isomorphic to

⊕λ(Iλ)R.

Deﬁnition 2.2.3 (Barcode). A barcode is a multiset of intervals. By Theorem 2.2.1, there exists a barcode associated to each pointwise ﬁnite dimensional persistence module. It is called the persistence barcode associated to the persistence module.

Deﬁnition 2.2.4 (Bottleneck distance). A partial matching between two multisets X and

Y is a pair of sub-multisets X0 and Y 0 of X and Y respectively and a bijection from X0 to

Y 0.

Let B and B0 be two barcodes and M be a matching between them. Let ha, bi denote an interval with endpoints a and b. The cost of M, denoted by cost(M), is the maximum of the following three numbers

sup{(b − a)/2 : ha, bi ∈ B, not matched}

sup{(b − a)/2 : ha, bi ∈ B0, not matched}

sup{max(|a − a0|, |b − b0|): ha, bi matched with ha0, b0i}.

0 The bottleneck distance db(B,B ) is deﬁned by

0 0 db(B,B ) := inf{cost(M): M a matching between B,B }.

One way of assigning barcodes to metric spaces is through Vietoris-Rips ﬁltration which is deﬁned as follows:

Deﬁnition 2.2.5 (Vietoris-Rips ﬁltration). Let X be a metric space and r > 0. The open

Vietoris-Rips complex VRr(X) of X is the simplicial complex whose vertices are the points

20 of X and whose simplices are ﬁnite subsets of X with diameter strictly less then r. Note

that if r < s, then VRr(X) is included in VRs(X). Hence, the family VR∗(X) is a filtration, which is called the open Vietoris-Rips filtration of X. The closed Vietoris-Rips filtration is defined similarly by letting the diameter be less than or equal to r.

Definition 2.2.6 (Cechˇ filtration). Let X be a metric space and r > 0. The open Cechˇ complex Cˇ r(X) is the simplicial complex whose vertices are the points of X and whose simplices are finite subsets {x0, xi, . . . , xn} such that the intersection of the open balls Br(xi) is non-empty. Note that if r < s, then Cˇ r(X) is included in Cˇ s(X). Hence Cˇ ∗(X) is a

filtration, which is called open Cechˇ filtration of X. The closed Cechˇ filtration is defined similarly by using the closed balls.

r Remark 2.2.2. H∗(VR (X)) forms a persistence module where the morphisms are induced by inclusions. As a persistence module, it is denoted by PH(VR∗(X)). The persistent Betti

X r number βn (r) is deﬁned by βn(VR (X)).

For the following theorem, see [24, Lemma 4.3].

Theorem 2.2.3. Let X and Y be metric spaces so that the persistence homology of their open Vietoris-Rips ﬁltrations have an interval decomposition. Let BX and BY denote the

corresponding barcodes. Then

db(BX ,BY ) ≤ 2 dGH(X,Y ).

Now, we can state the following theorem which shows how injective metric spaces can be utilized for understanding persistence.

Proposition 2.2.4. Let (X, dX ) be a subspace of a hyperconvex space (E, dE). For ≥ 0,

let B(X) denote the open -neighborhood of X in E. Then the open Vietoris-Rips complex

2 VR (X) is homotopy equivalent to B(X).

21 Proof of Proposition 2.2.4. Let

U := {B(x) ⊆ E : x ∈ X}

. By Lemma 2.1.14, U is a good cover of B(X). By nerve lemma, B(X) is homotopy

equivalent to the Cechˇ complex of U. By the deﬁnition of hyperconvexity, given a ﬁnite set

x1, . . . , xn in X, B(xi) ⊆ E has non-empty intersection if and only if dX (xi, xj) < 2 for

each i, j. Hence the Cechˇ complex of U is same with VR2(X).

The following result can be found in [42, Lemma 1.7.A, p. 96]. Here we give a diﬀerent

proof.

Corollary 2.2.5. Let X be a geodesic space. Then, VRr(X) is contractible for r > 4 hyp(X).

Proof. Let E(X) be the tight span of X (see [35, 34]), which is a hyperconvex space (see [53]).

Then X is a subspace of E(X) and by [53, Proposition 1.3] dH(X,E(X)) ≤ 2 hyp(X) (note that the deﬁnition of δ-hyperbolicity in [53] corresponds to our δ/2-hyperbolicity). Hence, by Proposition 2.2.4, VRr(X) is homotopy equivalent to E(X), which is contractible.

2.2.1 Persistence Sequence

The following theorem shows that the 1-dimensional persistence is determined by the

right endpoints of each interval in the barcode, hence allows us to construct a sequence of

real numbers representing the barcode.

Theorem 2.2.6. Let X be a compact geodesic space with ﬁnite ﬁrst Betti number. Let B be

the ﬁrst persistence barcode of X. Then

i) The cardinality of B is less than or equal to β1(X).

ii) Each interval in B has an endpoint 0.

22 The second part of Theorem 2.2.6 follows from [24, Theorem 6.3] and it is derived from

the following lemma (cf. [24, Corollary 6.2]):

r s Lemma 2.2.7. Let X be a geodesic space. The map H1(VR (X)) → H1(VR (X)) induced

by the inclusion VRr(X) → VRs(X) is surjective for all 0 < r < s.

Since it is quite short, and for completeness, here we provide a proof of Lemma 2.2.7.

This proof is a slight modiﬁcation of that of [24, Lemma 6.1].

r Proof of Lemma 2.2.7. It is enough to show that for each r > 0, the induced map H1(VR (X)) →

2r s/2n H1(VR (X)) is surjective, since for n ∈ N large enough, the map H1(VR (X)) →

s r s H1(VR (X)) splits through H1(VR (X)) → H1(VR (X)).

P 0 2r Now let c = i λi[xi, xi] be a 1-cycle in VR (X). For each i, let yi be a midpoint

0 0 P 0 0 r between xi, xi. Let c := i λi([xi, yi] + [yi, xi]). Note that c is a 1-cycle in VR (X). It is enough to show that c is homologous to c0 in VR2r(X). Let z be a 2-chain in VR2r(X)

P 0 0 deﬁned by z := i λ[xi, yi, xi]. Then, ∂z = c − c.

We need one more lemma before proving Theorem 2.2.6.

r Lemma 2.2.8. Let X be a compact geodesic space. For each r > 0, β1(VR (X)) ≤ β1(X).

Proof. Let E be a hyperconvex metric space which contains X as a subspace. Then, by

Proposition 2.2.4, VRr(X) is homotopy equivalent to r/2 neighborhood of X in E. Let us denote this neighborhood by Br/2(X). Hence it is enough to show that for each r > 0, the inclusion map X → Br(X) induces a surjective map on fundamental groups.

0 Let γ : [0, 1] → Br(X) be a continuous path with endpoints x and x in X. It is enough to show that γ is homotopy equivalent to a path in X relative to its endpoints. Choose

0 = t0 < t1 < ··· < tn = 1 such that for each i ≥ 1 there exists xi ∈ X such that

23 γ([ti−1, ti]) ⊆ Br(xi). Let yi = γ(ti) for i ≥ 0. Let αi, βi be continuous paths in Br(xi) such that αi is from yi−1 to xi and βi is from xi to yi respectively. As Br(xi) is contractible,

γ|[ti−1,ti] is homotopy equivalent to αi · βi relative to endpoints. Hence we have

γ ' (α1 · β1) ····· (αn · βn)

relative to endpoints. Note that α1 and βn can be chosen as geodesics in X as they connect

0 x, x1 and xn, x in Br(x1),Br(xn) respectively. Hence it is enough to show that

(β1 · α2) ····· (βn−1 · αn)

is homotopy equivalent to a path in X relative to endpoints. Let us show that βi · αi+1 is

homotopy equivalent to a path in X for each i. Let p be a midpoint of xi, xi+1 in X. Note

that p, yi is contained in Br(xi) ∪ Br(xi+1), which is contractible. Let θ be a path in that

intersection from yi to p. Let γxi,p be a geodesic in X from x to p and γp,xi+1 be a geodesic in ¯ X from p to xi+1. Note that γxi,p ·θ is contained in Br(xi) and has endpoints xi, yi hence it is

homotopy equivalent to βi relative to endpoints. Similarly θ · γp,xi+1 is homotopy equivalent

to αi+1 relative to endpoints. Hence

¯ βi · αi+1 ' γxi,p · θ · θ · γp,xi+1

' γxi,p · γp,xi+1

relative to endpoints. This completes the proof.

Now, we can give a proof of Theorem 2.2.6.

Proof of Theorem 2.2.6. By Lemma 2.2.7, every interval in the ﬁrst persistence barcode

starts from 0. By Lemma 2.2.8, the persistent ﬁrst Betti numbers are less than β1(X), hence

there are at most β1(X) intervals in B.

24 As the left points of the intervals in the 1-dimensional barcode of a geodesic space are

all 0, we can encode the barcode as a sequence as follows:

Definition 2.2.7 (Persistence sequence). Let X be a compact geodesic space with finite first

Betti number. Let B = I1,I2,...,IN be its ﬁrst persistence barcode such that the length

X of Ii is ai > 0 and a1 ≥ a2 ≥ · · · ≥ aN . The persistence sequence (an )n of X is deﬁned as

X X an = an for n ≤ N and an = 0 for n > N. Note that it is a non-increasing sequence.

X Remark 2.2.9. For n > β1(X), an = 0.

The following proposition shows how the persistence sequence interact with 1-Lipschitz functions.

Proposition 2.2.10. Let X and Y be compact geodesic spaces and f : X → Y be a 1-

X Y Lipschitz map such that it induces a surjection on ﬁrst homology groups. Then an ≥ an for each n.

We need the following lemma:

X Lemma 2.2.11. Let X be a compact geodesic space. If 0 ≤ r < an , then the persistent Betti

X X X number β1 (r) ≥ n (see Remark 2.2.2). If r > an , then β1 (r) < n.

Proof. The ﬁrst persistence barcode of X is {(0, an) or (0, an]}n. Note that if 0 ≤ r < an,

X then r is contained in (0, a1), (0, a2),..., (0, an), hence β1 (r) ≥ n. If r > an then r is not

X contained in (0, an], (0, an+1,... , hence β1 (r) < n.

Proof of Proposition 2.2.10. Let EX and EY be hyperconvex spaces containing X and Y respectively. By injectivity property of hyperconvex spaces, for each r > 0, f : X → Y extends to a map fr : Br(X) → Br(Y ) where Br(X),Br(Y ) denotes r-neighborhoods of X

25 and Y in EX and EY respectively. Note that the map X → Br(Y ) induces a surjection on

ﬁrst homology groups by Lemma 2.2.8, and as it splits through fr, fr induces a surjection on

ﬁrst homology groups. Therefore one dimensional persistent Betti numbers of X are greater than or equal to those of Y .

Y Y X If r < an , then by Lemma 2.2.11, β1 (r) ≥ n, therefore β1 (r) ≥ n. Then, by Lemma

X X Y 2.2.11 r ≤ an . Hence an ≥ an .

The stability result for barcodes implies the following stability result for persistence

sequences:

Proposition 2.2.12. Let X and Y be compact geodesic spaces with ﬁnite ﬁrst Betti numbers.

Then

X Y ||(an ) − (an )||∞ ≤ 4 dGH(X,Y ).

Proof. Let BX and BY be the ﬁrst persistence barcodes of X and Y respectively (for def-

initions and facts about barcodes see Section 2.2). Let M be a partial matching between

X Y X B ,B . We can ﬁnd an injective function ϕ : N → N such that if an is the right endpoint of an interval in BX , which is matched with an interval in BY whose right endpoint a then

Y Y X Y aϕ(n) = b, and in all other cases aϕ(n) = 0. Note that |an −aϕ(n)| ≤ 2 cost(M) for each n in N.

X Y Similarly, we can ﬁnd an injective function ψ : N → N such that |aψ(m) − am| ≤ 2 cost(M).

Given n in N, by the injectivity of ϕ, there exists n0 ≤ n and m0 ≥ n such that ϕ(n0) = m0. Hence, we have

X Y X Y an − an ≤ an0 − am0 ≤ 2 cost(M).

Similarly,

Y X an − an ≤ 2 cost(M).

26 Since this is true for all n in N, we get

X Y ||(an ) − (an )||∞ ≤ 2 cost(M).

Inﬁmizing over partial matchings, we get

X Y X Y ||(an ) − (an )||∞ ≤ 2 dB(B ,B ) ≤ 4 dGH(X,Y ).

2.3 Interleaving Distance

In this Section, we are going to review interleaving distance in the categorical sense,

which is going to be relevant especially in Chapter 3 and Chapter 8. For more details about

the interleaving distance, see [17].

Deﬁnition 2.3.1. Let C be a category F and G be functors from the poset category (R, ≤) to C.

• For ≥ 0, a degree -morphism from F to G is a family of maps (ϕr)r∈R such that

ϕr : F (r) → G(r + ) and

ϕs ◦ F (r → s) = G(r + → s + ) ◦ ϕr,

for each r ≤ s in R. F (r→s) F (r) F (s)

ϕr ϕs G(r+→s+) G(r + ) G(s + ).

• F , G are called -interleaved if there exists a degree morphisms ϕ and ψ from F to

G and G to F respectively such that

ϕr+ ◦ ψr = G(r → r + 2) and ψr+ ◦ ϕr = F (r → r + 2),

27 for each r in R.

F (r→r+2) G(r→r+2) F (r) F (r + 2) G(r) G(r + 2)

ϕr ψr+ ψr ϕ(r+)

G(r + ) F (r + ) .

• The interleaving distance dI(F,G) is deﬁned by

dI(F,G) := inf{ > 0 : F and G are − interleaved}.

The following proposition is straightforward.

Proposition 2.3.1. Let C be a category and F,G and H be functors from (R, ≤) to C. Then,

• dI(F,F ) = 0.

• dI(F,G) = dI(G, F ).

• dI(F,H) ≤ dI(F,G) + dI(G, H).

2.4 Metric Graphs

Metric graphs are simplest type of geodesic spaces. As finite metric spaces are dense among all compact metric spaces, finite metric graphs are Gromov-Hausdorff dense among all compact geodesic spaces. This motivates one to study metric graph representations of geodesic spaces, which is one of the main focuses of this thesis. We start this section by reviewing some basic properties of metric graphs.

Deﬁnition 2.4.1 (Metric graph). A metric graph is a geodesic space homeomorphic to a topological graph.

28 Figure 2.1: A metric graph

The following proposition follows from the Euler formula.

Proposition 2.4.1. Let G be a connected graph, V denote its number of vertices and E denote its number of edges and β := β1(G). Then β = 1 − V + E.

Example 2.4.2. A connected graph G is a tree if and only if β1(G) = 0.

Proposition 2.4.3. Let (G, l) be a metric graph with a vertex set V containing p. Adding at most one vertex from each edge to V if necessary, we can guarantee that for each edge

{v, w} we have

l(v, w) = dl(v, w) = |dl(p, v) − dl(p, w)|.

Proof. Without loss of generality we can assume that dl(p, v) ≤ dl(p, w). Identify the geometric realization of the edge (v, w) with [0, l(v, w)]. Let t be the maximal element in that edge such that there is a length minimizing curve in the geometric realization of G from p to t passing through v. Note that t ≥ 0 and for any t0 > t, all length minimizing curves from p to t0 passes through w. Hence, there are length minimizing curves α, α0 from p to t such that α passes through v and α0 passes through w. Since these curves cover [v, w] and every point in these curves other than t has distance strictly smaller than dl(p, t) to p, then t is the

29 unique point on the edge where dl(p, ·) takes its maximum. Extend the vertex set adding all such points. Hence all local maximums of dl(p, ·) in the geometric realization of G is contained in the vertex set. Furthermore, given an edge (v, w) such that dl(p, v) ≤ dl(p, w), there exist a length minimizing curve in the geometric realization from p to w containing the edge, therefore dl(v, w) = l(v, w) = |dl(p, v) − dl(p, w)|.

In the remaining part of this section, we analyze paths in metric graphs. The main theorem we are going to prove is the following:

Theorem 2.4.4. Let (G, dG) be a metric graph and p be a point in G. Let γ : [0, 1] → G be a continuous path in G from x to x0.

i) There exists 0 = t0 < t1 < ··· < tm = 1 such that m ≤ 2 β1(G) + 2 and

0 X dG(x, x ) ≤ |dG(p, γ(ti)) − dG(p, γ(ti−1))|. i≥1 ii) If γ is a length minimizing geodesic, then there exists 0 = t0 < t1 < ··· < tm = 1 such

that m ≤ 2 β1(G) + 2 and

0 X dG(x, x ) = |dG(p, γ(ti)) − dG(p, γ(ti−1))|. i≥1

Furthermore, for any reﬁnement 0 = s0 < s1 < ··· < sn = 1 of (ti)i,

0 X dG(x, x ) = |dG(p, γ(sj)) − dG(p, γ(sj−1))|. j≥1

The main concept we are going to use for proving Theorem 2.4.4 is the following:

Deﬁnition 2.4.2 (Path preorder). Let α, β be continuous paths in a topological space

X with the same domain [a, b] and with the same endpoints. We say α ≤ β if for each a = t0 < t1 < ··· < tn = b, there exists a = s0 < s1 < ··· < sn = b such that α(ti) = β(si) for each i. Note that this is a preorder, i.e. a reﬂexive and transitive relation.

30 The following remark is obvious.

Remark 2.4.5. If X is a length space with a length structure L and α, β are paths in X

such that α ≤ β, then L(α) ≤ L(β).

Remark 2.4.6. Let T be a topological tree and γ :[a, b] → T be a continuous path from x

0 0 to x . Let γx,x0 :[a, b] → T be the unique (up to reparametrization) simple path from x to x .

Then γx,x0 ≤ γ.

Proof. Note that the image of γx.x0 is contained in the image of γ since for any point q in

0 the γx,x0 ((a, b)), x and x are in diﬀerent path components of T − {q}. Let a = t0 < t1 <

··· < tn = b. Let s0 = a, sn = b and for 0 < i < n let

si := max{s : γ(s) = γ(ti)}.

0 0 Note that si+1 > si since γ|[si,b] is a path from γx,x (ti) to x .

Deﬁnition 2.4.3 (Edge path). Let G be a topological graph and γ :[a, b] → G be a

continuous path from x to x0 in G. γ is called an edge path if there exists a vertex set V

0 containing x and x and a partition a = t0 < t1 < ··· < tn = b such that γ([ti−1, ti]) is an

edge with respect to V and γ maps [ti−1, ti] homeomorphically onto its image. Note that if

γ satisﬁes the property described above for the vertex set V then it satisﬁes that property

for any reﬁnement V 0 of V .

Remark 2.4.7. Let G be a topological graph and π : T → G be its universal cover (which is a tree). If γ is an edge path in T , then π ◦ γ is an edge path in G.

We give a proof of Proposition 2.4.8 after the following lemma.

Proposition 2.4.8. Let G be a topological graph. Then for any continuous path γ in G, there exists a simple edge path γ0 such that γ0 ≤ γ.

31 Lemma 2.4.9. Let G be a topological graph. Then for any continuous path γ in G, there

exists an edge path γ0 such that γ0 ≤ γ.

Proof of Lemma 2.4.9. Let π : T → G be the universal cover of G. Let γ be a continuous

path in G and let α be a lift of γ to T , i.e. π ◦ α = γ. Let αx,x0 be the unique simple path

0 between the endpoints x and x of α. Note that αx,x0 is an edge path and by Remark 2.4.6

0 0 0 αx,x0 ≤ α. Hence if we let γ := π ◦ αx,x0 , then γ ≤ γ and by Remark 2.4.7 γ is an edge

path.

Proof of Proposition 2.4.8. By Lemma 2.4.9, without loss of generality we can assume that

γ is an edge path. If γ is a loop, we can let γ0 to be the constant path. Hence, assume that

γ is not a loop. We proceed by induction on the number of edges in γ. If it is one, then γ is

0 0 already simple. Now, if γ is not simple, i.e. there are t < t such that γ(t) = γ(t ) deﬁne γ0

0 by removing the edges in between t and t . Note that γ0 ≤ γ. By the inductive hypothesis,

0 there exists a simple edge path γ ≤ γ0. This completes the proof.

The following proposition gives a further analysis of the structure of a simple edge path.

Proposition 2.4.10. Let (G, dG) be a metric graph, p be a point in G and γ be a simple

edge path. Then γ has a decomposition γ = γ1 · γ2 ····· γm where m ≤ 2 β1(G) + 2 and dG(p, ·) ◦ γi is strictly monotonous for each i.

Lemma 2.4.11. Let (G, dG) be a metric graph and p be a point in G. There exists a vertex set V such that with respect to the graph structure given by V , if x and x0 are contained in an

0 0 edge, then dG(x, x ) = |dG(p, x) − dG(p, x )|. Furthermore, for such V each edge is a dG(p, ·)

strictly monotonous geodesic. Note that the same properties hold for any reﬁnement of V .

32 Proof. Let W be a set of vertices. Let e be an edge with respect to W . Identify e with [0, 1]

and let

t0 := max{t : there exists a length minimizing geodesic from p to t containing 0}.

Note that for any t > t0, any length minimizing geodesic from p to t contains 1. This implies

0 0 0 that if x and x are both in [0, t0] or [t0, 1], then dG(x, x ) = |dG(p, x) − dG(p, x )|. Also, there

0 are length minimizing geodesics γ, γ from p to t0 containing 0, 1 respectively. Hence, if we

extend W by adding a point from each edge corresponding to t0 as described above, we get

a vertex set V satisfying the conditions given in the statement.

Corollary 2.4.12. Let (G, dG) be a metric graph, p be a point in G and γ be a continuous

path in G. Then

LG(γ) = LR(dG(p, ·) ◦ γ).

Proof. A length structure on a graph is determined by its restriction to edges. By Lemma

2.4.11, both of the length structures described above coincide on an edge with respect to a vertex set V as it is described in Lemma 2.4.11.

Proof of Proposition 2.4.10. Take a vertex set V such that γ can be realized as an edge path

with respect to V and V is a vertex set as in Lemma 2.4.11. Then γ can be decomposed as

γ1 . . . γm where each γi is dG(p, ·) strictly monotonous. Let us show that m ≤ 2 β1(G) + 2.

Without loss of generality we can assume that m > 2.

Let v0, v1, . . . , vm be the points in V where γi is a path from vi−1 to vi. Let us call a point

v in V a merging vertex if there are at least two dG(p, ·) increasing edges coming to v. Since

γ is simple, either v1 or v2 is a merging vertex and if vi is a merging vertex and m > i + 2,

then vi+2 is a merging vertex. Hence, if we let M be the number of merging vertices, then

m ≤ 2 M + 2. Now, it is enough to show that M ≤ β1(G).

33 Given v in V , let ι(v) be the number of dG(p, ·) increasing edges coming to v. Note that

ι(p) = 0, ι(v) ≥ 1 for v 6= p and ι(v) > 1 if and only if v is a merging vertex. By Euler’s

formula, we have

β1(G) = number of edges − |V | + 1 X X = ι(v) − 1 + 1 v∈V v∈V X = (ι(v) − 1) ≥ M v∈V −{p}

Proof of Theorem 2.4.4. “i)” By Proposition 2.4.8, there exists a simple edge path γ0 from x

0 0 0 to x such that γ ≤ γ. Note that if we can ﬁnd necessary 0 = t0 < ··· < tm = 1 for γ , then by the deﬁnition of path preorder we also get (ti) for γ. Hence without loss of generality we can assume that γ is a simple edge path. By Proposition 2.4.10, γ = γ1 . . . γm where m ≤ 2β1(G) + 2 and γi is dG(p, ·) strictly monotonous path. Let xi, xi−1 be the endpoints of

γi. By Corollary 2.4.12, LG(γi) = |dG(p, xi) − dG(p, xi−1)|. Take 0 = t0 < t1 < ··· < tm = 1 such that γ(ti) = xi. Then we have

0 dG(x, x ) ≤ LG(γ) X = LG(γi) i≥1 X = |dG(p, xi) − dG(p, xi−1)| i≥1 X = |dG(p, γ(ti)), dG(p, γ(ti−1))|. i≥1

“ii)” In this case, γ is already a simple edge path and decomposition of γ as above gives us the required equalities.

34 X Xh

Figure 2.2: Reeb graph Xh of the height function h on X.

2.5 Reeb Graphs

The Reeb graph construction is a construction which takes a ﬁltered topological spaces

(i.e. a topological space with a real valued continuous function) and assigns (in many cases) a topological graph. Hence, it is a good candidate for obtaining metric graph representations of geodesic spaces, which we investigate in Chapters 5 and 6. Let us start by reviewing its deﬁnition.

Deﬁnition 2.5.1 (Reeb graph). Given a topological space X and a real valued function

0 f : X → R, the Reeb graph Xf is deﬁned as the quotient space X/ ∼ where x ∼ x if there

0 exists a f-constant continuous path between x and x . When (X, dX ) is a compact geodesic space and f = dX (p, ·) where p is a point in X, we denote the Reeb graph by Xp.

35 When f : X → R is a Morse function deﬁned on a compact manifold, then we already know that the Reeb space is a graph. The following proposition shows that when f is an excellent Morse function, the Reeb graph satisﬁes an extra condition.

Proposition 2.5.1. Let X be a compact Riemannian manifold of dimension n ≥ 2, and suppose f : X → R is a proper excellent Morse function. Then the vertices of the Reeb graph

Xf have degrees at most 3.

The following statement can be found in [25, p. 623].

Proposition 2.5.2. Let X be a compact geodesic space and p be a point in X such that Xp is a ﬁnite graph. Then the map π1(X, p) → π1(Xp, p) induced by the quotient map X → Xp is surjective. In particular β1(X) ≥ β1(Xp). Same results holds for X and Xf where X is a compact manifold and f : X → R is Morse.

Proposition 2.5.3 ([73, Proposition 1.2.]). The preimage of a connected set under the Reeb quotient map is connected.

We can endow the Reeb graph with an intrinsic metric as follows:

Proposition 2.5.4. Let (X, dX ) be a compact geodesic space and p be a point in X such that the Reeb graph Xp is a ﬁnite graph. Then dp : Xp × Xp → R deﬁned by

0 0 dp([x], [x ]) := inf{LR(dX (p, ·) ◦ γ): γ is a continuous path from x to x in X}.

is an intrinsic metric on Xp. Note that dp is well deﬁned since diﬀerent representatives of a

0 point in Xp can be connected by a dX (p, ·) constant path. Furthermore, for each x and x in

X, the inﬁmum is realized by a continuous path from x to x0.

36 Remark 2.5.5. Let (X, dX ) a geodesic space and p be a point in X. Then dp([p], [x]) = dX (p, x) for all x in X. This can be seen by a taking a length minimizing geodesic from p to x.

We prove Proposition 2.5.4 after proving the following lemma:

Lemma 2.5.6. Let (X, dX ) be a compact geodesic space and p be a point in X such that Xp is a ﬁnite graph. Then there is a vertex set V of Xp such that for each edge e with respect to V , there is a path γe which is a part of a length minimizing geodesic starting from p such that the quotient map π : X → Xp maps γe homeomorphically to e.

Proof. Choose a vertex set W for Xp. Over each edge e with respect to W , choose a point ve such that dX (p, ·) takes its maximum over e at ve. Let V be the vertex set obtained by adding W all ve. Note that, over each edge with respect to V , the maximum of dX (p, ·) is realized at one of the endpoints. Let us show that V satisﬁes the necessary property.

Let e be an edge with respect to V . Identify e with [0, 1] and without loss of generality

dX (p, 1) ≥ dX (p, 0) hence dX (p, ·) takes its maximum on e at 1. For each t ∈ (0, 1), chose a

point xt in X such that π(xt) = t. Let γt be a length minimizing geodesic from p to xt. Note that dX (p, ·) ◦ γt does not contain 1 since otherwise dX (p, t) > dX (p, 1). Hence dX (p, ·) ◦ γt is

an injective path in Xp whose last part contains [0, t]. By Arzela-Ascoli theorem γt converges to a length minimizing geodesic γ. Let γe be the part of γ after dX (p, ·) ◦ γ attains 0. Hence

γe maps homeomorphically onto e.

Proof of Proposition 2.5.4. Let V be a vertex set for Xp as in Lemma 2.5.6. Note that

dX (p, ·) is strictly monotonous on each edge with respect to V . Hence we can deﬁne an

intrinsic metric on Xp by

0 0 0 dp([x], [x ]) := inf{LR(dX (p, ·) ◦ γ): γ is a continuous path from [x] to [x ] in Xp}.

37 0 0 Let us show that dp = dp. It is obvious that dp ≤ dp since the quotient map π : X → Xp sends every continuous path between x and x0 to a continuous path between [x], [x0]. Let us

0 show that dp ≤ dp.

0 0 Let γ be a length minimizing path from [x] to [x ] in (Xp, dp). We can assume that γ is

an edge path with respect to V . Let γ = e1 · e2 ····· en. Let x0, . . . , xn be points in X such

0 that ei is from [xi−1] to [xi], x0 = x, xn = x . Note that we have

0 0 X dp([x], [x ]) = |dX (p, xi−1) − dX (p, xi)|. i≥1

Let γi be a path corresponding to ei as in Lemma 2.5.6. Let αi be a dX (p, ·) constant

path from xi−1 to the initial point of γi and βi be a dX (p, ·) constant path from the endpoint

of γi to xi. Let

γ := (α1 · γ1 · β1) ····· (αn · γn · βn).

Note that γ is a continuous path from x to x0. We have

X LR(dX (p, ·) ◦ γ) = LR(dX (p, ·) ◦ γi) i≥1 X 0 0 = |dX (p, xi−1) − dX (p, xi)| = dp([x], [x ]). i≥1

0 0 0 Therefore dp ≤ dp, so dp = dp and the inﬁmum in the deﬁnition of dp([x], [x ]) is realized by a continuous path from x to x0.

Definition 2.5.2 (Reeb metric graph). Let X be a geodesic space and p be a point in X such that Xp is a finite metric graph. We call the metric graph (Xp, dp) where dp is defined as in Proposition 2.5.4 the Reeb metric graph. We similarly define a metric on df on Xf by

pulling back the length structure of R to Xf by f.

Proposition 2.5.7. We have,

38 (i) [73, Proposition 1.1] If f :(X, dX ) → R is L-Lipschitz, then π : X → Xf is also L-Lipschitz.

(ii) f : Xf → R is 1-Lipschitz.

Proof. Given points x and y in X, let Γ(x, y) denote the set of continuous curves from x to

y in X. Let lR, lX denote the canonical length structures on R,X respectively. (i) Let x and y be points in X and r and s be their images under π respectively. Then

we have

df (r, s) = inf lR(f ◦ γ) ≤ L inf lX (γ) = L dX (x, y). γ∈Γ(x,y) γ∈Γ(x,y)

(ii) Let r, s be points in Xf and x and y be points in X mapped onto r, s under π : X →

Xf respectively. Then we have

|f(r) − f(s)| = |f(x) − f(y)| ≤ inf lR(f ◦ γ) = df (r, s). γ∈Γ(x,y)

2.5.1 Stability of Reeb Metric Graphs

In this subsection, we prove the following stability theorem:

Theorem 2.5.8 (Reeb stability). Let (X, dX ) and (Y, dY ) be compact geodesic spaces and p

and q be points in X and Y respectively such that Xp and Yq are ﬁnite metric graphs. Let

β = max(β1(Xp), β1(Yq)). Then

` dGH(Xp,Yq) ≤ (8β + 6) dGH (X, p), (Y, q) .

Proof. Let R be a path correspondence between X and Y with path subcorrespondence R0.

0 0 0 Assume x R0 y and x R0 y . Let α : [0, 1] → X be a dp realizing path from x to x . Let

39 0 γ : [0, 1] → Y be a path from y to y such that α(t) R γ(t) for each t. Let 0 = t0 < t1 <

··· < tk = 1 be a partition of [0, 1] for α as in the second part of Theorem 2.4.4. Let

0 0 0 0 = t0 < t1 < ··· < tl = 1 be a partition of [0, 1] for γ as in the ﬁrst part of Theorem 2.4.4.

0 Let 0 = s0 < ··· < sm = 1 be the coarsest common reﬁnement of the partitions (ti), (tj). Note that m ≤ 4β + 3. Now, we have

Similarly we can show that

0 0 dp(x, x ) − dq(y, y ) ≤ (8β + 6) dis(R).

If we denote the correspondence between Xp and Yq induced by R0 by S0, then we have

dGH(Xp,Yq) ≤ dis(S0)/2 ≤ (4β + 3) dis(R).

Inﬁmizing over path correspondences R between (X, p), (Y, q), we get

` dGH(Xp,Yq) ≤ (8β + 6) dGH (X, p), (Y, q) .

2.5.2 Smoothings

By Corollary 2.4.12, we know that the Reeb metric graph Gp assigned to a metric graph

G with a point p in it is G itself, hence the Reeb metric graph construction does not change a metric graph. Still, by modifying G in a certain way first, we can apply the Reeb construction to obtain a simpler space. One way of doing this is considering the -smoothings. Smoothings are introduced in [30, Section 4]. Here we use a slightly different definition.

40 Deﬁnition 2.5.3 (-smoothings). Let (G, dG) be a ﬁnite metric graph, p be a point in

X and ≥ 0. Note that the `1 product G × [0, ] is a length space and d((p, 0), ·) is a piecewise linear function on the product, therefore its Reeb graph is a topological graph. We

deﬁne the -smoothing Gp as the metric graph whose underlying graph is the Reeb graph of d((p, 0), ·): G × [0, ] → R and the length structure is deﬁned as in Proposition 2.5.4.

Remark 2.5.9. Let X be a compact geodesic space, p be a point in X such that Xp is a

metric graph. Then (X × [0, ])(p,0) is a metric graph isomorphic to (Xp)p, where the product

X × [0, ] denotes the `1 metric product. We denote this space also by Xp.

Our main reason for introducing -smoothings is the following Proposition:

Proposition 2.5.10. Let (G, dG) be ﬁnite metric graph and p be a point in G. Let ` be the length of kth largest interval in the 1-dimensional persistence barcode of X. If ≥ 3 `/2, then

β1(Gp) < k.

Proof. Let β1(G) = n. Note that if k > n, then by Proposition 2.5.2, β1(Gp) ≤ β1(G ×

[0, ]) = n < k. Now, assume that k ≤ n. By [72, Theorem 11.3], there are loops γ1, . . . , γn

G such that L(γi) = 3 ai and the one dimensional homology classes corresponding to γi freely

generate H1(G, R). Since G → Gp sending x 7→ (x, 0) induces a surjection on the ﬁrst homology groups, it is enough to show that for i ≥ k, the the loop (γi, 0) is homotopic to a

constant loop in Gp. Let i ≤ k. Note that

G diam(γi) ≤ L(γi)/2 ≤ 3 ai /2 ≤ .

Let r = maxt dG(p, γi(t)). For t in the domain of γi, let rt = r − dG(p, γi(t)). Note that 0 ≤

rt ≤ diam(γi) ≤ . If we deﬁne the loopγ ˜i in Gp by t 7→ (γi(t), rt), then (γi, 0) is homotopy

41 equivalent toγ ˜i through the homotopy (t, s) 7→ (γi(t), srt). Since d((p, 0), (γi(t), rt)) =

dG(p, γi(t)) + rt = r for each t,γ ˜i is a constant loop in Gp.

Corollary 2.5.11. Let X be a compact geodesic space and p be a point in X such that Xp

X is a ﬁnite graph. If ≥ 3 ak /2, then β1(Xp) < k.

Xp X Xp Proof. By Proposition 2.2.10, ak ≤ ak , hence ≥ 3 ak /2. The result follows from Propo-

sition 2.5.10 since by Remark 2.5.9 Xp is same with (Xp)p.

2.6 Diﬀerential Topology

We are going to use the following statements in Chapter 6, where we analyze the distortion

of the Reeb quotient map of Morse functions on Riemanian manifolds.

Recall that a Morse function is called excellent if there is a unique critical point for each critical value.

Proposition 2.6.1. Let f : X → R be an L-Lipschitz function from a compact Riemannian manifold. Then for each > 0 and δ > 0, there exists an (L + δ)-Lipschitz excellent Morse

0 0 function f : X → R such that ||f − f ||∞ ≤ .

Proof. By [41, Proposition 2.1.], there exists an L-Lipschitz smooth function g : X → R such

0 that ||f − g||∞ ≤ /3. By [6, Proposition 1.2.4.] there exists a Morse function g : X → R

0 0 such that g is (L + (δ/2))-Lipschitz and ||g − g ||∞ ≤ /3. As it is explained in [6, p. 40], there exists an excellent Morse function f 0 : X → R such that f is (L + δ)-Lipschitz and

0 0 0 ||g − f ||∞ ≤ /3. This f satisﬁes the desired properties.

In the rest of this section, we are going to prove the following statement about the level

sets of a Morse functions:

42 Proposition 2.6.2. Let X be a connected manifold, f : X → R be a proper Morse function,

−1 t0 be a regular value of f and Y = f (t0). Let s < t0 < t, A be a connected component of f −1(s) and B be a connected component of f −1(t) . Then there exists connected components

Y1,...,Yk, k ≤ β1(X) + 1 of Y separating A from B.

We need the following lemma for the proof of Proposition 2.6.2:

Lemma 2.6.3. Let Y be codimension 1 compact submanifold of X. If Y has at least β1(X)+1

connected components, then X − Y is disconnected.

Proof. During the proof we use Z/2 coeﬃcients for cohomology. By the long exact cohomology sequence for the pair (X,X − Y ), we have the following exact sequence:

H0(X,X − Y ) → H0(X) → H0(X − Y ) → H1(X,X − Y ) → H1(X).

For any open neighborhood N of Y , by excision H∗(X,X − Y ) ∼= H∗(N,N − Y ). Let N be

a tubular neighborhood of Y in X, hence (N,N − Y ) is homeomorphic to (NX Y,NX Y − Y )

where NX Y is the rank 1 normal bundle of Y in X [16, p. 100]. By the Thom Isomorphism

1 ∼ 0 0 Theorem [65, p. 106], H (NX ,NX − Y ) = H (Y ), H (NX Y,NX − Y ) = 0. Hence the exact

sequence above becomes

0 → H0(X) → H0(X − Y ) → H0(Y ) → H1(X),

which implies

0 0 dim H (Y ) ≤ dim H (X − Y ) − 1 + β1(X).

Proof of Proposition 2.6.2. Since s < t0 < t, Y separates A from B. Let Y1,...,Yk be

a minimal collection of components of Y separating A from B. We will show that k ≤

43 β1(X) + 1. Note that if k > β1(X) + 1, then the family Y2,...,Yk separates X into at least

two components by Lemma 2.6.3. Hence it is enough to show that this family does not

separate X.

0 0 Let Y = Y1 ∪· · ·∪Yk. Let us show that X −Y has exactly two components corresponding

to A and B. It has at least two components corresponding to A and B. By minimality, there

exist a curve γi from A to B intersecting Yi and not intersecting Yj for each i, j = 1, ˙,k, i 6= j.

Choose closed tubular neighborhoods Ni of Yi for each i = 1, . . . , k such that they are pairwise

disjoint and Ni does not intersect γj for i 6= j. The normal bundle NYi X is trivial since it is

1-dimensional and it has a nowhere zero section given by the gradient of f. Therefore Ni −Yi

has exactly two components. We can modify γi if necessary so that it intersects Yi once and

transversally. Let x and y be the entrance and exit points of γi to Ni respectively. Note

that x and y should be in diﬀerent components of Ni − Yi since otherwise we can modify γi

0 so it avoids Ni, hence Y altogether. Now, remove the part of γi between x and y and by

using triviality of the normal bundle replace it with one that intersects Yi at a single point

transversally. Let xi be this point. γi − {xi} has two components, each intersecting with

A B A diﬀerent components of Ni − Yi by transversality. Therefore Ni − Yi = Ni q Ni where Ni

0 B is the component of Ni − Yi which is in the same component of X − Y with A, and Ni is deﬁned similarly. Now for any point in q let γ be a path in X connecting q to A. If γ does not intersect Y 0 then q is in the same component with A in X − Y 0. Assume that γ

0 intersects Y . Let Ni be the ﬁrst tubular neighborhood that γ enters into. Then q is in the same component of X − Y 0 with the entrance point, which we have shown that either connected to A or B in X − Y 0. Therefore X − Y 0 has two components.

44 Any point in Y1 can be connected by a path to both B and A by using γi in X − (Y2 ∪

· · · ∪ Yk). Combining this with the previous paragraph, we see that X − (Y2 ∪ · · · ∪ Yk) is connected.

45 Chapter 3: A Geometric Characterization of Vietoris-Rips Filtration

3.1 Introduction

A particularly nice ambient space inside which one can isometrically embed a compact

metric space (X, dX ) is its Kuratowski space κ(X) consisting of all real valued functions

on X, together with the `∞ norm. In the case of ﬁnite metric spaces, that the Vietoris-

Rips ﬁltration of a metric space produces persistence modules isomorphic to the sublevel set

ﬁltration of the distance function δX : κ(X) → R+, κ(X) 3 f 7→ infx∈X kdX (x, ·) − fk∞ was used in [22] in order to prove the Gromov-Hausdorﬀ stability of Vietoris-Rips persistence.

We generalize this point of view signiﬁcantly by proving an isomorphism theorem between

−1 the Vietoris-Rips filtration of X and the filtration δX ((0, r)) r>0. We do so by constructing a filtered homotopy equivalence between (the geometric realization of) the Vietoris-Rips

ﬁltration and the sublevel set ﬁltration induced by δX . Furthermore, we prove that κ(X) above can be replaced with any injective metric space admitting an isometric embedding of

A certain well known construction that uses the isometric embedding X,→ κ(X) is that of the ﬁlling radius of a Riemannian manifold [43] deﬁned by Gromov. In that construction, given an n-dimensional Riemannian manifold M one studies for each r > 0 the inclusion

46 −1 ιr : M,→ δM ((0, r)), and seeks the inﬁmal r > 0 such that the map induced by ιr at n-th homology level maps the fundamental class [M] to zero. In a series of papers [51, 49,

48, 50] M. Katz studied both the problem of computing the ﬁlling radius of spheres and complex projective spaces, and the problem of understanding the change in homotopy type

−1 1 2 of δX ((0, r)) when X ∈ {S , S }. Of central interest in topological data analysis has been the complete characterization of the persistence diagrams of spheres of diﬀerent dimensions. Our isomorphism theorem permits applying Katz’s results in order to provide partial answers to these questions.

Comparison to the work of Adamaszek et al. Papers [2, 4] contain ideas similar to ours. The authors of those papers also point out that even though Vietoris-Rips complexes induced from sampling subset X from a manifolds M are an eﬀective tool to capture topology of underlying manifold, they cannot recover metric information of M. To remedy that situation, they embed X into the space of probability measures on M with 1-Wasserstein metric and consider thickenings of X in that large ambient space. Here, their “thickening” is not exactly the same as ours: whereas they choose elements of the thickening more carefully by using the convexity of the probability space. As a result, they could prove analogous version of Hausmann’s theorem [46] and Latschev’s theorem [54] and compute homotopy type of the thickenings of spheres.

Organization. In Section 3.2 we construct a category of metric pairs. This category will be the natural setting for our extrinsic persistence homology. Although being functorial is trivial in the case of Vietoris-Rips persistence, the type of functoriality which one is supposed

47 to expect in the case of metric embeddings is a priori not obvious. We address this question

in Section 3.2 by introducing a suitable category structure.

In Section 3.3, we show that the Vietoris-Rips ﬁltration can be (categorically) seen as a

special case of persistence obtained through metric embeddings.

In Section 3.4, we obtain new proofs of formulas about the Vietoris-Rips persistence of

product and wedge sum of metric spaces.

In Section 3.5, we give some applications of our ideas to the ﬁlling radius of Riemannian

manifolds. As a result, we will be able to prove, under some additional conditions, that if

two manifolds M and N are similar in the sense of Gromov-Hausdorﬀ distance, their ﬁlling radii are also close.

3.2 Persistence via Metric Pairs

One of the insights leading to persistence associated to metric spaces was considering neighborhoods of a metric space in a nice (for example Euclidean) embedding. In this section we formalize this idea in a categorical way.

Deﬁnition 3.2.1 (Category of metric pairs). • A metric pair is an ordered pair (X,E)

of metric spaces such that X is a metric subspace of E.

• Let (X,E) and (Y,F ) be metric pairs. A 1-Lipschitz map from (X,E) to (Y,F ) is a

1-Lipschitz map from E to F mapping X into Y .

• Let (X,E) and (Y,F ) be metric pairs and f and g be 1-Lipschitz maps from (X,E)

to (Y,F ). We say that f and g are equivalent if there exists a continuous family

(ht)t∈[0,1] of 1-Lipschitz maps from E to F and a 1-Lipschitz map φ : X → Y such that

h0 = f, h1 = g and ht|X = φ for each t.

48 • We deﬁne PMet as the category whose objects are metric pairs and whose morphisms

are deﬁned as follows. Given metric pairs (X,E) and (Y,F ), the morphisms from

(X,E) to (Y,F ) are equivalence classes of 1-Lipschitz maps from (X,E) to (Y,F ).

Deﬁnition 3.2.2 (Persistence families). A persistence family is a collection {(Ur), fr,s}0

map, fr,r = IdUr and fs,t ◦ fr,s = fr,t.

Given two persistence families (U∗, f∗,∗) and (V∗, g∗,∗) a morphism from the ﬁrst one to the second is a collection (φ∗)r>0 such that for each 0 < r ≤ s, φr is a homotopy class of maps from Ur → Vr and φs ◦ fr,s is homotopy equivalent to gr,s ◦ φr.

We denote the category of persistence families with morphisms speciﬁed as above by hTop∗

Remark 3.2.1. Let (X,E) and (Y,F ) be persistent pairs and f be a 1-Lipschitz morphism between them. Then f maps Br(X,E) into Br(Y,F ) for each r > 0. Furthermore, if g is equivalent to f, then they reduce to homotopy equivalent maps from Br(X,E) to Br(Y,F ) for each r > 0.

By the Remark above, we get the following functor from PMet to hTop∗.

Definition 3.2.3 (Persistence functor). Define the persistence functor B∗ : PMet → hTop∗ sending (X,E) to the persistence family obtained by the filtration (Br(X,E))r>0 and sending a morphism between metric pairs to the homotopy classes of maps it induces between the

ﬁltrations.

Let Met be the category of metric spaces where morphisms are given by 1-Lipschitz maps. There is a forgetful functor from PMet to Met mapping (X,E) to X and mapping

49 a morphism deﬁned on (X,E) to its restriction to X. Although forgetful functors tend to have left adjoints, we are going to see that this one has a right adjoint.

Theorem 3.2.2. The forgetful functor PMet to Met has a right adjoint.

First we need prove a few results. See the Section 2.1.3 for background on injective metric spaces.

Lemma 3.2.3. Let (X,E) and (Y,F ) be metric pairs such that F is an injective metric space. Let f and g be 1-Lipschitz maps from (X,E) to (Y,F ). Then f is equivalent to g if and only if f|X ≡ g|X .

Proof. The only if part obvious. Now assume that f|X ≡ g|X . By [53, Proposition 3.8], there exists a geodesic bicombing γ : F × F × [0, 1] → F such that for each x, y, x0 and y0 in

F and t in [0, 1],

0 0 0 0 dF (γ(x, y, t), γ(x , y , t)) ≤ (1 − t) dF (x, x ) + t dF (y, y ).

For t in [0, 1], deﬁne h : E → F by ht(x) = γ(f(x), g(x), t). Note that h0 = f, h1 = g and

(ht)|X is the same map for all t. The inequality above implies that ht is 1-Lipschitz for all t.

This completes the proof.

Lemma 3.2.4. Let (X,E) and (Y,F ) be metric pairs such that F is an injective metric

space. Then for each φ : X → Y , there exist a unique 1-Lipschitz map from (X,E) to (Y,F )

extending φ up to equivalence.

Proof. The uniqueness part follows from Lemma 3.2.3. The existence part follows from the

injectivity of F .

50 Proof of Theorem 3.2.2. Let κ : Met → PMet be the functor sending X to (X, κ(X)) where

κ(X) is the Kuratowski space of X (see Deﬁnition 2.1.12). A 1-Lipschitz maps f : X → Y is sent to the unique morphism (see Lemma 3.2.3) extending f. There is a natural morphsim

Hom((X,E), (Y, κ(Y ))) → Hom(X,Y ), sending a morphism to its restriction to X. By Lemma 3.2.4, this is a bijection. Hence κ is a right adjoint to the forgetful functor.

Remark 3.2.5. Note that in the above proof the Kuratowski space κ(·) can be replaced by the tight span E(·) (see Example 3.2.6 below and [35]) or any other construction which assigns an injective space containing a given space.

Deﬁnition 3.2.4 (Metric homotopy pairing). A functor η : Met → PMet is called a metric homotopy pairing if it is a right adjoint to the forgetful functor.

Example 3.2.6. Let (X, dX ) be a metric space. By κ(X) denote the Kuratowski space associated to X. Consider also the following additional spaces associated to X:

0 0 0 ∆(X) := {f ∈ `∞(X): f(x) + f(x ) ≥ dX (x, x ) for all x, x ∈ X},

E(X) := {f ∈ ∆(X) : if g ∈ ∆(X) and g ≤ f, then g = f},

∆1(X) := ∆(X) ∩ Lip1(X, R),

with `∞ metrics for all of them (cf. [53, Section 3]). Here E(X) is the tight span of X [35].

Then,

(X, κ(X)), (X,E(X)), (X, ∆(X)), (X, ∆1(X)) are all metric homotopy pairings since the second element in each pair is an injective metric space (see [53, Section 3]) into which X isometrically embeds.

51 3.3 Isomorphism

Recall that Met is the category of metric spaces with 1-Lipschitz maps as morphisms.

∗ We have a functor VR : Met → hTop∗ obtained by the (geometric realization of the) Vietoris-Rips ﬁltration. The main theorem we prove in this section is the following:

Theorem 3.3.1 (Isomorphism Theorem). Let η : Met → PMet be a metric homotopy pairing

(for example the Kuratowski functor). Then B∗ ◦ η : Met → hTop∗ is naturally isomorphic to VR2∗.

Deﬁnition 3.3.1. Let (X,E) be a metric pair and r > 0. Let Ur denote the open cover of

Br(X,E) consisting of open balls in E with center in X and radius r.

ˇ • We denote the geometric realization of the Cech complex of Ur by Sr(X,E). Note

2∗ that S∗(X,E) is a ﬁltration and we have an inclusion S∗(X,E) into VR (X) where

by abuse of notation we denote the geometric realization of VR2∗(X) by the same

notation.

• We denote the corresponding complex of spaces by (X,E)r. We denote the amalgama- ` tion of (X,E)r by r(X,E) and realization of (X,E)r by ∆r(X,E) (see [45, Chapter 4.G]).

• We have natural morphisms

a 2r Br(X,E) ← (X,E) ← ∆r(X,E) → Sr(X,E) → VR (X). r (see [45, Chapter 4.G])

Proposition 3.3.2. • Given 0 < r ≤ s, the inclusion of open balls Br(x, E) ⊆ Bs(x, E) ` ` induces morphisms r X → s X and ∆rX → ∆sX such that the diagram in the

52 deﬁnition above induces a diagram of ﬁltered topological spaces:

a 2∗ B∗(X,E) ← (X,E) ← ∆∗(X,E) → S∗(X,E) → VR (X). ∗

Furthermore, if E is an injective metric space, then all morphisms in the diagram are

homotopy equivalences.

• If (Y,F ) is a metric pair and f :(X,E) → (Y,F ) is 1-Lipschitz, then the following

diagram commutes

` 2∗ B∗(X,E) ∗(X,E) ∆∗(X,E) S∗(X,E) VR (X)

f∗ f∗ f∗ f∗ f∗ . ` 2∗ B∗(Y,F ) ∗(Y,F ) ∆∗(Y,F ) S∗(Y,F ) VR (Y )

Furthermore, if we change f with an equivalent map, then the homotopy types of the

vertical maps remain unchanged.

Proof. Commutativity of the diagrams in the statement follows easily from the deﬁnitions. ` ` Note that B∗(X,E) ← ∗(X,E) is a homeomoprhism since the r(X,E) is an amalga- ` mation corresponding to an open cover of Br(X,E). The morphism ∗(X,E) ← ∆∗(X,E) is a homotopy equivalence by [45, Proposition 4G.2]. If E is an injective metric space, then the open cover Ur of Br(X,E) is a good cover by Lemma 2.1.14, therefore ∆∗(X,E) → S∗(X,E)

is a homotopy equivalence by [45, Corollary 4G.3]. Finally, since E is hyperconvex, the

2∗ morphism S∗(X,E) → VR (X) is a homeomorphsim (see the proof of Proposition 2.2.4).

If f, g are equivalent, then the homotopy (ht) between f, g induces homotopy (ht)∗ for

all horizontal maps (note that the two rightmost vertical maps does not change at all).

Proof of Theorem 3.3.1. Since all metric homotopy pairings are naturally isomorphic, with-

out loss of generality we can assume that η = κ, the Kuratowski functor. By Proposition

53 ` 2∗ 3.3.2, B∗, ∗, ∆∗,S∗ and VR are functors from PMet to the category of persistence families with certain natural transformations between them. By the homotopy equivalence

part of Proposition 3.3.2, if we precompose these functors with the Kuratowski functor

κ : Met → PMet, they all become naturally isomorphic.

3.3.1 Stability of Metric Homotopy Pairings

In this section, we deﬁne a distance between metric pairs by using the homotopy in-

terleaving distance introduced by Blumberg and Lesnick [12] and then show that metric

homotopy pairings are 1-Lipschitz with respect to this distance and the Gromov-Hausdorﬀ

distance.

First, we need to review homotopy interleaving distance between R-spaces (see Section 2.3). For more details, please see [12, Section 3.3].

Deﬁnition 3.3.2 (R-space). An R-space is a functor from the poset (R, ≤) to the category of topological spaces.

Example 3.3.3. Given a metric pair (X,E), the ﬁltration of closed neighborhoods B∗(X,E)

is an R-space.

Deﬁnition 3.3.3 (Interleaving). Two R-spaces A∗ and B∗ are said to be δ-interleaved for

some δ > 0 if there are natural transformations f : A∗ → B∗+δ and g : B∗ → A∗+δ such that

f ◦ g and g ◦ f are equal to the structure maps B∗ → B∗+2δ and A∗ → A∗+2δ, respectively.

Deﬁnition 3.3.4 (Homotopy interleaving distance). A natural transformation f : R∗ → A∗ is called a weak homotopy equivalence if f induces a isomorphism between homotopy groups

0 at each index. Two R-spaces A∗ and A∗ are said to be weakly homotopy equivalent if there

0 0 exists an R-space R∗ and weak homotopy equivalences f : R∗ → A∗ and f : R∗ → A∗.

54 The homotopy interleaving distance dHI(A∗,B∗) is then deﬁned to be the inﬁmal δ such that

0 0 0 0 there exists δ-interleaved R-spaces A∗ and B∗ with the property that A∗ and B∗ are weakly homotopy equivalent to A∗ and B∗, respectively.

We now adapt this construction to metric pairs. Given metric pairs (X,E) and (Y,F ), we deﬁne the homotopy interleaving distance between them by

dHI((X,E), (Y,F )) := dHI((B∗(X,E)),B∗(Y,F )).

The main theorem that we are going to prove in this section is the following:

Theorem 3.3.4. Let η : Met → PMet be a metric homotopy pairing. Then, for any compact metric spaces X and Y ,

dHI(η(X), η(Y )) ≤ dGH(X,Y ).

Remark 3.3.5. Note that combining Theorem 3.3.4 and Theorem 3.3.1, we obtain another proof of the following classical stability result (see [22, Theorem 3.1] and [24, Lemma 4.3]): for any compact metric spaces X and Y,

∗ ∗ dI(PH(VR (X)), PH(VR (Y ))) ≤ 2 dGH(X,Y ).

Lemma 3.3.6. If (X,E) and (Y,F ) are isomorphic in PMet, then dHI((X,E), (Y,F )) = 0.

Proof. Let f :(X,E) → (Y,F ) and g :(Y,F ) → (X,E) be 1-Lipschitz maps such that f ◦ g

and g ◦ f are equivalent to the respective identities. Then, the result follows since f and g

induce isomorphism between the R-spaces B∗(X,E) and B∗(Y,F ).

Lemma 3.3.7. Let E and F be injective metric spaces containing X. Then (X,E) is

isomorphic to (X,F ) in PMet.

55 Proof. By injectivity of E and F , there are 1-Lipschitz maps f : E → F and g : F → E

such that f|X and g|X are equal to IdX . Hence, by Lemma 3.2.3, f ◦ g :(X,F ) → (X,F )

and g ◦ f :(X,E) → (X,F ) are equivalent to identity.

Proof of Theorem 3.3.4. Since all natural homotopy pairings are naturally isomorphic, by

Lemma 3.3.6, without loss of generality we can assume that η = κ, the Kuratowski functor.

Let r > dGH(X,Y ). Let us show that

dHI((X, κ(X)), (Y, κ(Y )) ≤ r.

By assumption, there exists a metric space Z containing X and Y such that the Hausdorﬀ

distance between X and Y as subspaces of Z is less than or equal to r. Hence, the R-

spaces B∗(X, κ(Z)) and B∗(Y, κ(Z)) are r-interleaved as B(X, κ(Z)) ⊆ Br+(Y, κ(Z)) and

B(Y, κ(Z)) ⊆ Br+(X, κ(Z)) for each . Now, by Lemma 3.3.7, we have

dHI((X, κ(X)), (Y, κ(Y ))) = dHI((X, κ(Z), (Y, κ(Z))) ≤ r.

3.4 Application to the Vietoris-Rips Filtration

In this section we see applications of our isomorphism theorem to establish properties of

the Vietoris-Rips ﬁltration and its persistent homology.

3.4.1 Products and Wedge Sums

The following statements are obtained at the simplicial level in [67], [3, Proposition 4],

[62, Proposition 13]. Here we give alternative proofs using neighborhoods in a hyperconvex

embedding.

56 Theorem 3.4.1. Let X and Y be metric spaces. Then,

(1) (Persistent K¨unnethformula) Let X × Y denote the `∞ product of X and Y . If X has

pointwise ﬁnite dimensional Vietoris-Rips persistent homology, then

PH(VR∗(X × Y )) ∼= PH(VR∗(X)) ⊗ PH(VR∗(Y )).

(2) Let p and q be points in X and Y respectively. Let X ∨ Y denote the wedge sum of

metric spaces X and Y along p and q. Then

PH(VR∗(X ∨ Y )) ∼= PH(VR∗(X)) ⊕ PH(VR∗(Y )).

Remark 3.4.2. Note that the tensor product of two irreducible persistence modules corresponding to intervals I,J is the simple persistence module corresponding to the interval I ∩J.

Therefore, the ﬁrst part of Theorem 3.4.1 implies that, as a multiset,

VR VR VR dgmn (X × Y ) = {I ∩ J : I ∈ dgmi (X),J ∈ dgmj (Y ), i + j = n}.

n 1 Example 3.4.3. For a given choice of α1, . . . , αn > 0, let X be the `∞-product Πi=1(αi · S ). Then, by [1, Main result, p. 3] and Remark 3.4.2, for every positive integer m we have: ( ) VR 2παili 2παi(li + 1) X dgmm (X) = max , min :(li)i=1,...,n, li ∈ Z≥0,, (2 li + 1) = m . li6=0 2li + 1 li6=0 2li + 3 i,li6=0 Note that we are deﬁning a multiset above, hence if an element appears more than once in

the deﬁnition, then it will appear more than one in the multiset. In particular, in the case

of X = S1 × S1, for all integers m ≥ 0 we have the following:

VR 1 1 dgm0 (S × S ) = {[0, ∞)}, 2πm 2π(m + 1) 2πm 2π(m + 1) dgmVR ( 1 × 1) = , , , , 2m+1 S S 2m + 1 2m + 3 2m + 1 2m + 3 2πm 2π(m + 1) dgmVR ( 1 × 1) = , , 4m+2 S S 2m + 1 2m + 3 VR 1 1 dgm4m+4(S × S ) = ∅.

57 To be able to prove Theorem 3.4.1, we need the following lemmas:

Lemma 3.4.4. If E and F are injective metric spaces, then so is their `∞ product.

Proof. Let X be a metric space. Note that (f, g): X → E × F is 1-Lipschitz if and only if f and g are 1-Lipschitz. Given such f and g and a metric embedding X into Y , we have 1-Lipschitz extensions f,˜ g˜ of f and g from Y to E and F respectively. Hence,

(f,˜ g˜): Y → E × F is a 1-Lipschitz extension of (f, g). Therefore E × F is injective.

Lemma 3.4.5. If (E, dE) and (F, dF ) are injective metric spaces, then so is their wedge sum

along any two points.

Proof. Let p and q be points in E and F respectively and E ∨ F denote the wedge sum of

E and F along p and q. We are going to show that E ∨ F is hyperconvex, hence injective.

Let (xi, ri)i, (yj, rj)j be such that xi is in E, yj is in F , ri ≥ 0, sj ≥ 0, dE(xi, xi0 ) ≤ ri +ri0 ,

0 0 dF (yj, yj0 ) ≤ sj + sj0 and d(xi, yj) ≤ ri + sj for each i, i , j, j . Deﬁne by

:= max inf ri − dE(xi, p), inf sj − dF (yj, q) . i j

Let us show that ≥ 0. If the second element inside the maximum is negative, then there

exists j0 such that dF (yj0 , q) − sj0 > 0. Since dE(xi, yj0 ) = dE(xi, p) + dF (q, yj0 ) for all i, we

have

ri − dE(xi, p) ≥ dF (yj0 , q) − sj0 > 0.

Therefore the ﬁrst element inside the maximum is positive. Hence > 0.

Without loss of generality let us assume that

:= inf ri − dE(xi, p) > 0. i

58 This implies that the non-empty closed ball b(q, F ) is contained in bri (xi,E ∨ F ) for all i.

Now, for each j, we have

+ sj = inf ri − dE(xi, p) + sj i

≥ inf d(xi, yj) − dE(xi, p) i

= dF (yj, q).

Therefore,

(∩ibri (xi,E ∨ F )) ∩ (∩jbsj (yj,E ∨ F )) ⊇ b(q, F ) ∩ (∩jbsj (yj,F ))

which is non-empty by hyperconvexity of F .

Proof of Theorem 3.4.1. (1) Let E and F be injective metric spaces containing X and Y respectively. Let E × F denote the `∞ product of E and F . Note that for each r > 0,

Br(X × Y,E × F ) = Br(X,E) × Br(Y,F ).

Hence, by the K¨unnethformula,

∼ H∗(Br(X × Y,E × F )) = H∗(Br(X,E)) ⊗ H∗(Br(Y,F )).

Now, the result follows from Lemma 3.4.4 and Proposition 3.3.2.

(2) Let E and F be as above and E ∨ F denote wedge product of E and F along p and

q. Note that

Br(X ∨ Y,E ∨ F ) = Br(X,E) ∨ Br(Y,F ).

Hence, by [45, Corollary 2.25],

∼ H∗(Br(X ∨ Y,E ∨ F )) = H∗(Br(X,E)) ⊕ H∗(Br(Y,F )).

Now, the result follows from Lemma 3.4.5 and Proposition 3.3.2.

59 The following statement can be found in [42, Lemma 1.7.A, p. 96], where the proof is

given in the simplicial level. Here, we include another proof since it easily follows from the

tools we developed up to this point.

Proposition 3.4.6. Let X be a geodesic space. Then, for r > 4 hyp(X), VRr(X) is contractible.

Proof. Let E(X) denote the injective hull (tight span) of X (see [53]). By [53, Prop 1.3] Prop

1.3 dH(X,E(X)) ≤ 2 hyp(X) (note that the deﬁnition of hyperbolicity in [53] is twice the conventional deﬁnition we gave above). As E(X) is an injective metric space, the injective hull functor E : Met → PMet is a metric homotopy pairing. Hence, by Theorem 3.3.1,

r VR (X) is homotopy equivalent to Br/2(X,E(X)) = E(X), which is contractible.

3.5 Applications to the Filling Radius

Here we prove a few statements about the ﬁlling radius of a Riemannian manifold [51]. We

also deﬁne a strong notion of ﬁlling radius which is akin to the so called maximal persistence

in the realm of topological data analysis.

3.5.1 Bounding Barcode Length via Spread

We recall a metric concept called spread. The following deﬁnition is a variant of the one

given in [51].

Deﬁnition 3.5.1 (k-spread). For every non-negative integer k, the k-th spread spreadk(X)

of a metric space (X, dX ) is the inﬁmal r > 0 such that there exists a subset A of X with

cardinality at most k such that

• diam(A) < r

60 • supx∈X infa∈A dX (x, a) < r.

Finally, the spread of X is deﬁned to be spread(X) := infk spreadk(X), i.e. the set A is allowed to have arbitrary cardinality.

Remark 3.5.1. Recall that the radius of a compact metric space (X, dX ) is rad(X) := infp∈X maxx∈X dX (p, x). Thus rad(X) = spread1(X).

VR Let (X, dX ) be a metric space. Then, for each integer k ≥ 0 we deﬁne dgmk (X) as the

r persistence barcode associated to Hk(VR (X)), the k-th persistence module induced by the

Vietoris-Rips ﬁltration of X.

VR Proposition 3.5.2. Let (X, dX ) be a metric space, k ≥ 1, and I ∈ dgmk (X). Then

length(I) ≤ spread(X).

Proof. Let r > spread(X). It is enough to show that for each s > 0, the map

s r+s H∗(VR (X)) → H∗(VR (X)) induced by the inclusion is zero. Let A be a subset realizing r in the deﬁnition of spread.

Let π : X → A be a map sending x to a closest point in A. Let ι : A → X be the inclusion map. These induces the following maps

VRs(X) → VRr(A) → VRr+s(X).

Note that it is enough to show that composition of these maps is contiguous to the inclusion, since then the inclusion maps splits through H(VRr(A)) = 0 in the homology level. Let

{x1, . . . , xk} be a subset of X with diameter less than s. Let ai := π(xi). Then we have

dX (xi, aj) ≤ dX (xi, xj) + dX (xj, aj) < r + s.

Hence the diameter of the subspace {x1, . . . , xk, a1, . . . , ak} is strictly less than r + s. This shows the desired contiguity and completes the proof.

61 3.5.2 Bounding the Filling Radius

For a given compact space (X, dX ) consider the the Kuratowski embedding ι : X → κ(X).

Recall that given a compact n-dimensional manifold M one deﬁnes a the ﬁlling radius [51]

of M as follows:

FillRad(M) := inf{ > 0| Hn(ι,A)([M]) = 0},

where ι : ι(M) ,→ B(ι(M), κ(M)), [M] is the fundamental class of M, and A = Z if M is orientable and A = Z2 otherwise. Note that the ﬁlling radius can be deﬁned not just for the Riemannian metric but for any metric on a compact manifold.

Remark 3.5.3. Note that relative ﬁlling radius can be deﬁned for every metric pair (M,E) by considering -neighborhoods of M in E. The resulting invariant is denoted by FillRad(M,E).

Gromov [43] showed that we obtain the minimal possible relative ﬁlling radius through the

Kuratowski embedding. This also follows from our work but in greater generality in the

context of embeddings into injective metric spaces. If M can be isometrically embedded into

an injective metric space I, then the embedding of M into E can be extended to a 1-Lipschitz

map f : E → I, which induces a map of ﬁltrations f : B∗(M,E) → B∗(M,I). Hence, if

the fundamental class of M vanishes in B(M,E) for some > 0, then it also vanishes in

B(M,I). Therefore, FillRad(M,I) ≤ FillRad(M,E).

n 1 1 Remark 3.5.4. In [51, Theorem 2] Katz proved that FillRad(S ) = 2 arccos − n+1 . More-

n n over, in a remark right after the proof of Theorem 2 he shows that Br(S , κ(S )) is homotopy

equivalent to Sn if r ∈ (0, FillRad(Sn)]. One might then conjecture that FillRad(M) is the

ﬁrst point where the homotopy type of Br(M, κ(M)) changes. In general, however, this is

not true as the following example show.

62 Figure 3.1: A big sphere X with a small handle. In this case, as r > 0 increases, Br(X, κ(X)) 2 changes homotopy type from that of X to that of S as soon as r > r0 for some r0 < FillRad(X).

The following example provides a geometric intuition for how homotopy type of Kura- towski neighborhoods may change before the filling radius. Consider a big sphere with a small handle attached (see Figure 3.1). Since the top dimensional hole in this space is big, we expect the filling radius to be big. On the other hand, 1-dimensional homology class coming from the small handle dies in a small Kuratowski neighborhood, hence the homotopy type changes. We can prove the fact that the filling radius of this space is big as follows: Embed a big sphere into the interior of this space in the Euclidean space and consider the radial projection onto the sphere. This is a degree 1, 1-Lipschitz map. The existence of such a map guarantees that the filling radius of the original space is greater than or equal to the

ﬁlling radius of the embedded sphere (this follows from the functoriality of the Vietoris-Rips

ﬁltration and the Vietoris-Rips characterization of the ﬁlling radius (Proposition 3.5.5).1

Proposition 3.5.5. Let M be a compact n-dimensional Riemannian manifold and let B be the n-th persistence barcode of the open Vietoris-Rips ﬁltration of X. Then B contains a unique interval I with left endpoint 0. Furthermore, the right endpoint of I is 2 FillRad(M).

1Another proof can be found in [57, Theorem 1.1.]).

63 Proof. By Haussmann [46], the persistent homology of the Vietoris-Rips filtration of M contains a unique interval with left endpoint 0. Now the result follows from the definition of the filling radius and Theorem 3.3.1.

The following proposition is proved in [51]. Here we present diﬀerent proof which easily

follows from what we have done until now from the persistence homology perspective.

Proposition 3.5.6. Let M be a compact Riemannian manifold. Then,

FillRad(M) ≤ spread(X)/2.

Proof. Follows from Proposition 3.5.5 and Proposition 3.5.2.

Remark 3.5.7. Let M and N be compact Riemannian manifolds. Let M × N denote the

`∞-product of M and N. By Proposition 3.5.5 and Theorem 3.4.1,

FillRad(M × N) = min(FillRad(M), FillRad(N)).

A similar result is true if we take `∞ product of more than two Riemannian manifolds.

3.5.3 Stability

Deﬁnition 3.5.2 (Strong ﬁlling radius). Given a compact n-dimensional Riemmannian

manifold M, we deﬁne the strong ﬁlling radius sFillRad(M) as half the length length of the largest interval in the n-th Vietoris-Rips persistence barcode of M:

1 sFillRad(X) := max length(I),I ∈ dgmVR(M) . 2 n

The following remark follows directly from Proposition 3.5.2 and Proposition 3.5.5.

Remark 3.5.8. FillRad(M) ≤ sFillRad(M) ≤ spread(M)/2.

64 Deﬁnition 3.5.3 (Regularly ﬁlled manifold). Let M be a compact Riemannian manifold.

We say that M is regularly ﬁlled if FillRad(M) = sFillRad(M).

Proposition 3.5.9. For each n ≥ 1, the n-dimensional unit sphere with the intrinsic metric is regularly ﬁlled.

Proof. By [51, Proof of Theorem 2], FillRad(X) = spread(X)/2. Hence the result follows from Remark 3.5.8.

Remark 3.5.10 (A non regularly ﬁlled geodesic metric space). Let r > 1 and X be the `∞ product S1 × S1 × rS1. By Remark 3.5.7,

2π FillRad(X) = FillRad( 1) = . S 3

VR By Example 3.4.3, dgm3 (X) contains the interval (2πr/3, 4πr/5], which has length 2πr/15. Hence, if r > 5, X is not regularly ﬁlled.

By invoking the relationship between the Vietoris-Rips persistent homology and the filling radius, we can prove that the filling radii of two n-dimensional Riemannian manifolds M and N are close if these two manifolds are similar in the Gromov-Hausdorff distance sense. .

Proposition 3.5.11. Let M,N be two n-dimensional regularly ﬁlled compact Riemannian manifolds. Then

|FillRad(M) − FillRad(N)| ≤ 2 dGH(M,N).

Proof. Let BM and BN be the n-dimensional persistence barcode of the Vietoris-Rips ﬁltra- tions of M and N respectively. Hence, by Proposition 3.5.5, BM contains an interval IM with endpoints 0, 2FillRad(M) and BN contains an interval IN with endpoints 0, 2FillRad(N).

Without loss of generality assume that FillRad(M) ≥ FillRad(N).

65 Take a partial matching M between the barcodes BM and BN . Let us show that

|FillRad(M) − FillRad(N)| ≤ cost(M).

If M does not match IM , then we have

|FillRad(M) − FillRad(N)| ≤ FillRad(M)

≤ cost(M).

Assume that M matches IM with I with endpoints a ≤ b. Since N is regularly ﬁlled, b − a ≤ 2 FillRad(N). Hence we have

cost(M) ≥ max(a, |2FillRad(M) − b|)

≥ FillRad(M) − FillRad(N).

Hence, in any case

|FillRad(M) − FillRad(N)| ≤ cost(M).

Now, the result follows from inﬁmizing over M and Theorem 2.2.3.

66 Chapter 4: Finite Approximations of Compact Metric Spaces

4.1 Introduction

Every compact metric space can be Gromov-Hausdorﬀ approximated arbitrarily well by

ﬁnite metric spaces, which can be seen by taking ﬁnite -nets. To control the cardinality of

X the approximating spaces, we deﬁne the following sequence (compare with δn in Chapter 5)

Deﬁnition 4.1.1. Given a compact metric space X, we deﬁne

X φn := inf{dGH(X,M): M is a ﬁnite metric space such that |M| ≤ n}.

Note that this is a nonincreasing sequence converging to 0.

X The quantity φn is also known as the sketch Sketchn(X) and studied in detail in [64].

X We use the notation (φn ) to highlight the similarity of cases of metric graph approximations (see Chapter 5) and ﬁnite approximations.

In this chapter we assume a given metric space X is compact, locally path connected and

0 has ﬁnitely many connected components. Given such a metric space X, let (X , d0) denote the quotient metric space X/ ∼ where x ∼ x0 if x and x0 are in the same path component.

The main theorem we are going to prove in this chapter is the following (compare with the main results of Chapter 5):

67 X Theorem 4.1.1. Let X be a compact metric space with β = β0(X) ﬁnite. Let `n be the length of the nth largest interval in the 0-dimensional persistence barcode of X. Then

i) For n ≥ β, d (X,X0) GH ≤ φX ≤ d (X,X0). 3n − 2 n GH

ii) For n < β, d (X,X0) (β − 1)`X GH ≤ φX ≤ d (X,X0) + n . 3β − 2 n GH 2

This result has a striking similarity with Theorem 5.1.2, where the role of the Reeb

graph is taken by X0. Hence, this result can be considered as a preamble of the graph

approximation result we are going to get in Chapter 5, and also suggests that there may be

higher dimensional generalizations. Note that when X is a path connected space (for example

a geodesic space), then X0 is a single point and Theorem 4.1.1 is almost straightforward in

that case, hence the result becomes more interesting when X has several path connected

components.

In Section 4.2, we introduce discrete length structures on sets. In Section 4.3, we intro-

duce some discrete length structure constructions on metric space and prove some stability

properties about them. Lastly, in Section 4.4, we give a proof of Theorem 4.1.1.

4.2 Discrete Length Structures

We are going to use discrete length structures to construct and analyze ﬁnite approximations of compact spaces.

Deﬁnition 4.2.1 (Discrete length structure). A discrete length structure on a set X is a non-negative symmetric function ` : X × X → R.

68 Deﬁnition 4.2.2 (Discrete paths). Let X be a set and x and x0 be points in X. A discrete

0 path γ from x to x is an (n + 1)-tuple (x0, . . . , xn) of elements of X such that x0 = x and

0 0 xn = x . The concatenation of paths γ · γ and transverse of a pathγ ¯ is deﬁned as expected.

Deﬁnition 4.2.3 (Lengths of discrete paths). Let X be a set with a length structure `. Let

γ = (x0, . . . , xn) be a discrete path in X. Deﬁne the length `(γ) of γ by

n X `(γ) := `(xi, xi−1). i=1 Remark 4.2.1. • `(¯γ) = `(γ).

• `(γ · γ0) = `(γ) + `(γ0).

Deﬁnition 4.2.4. Let (X, `) be a discrete length space. Deﬁne the induced metric dl :

X × X → R by

0 d`(x, x ) := inf `(γ) where the inﬁmum is taken along all discrete paths between x and x0.

The following proposition is straightforward:

Proposition 4.2.2. Let (X, `) be a discrete length space, Then (X, d`) is a pseudo-metric space.

4.3 Discrete Length structures on Metric Spaces

A nice way discrete length structures arise on metric spaces is as follows:

Deﬁnition 4.3.1. Let (X, dX ) be a metric space with an equivalence relation ∼. Deﬁne the discrete length structure `∼ by

0 0 `∼(x, x ) = dX (x, x ) if x is not equivalent to x0 and zero otherwise.

69 The following proposition is obvious:

Proposition 4.3.1. Let X be a metric space with an equivalence relation ∼. Let d∼ denote

the quotient metric on X. Then d∼ = d`∼

Deﬁnition 4.3.2. Let (X, dX ) be a metric space and r > 0. Deﬁne the discrete length

0 0 0 structure `r on X by `r(x, x ) = dX (x, x ) if dX (x, x ) > r and 0 otherwise. We denote the

r metric space induced by the pseudo-metric d`r by (X , dr).

r Proposition 4.3.2. Let (X, dX ) be a metric space and r > 0. Then X is isometric to

(X0)r

0 Proof. Let `0,r denote the r-length structure on (X, d0). Note that if dX (x, x ) ≤ r, then

0 0 0 0 0 0 d0(x, x ) ≤ dX (x, x ) ≤ r, hence `0,r(x, x ) = 0. If dX (x, x ) > r, then `0,r(x, x ) ≤ d0(x, x ) ≤

0 0 dX (x, x ) = `r(x, x ). Hence `0,r ≤ `r and d0,r ≤ dr.

0 Now, let us show that dr ≤ d0,r. It is enough to show that dr ≤ `0,r. If d0(x, x ) ≤ r,

1 2 1 2 0 1 2 then there exists a path (x0, x0, . . . , xk, xk) realizing d0(x, x ) such that dX (xi , xi−1) ≤ r

1 2 and xi , xi are in the same path component for all i. Take αi to be a continuous path

1 2 from xi to xi and take a partition of αi such that each part has diameter less than r.

1 2 1 2 Existence of such a partition shows that dr(xi , xi ) = 0. We also have dr(xi , xi−1) = 0.

0 0 0 0 0 0 Hence dr(x, x ) = 0 ≤ `0(x, x ). If d0(x, x ) > r, then dr(x, x ) ≤ d0(x, x ) = `0,r(x, x ).

We have the following stability result:

Proposition 4.3.3. Let (X, dX ) and (Y, dY ) be ﬁnite metric spaces. Then, for all r, s ≥ 0

max(r, s) d (Xr,Y s) ≤ max(|X| − 1, |Y | − 1) d (X,Y ) + . GH GH 2

Proof. Let R be a correspondence between X and Y . Let R0 be the induced correspondence

r s 0 0 0 0 between X and Y . Given x R y and x R y , let us upper bound dr(x, x ) − ds(y, y ).

70 0 Let y0, . . . , yn be the `r minimizing discrete path from y to y with n ≤ |Y | − 1. Choose

0 x0, . . . , xn such that x0 = x, xn = x and xi R yi for all i. Let us upper bound `r(xi, xi−1) −

`s(yi, yi−1). If dX (xi, xi−1) ≤ r, then `r(xi, xi−1) − `s(yi, yi−1) ≤ 0. Hence, assume that dX (xi, xi−1) > r If dY (yi, yi−1) ≤ s, then dX (xi, xi−1) ≤ dis(R) + s, hence `r(xi, xi−1) −

`s(yi, yi−1) ≤ dis(R) + s. Lastly, if dY (yi, yi−1) > s, then `r(xi, xi−1) − `s(yi, yi−1) ≤ dis(R).

Therefore, in all cases, `r(xi, xi−1) − `s(yi, yi−1) ≤ dis(R) + s.

Now, we have

n 0 0 X dr(x, x ) − ds(y, y ) ≤ `r(xi, xi−1) − `s(yi, yi−1) i=1 ≤ (|Y | − 1)(dis(R) + s).

0 0 We can get a similar result for ds(y, y ) − dr(x, x ), which implies the desired result.

Proposition 4.3.4. Let X be a compact metric space with ﬁnitely many connected compo-

nents and r > 0. There exists δ > 0 such that if r ≤ s ≤ r + δ, then dr = ds.

0 Proof. Let r1 be the minimal distance in X strictly larger than r. Choose δ > 0 such that

0 0 r + δ < r1. For any s in [r, r + δ], we have d0(x, x ) ≤ r if and only if d0(x, x ) ≤ s. Hence, by Proposition 4.3.2 dr = (d0)r = (d0)s = ds.

We immediately get the following corollary:

Corollary 4.3.5. Let X be a compact metric space with ﬁnitely many connected components.

Then,

r s lim dGH(X ,X ) = 0. s→r+

Proposition 4.3.6. Let (X, dX ) be a compact metric space with ﬁnitely many connected

components and (Y, dY ) be a ﬁnite metric space. Let R be a correspondence between X,Y

71 such that dis(R) ≤ r. Then

r d (X0,Y r) ≤ max(|X0| − 1, 2|Y | − 2) . GH 2

0 0 0 Proof. Let x R y and x R y . Let y0, . . . , yn be the `r minimizing discrete path from y to y

0 with n ≤ |Y | − 1. Choose x0, . . . , xn such that x0 = x, xn = x and xi R yi for all i. Let us upper bound `0(xi, xi−1) − `r(yi, yi−1). If xi and xi−1 are in the same path component, then `0(xi, xi−1) − `r(yi, yi−1) ≤ 0. Assume xi and xi−1 are in diﬀerent components. If dY (yi, yi−1) > r, then `0(xi, xi−1) − `r(yi, yi−1) ≤ dis(R) ≤ r. If dY (yi, yi−1) ≤ r, then dX (xi, xi−1) ≤ r + dis(R) < 2r, hence `0(xi, xi−1) − `r(yi, yi−1) < 2r. Therefore, we have

n 0 0 X d0(x, x ) − dr(y, y ) ≤ `0(xi, xi−1) − `r(yi, yi−1) ≤ (|Y | − 1) 2r. i=1

1 2 1 2 0 1 2 Take the discrete path (x0, x0, . . . , xk, xk) realizing d0(x, x ) such that xi , xi are in the

0 1 2 1 2 same path component for all i and k ≤ |X | . Chose a discrete path (y0, y0, . . . , yk, yk) from

0 a a y to y such that yj R xj for all a, j. Let δ > 0 be such that dis(R) + δ ≤ r Take αi to be a

1 2 continuous path from xi to xi and take a partition of αi such that each part has diameter

1 2 less than δ. Existence of such a partition implies that dr(yi , yi ) = 0. Hence,

k 0 0 X 1 2 1 2 dr(y, y ) − d0(x, x ) ≤ dr(yi , yi−1) − dX (xi , xi−1) i=2 ≤ (|X0| − 1)dis(R) ≤ (|X0| − 1) r.

Hence r d (X0,Y r) ≤ max(|X0| − 1, 2|Y | − 2) . GH 2

The following corollary follows from Corollary 4.3.5 and Proposition 4.3.6:

72 Corollary 4.3.7. Let X be a compact metric space with ﬁnitely many connected components and Y be a ﬁnite metric space. Then

0 2dGH(X,Y ) 0 dGH(X ,Y ) ≤ max(|X | − 1, 2|Y | − 2) dGH(X,Y ).

4.4 Proof of Theorem 4.1.1

We ﬁrst need the following lemma

X r Lemma 4.4.1. Under the notation of Theorem 4.1.1, if r > `i , then |X | ≤ i

Proof. Note that x and y are identiﬁed in under d`r if and only if they are in the same path

component of VRr(X). Hence

r r |X | = dim(H0(VR (X))) ≤ i.

Proof of Theorem 4.1.1. Let Y be a ﬁnite metric space of cardinality n and r = dGH(X,Y ).

Let Then, by Proposition 4.3.3 and Corollary 4.3.7 we have

0 2r 2r 0 dGH(X,X ) ≤ dGH(X,Y ) + dGH(Y,Y ) + dGH(Y ,X )

≤ r + (n − 1)r + max(β − 1, 2n − 2) r

≤ max(3β − 2, 3n − 2) r

Inﬁmizing this inequality over all Y with cardinality less than or equal to n, we get

d (X,X0) φX ≥ GH . n 3 max(β, n) − 2

This ﬁnishes part i) and the ﬁrst inequality in part ii).

73 Now, let n < β and r > an. By Lemma 4.4.1 and Proposition 4.3.3, we have

X r 0 0 r φn ≤ dGH(X,X ) ≤ dGH(X,X ) + dGH(X ,X )

0 0 0 r = dGH(X,X ) + dGH(X , (X ) )

0 ≤ dGH(X,X ) + (β − 1)r/2

We get the second inequality in part ii) by letting r go to an.

74 Chapter 5: Metric Graph Approximations of Geodesic Spaces

5.1 Introduction

Every compact metric space can be Gromov-Hausdorﬀ approximated by ﬁnite metric

spaces, which can be seen by taking ﬁnite -nets. A length metric space version of this statement is that every compact geodesic space can be approximated by ﬁnite metric graphs

[18, Proposition 7.5.5], where a metric graph is deﬁned to be a length space which is homeomorphic to a topological graph. The proof similarly proceeds by picking a ﬁnite -net and

constructing a metric graph out of it. To analyze this result more deeply, we introduce the

following sequence. Given a compact geodesic space X and an integer n ≥ 0, we deﬁne

X δn := inf{dGH(X,G): G a ﬁnite metric graph, β1(G) ≤ n}.

X Note that by the approximation result mentioned above, (δn ) is a non-increasing sequence

X converging to 0. Now we can ask quantitative questions such as: how fast (δn ) converges to

X X 0, is there a relation between δn and δm , or is there a relation between this sequence and other metric invariants of X?

One metric invariant we are interested in is the ﬁrst persistent barcode of the open

Vietoris-Rips ﬁltration of X, which intuitively measures the size of one dimensional holes

X X in X. Let a1 ≥ a2 ≥ ... be the lengths of the intervals in the ﬁrst persistent barcode of

75 X the open Vietoris-Rips filtration of X, where ai is defined to be zero if i is greater than the number of intervals in the barcode. Our first result is the following:

Theorem 5.1.1. Let X be a compact geodesic space with β = β1(X) ﬁnite. Then

i) For each n ≥ 0, aX n+1 ≤ δX . 4 n ii) For n > β, δX β ≤ δX ≤ δX . 24n + 19 n β iii) For n < β, aX δX ≤ δX ≤ (24 β + 19) δX + n+1 . β n β 4

X Item i) provides a precise lower for δn based on the information contained in the ﬁrst persistent barcode of X. Item ii) indicates that increasing n beyond β does not result in

X superlinear improvement in the approximation as measured by δn . Item iii) provides a

X X th precise relationship between δn on the one hand, and δβ and the (n + 1) -bar in the ﬁrst persistent barcode of X in the regime when n is less than β.

Furthermore, if p is a point in X such that the Reeb metric graph (with an intrinsic metric) associated to the distance function to p,

Xp := R(dX (p, ·): X → R) is a ﬁnite metric graph , then we have the following statement.

Theorem 5.1.2. Let (X, dX ) be a compact geodesic space such that β = β1(X) is ﬁnite and

X,p p be a point in X such that Xp is a ﬁnite graph. Let ρ := dGH(X,Xp). Then

76 i) For n ≥ β, ρX,p ≤ δX ≤ ρX,p. 16n + 13 n

ii) For n < β, ρX,p ≤ δX ≤ ρX,p + (6 β + 6) aX . 16 β + 13 n n+1

G Note that for a ﬁnite metric graph G, δn = 0 for n ≥ β1(G). For n < β1(G), we have the following statement.

Proposition 5.1.3. Let G be a ﬁnite metric graph and n < β1(G). Then

aG n+1 ≤ δG ≤ (6 β (G) + 6) aG . 4 n 1 n+1

Note that

X δ0 = inf{dGH(X,T ): T is a ﬁnite metric tree}.

A metric space is called a tree metric if it can be isometrically embedded into a metric tree.

The approximation of a ﬁnite metric space M of cardinality n by a tree metric is studied by

Gromov in [42, Proposition 6.1.B], where he found the upper bound (log2(2n + 2)) hyp(M) where hyp(M) is the hyperbolicity constant of M (also see [63]). He further remarked that

[42, Remark 6.1.C] this result still holds for spaces which can be covered by at most n geodesics. However, for a ﬁnite graph, the number of geodesics that covers the whole graph can be much larger than its ﬁrst Betti number. Let us compare the structure of our upper

X X bounds for δ0 with Gromov’s bound. By Corollary 2.2.5, an ≤ 4 hyp(X) for each n, this result is obtained by appealing to the notion of tight span [35, 34, 53]. Hence, Theorem 5.1.1 implies that

X X δ0 ≤ (24 β1(X) + 19)(δβ + hyp(X)).

77 In particular, if X = G is a ﬁnite metric graph, then

G δ0 ≤ (24 β1(G) + 19) hyp(G).

Given a metric space X, Gromov [42] introduces a construction which produces a tree metric

space TpX. By reinterpreting Gromov’s construction in the setting of geodesic spaces, we

are able to prove the following improvements of the upper bounds we gave above.

Theorem 5.1.4. Let X be a compact geodesic space with ﬁnite ﬁrst Betti number β = β1(X)

X,p and p be a point in X. Let τ := dGH(X, TpX). Then,

τ X,p X X,p i) 6 ≤ δ0 ≤ τ .

X X X ii) δn ≤ δ0 ≤ log2(4n + 4) 5 δn + hyp(X) .

iii) If G = X is a ﬁnite metric graph, then

hyp(G) ≤ δG ≤ log (4β + 4) hyp(G). 4 0 2

A proof of Theorem 5.1.4 is given in Section 5.3. One of the main tools in the proof of

Theorem 5.1.4 is the following novel stability result for Gromov’s tree construction:

Proposition 5.1.5 (Gromov tree stability). Let X and Y be geodesic spaces and p and q be points in X and Y respectively. Then

dGH(TpX, TqY ) ≤ 5 dGH (X, p), (Y, q) .

In Section 5.2, we prove Theorem 5.1.1, Theorem 5.1.2 and Proposition 5.1.3, which are

results about graph approximations. In Section 5.3, we prove Theorem 5.1.4, which is about

tree approximations. Previous sections develop necessary tools and results for Section 5.2

and Section 5.3. They also contain results which are interesting by themselves.

78 Related work. Approximating metric spaces by lower dimensional spaces (such as metric graphs) is considered by Gromov in [44] and in [43, Appendix I]. In [43, Appendix I],

Gromov utilized a Reeb graph construction which was later elaborated by Zinov’ev [73] for approximating metric surfaces. Zinov’ev’s result was generalized to arbitrary Riemannian manifolds in [60]. The approximation of a geodesic space by Reeb graphs is studied in [25].

X,p In these papers, diﬀerent upper bounds for ρ = dGH(X,Xp) are obtained, which in our terminology imply upper bounds for δX . In this chapter, we are not concerned with upper β1(X) bounds on ρX,p, but with how ρX,p controls graph approximations in general.

The main tools we use in this chapter for graph approximations in general are i) Reeb graph of a distance function, ii) stability of the Reeb graph construction, iii) -smoothings and iv) interpretation of the ﬁrst persistence barcode of the Vietoris-Rips ﬁltration of a metric graph in terms of the sizes of loops. Item i) is studied in [25], item ii) [8, 11, 30, 9], item iii) in [30] and item iv) in [72].

Approximations with metric trees are studied in [42, 15, 28]. However, to best of our knowledge, the continuous interpretation of the Gromov tree construction and its use for

G,p obtaining an upper bound for τ = dGH(G, TpG) in terms of the ﬁrst Betti number of the metric graph G is done here for the ﬁrst time.

5.2 Graph Approximations

Lemma 5.2.1. Let G be a ﬁnite metric graph. Then

dGH(G, Gp) ≤ (4 β1(G) + 3) .

79 Proof. Note that β1(Gp) ≤ β1(G) and G = Gp by Corollary 2.4.12. Hence, by Theorem 2.5.8, we have

` dGH(G, Gp) = dGH(Gp,Gp) ≤ (8 β1(G) + 6) dGH (G, p), (G × [0, ], (p, 0)) .

Let R be the correspondence between (G, p) and the `1-product (G × [0, ], (p, 0)) given

by x R (x, t) for each x in G and t in [0, ]. Let us show that R is a path subcorrespondence

of itself. Take xand x0 in G and t and t0 in [0, ]. Let α be a path from x to x0 in G, and

(β, γ) be a path from (x, t) to (x0, t0). Letα ˜ = β from x to x0. Thenα ˜(s) R (β, γ)(s) for each

s. Now, deﬁne the path β˜ from (x, t) to (x0, t0) in G × [0, ] by

s 7→ (α(s), (1 − s)t + st0).

Then α(s) R β˜(s) for each s. Hence R is a path subcorrespondence of itself, so R is a path

` correspondence. Note that dis(R) = . Hence dGH (G, p), (G × [0, ], (p, 0)) ≤ /2.

Lemma 5.2.2. Let X be a compact geodesic space with ﬁnite ﬁrst Betti number. Let G be

r a ﬁnite metric graph and p be a point in G. If r ≥ 6 dGH(X,G), then β1(Gp) ≤ β1(X).

X Proof. Note that by Remark 2.2.9, an = 0 for n > β1(X). By Proposition 2.2.12, for n > β (X), aG ≤ 4 d (X,G). Hence, r ≥ 3 aG /2 and by Proposition 2.5.10, β (Gr) ≤ 1 n GH β1(X)+1 1 p

β1(X).

Proof of Theorem 5.1.1. “i)” Let G be a ﬁnite metric graph with β1(G) = n. By Theorem

G X 2.2.6, an+1 = 0. Hence, by Proposition 2.2.12, an+1 ≤ 4 dGH(X,G). We get the desired result by inﬁmizing over G.

80 “ii)” Let G be a graph with β1(G) = n > β1(X) = β. Let r = 6 dGH(X,G). Let p be a

point in G. By Lemma 5.2.1 and Lemma 5.2.2 we have

X r r δβ ≤ dGH(X,Gp) ≤ dGH(X,G) + dGH(G, Gp)

≤ dGH(X,G) + (4n + 3)r = (24n + 19) dGH(X,G).

Inﬁmizing over G with β1(G) = n, we get

δX δX ≥ β . n 24n + 19

“iii)” Let G be a graph with β1(G) = β and p be a point in G. Let n < β and

G G X G G rn rn = 3(an+1 + 4 dGH(X,G))/2. Then by Proposition 2.2.12 rn ≥ 3 an+1/2 and β1(G ) ≤ n. By Lemma 5.2.1, we have

G G X rn rn δn ≤ dGH(X,G ) ≤ dGH(X,G) + dGH G, G

G ≤ dGH(X,G) + (4 β + 3) rn .

Inﬁmizing over G with β1(G) = β, we get

aX δX ≤ δX + (4 β + 3) 3 (aX + 4 δX )/2 ≤ (24 β + 19) δX + n+1 . n β n+1 β β 4

Proof of Theorem 5.1.2. Let G be a graph with β1(G) = n. Let g = max(β, n). Then, for

each q in G by Proposition 2.1.7 and Theorem 2.5.8

` dGH((X, p), (G, q)) ≥ dGH((X, p), (G, q))/2

≥ dGH(Xp,G)/(16 g + 12).

81 Inﬁmizing over q in G, we get

dGH(X,G) ≥ dGH(Xp,G)/(16 g + 12) ≥ dGH(Xp,X) − dGH(X,G) /(16 g + 12)

Once more inﬁmizing over G with β1(G) = n, we get

ρX,p δX ≥ . n 16 max(β, n) + 13

This proves i) and the ﬁrst inequality in ii).

X Now, let n < β. Let r = 3 an+1/2. By Corollary 2.5.11 and Lemma 5.2.1, we have

X r r δn ≤ dGH(X,Xp ) ≤ dGH(X,Xp) + dGH(Xp,Xp )

X,p X,p X ≤ ρ + (4 β + 3) r ≤ ρ + (6β + 6) an+1.

Proof of Proposition 5.1.3. The upper bound follows from Theorem 5.1.2. If G0 is a ﬁnite

0 G0 metric graph such that β1(G ) = n, then an+1 = 0 by Remark 2.2.9. By Proposition 2.2.12,

G 0 a d (G, G0) ≥ (aG − aG )/4 = n+1 . GH n+1 n+1 4

0 0 We get the lower bound by inﬁmizing the inequality above over G with β1(G ) ≤ n.

5.3 Tree Approximations

In [42], Gromov gives a construction of a tree metric (i.e. a metric with zero hyperbolicity)

on every metric space. More speciﬁcally, given a metric space (X, dX ) and a point p in X,

tree metric tp is deﬁned by

0 0 ∞ 0 tp(x, x ) := dX (p, x) + dX (p, x ) − 2gp (x, x ),

82 where

∞ 0 gp (x, x ) := sup min gp(xi, xi−1). 0 i=1,...,n x=x0,...,xn=x 0 For the deﬁnition of the Gromov product gp(x, x ), see Section 2.1.2.

Deﬁnition 5.3.1 (TpX). Let (X, dX ) be a geodesic space and p be a point in X. (TpX, tp)

0 0 is deﬁned as the metric space associated to the pseudo-metric (x, x ) 7→ dX (p, x)+dX (p, x )−

∞ 0 2 gp (x, x ) on X. Note that hyp(TpX) = 0.

∞ We have the following characterization of gp for geodesic spaces.

0 0 Deﬁnition 5.3.2 (mp(x, x )). Let (X, dX ) be a geodesic space and p, x and x be points in

X. Let Γ(x, x0) denote the set of continuous paths from x to x0. We deﬁne

0 mp(x, x ) := sup min dX (p, ·) ◦ γ. γ∈Γ(x,x0)

Proposition 5.3.1. Let (X, dX ) be a connected geodesic space and p be a point in X. Then

∞ gp ≡ mp.

0 Lemma 5.3.2. Let (X, dX ) be a geodesic space and p be a point in X. Let x = x0, . . . , xn = x be points in X. Then

0 mp(x, x ) ≥ min mp(xi, xi−1). i=1,...,n

00 0 00 00 0 Proof. It is enough to show that given x in X, mp(x, x ) ≥ min(mp(x, x ), mp(x , x ). Let α be a path from x to x00 and β be a path from x00 to x0. Then we have

0 mp(x, x ) ≥ min dX (p, ·) ◦ (α · β) = min(min dX (p, ·) ◦ α, min dX (p, ·) ◦ β).

Maximizing over α, β, we get

0 00 00 0 mp(x, x ) ≥ min(mp(x, x ), mp(x , x )).

83 Lemma 5.3.3. Let (X, dX ) be a geodesic space and p be a point in X. Then mp ≥ gp.

Proof. Let x and x0 be points in X and α be a length minimizing geodesic from x to x0. Let q be a point in the image of α where dX (p, ·) takes its minimum. Then we have

0 0 0 2 gp(x, x ) = dX (p, x) + dX (p, x ) − dX (x, x )

0 0 = dX (p, x) − dX (q, x) + dX (p, x ) − dX (q, x )

0 ≤ 2 dX (p, q) ≤ 2 mp(x, x ).

∞ 0 0 Proof of Proposition 5.3.1. “mp ≥ gp ”: Let x and x be points in X. Let x = x0, . . . , xn = x be points in X. By Lemma 5.3.2 and Lemma 5.3.3 we have

0 mp(x, x ) ≥ min mp(xi, xi−1) i=1,...,n

≥ min gp(xi, xi−1). i=1,...,n

Maximizing over (xi), we get

0 ∞ 0 mp(x, x ) ≥ gp (x, x ).

∞ 0 “gp ≥ mp ” Let > 0 Let α : [0, 1] → X be a continuous path from x to x . Choose

0 = t0 < ··· < tn = 1 such that diam(α([ti−1, ti])) ≤ . Let xi = α(ti) for i = 0, . . . , n. We have

∞ 0 gp (x, x ) ≥ min gp(xi, xi−1) i=1,...,n = min dX (p, xi) + dX (p, xi−1) − dX (xi, xi+1) /2 i=1,...,n

≥ min min(dX (p, xi), dX (p, xi−1)) − (/2) i=1,...,n

= min dX (p, xi) − (/2) i=0,...,n

≥ min dX (p, ·) ◦ α − (3 /2).

84 Since > 0 is arbitrary, we have

∞ 0 gp (x, x ) ≥ min dX (p, ·) ◦ α.

Maximizing over α, we get

∞ 0 0 gp (x, x ) ≥ mp(x, x ).

Proposition 5.3.4. If (X, dX ) is a compact geodesic space and p be a point in X, then TpX is a metric tree.

Lemma 5.3.5. Let (X, dX ) be a geodesic space, p be a point in X and α : [0, 1] → X be a continuous path. Let q be a point on α where dX (p, ·) takes its minimum on α. Then for any x on α,

tp(q, x) = dX (p, x) − dX (p, q).

Proof. Let q = α(t) and x = α(t0). Let I be the subinterval of [0, 1] between t and t0. Then over I, dX (p, ·) ◦ α ≥ dX (p, q). Hence mp(q, x) = dX (p, q). Therefore

0 tp(q, x) = dX (p, q) + dX (p, x) − 2 mp(q, q ) = dX (p, x) − dX (p, q).

Lemma 5.3.6. Let (X, dX ) be a geodesic space and p be a point in X. Let α : [0, 1] → X

be a continuous path in X with α(0) = x. Let q be a point on α where dX (p, ·) takes its minimum on α. Then there is a continuous path γ in TpX from [q] to [x], such that

Ltp (γ) = dX (p, x) − dX (p, q).

Proof. Let := dX (p, x) − dX (p, q). Deﬁne ϕ : [0, ] → [0, 1] by

ϕ(s) = min{t : dX (p, α(t)) = dX (p, q) + s}.

85 ϕ is not necessarily continuous. Since dX (p, α(ϕ(s)) = dX (p, q) + s and dX (p, α(0)) = r + ,

by intermediate value theorem ϕ is strictly decreasing. Note that ϕ() = 0. Let us show

that γ : [0, ] → TpX deﬁned by [α ◦ ϕ] is a continuous curve with the desired properties.

0 Note that q = α(ϕ(0)) is a point where dX (p, ·) takes its minimum over α, by Lemma 5.3.5

0 tp(q, q ) = 0. Hence γ(0) = [q] and γ() = [x].

Let s < s0. Then we have ϕ(s0) < ϕ(s) and by the deﬁnition of ϕ, over α restricted to

0 [ϕ(s ), ϕ(s)], dX (p, ·) is greater than or equal to dX (p, q) + s. Hence

0 mp(α(ϕ(s), α(ϕ(s )) = dX (p, α(ϕ(s))) = dX (p, q) + s.

Therefore,

0 0 0 tp(α(ϕ(s), α(ϕ(s )) = dX (p, q) + s + dX (p, q) + s − 2 (dX (p, q) + s) = s − s.

0 0 In other words tp(γ(s), γ(s )) = s − s . Hence γ is the desired path.

Proof of Proposition 5.3.4. Since tp coincides with the metric induced by Gromov’s construc-

tion, we know that hyp(TpX) = 0. Hence, by [15, Proposition 3.4.2] it is enough to show

that tp is intrinsic. Let t˜p be the intrinsic metric induced by tp. Let us show that tp = t˜p.

Since tp ≤ t˜p, it is enough to show that t˜p ≤ tp.

Let x and x0 be points in X and α be a continuous path from x to x0. Let q be a point on α where dX (p, ·) takes its minimum over α. Let γ be a curve for q, α as in Lemma 5.3.6.

0 0 0 Let γ be a curve for q, α¯ as in Lemma 5.3.6. Thenγ ¯ · γ is a path in TpX from x to x and

0 it is tp length is dX (p, x) − dX (p, q) + dX (p, x ) − dX (p, q). Therefore,

0 0 t˜p(x, x ) ≤ dX (p, x) + dX (p, x ) − 2 min dX (p, ·) ◦ α.

Inﬁmizing over α, we get

0 0 t˜p(x, x ) ≤ tp(x, x ).

86 Here we recall Proposition 5.1.5.

Proposition 5.1.5 (Gromov tree stability). Let X and Y be geodesic spaces and p and q be points in X and Y respectively. Then

dGH(TpX, TqY ) ≤ 5 dGH (X, p), (Y, q) .

0 Lemma 5.3.7. Let (X, dX ) be a geodesic space and p be a point in X. Let x = x0, . . . , xn = x

be points in X such that dX (xi, xi−1) ≤ r for some r > 0. Then

0 mp(x, x ) ≥ min dX (p, xi) − (r/2). i=0,...,n

Proof. For i = 1, . . . , n, by Lemma 5.3.3 we have

mp(xi, xi−1) ≥ gp(xi, xi−1)

0 = dX (p, xi) + dX (p, xi−1) − dX (x, x ) /2

≥ min(dX (p, xi), dX (p, xi−1)) − (r/2).

The result then follows from Lemma 5.3.2.

Proof of Proposition 5.1.5. Let R be a correspondence between (X, p), (Y, q). Let x and x0

be points in X and y and y0 be points in Y such that x R y and x0 R y0. Let > 0. Let

0 0 α : [0, 1] → X be a continuous path from x to x such that mp(x, x ) ≤ min(dX (p, ·) ◦ α) + .

Choose 0 = t0 < t1 < ··· < tn such that diam(α([ti−1, ti])) ≤ for i = 1, . . . , n. For

0 i = 0, . . . , n, let xi = α(ti) and yi be a point in Y such that xi R yi, y0 = y, yn = y . Note

87 that dY (yi, yi−1) ≤ dis(R) + by Lemma 5.3.7, we have

0 0 mp(x, x ) − mq(y, y ) ≤ (min dX (p, ·) ◦ α) + − min dY (q, yi) + (dis(R) + )/2 i=0,...,n

≤ min dX (p, xi) − min dY (q, yi) + (dis(R)/2) + (3 /2) i=0,...,n i=0,...,n ≤ 3 (dis(R) + )/2.

Since > 0 is arbitrary

0 0 mp(x, x ) − mq(y, y ) ≤ 3 dis(R)/2.

Similarly we can show that

0 0 mq(y, y ) − mp(x, x ) ≤ 3 dis(R)/2.

Therefore,

0 0 0 0 0 0 |tp(x, x ) − tq(y, y )| ≤ |dX (p, x) − dY (q, y)| + |dX (p, x ) − dY (q, y )| + 2|mq(y, y ) − mp(x, x )|

≤ 5 dis(R).

If we denote the correspondence between TpX, TqY induced from R by S, then we have

dGH(TpX, TqY ) ≤ dis(S)/2 ≤ 5 dis(R)/2.

Inﬁmizing over R, we get

dGH(TpX, TqY ) ≤ 5 dGH (X, p), (Y, q) .

In [42, Remark 6.1.C], Gromov shows that if a geodesic space X can be decomposed

k+1 into less than 2 + 1 geodesics, then ||d − tp||∞ ≤ 2k hyp(X). Note that if X is a metric graph, then k can be much larger than log(β1(X) + 1). We have the following reﬁnement of

Gromov’s result.

88 Proposition 5.3.8. Let (G, dG) be a ﬁnite metric graph with and p be a point in G. Then,

||d − tp||∞ ≤ 2(log2(4β1(G) + 4)) hyp(G).

In particular,

dGH(G, TpG) ≤ (log2(4β1(G) + 4)) hyp(G).

Proof. Let x and x0 be points in G. Note that

0 0 0 0 dG(x, x ) − tp(x, x ) = 2(mp(x, x ) − gp(x, x )) ≥ 0

Given two paths α, α0 from x to x0 such that α ≤ α0 (see the deﬁnition of path preorder in Section 2.4), we have

0 min dG(p, ·) ◦ α ≥ min dG(p, ·) ◦ α .

Hence, by Proposition 2.4.8, mp can be deﬁned as the supremum over simple edge paths.

Since (up to reparametrization) there are ﬁnitely many simple edge path between x and x0,

0 there exists an edge path α realizing mp(x, x ). By Proposition 2.4.10, α = γ1·····γn such that dG(p, ·) is strictly monotonous on γi for each i and n ≤ 2 β1(G) + 2. Let x = x0, x1, . . . , xn

be points such that γi is from xi−1 to xi. By Corollary 2.4.12, dG(xi, xi−1) = |dG(p, xi) −

dG(p, xi−1)| and this implies that

gp(xi, xi−1) = min(dG(p, xi), dG(p, xi−1)).

Therefore,

0 mp(x, x ) = min dG(p, xi) = min gp(xi, xi−1). i=0,...,n i=1,...,n

If we let k = dlog2 ne, then by [28, p. 91, Lemme 2]

0 0 mp(x, x ) − gp(x, x ) ≤ k hyp(G).

89 The result follows since

k ≤ log2 2n ≤ log2(4β1(G) + 4).

Proof of Theorem 5.1.4. “i)” Since TpX is a compact metric tree by Proposition 5.3.4 and

X X,p every compact metric tree can be approximated by ﬁnite metric trees, δ0 ≤ τ . Now let T

∞ be a metric tree. Note that for any q in T ,TqT = T (since for metric trees gp = gp ), hence by Proposition 5.1.5 we have

dGH(X,T ) ≥ dGH(X, TpX) − dGH(TpX, TqT )

≥ dGH(X, TpX) − 5 dGH((X, p), (T, q))

Inﬁmizing over q in T , we get

X,p 6 dGH(X,T ) ≥ τ , and further inﬁmizing over T , we get

X X,p δ0 ≥ τ /6.

“ii)” Let G be a ﬁnite metric graph with β1(G) = n. Let q be a point in G. By Proposition

5.3.8 and Proposition 2.1.11, we have

X δ0 ≤ dGH(X, TqG)

≤ dGH(X,G) + dGH(G, TqG)

≤ dGH(X,G) + (log2(4n + 4))hyp(G)

≤ dGH(X,G) + (log2(4n + 4))(hyp(X) + 4 dGH(X,G))

Inﬁmizing over G with β1(G) = n, we get

X X δ0 ≤ (log2(4n + 4))(5 δn + hyp(X)).

90 “iii)” The upper bound follows from Proposition 5.3.8. The lower bound follows from Propo- sition 2.1.11.

91 Chapter 6: The Distortion of the Reeb Quotient Map on Riemannian Manifolds

6.1 Introduction

In this chapter, we consider the distortion of the Reeb quotient map of a Morse function on a compact Riemannian manifold. We generalize the results of [43, 73] about compact

Riemannian surfaces. To be able to do this we introduce a metric invariant thickness Tf of a function f : X → R deﬁned on a metric space (see Section 6.3). Our main result is the following:

Theorem 6.1.1. Let (X, dX ) be a compact connected n-dimensional Riemannian manifold, n ≥ 2, and f : X → R be an L-Lipschitz excellent Morse function. Let p ∈ X and

p := ||f − dX (p, ·)||∞. Let Xf be the Reeb graph of f and π : X → Xf the Reeb quotient map. Then the metric distortion of π is less than or equal to

1 ! 2n+1 L µn(X) n 1 n−1 2 n n 2(β1(X) + 1) · + 16 diam(X) p + p + |L − 1| diam(X), (β1(X) + 1) Tf

where µn denotes the n-dimensional Hausdorﬀ measure on X.

Theorem 6.1.1 follows directly from Proposition 6.2.4 and Proposition 6.4.2.

92 Remark 6.1.2. Note that if in the statement above we replace p by p := infc∈R kf − dX (p, ·) + ck∞ (which is a smaller quantity) the theorem is still correct. Indeed, if we take g = f + c where c is the constant realizing p, then X → Xf coincides with X → Xg as a

map of metric spaces.

Remark 6.1.3. At ﬁrst sight it may appear that our upper bound is not tight in the case where f is constant, since in that case the actual distortion of any such map is diam(X), the diameter of X. Indeed, note that, since when L = 0 we have p ≤ diam(X), our bound

2 becomes diam(X) · 64(β1(X) + 1) + 1 . However, note that the constant function is not

Morse and therefore our analysis does not apply to it.

Interpretation of the upper bound. Let us interpret this upper bound as follows. By distributing the product over the sum and combining topological constants depending on the ﬁrst Betti number β := β1(X) of X properly as C(β), D(β), and E(β), the upper bound can now be written as:

1 n n 1 µ (X) 1/n (n−1)/n C(β) L n + D(β) diam(X) + E(β) + diam(X) |L − 1|, T p p f | {z } | {z } | {z } (II) (III) (I) which consists of three parts. The last part (III) shows that in order to decrease the value

of the upper bound, this term favors values of the Lipschitz constant L of f to be close to 1.

The second term (II) shows that smaller values of p are favored in order to obtain a smaller

upper bound. These two observations can be combined to say that the last two terms of

our upper bound become smaller for distance-like functions, where max(|L − 1|, p) can be

regarded as a ﬁrst degree Sobolev type distance of f to dX (p, ·): X → R. Note that by

Proposition 2.6.1, the distance function dX (p, ·): X → R can be approximated (in terms of

max(|L − 1|, p)) by excellent Morse functions.

93 1 1 µn(X) n Now let us look at the ﬁrst term (I) of the upper bound: C(β) L n . Because of Tf its eﬀect on (III), we already mentioned that L is favored to be around 1. The structure of

(I) shows that smaller total volume guarantees a smaller upper bound, as is expected from

the situation arising from considering the boundary of a tubular neighborhood around an

embedded metric graph (see Section 6.5). In addition, (I) also shows that small values of

Tf make the upper bound larger. In the case where X is a surface one has Tf ≥ 1 (see

Remark 6.3.4). However, in general, no such lower bound can be guaranteed for manifolds of arbitrary dimensions. Hence, dimensions greater than 2 require extra care in the selection of f: in addition to being distance like, we also want control on its thickness Tf . In Section

6.3.1, we construct rich families of Riemannian manifolds, called thickened graphs, where we have such control.

In Section 6.3 we see that for n = 2 it holds that Tf is always greater than or equal to 1. Then, since the distance function can be approximated with Morse functions nicely

(see Section 2.6), in the case n = 2 Theorem 6.1.1 gives us that the distortion of the Reeb √ 3/2 1/2 quotient map is bounded above by 4 2(β1(X)+1) · area(X) , which is the main result of [73].

Outline of the chapter. In Section 6.2, we prove an upper bound on the distortion of the

Reeb quotient map in terms of the maximal diameter of a connected component of a ﬁber.

Sections 6.3 and 6.4 are devoted to ﬁnding an upper bound to this diameter. We introduce an invariant called thickness in Section 6.3 and in Section 6.4 we use it to obtain the desired upper bound. In Section 6.3 we also introduce a family of metric spaces which are called thickened graphs and calculate the thickness invariant for such spaces. Finally, in Section

6.5 we calculate the bound of Theorem 6.1.1 for thickened graphs.

94 6.2 Distortion of the Reeb Quotient Map

In this section we follow [73, Section 3] with suitable modiﬁcations to reach the level of generality we need.

Recall that the distortion of a function φ : X → Y between metric spaces (X, dX ) and

(Y, dY ) is deﬁned as

0 0 dis(φ) := sup |dX (x, x ) − dY (φ(x), φ(x ))|. x,x0∈X

If φ is surjective, then the Gromov-Hausdorﬀ distance between X and Y satisﬁes (see [18])

dGH(X,Y ) ≤ dis(φ)/2.

Let (X, dX ) be a compact Riemannian manifold and f : X → R be an L-Lipschitz excellent Morse function. Let p be a point in X and let θ := π(p) ∈ Xf . Let p :=

||f − dX (p, ·)||∞.

Given a subset Y of Xf let us denote its preimage in X under π by CY . For r ∈ Xf let

Cr := C{r} and for x ∈ X let Cx := Cπ(x). Note that if Y is connected then CY is connected by Proposition 2.5.3. In particular for any x in X and any r ∈ Xf , Cx and Cr are connected.

Given three subspaces A, B, C of a topological space we say that B separates A and C if every path from A to C intersects B.

Proposition 6.2.1 ([73, Lemma 3.6.]). Let r, s be points in Xf so that s separates r and θ.

Then for any x such that π(x) = r, we have

dist(x, Cs) ≤ df (r, s) + 2p.

Proof. Since s separates r and θ, Cs separates Cr from p. Let x be a point in Cr, i.e.

π(x) = r. Then there is a point y in Cs such that there is a length minimizing geodesic from

95 x to p containing y. Hence we have

dist(x, Cs) ≤ dX (x, y) = dX (x, p) − dX (y, p)

≤ f(x) + f(y) + 2p

= f(r) − f(s) + 2p ≤ df (r, s) + 2p (By Proposition 2.5.7).

Proposition 6.2.2 ([73, Lemma 3.8.]). Let r, s be points in Xf and γ be a continuous curve

between r, s such that r, s separates γ from θ. Then we have

dist(Cr,Cs) ≤ length(γ) + 4p.

Proof. Cγ is a connected compact subset of X. Deﬁne Fr to be the subset of Cγ consisting of

points such that there is a length minimizing geodesic from that point to p intersecting Cr.

Deﬁne Fs similarly. Note that Fr,Fs are closed and they are nonempty since they contain

Cr,Cs respectively. Furthermore, they cover Cγ since Cr ∪ Cs separates Cγ and p. Hence

0 their intersection is nonempty. Let x ∈ Fr ∩ Fs. There exists y ∈ Cr (resp. y ∈ Cs) such that there is a length minimizing geodesic from x to p containing y (resp. y0). Now we have

0 dist(Cr,Cs) ≤ dX (y, y )

0 0 ≤ dX (x, y) + dX (x, y ) = dX (x, p) − dX (y, p) + dX (x, p) − dX (y , p)

0 ≤ f(x) − f(y) + f(x) − f(y ) + 4p = f(π(x)) − f(r) + f(π(x)) − f(s) + 4p

≤ df (r, π(x)) + df (π(x), s) + 4p ≤ length(γ) + 4p.

Proposition 6.2.3 ([73, Lemma 3.9]). Let (G, θ) be a pointed ﬁnite topological graph, whose

vertices have degrees at most three. Let γ be a simple path from a point r to a point s in G.

96 Then there are points r1, . . . , rn, n ≤ 2 β1(G) + 1 on γ such that r0 separates r from θ, rn separates s from θ and {ri, ri+1} separates the interior of the segment of γ from ri to ri+1 from θ.

Proposition 6.2.4. Assume that the level sets of π : X → Xf have diameter less than or equal to a constant C > 0. Then

dis(π) ≤ (2β1(X) + 1)(C + 4p) + |L − 1| diam(X).

Proof. Let x and y be points in X and r and s be their images under π respectively. By

Proposition 2.5.7 we have

df (r, s) − dX (x, y) ≤ (L − 1) dX (x, y) ≤ |L − 1| diam(X).

Let γ be a length minimizing curve between r, s. By Proposition 2.5.1, Xf is a graph

satisfying the conditions of Proposition 6.2.3, hence we can choose r1, . . . , rn as in Proposition

6.2.3. Let γi denote the part of γ between ri, ri+1 for 1 ≤ i < n. Note that df (ri, ri+1) =

length(γi). Now, by Proposition 6.2.1 and 6.2.2 we have

n−1 X df (r, s) = length(γ) = df (r, r0) + length(γi) + df (rn, s) i=1 n−1 X ≥ dist(x, Cr1 ) − 2p + dist(Cri ,Cri+1 ) − 4p + dist(Crn , s) − 2p i=1

≥ dX (x, y) − n C − 4 n p ≥ dX (x, y) − (2β1(X) + 1)(C + 4p).

6.3 Thickness

Given k ∈ N, we deﬁne the kth-thickness of a path connected metric space A by

k k µ (A) TA := . diam(A)k

97 a A

2 ab a a Figure 6.1: TA = a2+b2 depends only on b and converges to 0 as b → 0.

Deﬁne it as 1 if the diameter is 0. Note that low values of this invariant intuitively indicate

that the space has very thin parts: That is, parts with large diameter but small kth dimen-

sional Hausdorﬀ measure (see Figure 6.1). For a metric space with several path components,

we deﬁne the thickness by taking as the inﬁmum thickness across its components. We will

elaborate on the interpretation of the thickness invariants thorough the examples that follow

and the situation described in Section 6.3.1.

th Example 6.3.1. Let sn be the total n Hausdorﬀ measure of the standard n-dimensional

unit sphere. It is known that 2π(n+1)/2 s = . n Γ((n + 1)/2)

Let Sn(r) be an n-dimensional sphere of radius r with the standard Riemannian metric.

n n n n Then µ (S (r)) = snr and diam(S (r)) = π r. Therefore,

−(n−1)/2 n sn 2π T n = = , S (r) πn Γ((n + 1)/2)

which is independent of r.

1 Remark 6.3.2. TA ≥ 1 since length is always greater than or equal to diameter.

98 Figure 6.2: A 2-dimensional thickened ﬁltered graph.

Let X be a metric space with Hausdorff dimension n. Given a function f : X → R , we define the thickness Tf of f as the following essential infimum:

n−1 Tf := ess inft∈range(f)Tf −1(t).

Example 6.3.3. If f is the constant function, then the Tf = ∞ as the inﬁmum of the empty set.

Remark 6.3.4. If X is a surface and f is Morse, then Tf ≥ 1.

6.3.1 A Calculation: Thickened Filtered Graphs

Here we first introduce thickened filtered graphs and then find a lower bound for the thickness of this type of filtered spaces. We first need to define how to glue filtered spaces.

A compact ﬁltered space f : X → R has maximum and minimum level sets which we denote by Xt,Xb respectively. Let us denote the value of f on Xt,Xb by tf , bf respectively. If we are given two ﬁltered spaces f : X → R and g : Y → R together with a homeomorphism

ϕ : Xt ' Yb, we can glue these two topological spaces through ϕ and then deﬁne a ﬁltration ` h : X ϕ Y → R by h|X := f and h|Y := g−bg +tf . See Figure 6.2 for how the gluing is done

99 Figure 6.3: A 2-dimensional thickened 3-fork.

along circles. Furthermore, if in addition X and Y are metric spaces endowed with metrics

dX and dY , respectively, such that the homeomorphism ϕ : Xt → Yb is an isometry, then we ` deﬁne a metric d on X ϕ Y as follows. The restriction of d to X is dX , the restriction of d to Y is dY and for all x ∈ X, y ∈ Y ,

d(x, y) := inf dX (x, z) + dY (ϕ(z), y) . z∈Xt

` Note that if X and Y are length spaces, then so is X ϕ Y . We then have:

Lemma 6.3.5. Let h : Z → R be the ﬁltered metric space which is obtained by gluing ﬁltered

metric spaces f : X → R, g : Y → R, described as above. Then Th = min(Tf , Tg).

Proof. Recall that Tf is deﬁned through taking a minimum through the levels sets of f. The

result follows from the fact that the level sets of h can be seen as the union of the level sets

of f, g.

Since the thickness of a filtered metric space is defined as infimum across its connected

components, we can perform the gluing through a collection of connected components of the

maximum level set of the ﬁrst ﬁltered space with a collection of connected components of

the minimum level set of the second ﬁltered space, and in this case Lemma 6.3.5 still holds.

100 0 A k-fork is a directed graph with k + 2 vertices v1, . . . , vk, w, w where the directed edges

0 are given by (w , w) and (w, vi) for all i. Note that its geometric realization can be embedded

into R2 with coordinates (x, y) where the height y is increasing along the directed edges and the points on each level set of the height function are collinear. We call such a realization a

n+1 ﬁltered k-fork. It can be thickened and smoothed in R with coordinates (x = x1, . . . , xn, y)

such that it becomes an n-dimensional submanifold of Rn+1 and the level sets with respect

to the height function h is the boundary of the union of balls of same radius in Rn with

coordinates (x1, . . . , xn) where the centers of the balls are given by the corresponding level

sets of the fork. We call such a metric space an n-dimensional ﬁltered thickened k-fork (see

Figure 6.3).

Remark 6.3.6. By a general position argument, we can thicken a ﬁltered k − fork so that

the ﬁltering on the thickened fork is excellent Morse.

th Lemma 6.3.7. Let sn be the total n Hausdorﬀ measure of the standard n-dimensional unit

sphere. Then for any n-dimensional ﬁltered thickened k-fork f : F → R we have

n−1 s T n−1 T ≥ n−1 = S . f (k π)n−1 kn−1

Proof. By deﬁnition of a ﬁltered thickened fork, a level set of f is the boundary of the union of balls of radius r with collinear centers in Rn. Let S be a connected component

of such a level set, and let p1, . . . , pm be the centers given in a linear order. Note that by

linearity, S contains the two half (n − 1)-dimensional spheres with centers p1 and pm. Hence

µn−1(S) ≥ µn−1(Sn−1(r)). Since m ≤ k, diam(S) ≤ k diam(Sn−1(r)) by Proposition 2.1.1. Therefore µn−1( n−1(r)) s Tn−1 ≥ S = n−1 (see Example 6.3.1). S n−1 n−1 k diam(Sn−1(r)) (k π)

101 Figure 6.4: An inverse 2-dimensional thickened fork.

An inverse filtered thickened fork is defined similarly but with the negative filtration (see

Figure 6.4): the negative filtration of a filtered space f : X → R is defined as −f : X → R. Lemma 6.3.7 still holds for this case, since taking the negative filtration does not change the thickness invariant.

Definition 6.3.1. (Thickened graph) An n-dimensional filtered thickened graph is obtained by gluing n-dimensional thickened forks and inverse forks, and filtered half-spheres filtered by the height function in Rn+1 (see Figure 6.2). Here, to guarantee that the resulting space is a Riemannian manifold, we are taking (topological) filtered half spheres in Rn+1 so that its boundary C has a neighborhood which is isometric to C × [0, ] for some > 0, where this isometry is compatible with the height filtering in the expected sense.

Remark 6.3.8. A thickened graph is a Riemannian manifold. This is because we only allow gluings along isometric boundaries with isometric neighborhoods. By a general position argument, we can assume that the height function is excellent Morse.

The following proposition follows from Lemma 6.3.5 and Lemma 6.3.7.

102 Proposition 6.3.9. Let f : G → R be an n-dimensional ﬁltered thickened graph and let K be the maximal k where a k-fork is used in the construction of G. Then

Tn−1 T ≥ Sn−1 . f Kn−1

6.4 The Diameter of a Fiber of the Reeb Quotient Map

Let (X, dX ) be a compact connected n-dimensional Riemannian manifold, n ≥ 2 and f : X → R be a L-Lipschitz smooth function. For any p ∈ X, let p := ||f − dX (p, ·)||∞.

2 Lemma 6.4.1. Let t0 be a regular value of f and let p ∈ X be a point such that t0 > f(p).

−1 Let t > t0 and A be a connected component of f (t). Then we have

n−1 −1 Tf n−1 µ f (t0) ≥ n−2 diam(A) − 2(β1(X) + 1)(t − t0 + 2p) . (β1(X) + 1)

−1 Proof. By Proposition 2.6.2 there are components Y1, ··· ,Yk, k ≤ β1(X) + 1 of Y := f (t0)

separating A from p. Let Ai be the subset of A consisting of points which can be connected

to p through Yi by a length minimizing geodesic. Note that {Ai}i=1,...,k is a ﬁnite closed cover of A, hence by Proposition 2.1.1

k X diam(A) ≤ diam(Ai). i=1

For each x in Ai, there is a point xi such that there is a length minimizing geodesic from x to p containing xi. Given x and y in Ai we have

dX (x, y) ≤ dX (x, xi) + dX (xi, yi) + dX (yi, y)

= dX (x, p) − dX (xi, p) + dX (xi, yi) + dX (y, p) − dX (yi, p)

≤ f(x) − f(xi) + 2p + diam(Yi) + f(y) − f(yi) + 2p

= diam(Yi) + 2(t − t0 + 2p),

2Cf. [73, Lemma 2.2].

103 hence k X diam(A) ≤ (diam(Yi) + 2(t − t0 + 2p)) i=1 k X ≤ 2(β1(X) + 1)(t − t0 + 2p) + diam(Yi). i=1 Now we have k k n−1 −1 X n−1 X n−1 µ (f (t0)) ≥ µ (Yi) ≥ Tf diam(Yi) i=1 i=1 k !n−1 Tf X ≥ diam(Y ) (By convexity of λ 7→ λn−1 over [0, ∞)) kn−2 i i=1 Tf n−1 ≥ n−2 diam(A) − 2(β1(X) + 1)(t − t0 + 2p) . (β1(X) + 1)

Proposition 6.4.2. 3 Let A be a connected component of f −1(t). Then for any p in X, the diameter of A is less than or equal to 1 ! 2n+1L(β (X) + 1)n−1µn(X) n 1 n−1 1 n n max 8(β1(X) + 1) p, + 8(β1(X) + 1) diam(X) p . Tf

diam(A) Proof. Assume diam(A) > 8(β1(X) + 1) p. Let δ = . Note that diam(A) ≤ 4(β1(X)+1)

2 supx∈A dX (x, p) ≤ 2t + 2p. Then we have diam(A) diam(A) t − δ ≥ − p − 2 4(β1(X) + 1) diam(A) ≥ − > 2(β (X) + 1) − ≥ ≥ f(p), 4 p 1 p p p hence for any regular value s ∈ [t − δ, t] we can apply Lemma 6.4.1 to obtain a lower bound for µn−1(f −1(s)). Note that for such an s

diam(A) − 2(β1(X) + 1)(t − s + 2p) ≥ diam(A) − 2(β1(X) + 1)(δ + 2p) diam(A) = − 4(β (X) + 1) > 0, 2 1 p 3Cf. [73, Proposition 2.3].

104 Hence by Lemma 6.4.1 we get

n−1 n−1 −1 Tf diam(A) µ (f (s)) ≥ n−2 − 4(β1(X) + 1) p . (β1(X) + 1) 2

Since almost all values of a smooth function are regular, by the coarea formula (Propo- sition 2.1.2), we get

Z t n−1 n n−1 −1 Tf diam(A) diam(A) L µ (X) ≥ µ (f (s)) ds ≥ n−1 − 4(β1(X) + 1) p . t−δ 4(β1(X) + 1) 2

This gives us

1 n+1 n−1 n n−1 2 L(β1(X) + 1) µ (X) diam(A) ≤ + 8(β1(X) + 1) p, Tf diam(A)

which implies

1 n 2n+1L(β (X) + 1)n−1µn(X) n−1 1 n−1 1 n−1 diam(A) ≤ + 8(β1(X) + 1) diam(X) p, Tf

hence

1 2n+1L(β (X) + 1)n−1µn(X) n 1 n−1 1 n n diam(A) ≤ + 8(β1(X) + 1) diam(X) p . Tf

Note that here we used the fact that for x and y positive and 0 ≤ r ≤ 1, we have (x + y)r ≤ xr + yr.

6.5 The Bound of Theorem 6.1.1 for Thickened Graphs

Throughout this section, we assume that an r-thickened k-fork (resp. inverse k-fork)

is constructed so that before thickening the Euclidean distance between the consecutive

collinear points on the highest level (resp. the lowest level) is 3r, see Figure 6.5. We

also assume that the construction of the graph starts with a lower cap and new pieces are

glued consecutively along the whole of their lowest level, see Figure 6.2. We impose these

105 3r 3r

Figure 6.5: A vertical fork.

requirements to prevent the graph being wide. Note that such a thickened graph has a single point in its lowest level, which is the lowest point of the initial lower cap. We call such a pointed ﬁltered thickened graph (G, p) a vertical graph.

Let us calculate the bound of Theorem 6.1.1 for an r-thickened n-dimensional vertical

m graph (G, p). Denote its height ﬁltering by f : G → R. Assume it consists of pieces (Pi)i=1.

Claim 1. Let hi be the diﬀerence between the highest and the lowest value of the ﬁltering on

Pi for each i. Let K be the maximal k for which a k-fork or an inverse k-fork is used in the construction of G. Then there exists a constant cK > 0 depending on K such that m n X n−1 µ (G) ≤ cK hi r . i=1

Proof. This follows from the obvious fact that for each piece Pi for i = 1, . . . , m, there exists

n n−1 a constant cK > 0 such that µ (Pi) ≤ cK hi r .

Claim 2. For i = 1, . . . , m, deﬁne

i := sup{dX (x, y) − |f(x) − f(y)| : x ∈ highest level of Pi, y ∈ lowest level of Pi}.

Then, there exists a constant dK > 0 depending on K such that

m X p ≤ i ≤ m dK r. i=1 106 Recall that p is deﬁned as the l∞ distance between the ﬁltration and dX (p, ·),

Proof. Obviously, there exists a constant dK > 0 such that i ≤ dK r. Note that the condition

that consecutive points on the end of a fork are at distance 3r is crucial here. Furthermore,

since the pieces are added consecutively so that a new piece is glued along the whole of its

lowest level, for every point x in G, there is a length minimizing curve γ from p to x so that

the ﬁltering f is monotone on that curve. By using the points x0 = p, x1, . . . , xn = x on

this curve where xjs are the intersection points of the curve with the boundaries of pieces

for j = 1, ˙,n − 1, we have

n n X X 0 ≤ dX (p, x) − f(x) = dX (xj, xj−1) − f(xj) − f(xj−1) ≤ m dK r. j=1 j=1

Claim 3. There exists a constant eK > 0 depending on K such that

m X diam(G) ≤ hi + m eK r, i=1

where his are deﬁned as in the previous claim.

Proof. For each piece Pi, there exists a constant eK > 0 depending on K such that diam(Pi) ≤

hi + eK r. The result follows from this and the fact that G is connected.

Let us combine and summarize all of these calculations in the following Corollary.

Corollary 6.5.1. Let (G, p) be an n-dimensional r-thickened vertical graph consisting of

m m pieces (Pi)i=1. Let hi be the diﬀerence between the highest and the lowest value of the height ﬁltering on Pi for each i ∈ {1, . . . , m}, and let H be the sum of all his. Let K be the maximal k where a k-fork or an inverse k-fork is used in the construction of G. Then there

107 exist constants cK , dK , eK depending on K such that the distortion of the Reeb quotient map from G to its Reeb graph is bounded above by

n−1 1/n cK H(K r) 1/n (n−1)/n C(β) + D(β)(H + m ek r) (m dk r) + E(β)m dk r, TSn−1 where C(β),D(β),E(β) are positive constants depending on the ﬁrst Betti number β of G.

Note that this bound is of order O(r(n−1)/n).

108 Chapter 7: Reeb Posets and Metric Tree Approximations

Trees, as combinatorial structures which model branching processes arise in a multitude

of ways in computer science, for example as data structures that can help encoding the

result of hierarchical clustering methods [47], or as structures encoding classiﬁcation rules

in decision trees [36]. In biology, trees arise as phylogenetic trees [70], which help model

evolutionary mechanisms. In computational geometry and data analysis trees appear for

instance as contour/merge trees of functions deﬁned on a manifold [20, 66].

7.1 Introduction

Gromov shows that for each ﬁnite metric space (X, dX ), there exists a tree metric tX on

X such that

||dX − tX ||∞ ≤ Υ(X) := 2 hyp(X) log(2|X|), where |X| is the cardinality of X. Although this bound is known to be asymptotically tight, one expects a better upper bound if the space has more regularity. For example, if X is a

large ﬁnite sample from a space, then tree approximations of X can be expected to depend

on some geometric or topological properties of the ambient space instead of the cardinality

of X.

109 Figure 7.1: A ﬁnite metric space embedded in a metric graph

For a metric graph G, we deﬁne

φ(G) := 2 hyp(G) log(4β1(G) + 4).

For a finite metric space X, Φ(X) is defined to be the infimal value of φ(G) amongst all

graphs G inside which X can be isometrically embedded. We then obtain the following

theorem:

Theorem 7.1.1. For any ﬁnite metric space (X, dX ) there exists a tree metric tX such that

kdX − tX k∞ ≤ Φ(X).

Example 7.1.2. Consider the case when Xn is a finite sample consisting of n points from a fixed metric graph G such as in the figure above. Assume that as n grows the sample becomes denser and denser inside G. In this case, as hyp(Xn) ' hyp(G) since hyp is stable

[27], we have Υ(Xn) ' 2 hyp(G) log(2n) → ∞ as n → ∞. On the other hand, Φ(Xn) is bounded by a constant/independent of n (more precisely it will be bounded by φ(G)).

Remark 7.1.3. Since Gromov’s bound is known to be tight [27] one would expect that there exists a sequence (Zn) of ﬁnite metric spaces such that both Υ(Zn) and Φ(Zn) have the same

growth order. Such a construction is given in Section 7.7.

110 The underlying idea: Reeb posets. To obtain our bounds, we consider the case where

X is a metric space arising from a filtered poset. More precisely, given a poset (X, ≤) with an order preserving filtration f : X → R, the filtration induces a distance df on X given by

( n ) X df (x, y) := min |f(xi) − f(xi−1)| : x0 = x, xn = y, xi is comparable with xi+1 ∀i . i=1

A large class of metric spaces arise in this way. For example every metric graph, with possible addition of new vertices, can be realized this way. Hence, by embedding ﬁnite metric spaces into metric graphs, our methods can be applied to ﬁnite metric spaces.

Given a a filtered poset (X, ≤, f), we give a Reeb [68] type construction to obtain a tree, which gives a metric tf on X. To obtain an upper bound for kdf − tf k∞, we define a filtered

≤ ≤ poset version hypf of hyperbolicity and show that ||df −tf ||∞ ≤ 2 hypf log(2MF ), where MF is the poset theoretic constant given by the length of the largest fence in (X, ≤). A fence is a

ﬁnite chain of elements such that consecutive elements are comparable and non-consecutive elements are non-comparable. Note that the cardinality in the Gromov’s result is replaced by MF , which can be signiﬁcantly smaller than the cardinality.

In Section 7.2 we review and give some useful results about posets. In Section 7.3 we introduce Reeb poset and Reeb tree poset constructions for ﬁltered posets. In Section 7.4 we introduce poset hyperbolicity for Reeb posets. In Section 7.5 we consider tree metric approximations of Reeb posets. In Section 7.6 we gave an application of Reeb poset constructions to ﬁnite graphs and metric spaces.

7.2 Posets

In this section we review some basic concepts for posets and give some results that we need later. For simplicity we are assuming that all posets we consider are ﬁnite and connected

111 (i.e. each pair of points can be connected through a ﬁnite sequence (x0, . . . , xn) of points

such that xi is comparable to xi+1).

Deﬁnition 7.2.1 (Covers and merging points). Let X be a poset. Given x and y in X, we say that x covers y if x > y and there is no z such that x > z > y. A point is called a merging point if it covers more than one elements. Given a point x, the number of points covered by x is denoted by ι(x). Hence x is a merging point if and only if ι(x) > 1.

Lemma 7.2.1. Let X be a poset and y, y0 be non-comparable points in X. If there is a point x such that x > y, y0, then there exists a merging point x0 such that x ≥ x0 > y, y0.

Proof. Let x0 be a minimal element among all points satisfying x ≥ x0 > y, y0. We show that x0 is a merging point. Let z be a maximal point satisfying x0 > z ≥ y and z0 be maximal satisfying x0 > z0 ≥ y0. Note that x covers both z, z0 and z 6= z0 by the minimality of x0.

Deﬁnition 7.2.2 (Chains and fences). A totally ordered poset is called a chain. A fence is

a poset whose elements can be numbered as {x0, . . . , xn} so that xi is comparable to xi−1

for each i = 1, . . . , n and no other two elements are comparable. Note that a fence looks

like a zigzag as its elements are ordered in the following fashion: x0 < x1 > x2 < x3 > . . .

or x0 > x1 < x2 > x3 < . . . . The length of a chain or a fence is deﬁned as the number of

elements minus one.

Proposition 7.2.2. Let X be a poset and F be a fence with length l in X. Then X has at

l−1 least b 2 c merging points.

l−1 l−2 Proof. Let us start with the case l is even. Then b 2 c = 2 . By removing two endpoints

if necessary, we get a fence of the form x0 < x1 > x2 < ··· > x2k−2 < x2k−1 > x2k,

l−2 where l ≤ 2k + 2. Let us show that X has k ≥ 2 merging points. For i = 1, . . . , k 112 let yi be a merging point such that x2i−1 ≥ yi ≥ x2i−2, x2i whose existence is given by

Lemma 7.2.1. It is enough to show that yi’s are distinct. Assume i ≤ j and yi = yj. Then

x2i−2 ≤ yi = yj ≤ x2j−1, so 2i − 1 ≥ 2j − 1 > 2i − 2 and we have i = j. This completes l is

even case.

Now assume that l is odd. Then by removing one of the endpoints, we get a fence of the

form x0 < x1 > x2 < ··· > x2k−2 < x2k−1 > x2k, where l = 2k + 1. Note that k is exactly

l−1 b 2 c and by the analysis above X has k merging points.

Deﬁnition 7.2.3 (Covering graph). The covering graph of a poset X is the directed graph

(V,E) whose vertex set is X and a directed edge is given by (x, x0) where x0 covers x.

Recall that the ﬁrst Betti number β1 of a graph is deﬁned as the minimal number of edges one needs to remove to obtain a tree.

Proposition 7.2.3. Let X be a poset with a smallest element 0 and G be the covering graph P of X. Then β1(G) = x:ι(x)≥1(ι(x) − 1).

Proof. By Euler’s formula, the ﬁrst Betti number of a graph is equal to 1 + e − v, where e is

the number of edges and v is the number of vertices. Note that since edges in G are given P by the covering relations in X, the number of edges e of G is equal to e = x:ι(x)≥1 ι(x). P Since the only vertex with ι(x) = 0 is x = 0, we have v − 1 = ι(x)≥1 1. Hence β1(G) = P 1 + e − v = e − (v − 1) = x:ι(x)≥1(ι(x) − 1).

Corollary 7.2.4. Let X be a poset with a smallest element 0 and β be the ﬁrst Betti number

of the covering graph of X. Then the length of a fence in X is less than or equal to 2β + 2.

Proof. Let F be a fence of length l in X. Let m be the number of merging points in X.

l l−1 Then by Proposition 7.2.2 2 − 1 ≤ b 2 c ≤ m, so l ≤ 2m + 2. By Proposition 7.2.3, P m ≤ ι(x)>1(ι(x) − 1) = β.

113 Deﬁnition 7.2.4 (Tree). A (connected) poset (T, ≤) is called a tree if for each x in the

the set of elements less than or equal to x form a chain. In other words, non-comparable

elements does not have a common upper bound.

Remark 7.2.5. A tree has a smallest element.

Proof. Let x be a minimal element. Let us show that it is the smallest element. Let x0 be

a point in T distinct from x. By connectivity, there exists a minimal chain (x0, . . . , xn) of

0 elements so that x0 = x, xn = x . By minimality of the chain, this sequence is a fence. By

minimality of x0 = x, x0 < x1. This implies n = 1, since otherwise x1 is an upper bound for

0 non-comparable elements x0, x2. Hence x = x0 < x1 = x .

Proposition 7.2.6. A poset T is a tree if and only if it does not contain any merging point.

Proof. “ =⇒ ” Note that if x is a merging point then x covers non-comparable elements,

hence a tree does not contain merging points.

“ ⇐= ” By Lemma 7.2.1, if non-comparable elements have a common upper bound, then

there exists a merging point. Hence if there are no merging points, then the set {x0 : x0 ≤ x}

is a chain for all x, hence T is a tree.

The following proposition gives some other characterizations of tree posets.

Proposition 7.2.7. Let T be a poset with the minimal element 0. Then the following are

equivalent

(i) T is a tree.

(ii) If F is a fence in T , then the length of F is less than or equal to two and if it is two

then F = x > y < z.

114 (iii) The covering graph of T is a tree.

Proof. (i) =⇒ (ii) A tree does not contain a fence of the form x < y > z. Any fence of length greater than two contains a sub-fence of the form x < y > z.

(ii) =⇒ (iii) If y is a merging point, then there are non-comparable elements x, z such that x < y > z, hence T does not contain any merging point. By Proposition 7.2.3, the ﬁrst Betti number of the covering graph is 0, hence it is a tree.

(iii) =⇒ (i) By Proposition 7.2.3, T does not contain any merging points. By Proposition

7.2.6, T is a tree.

7.3 Reeb Constructions

In this section we generalize the deﬁnition of Reeb graphs [68] to posets.

7.3.1 Poset Paths and Length Structures

Two basic concepts used in deﬁning Reeb graphs (as metric graphs) for topological spaces are those of paths and length [8]. We start by introducing these concepts in the poset setting.

Deﬁnition 7.3.1 (Poset path). Let X be a poset and x and y be points in X.A poset path from x to y is an n-tuple (x0, . . . , xn) of points of X such that x0 = x, xn = y and xi−1 is comparable with xi for i = 1, . . . , n. We denote the set of all poset paths from x to y

≤ ≤ ≤ by Γ (x, y). By Γ we denote the union ∪x,y∈X Γ (x, y). Let us call a poset path simple if xi 6= xj for i 6= j. The image of the path (x0, . . . , xn) is {x0, . . . , xn}. Note that a ﬁnite chain is the image of a simple path which is monotonous and fence is the image of a simple path

(x0, . . . , xn) where xi is not comparable with xj if j 6= i − 1, i, i + 1.

115 Note that a poset path as deﬁned above corresponds to an edge path in the comparability graph of the poset. Recall that the comparability graph of a poset is the graph whose set of vertices is the elements of the poset and the edges are given by comparable distinct vertices.

Deﬁnition 7.3.2 (Inverse path). Given a poset path γ = (x0, . . . , xn), the inverse path γ¯ is deﬁned asγ ¯ := (xn, . . . , x0).

0 Deﬁnition 7.3.3 (Concatenation of paths). If γ = (x0, . . . , xn) and γ = (y0, . . . , ym) are

0 poset paths such that the terminal point xn of γ is equal to the initial point y0 of γ , then

0 0 we deﬁne the concatenation of γ, γ by γ · γ := (x0, . . . , xn, y1, . . . , ym).

Deﬁnition 7.3.4 (Length structure over a poset). A length structure l over a poset X assigns a non-negative real number to each poset path, which is additive under concatenation, invariant under path inversion and deﬁnite in the sense that non-constant paths have non- zero lengths.

Remark 7.3.1. If X is a poset with a length structure, then its covering graph becomes a metric graph in a canonical way where the length of an edge is given by its length as a poset path.

Deﬁnition 7.3.5 (Induced metric). Given a length structure l over a poset X, we deﬁne dl : X × X → [0, ∞] as

dl(x, y) = inf l(γ). γ∈Γ≤(x,y)

This is an extended metric over X and called the metric induced by l.

Deﬁnition 7.3.6 (Length minimizing paths). Let X be a poset with a length structure l.

≤ A poset path γ ∈ Γ (x, y) is called length minimizing if dl(x, y) = l(γ).

116 We will later see that fences play an important role in minimization problems, including

distance minimization.

7.3.2 Reeb Posets

Let X be a ﬁnite poset and f : X → (R, ≤) be an order preserving function, i.e. x ≤ y implies that f(x) ≤ f(y). Let us deﬁne a relation ∼ on X as follows: We say x ∼ y if

≤ there exists γ ∈ Γ (x, y) such that f is constant along γ, more precisely if γ = (x0, . . . , xn),

then f(xi) = f(xj). This is an equivalence relation: reﬂexivity x ∼ x follows from the

constant path (x), symmetry follows from considering inverse path, and transitivity follows from concatenation. Let us denote the equivalence class of x under this relation byx ˜ and

the quotient set X/ ∼ by Rf (X).

≤ We deﬁne ≤ on Rf (X) as follows:x ˜ ≤ y˜ if there exists γ ∈ Γ (x, y) such that f is

non-decreasing along γ. This is well deﬁned since diﬀerent representatives of the same equivalence class can be connected through an f-constant poset path. Transitivity follows

from concatenation. Also note that ifx ˜ ≤ y˜ andy ˜ ≤ y˜ then f-nondecreasing paths connecting

x to y and y to x have to be f-constant, sox ˜ =y ˜. Therefore (Rf (X), ≤) is a poset. Also

note that f is still well deﬁned on Rf (X) since f is constant inside equivalence classes.

Deﬁnition 7.3.7 (Reeb poset of f : X → R). We call (Rf , ≤, f :Rf → R) described above

the Reeb poset of the order preserving map f : X → R.

Remark 7.3.2. i) The quotient map X → Rf (X) is order preserving.

ii) f : Rf → R is strictly order preserving.

Proof. “i)” If x ≤ y, just take the f-nondecreasing path (x, y).

117 ≤ “ii)” Letx ˜

Remark 7.3.3. If f : X → R is strictly order preserving, then Rf (X) = X as a poset.

Proof. Since f is strict, no non-constant path is f-constant. Hencex ˜ = {x}. Therefore the order preserving quotient map X → Rf (X) is an isomorphism.

Corollary 7.3.4. Rf (Rf (X)) = Rf (X).

Remark 7.3.5. Let R be a poset with an order preserving ﬁltration f : R → R. (R, f) is the Reeb poset of some order preserving map f 0 :(X, ≤) → R if and only if f is strictly increasing.

Inspired by this remark, we give the following deﬁnition.

Deﬁnition 7.3.8 (Reeb poset, Reeb tree poset). A Reeb poset (R, f) is a poset R with a strictly order preserving map f : R → R. A Reeb poset is called a Reeb tree poset if R is a tree. (Recall that a poset is called a tree if for each element x the elements less than or equal to x form a chain.)

Deﬁnition 7.3.9 (Reeb metric). A Reeb poset (R, f) carries a canonical length structure

≤ lf deﬁned as follows: For every poset path γ = (x0, . . . , xn) ∈ Γ

n X lf (γ) := |f(xi) − f(xi−1)|. i=1

Additivity and invariance with respect to inversion are obvious. Deﬁniteness follows from the strictness of f. The metric structure induced by this length structure is denoted by df and is called the Reeb metric induced by f.

118 ≤ Remark 7.3.6. If γ ∈ Γ (x, y), then lf (γ) ≥ |f(x) − f(y)|. Hence, df (x, y) ≥ |f(x) − f(y)|.

Therefore, if x and y are comparable, then df (x, y) = |f(x) − f(y)|. Furthermore, for a Reeb

graph (R, f), df (x, y) = |f(x) − f(y)| if and only if x and y are comparable since the equality

implies that there exists an f-increasing path between x and y and in that case x and y are

comparable by the equality Rf (R) = R.

Remark 7.3.7. The distance df (x, y) can be realized as the f-length lf (γ) where γ is a fence.

This can be done by taking a length minimizing curve and removing points until one gets a

fence (i.e. if i + 1 < j and xi comparable to xj remove all points between xi, xj).

7.3.3 Reeb Tree Posets

In this subsection we focus on the construction, characterization and properties of Reeb

tree posets.

Let X be a poset with an order preserving function f : X → R. Deﬁne an equivalence

≤ relation ∼ on X as follows: x ∼ y if there exists γ in Γ (x, y) such that for each xi in γ

f(xi) ≥ max(f(x), f(y)). Note that this implies that if x ∼ y, then f(x) = f(y). The

fact that this is an equivalence relation follows from concatenation of paths. Let us denote

the equivalence class of a point x byx ˜ and the quotient set by Tf (X). Note that f is still

well deﬁned on Tf (X). Let us deﬁne a partial order ≤ on Tf (X) as follows:x ˜ ≤ y˜ if there

≤ exists γ in Γ (x, y) such that for each xi in γ, f(xi) ≥ f(x). This is well defined since different representatives of an equivalence class can be connected by a poset path on which f takes values greater than or equal to that of the representatives. Let us show that this is a partial order. Reflexivity (i.e.x ˜ ≤ x˜) follows from the constant path and transitivity (i.e. x˜ ≤ y˜ ≤ z˜ =⇒ x˜ ≤ z˜) follows from concatenation of paths. Ifx ˜ ≤ y˜ then f(x) ≤ f(y), hence ifx ˜ ≤ y˜ andy ˜ ≤ x˜, then f(x) = f(y) thus the path givingx ˜ ≤ y˜ also givesx ˜ =y ˜. Note

119 that this also shows that f : Tf (X) → R is strictly order preserving. Note that X → Tf (X) is order preserving.

Proposition 7.3.8. (Tf (X), ≤) is a tree.

Proof. Assumex, ˜ x˜0 ≤ y˜. Let us show thatx, ˜ x˜0 are comparable. Without loss of generality assume that f(x) ≤ f(x0). Let γ (resp. γ0) be a poset path from x (resp. x0) to y giving x˜ ≤ y˜ (resp.x ˜0 ≤ y˜). Then γ · γ¯0 is a path on which f takes greater values than f(x), hence x˜ ≤ x˜0.

Deﬁnition 7.3.10 (Reeb tree poset of f : X → R). (Tf (X), ≤, f) is called the Reeb tree poset of f : X → R. We denote its Reeb metric by tf .

Remark 7.3.9. The minimal element of Tf (X) is the equivalence class of x where f takes its minimal value.

Remark 7.3.10. Tf (Rf (X)) = Tf (X) since the equivalence the relation used to deﬁne

Rf (X) is stronger than that of Tf (X).

Proposition 7.3.11. If (T, ≤, f) is a Reeb tree poset, then Tf (T ) = T .

Proof. As T → Tf (T ) is order preserving, it is enough to show thatx ˜ =y ˜ if and only if x = y. Let γ = (x0, . . . , xn) be a path from x to y givingx ˜ =y ˜, i.e. f(xi) ≥ f(x) = f(y) for each i. By removing elements from γ if necessary, we can assume that γ is a fence. By

Proposition 7.2.7, n is at most 2. If n = 2, then by Proposition 7.2.7 γ = (x > x1 < y), which is not possible by the strictness of f. If n = 1, then x and y are diﬀerent but comparable, which is again not possible by the strictness of f. Hence n = 0 and x = y.

120 Now let us study properties of the Reeb tree metric tf . Note that if T is a tree poset and

x and y are points in T , than the intersection of the chains {z : z ≤ x}, {z : z ≤ x0} is itself

a chain hence it has a unique maximal element, which we denote by px,y.

Proposition 7.3.12. Let (T, f) be a Reeb tree poset. Then for all x and y in T

tf (x, y) = f(x) + f(y) − 2f(px,y).

Proof. Without loss of generality f(x) ≤ f(y). If x and y are comparable, then px,y = x by

the strictness of f. By Remark 7.3.6 tf (x, y) = f(y)−f(x) = f(x)+f(y)−2f(px,y). If x and y

are not comparable, then by Proposition 7.2.7 (x > px < y) is the only fence from x to y. By

Remarks 7.3.6 and 7.3.7, tf (x, y) = f(x)−f(px,y)+f(y)−f(px,y) = f(x)+f(y)−2f(px,y).

Now, let us give a characterization of f(px,˜ y˜) for an order preserving map f : X → R. We ﬁrst introduce the following deﬁnition:

Deﬁnition 7.3.11 (Merge value). Let f : X → R be an order preserving function. Deﬁne

≤ mf (x, y) := max min f ◦ γ. γ∈Γ≤(x,y)

Proposition 7.3.13. Let f : X → R be an order preserving map. Let the map X → Tf (X)

be the map given by x 7→ x˜. Let px,˜ y˜ be the maximal element in tree Tf (X) which is less than

or equal to both x,˜ y˜. Then

≤ f(px,˜ y˜) = mf (x, y).

≤ We denote the right hand side of the equality above by mf (x, y).

≤ Proof. Let px,y be a point in the preimage of px,˜ y˜ in X. Let γ be a path realizing mf (x, y)

and qx,y be the point on it where f takes its minimal value. Note thatq ˜x,y ≤ x,˜ y˜, hence by the deﬁnition of px,˜ y˜ we haveq ˜x,y ≤ px,˜ y˜. So f(qx,y) = f(˜qx,y) ≤ f(px,˜ y˜). Note that there are

121 ≤ 0 ≤ paths γ ∈ Γ (x, px,y), γ ∈ Γ (px,y, y) such that on both curves f takes values greater than

0 or equal to px,y. Hence f(qx,y) ≥ min f ◦ (γ · γ ) = f(px,y) = f(px,˜ y˜).

As a corollary of Proposition 7.3.12,7.3.13, we have the following.

Corollary 7.3.14. Let f : X → R be an order preserving map. Then

≤ tf (˜x, y˜) = f(x) + f(y) − 2 mf (x, y).

≤ Remark 7.3.15. mf (x, y) can be realized by a fence since if γ is a poset path realizing

≤ mf (x, y), we can remove points until it becomes a fence without decreasing the minimal f- value.

7.4 Hyperbolicity for Reeb Posets

Metric hyperbolicity is a metric invariant which determines if a metric space is metric tree or not [42, 18]. In this section, we introduce a similar invariant which determines if a

Reeb poset is a Reeb tree poset or not.

Deﬁnition 7.4.1. (Gromov product for Reeb posets) Let (R, f) be a Reeb poset. We deﬁne the Gromov product f(x) + f(y) − d (x, y) g≤(x, y) = f . f 2

Deﬁnition 7.4.2. (Hyperbolicity for Reeb posets) Let (R, f) be a Reeb poset. We deﬁne

≤ the hyperbolicity hypf (R) as the minimal ≥ 0 such that for each x, y and z in X,

≤ ≤ ≤ gf (x, z) ≥ min(gf (x, y), gf (y, z)) − .

The main statement of this section is the following.

≤ Proposition 7.4.1. A Reeb poset (T, f) is a Reeb tree if and only if hypf (T ) = 0.

122 We give the proof after some remarks and lemmas.

Remark 7.4.2. If x ≤ y, then by Remark 7.3.6 gf (x, y) = (f(x) + f(y) − (f(y) − f(x))/2 = f(x).

Lemma 7.4.3. Let (R, f) be a Reeb poset. Then, gf (x, y) ≤ min(f(x), f(y)) for all x and y in R. Equality happens if and only if x and y are comparable.

Proof. Without loss of generality assume that f(x) ≤ f(y). Then, by Remark 7.3.6,

≤ gf (x, y) − f(x) = (f(y) − f(x) − df (x, y))/2 ≤ 0 and equality happens if and only if x and y are comparable.

Remark 7.4.4. If (T, f) is a tree poset and px,y is the maximal point which is smaller than both x and y. Then, by Proposition 7.3.12,

1 g≤(x, y) = (f(x) + f(y) − (f(x) + f(y) − 2(p ))) = f(p ). f 2 x,y x,y

Proof of Proposition 7.4.1. “ =⇒ ” Let x, y and z be points in T . Then px,y and py,z are comparable since they are both less than y and furthermore px,z ≥ min(px,y, py,z) since that minimum is less than or equal to both x and z. Therefore, by Remark 7.4.4

≤ ≤ ≤ gf (x, z) = f(px,z) ≥ min(f(px,y), f(py,z)) = min(gf (x, y), gf (y, z)).

“ ⇐= ” Assume y ≥ x, z. Let us show that x, z are comparable. By Lemma 7.4.3 and Remark

≤ ≤ ≤ 7.4.2, we have min(f(x), f(z)) ≥ gf (x, z) ≥ min(gf (x, y), gf (y, z)) = min(f(x), f(z)). Hence

≤ gf (x, z) = min(f(x), f(z)) and by Lemma 7.4.3 x, z are comparable.

123 7.5 Approximation

In this section we consider tree approximations of Reeb posets. Our approximation result includes the following poset invariant.

Deﬁnition 7.5.1 (Maximal fence length MF ). Given a poset X, we deﬁne

MF (X) := max |F | − 1, F a fence in X where |F | is the number of elements in F .

Theorem 7.5.1. Let (R, f) be a Reeb poset. Let π : R → Tf (R) be the projection map.

Then

0 0 ≤ |df (x, x ) − tf (π(x), π(x ))| ≤ 2 log(2 MF (R)) hypf (R).

Here we are considering logarithm base 2.

We ﬁrst prove following two lemmas.

≤ ≤ Lemma 7.5.2. Let f : X → R be an order preserving map. Then mf (x, y) ≥ gf (x, y).

≤ Proof. Let γ in Γ (x, y) be the path realizing df (x, y). Let z be the point on γ where f takes its minimal. Then by Remark 7.3.6 we have

df (x, y) = df (x, z) + df (z, y) ≤ f(x) − f(z) + f(y) − f(z) = f(x) + f(y) − 2f(z).

≤ ≤ Hence we have gf (x, y) = (f(x) + f(y) − df (x, y))/2 ≤ f(z) = min f ◦ γ ≤ mf (x, y).

Lemma 7.5.3. Let f : X → R be an order preserving map and x0, . . . , xn be a family of elements in a poset X. Then

≤ ≤ ≤ g (x0, xn) ≥ min g (xi, xi+1) − dlog ne hypf (X). f i f 124 Proof. Let us prove by induction on n. The case n = 1 is trivial and n = 2 follows from the

≤ deﬁnition of hypf . Now assume that the statement is true up to n and n > 3. Let k = dn/2e, then k ≥ n − k = bn/2c and dlog 2ke = dlog ne. By the inductive hypothesis we have

≤ ≤ ≤ ≤ gf (x0, xn) ≥ min(gf (x0, xk), gf (xk, xn)) − hypf (X)

≤ ≤ ≥ min g (xi, xi+1) − (dlog ke + 1) hypf (X) i f

≤ ≤ = min g (xi, xi+1) − dlog n ehypf (X). i f

Now, we can give the proof of Theorem 7.5.1.

Proof of Theorem 7.5.1. Let π : R → Tf (R) be the quotient map x 7→ x˜. Since π is surjec-

tive, it is enough to prove that the metric distortion dis(π) = maxx,y∈R |df (x, y) − tf (˜x, y˜)|

satisﬁes

≤ dis(π) ≤ 2 log(2 MF (R)) hypf (R).

By Corollary 7.3.14, we have

≤ ≤ ≤ df (x, y) − tf (x, y) = df (x, y) − f(x) + f(y) − 2 mf (x, y) = 2 mf (x, y) − gf (x, y) .

Hence, by Lemma 7.5.2, it is enough to show that for each x, y in X, we have

≤ ≤ ≤ mf (x, y) − gf (x, y) ≤ log(2 MF (R)) hypf (R).

≤ By Remark 7.3.15, there exists a fence (x = x0, . . . , xn = y) which realizes mf (x, y), i.e.

≤ ≤ mf (x, y) = mini f(xi). Since xi and xi+1 are comparable, by Remark 7.4.2, gf (xi, xi+1) =

≤ ≤ min(f(xi), f(xi+1)). Therefore, mf (x, y) = mini gf (xi, xi+1). Since n ≤ MF (R), by Lemma

≤ ≤ ≤ ≤ 7.5.3 mf (x, y) − gf (x, y) ≤ dlog n ehypf (X) ≤ log(2 MF (R)) hypf (X).

125 7.6 An Application to Metric Graphs and Finite Metric Spaces

In this section, we show how our poset theoretic ideas can be used to prove certain results for metric graphs and finite metric spaces. In particular we show that a metric graph naturally induces a poset with an order preserving filtration given by a distance function, and the induced metric from this filtration coincides with the original one. These observations makes our results for filtered posets applicable to graphs.

Deﬁnition 7.6.1 (p-regularity). Let G = (V, E, l) be a simple metric graph with the length structure l and p be a vertex. We call G p-regular if it satisﬁes the following:

i) Each edge (v, w) satisﬁes l(v, w) = dl(v, w) = |dl(p, v) − dl(p, w)|,

ii) Edges are the only length minimizing paths between their endpoints.

Note that by Proposition 2.4.3, each metric graph can be extended by adding at most one vertex from the geometric realization of each edge so that the property “i)” is satisﬁed.

We can further extend the vertex set by adding the midpoints of edges which are not the only length minimizing path between their endpoints. After this extension, property “ii)” is also satisﬁed.

In this section we prove the following:

Theorem 7.6.1. Let G = (V, E, l) be a p-regular metric graph with the ﬁrst Betti number

β. Then there exists a tree metric space (T, tT ) and a surjective map π : X → T such that

0 0 max |dl(x, x ) − tT (π(x), π(x ))| ≤ 2 log(4β + 4) hyp(V, dl), x,x0∈V

where dl is the metric induced by the length structure.

126 Note that any ﬁnite metric space (X, d) can be isometrically embedded into a ﬁnite metric graph. The simplest example is the complete graph with the vertex set X where the edge length is given by d. It is possible to obtain simpler embeddings [70, Chapter 5.4]. Here, we do not go into that path but assume that an embedding is already given.

Before providing the proof of Theorem 7.6.1 we go ahead and provide rthat of Theorem

7.1.1.

Proof of Theorem 7.1.1. Let p be a vertex of G. Without loss of generality we can assume that G is p-regular since otherwise we can add some new vertices from its geometric realization to make it p-regular, as it is explained in the beginning of this section. Let

π : G → (T, tT ) be the map given in Theorem 7.6.1. Let t : X × X → R be the pseudo-

0 0 metric given by t(x, x ) = tT (π(x), π(x )). Then t is a tree like metric on X and the upper bound that we are trying to prove follows from 7.6.1.

Remark 7.6.2. Note that for any ﬁnite metric space X which can be isometrically embedded

in the geometric realization of a metric graph G, the upper bound given in Theorem 7.1.1 still holds. This shows that |X| can possibly be much larger than 4β + 4. In particular, the upper bound in Theorem 7.1.1 can be much smaller than the one given by Gromov, i.e. log(2|X|) hyp(X).

Through the following lemmas, we will carry this problem to the Reeb poset setting.

Through this section assume that G = (V, E, l) is a p-regular metric graph.

Lemma 7.6.3. Deﬁne a relation ≤p on V by x ≤p y if dl(p, y) − dl(p, x) = dl(x, y). Then

≤p is a partial order.

Proof. Note that x ≤p x and x ≤p y implies that dl(p, x) ≤ dl(p, y). Hence, if x ≤p y, y ≤p x,

then dl(x, y) = dl(p, x) − dl(p, y) = 0, which means that x = y. It remains to show the

127 transitivity. Assume that x ≤p y ≤p z. Then we have

dl(x, z) ≤ dl(x, y) + dl(y, z)

= dl(p, y) − dl(p, x) + dl(p, z) − dl(p, y) = dl(p, z) − dl(p, x) ≤ dl(x, z),

so dl(x, z) = dl(p, z) − dl(p, x) which means that x ≤p z.

Lemma 7.6.4. (V, ≤p, dl(p, ·): V → R) is a Reeb poset.

Proof. Note that dl(p, ·) is strictly order preserving on (V, ≤p) since if x

dl(p, x) = dl(x, y) > 0.

Note that the covering graph of a Reeb poset has a canonical metric graph structure

given by (v, w) 7→ lf (v, w) = |f(v) − f(w)|.

Lemma 7.6.5. G = (V, E, l) is the covering graph of the Reeb poset (V, ≤p, dl(p, ·)) as a

metric graph (see Remark 7.3.1).

0 0 0 Proof. Let G = (V,E , l ) be the covering graph of the Reeb poset (V, ≤p, dl(p, ·)). Let us

show that E = E0 and l = l0. Note that if (v, w) is an edge contained in E ∩ E0, then by the

property of G with respect to p described in Theorem 7.6.1, l(v, w) = |dl(p, v) − dl(p, w)|,

which is equal to l0(v, w) by deﬁnition. Hence it remains to show that E = E0.

Let v, w be a pair of distinct vertices. Without loss of generality we can assume that

dl(p, v) ≤ dl(p, w). Let us show that w covers v if and only if {v, w} is an edge.

“ =⇒ ” Take a length minimizing path (v = v0, . . . , vn = w) of edges in G. Then we have

X X dl(v, w) = dl(vi, vi−1) ≥ dl(p, vi) − dl(p, vi−1) = dl(p, w) − dl(p, w) = dl(v, w), i i

128 so for each i we have dl(vi, vi−1) = dl(p, vi) − dl(p, vi−1), which means vi ≥ vi−1. Since w

covers v, this means the path consists of two vertex. Since the path was arbitrary, this

implies that (v, w) is an edge.

“ ⇐= ” By p-regularity dl(v, w) = dl(p, w) − dl(p, v), hence v

sequence of vertices such that vi+1 covers vi. Note that w covers v if and only if n = 1. By

the previous part (vi−1, vi) is an edge, hence (v0, . . . , vn) is a path of edges in G. It is length P P minimizing since dl(v, w) = dl(p, w) − dl(p, v) = i dl(p, vi) − dl(p, vi−1) = i dl(vi−1, vi). Since the edge is the only length minimizing path between its vertices, n=1. This completes

the proof.

≤ Lemma 7.6.6. hypf (V, ≤p) = hypp(V, dl), where f := dl(p, ·): V → R.

Proof. Note that by Lemma 7.6.5 df = dl. Now the result follows since

≤ gf (v, w) = (f(v) + f(w) − df (v, w))/2 = (dl(p, v) + dl(p, w) − dl(v, w))/2 = gp(v, w).

Now we can give the proof of Theorem 7.6.1.

Proof of Theorem 7.6.1. Let (T, tf ) be the Reeb poset tree of the Reeb poset (V, ≤p, f :=

dl(p, ·)). Note that (T, tf ) is a tree metric since it can be isometrically embedded into its covering graph, which is a metric tree by Proposition 7.2.7. By Lemma 7.6.5 and Corollary

≤ 7.2.4 df = dl, MF (V, ≤p) ≤ 2β + 2 and by Lemma 7.6.6, hypf (V, ≤p) = hypp(V, dl) ≤

hyp(V, dl). . Now, the result follows from Theorem 7.5.1.

7.7 Example where Φ ∼ Υ

Let Gn be the metric graph with the vertex set Zn as it is described in Figure 7.2. The

Gromov bound for Zn is Υ(Zn) = 2 log(4n + 4) hyp(Zn).

129 Figure 7.2: Let R ≥ r > 0 and consider the metric graph from the ﬁgure. Let Zn be the ﬁnite subset {p, x0, . . . , xn, y1, . . . , yn}. We show that Φ(Zn) ∼ 2 log(4n) hyp(Zn) and Υp(Zn) = 2 log(4n + 4) hyp(Zn).

130 Assume Zn is isometrically embedded in the geometric realization of metric a graph

G = (V, E, l). We can assume that G is p-regular and V contains Z. Consider the poset structure ≤p on V described in Lemma 7.6.3. Under this poset structure (x0, y1, x1, . . . , xn) becomes a fence with length 2n (note that this is true independent of the embedding, since ≤p is completely determined by the metric). By Lemma 7.6.5 and Corollary 7.2.4, β1(G) ≥ n−1.

Since Zn is a subspace of G, hyp(G) ≥ hyp(Zn). Therefore, φ(G) ≥ 2 log(4n) hyp(Zn). Since

G was arbitrary, we have Φ(Zn) ≥ 2 log(4n) hyp(Zn)

If R = r, then one can show that hyp(Gn) = hyp(Zn). Also note that β(Gn) = n. In this

case we get the upper bound Φ(Zn) ≤ φ(Gn) = 2 log(4n + 4)hyp(Zn). Therefore in this case

Φ(Zn), Υ(Zn) have the same growth rate.

131 Chapter 8: Quantitative Simpliﬁcation of Filtered Simplicial Complexes

8.1 Introduction

In this chapter, we introduce a distance between filtered simplicial complexes and an invariant called codensity on the set of vertices of the filtered simplicial complex. This invariant controls the effect of removing a vertex and all cells containing it from a filtered simplical complex. Applying these ideas to Vietoris-Rips complexes of finite metric spaces, in Section

8.4 we show in particular that the Vietoris-Rips ﬁltrations of the two ﬁnite metric spaces

M and M 0 in Figure 8.1 have the same Vietoris-Rips persistent homology in dimensions 1

0 and higher: PH≥1(VR(M)) = PH≥1(VR(M )). This result does not follow from the standard stability of Vietoris-Rips persistence result [22]: In fact by increasing the length of the ﬂares in M one can make the Gromov-Hausdorﬀ distance between M and M 0 grow without bound.

Note that here M,M 0 are not equipped with the Euclidean metric but with the path length metric on the circle with ﬂares.

For simplicity, in this chapter we assume that all families of simplicial complexes are pointwise ﬁnite dimensional, indexed over R, and constructible (i.e. changes happen at ﬁnitely many indices and the births of cells are realized). By the functoriality of homology

(with coefficients in a field), taking the homology of a filtered simplicial complex yields a

132 Figure 8.1: These two ﬁnite spaces have the same Vietoris-Rips PH≥1, see Example 8.4.3.

persistence module, and hence a persistence barcode. As we have several notions of similarity for barcodes like the bottleneck or Wasserstein distances, a natural question to ask is what type of notions of similarity can be deﬁned for ﬁltered simplicial complexes so that they interact nicely with the desired distance for barcodes.

We start Section 8.2 by reviewing the generalization of the Gromov-Hausdorﬀ distance to ﬁltered simplicial complexes given in [59]. We show that it generalizes the Gromov-

Hausdorff distance between metric spaces in the sense that the Gromov-Hausdorff distance between metric spaces is equal to the Gromov-Hausdorff distance between their Vietoris-Rips complexes, using the ideas in [59, Proposition 5.1]. We then introduce an interleaving type

F pseudo-distance dI for ﬁltered simplicial complexes. By its categorical nature, interleaving type distances appear in many diﬀerent settings [23, 11, 17, 56, 66, 12].

133 It is known that the interleaving distance between the persistent homology of Vietoris-

Rips complexes of metric spaces is less than or equal to twice the Gromov-Hausdorﬀ distance

between the spaces, see [22] and [24, Lemma 4.3]. However, we have the following general

F theorem which establishes that dI mediates between the interleaving distance and (twice) the Gromov-Hausdorﬀ distance but in general diﬀers from both:

Theorem 8.1.1 (Stability). Let X∗ and Y ∗ be constructible ﬁltered simplicial complexes.

Then, for every k ∈ N we have

∗ ∗ F ∗ ∗ ∗ ∗ dI PHk(X ), PHk(Y ) ≤ dI (X ,Y ) ≤ 2 dGH(X ,Y ).

F The proof of this theorem is given in Section 8.2.4. In the construction of the metric dI we had to pay special attention to the notion of contiguity, a coarse way in which homotopy

arises between simplicial maps; related studies appear in [13, 71, 12].

∗ In Section 8.3, given a ﬁltered simplicial complex X , we introduce an invariant δX (v, w),

called the vertex quasi-distance of X∗, deﬁned for each pair of vertices v and w. We then

deﬁne δX (v), the codensity of the vertex v, as the minimal of δX (v, w) as w ranges over all

vertices distinct from v. We show that this invariant controls the contribution of a vertex to

the persistent homology, in a way described in the following proposition:

Proposition 8.1.2 (Removal of a vertex). Let v be a vertex of X∗ and (X − {v})∗ be the

full ﬁltered subcomplex of X∗ obtained by removing the vertex v. Then,

F ∗ ∗ dI X , (X − {v}) ≤ δX (v).

This proposition shows that by computing δX (v, w) for all v, w we have a method for

simplifying a ﬁltered simplicial complex while keeping deﬁnite guarantees in terms of the

approximation error of the persistent homology. We then discuss how we can make the

134 calculation of δX (v, w) simpler and how to make δX (v, w) smaller if we are only interested

in persistent homology of certain degrees only (e.g. PH1).

In Section 8.4 we then show what our constructions correspond to for Vietoris-Rips

complexes of ﬁnite metric spaces and give an example showing the advantages of our simpli-

ﬁcation guarantees to those given by the Gromov-Hausdorﬀ based bounds of [22]. We then

introduce a class of metric graphs which we call simple graphs and as an application of our

results we characterize their persistent homology.

In Section 8.5, we introduce simple ﬁltered simplicial complexes: We call a ﬁltered sim-

∗ plicial complex X simple if for each vertex v its condensity δX (v) > 0. Proposition 8.1.2

implies that any non-simple ﬁltered simplicial complex can be reduced in size without chang-

ing its persistent homology. Then we show that this observation can be strengthened in the

following way:

Theorem 8.1.3 (Classification via cores). For each filtered finite simplicial complex X∗,

∗ F ∗ ∗ there exists a unique (up to isomorphism) simple ﬁltered complex C such that dI (X , C ) = 0. Furthermore, C∗ is a full subcomplex of X∗.

F Hence simple ﬁltered complexes classify ﬁltered complexes with respect to dI . We denote C∗ described in Theorem 8.1.3 by C(X∗) ⊆ X∗ and call it the core of X∗.

F Theorem 8.1.3 above can then be interpreted as follows. Equivalence (i.e. dI = 0) between ﬁltered simplicial complexes coincides with isomorphism between their respective

F ∗ ∗ ∗ ∗ cores: Namely dI (X ,Y ) = 0 if and only if C(X ) and C(Y ) are isomorphic. In particular this implies that the number of elements in the core is a well deﬁned invariant. More precisely,

the core of a ﬁltered simplicial complex X∗ coincides with the minimal cardinality ﬁltered

F ∗ simplicial complex at zero dI distance from X (Corollary 8.5.4).

135 We obtain Theorem 8.1.3 as a corollary of a more general statement (Proposition 8.5.2)

which says that between simple ﬁltered simplicial complexes, for small enough distances,

F 2dGH and dI coincide and furthermore this coincidence is realized through speciﬁc bijective maps.

In Section 8.6, we give a construction depending on a parameter r ≥ 0 which extends a

F ﬁltered simplicial complex so that its dI distance to the original space is 0, while the dGH

F distance is at least r/2. This shows that dI can be much smaller than dGH. In Section 8.7, we show given an arbitary family of simplicial complexes, how one can obtain a ﬁltered simplicial complex with the same persistent homology.

Related work. Given an n-point metric space (X, d), Sheehy [71] constructs a ﬁltered simplicial complex of size O(n), whose persistence diagram is an approximation of the Vietoris-

Rips ﬁltration. Here the approximation is between the respective persistence diagrams, but in a (multiplicative) sense diﬀerent from the one provided by the usual bottleneck distance.

He calls this type of approximation a c-approximation, where c is the multiplicative constant.

Using a net-tree, Sheehy assigns a non-negative deletion time to each point of the metric

space (X, d). For a scale α ≥ 0, he deﬁnes Nα as the subset of points with deletion time

greater than α. Furthermore, for each scale α, he deﬁnes a relaxed metric dα on Nα such

that d0 = d and the distance between two points is determined by the scale α, and the

deletion times of points. Through this relaxed family of metric spaces he obtains a zigzag

ﬁltration, whose persistence module is non zig-zag, i.e. the arrows in the backwards direction

are isomorphisms. Then he constructs a non-zigzag ﬁltration S with the same persistent

homology. He shows that S is a c = 1/(1 − 2) approximation of the Vietoris-Rips ﬁltration

of X, where 0 < ≤ 1/3 is a user deﬁned precision constant. He also shows that S is of

136 size O(n) where the constant depends on the precision and the doubling dimension of the

metric space.

In [52], the authors extend the work in [71] to Cechˇ complexes of metric spaces. They con-

struct an approximate family of simplicial complexes using a net-tree and a structure which

they called Well-Separated Simplicial Decomposition (WSSD). This approximate family is

of size O(n) where n is the number of points in the original metric space. As in [71], they

use a multiplicative approximation for persistence diagrams. In the last part of the chapter,

they give a generalization of the multiplicative approximation relation between Cechˇ and

Vietoris-Rips complexes. First, they observe that the Vietoris-Rips complex can be thought

as the 1-completion of the Cechˇ complex and a similar completion can be applied in higher

dimensions. Here k-completion of a simplicial complex means the largest simplicial complex

containing the k-skeleton of the originial one. Then they show that the (d1/(2+2)+1e−1)-

completion of the Cechˇ complex is a (1 + )-approximation of the Cechˇ ﬁltration.

In [31] the authors study the general problem of computing the persistent homology of a given family of simplicial complexes connected by simplicial maps. It is well known that given a simplicial map between simplicial complexes, the induced map between homology groups can be calculated more efficiently if the map is an inclusion. Here, the authors describe a way of calculating the induced homology map of a general simplicial map by using only simplicial inclusions. They introduce two elementary types of simplicial maps: elementary inclusions and elementary collapses. They first show that every simplicial map can be expressed as the composition of such maps. Then they study how to compute the induced homology maps of elementary simplicial maps effectively. In particular, they show how one can calculate the induced homology map of a simple collapse from a simplicial inclusion. The authors then note that the simplicial maps introduced in the approximations given by [71] are not

137 necessarily inclusions, hence their methods can be useful here. Furthermore, they introduce

two more approximation schemes for Vietoris-Rips ﬁltration similar to [71].

In [33] the authors modify their sparsiﬁcation method for approximating the Vietoris-Rips

complex described in [31] with a more strict one which they call Batch-Collapse. They obtain

a similar approximation bound while obtaining a smaller family of simplicial complexes.

The authors of [14] consider approximating Cechˇ complex of a ﬁnite set of points in the

Euclidean space. They first prove a sandwich theorem for sequences of covers of space giving a multiplicative approximation relation between the sequence of nerves of given sequences of covers. Using this theorem, first they obtain an approximation for the Cechˇ filtration

by using a net-tree construction as in [71]. Then they develop a method of coarsening a

sequence of covers and apply this to get an approximation to Cechˇ ﬁltration. Although the

latter construction does not have the theoretical guarantees that the ﬁrst have in terms of

size, the authors report that in practice it works much faster.

In [71, 52, 31, 14], approximation results are proved by constructing somewhat intri-

cate simplicial maps. Furthermore, the cases of Vietoris-Rips and Cechˇ complexes required

diﬀerent treatment. In [21], although not proving new approximation results, the authors

unify previous results through the notion of convex metrics and also give much simpler ap-

proximation proofs by using the neighborhoods in the ambient space instead of simplicial

complexes.

138 8.2 Gromov-Hausdorﬀ and Interleaving Type Distances between Filtered Simplicial Complexes

Given a ﬁnite set V we denote the power set of V minus the empty set by P(V ). Given

a metric space (X, dX ) the diameter function is deﬁned by diamX : P(X) → R+, where

0 σ 7→ maxx,x0∈σ dX (x, x ). By R we will mean the extended reals R ∪ {−∞, +∞}.

8.2.1 Gromov-Hausdorﬀ Distance between Filtered Simplicial complexes

We deﬁne the vertex set of a ﬁltered simplicial complex as the union of the vertex sets of its components (i.e. individual Xt’s).

Definition 8.2.1 (Size function). Given a finite filtered simplicial complex X∗ with vertex

r set V deﬁne the size function DX : P(V ) → R as follows: DX (α) := inf{r : α ∈ X }.

r Note that if α is not contained in X for any r then DX (α) = ∞ and if it is contained in

r all X then DX (α) = −∞. Also, by the constructibility the condition DX (α) is realized as

0 0 the minimum if it is ﬁnite. Note that if α ⊆ α , then DX (α) ≤ DX (α ).

∗ Remark 8.2.1. If X is the Vietoris-Rips complex of a metric space, then DX ≡ diamX .

Conversely, if we have D : P(V ) → R monotonic with respect to inclusion, then we can

∗ r deﬁne a ﬁltered simplicial complex XD with the vertex set V by XD := {α : D(α) ≤ r}.

Remark 8.2.2. These constructions are inverses of each other, more precisely D ≡ DXD and

X∗ = X∗ . Hence a ﬁltered simplicial complex is uniquely determined by its size function. DX

We now review a notion of distance between ﬁltered simplicial complexes [59].

139 Deﬁnition 8.2.2 (Tripods and distortion). A tripod between X∗ and Y ∗ with vertex sets V and W respectively is a ﬁnite set Z with surjective maps pX : Z → V and pY : Z → W . The

distortion dis(Z) of a tripod (Z, pX , pY ) is deﬁned by maxα∈P(Z) |DX (pX (α)) − DY (pY (α))|,

where the convention ∞ − ∞ = 0 is assumed.

Definition 8.2.3 (Gromov-Hausdorff distance between filtered simplicial complexes). The

Gromov-Hausdorﬀ distance between the ﬁltered simplicial complexes X∗ and Y ∗ is

1 d (X∗,Y ∗) := inf {dis(Z): Z a tripod between X∗ and Y ∗} . GH 2

Note that the product of vertex sets with the projection maps gives a tripod and if the size

functions are finite then the distortion of this tripod is finite. Hence, the Gromov-Hausdorff

distance between filtered simplicial complexes with finite size functions is finite.

Remark 8.2.3. Given a tripod (Z, pX , pY ), let R = {(pX (z), pY (z)) : z ∈ Z} ⊆ V × W . If

we denote the projection maps V × W → V,W by π1, π2, then (R, π1, π2) is a tripod between

X∗ and Y ∗. Furthermore, dis(Z) = dis(R). Since the vertex sets V,W are assumed to be

finite, there are finitely many such R’s. Therefore, the infimum in the definition of dGH is

realized.

The definition of the Gromov-Hausdorff distance between filtered spaces generalizes the

Gromov-Hausdorﬀ distance between metric spaces (see [18, Section 7.3]):

Proposition 8.2.4 (Extension). Let M and N be ﬁnite metric spaces and X∗ and Y ∗ be

∗ ∗ their Vietoris-Rips complexes respectively. Then dGH(M,N) = dGH(X ,Y ).

Proof. Let R be a correspondence between M and N (i.e. R ⊆ M × N and πM (R) = M,

∗ ∗ πN (R) = N). Note that R can be considered as a tripod between X and Y . By Remark

140 8.2.3, it is enough to show that the distortion of R as a metric correspondence between M

and N is same with the distortion of R as a tripod between X∗ and Y ∗. Let us denote the

ﬁrst one by dismet(R) and the second one by distri(R).

Claim 4. distri(R) ≥ dismet(R).

Proof. By Remark 8.2.1, the size functions of X∗ and Y ∗ are given by the diameter. Hence

we have:

tri 0 0 dis (R) ≥ max |diamM (x, x ) − diamN (y, y )| (x,y),(x0,y0)∈R

0 0 = max |dM (x, x ) − dN (y, y )| (x,y),(x0,y0)∈R = dismet(R).

Claim 5. distri(R) ≤ dismet(R).

Proof. Let α ∈ P(R). Let α ∈ P(R) such that

tri dis (R) = |diamM (πM (α)) − diamN (πN (α))|.

Without loss of generality, we can assume that

diamM (πM (α)) ≥ diamN (πN (α)).

0 Let x and x be points in πM (α) so that

0 diamM (πM (α)) = dM (x, x ).

141 There exists points y and y0 in N such that (x, y), (x0, y0) ∈ α. Then we have

tri dis (R) = diamM (πM (α)) − diamN (πN (α))

0 = dM (x, x ) − diamN (πN (α))

0 0 ≤ dM (x, x ) − dN (y, y )

≤ dismet(R).

We also have:

Proposition 8.2.5. dGH is a (pseudo-)metric between ﬁltered simplicial complexes.

∗ ∗ Proof. Non-negativity and symmetry properties follows from the deﬁnition. dGH(X ,X ) =

0 since the distortion of the identity tripod on the vertex set of X is 0. Let us show the triangle inequality. Let (Z, p, p0) be a tripod between X∗,X0∗ and (Z0, q0, q00) be a tripod between X0∗,X00∗. Let Z00 be the ﬁber product

00 0 Z = Zp0 ×q0 Z .

00 00 ∗ 00∗ 00 Then (Z , p ◦ πZ , q ◦ πZ0 ) is a tripod between X ,X . Given α ∈ P(Z ), we have

00 0 |DX (p ◦ πZ (α)) − DX00 (q ◦ πZ0 )| ≤ |DX (p ◦ πZ (α)) − DX0 (p ◦ πZ (α))|

0 00 + |DX0 (p ◦ πZ (α)) − DX00 (q ◦ πZ0 (α))|

0 = |DX (p ◦ πZ (α)) − DX0 (p ◦ πZ (α))|

0 00 + |DX0 (q ◦ πZ0 (α)) − DX00 (q ◦ πZ0 (α))|

≤ dis(Z) + dis(Z0).

142 Since α ∈ P(Z00) was arbitrary, dis(Z00) ≤ dis(Z0) + dis(Z00). Since the tripods Z,Z0 were

∗ 00∗ ∗ 0∗ 0∗ 00∗ arbitrary, dGH(X ,X ) ≤ dGH(X ,X ) + dGH(X ,X ).

Deﬁnition 8.2.4 (Isomorphism and weak isomorphism). We call two ﬁltered simplicial

complexes isomorphic if there exists a size preserving bijection between their vertex sets. We

call two ﬁltered simplicial complexes weakly isomorphic if their Gromov-Hausdorﬀ distance

is 0.

Note that an isomorphism between ﬁltered simplicial complexes means they only diﬀer

by a relabeling of their vertices. Hence we have the following Remark.

Remark 8.2.6. Isomorphism implies weak isomorphism. This can be seen by taking the

tripod (R, p, q) where R is the graph of the size preserving bijection between the vertex sets

of the initial ﬁltered simplicial complexes and p and q are the natural projection maps.

Now let us see that the converse is not true, i.e. there are weakly isomorphic ﬁltered

simplicial compelexes which are not isomorphic.

Example 8.2.7 (A pair of non-isomorphic but weakly isomorphic ﬁltered simplicial com-

plexes). Given a positive integer n and a real number c, let X∗(n, c) be the ﬁltered simplicial

complex with vertex set {1, . . . , n} and the constant size function equal to c. Note that for

any n, m ∈ N (possibly diﬀerent), the tripod between X∗(n, c) and X∗(m, c) given by the

∗ ∗ product of their vertex sets has zero distortion. Hence, dGH(X (n, c),X (m, c)) = 0 which

means that X∗(n, c) and X∗(m, c) are weakly isomorphic.

F 8.2.2 The Interleaving Type Distance dI between Filtered Simpli- cial Complexes

Here we introduce an interleaving type distance between ﬁltered simplicial complexes.

For the interleaving distance between persistence modules see [10, Section 3.1].

143 Here, we introduce an interleaving type of distance between ﬁltered simplicial complexes

which interacts nicely with their persistent homology. We use the following notation/termi-

t nology: A persistence module (over R) is a family of vector spaces (V )t∈R with linear maps

t,t0 t t0 0 t,t 0 00 t,t00 t0,t00 t,t0 f : V → V for t ≤ t such that f = idV t and for each t ≤ t ≤ t , f = f ◦f . By

i the functoriality of homology, for k ∈ N, the homology groups Hk(X ) of a ﬁltered simplicial complex X∗ form a persistence module, where the linear maps are induced by the inclusion

Xt ,→ Xt0 . This persistence module is called the k-th persistent homology of X∗ and is

∗ denoted by PHk(X ).

A morphism between ﬁltered simplicial complexes is a function between their vertex sets.

Deﬁnition 8.2.5 (Degree). Let f be a morphism from X∗ to Y ∗. Given r ≥ 0, we say that f is r-simplicial if DY (f(α)) ≤ DX (α) + r for each α. We deﬁne the degree deg(f) of f by

deg(f) := inf{r ≥ 0 : f is r-simplicial}.

By the constructibility assumption, f is deg(f)-simplicial.

Hence the degree of a morphism can be thought as a measure of the failure of the morphism at being simplicial.

∗ ∗ Remark 8.2.8. If f : X → Y is r-simplicial, then it induces a morphism f∗ from the

∗ ∗+r t t+r persistence module PHk(X ) to PHk(Y ), induced by the simplicial maps X → Y ,

α 7→ f(α).

Deﬁnition 8.2.6 (Codegree). Let f, g be morphisms from X∗ to Y ∗. Given r ≥ 0, we say that f, g are r-contiguous if DY (f(α)∪g(α)) ≤ DX (α)+r for each α. We deﬁne the codegree codeg(f, g) of f, g by

codeg(f, g) := inf{r ≥ 0 : f, g are r-contiguous}.

144 By the constructibility assumption, f, g are codeg(f, g)-contiguous.

Remark 8.2.9. Let f, g : X∗ → Y ∗ be morphisms of ﬁltered simplicial complexes. Then,

1. deg(f) = codeg(f, f) ≤ codeg(f, g).

∗ ∗+r 2. If f, g are r-contiguous, then they induce the same maps PHk(X ) → PHk(Y ), as

the maps Xt → Y t+r given by α 7→ f(α), g(α) are contiguous as simplicial maps.

3. For each morphism h : Z∗ → X∗, we have codeg(f ◦ h, g ◦ h) ≤ codeg(f, g) + deg(h).

4. For each morphism h : Y ∗ → Z∗, we have codeg(h ◦ f, h ◦ g) ≤ codeg(f, g) + deg(h).

Assume we are given three morphsims f, g, h such that codeg(f, g) ≤ r and codeg(g, h) ≤ r. Although it is possible that codeg(f, h) > r, by part 1 of the remark above, f, g, h induce

∗ ∗+r the same maps PHk(X ) → PHk(Y ). The following deﬁnition is given to capture this type of situations, see Section 8.2.3 below.

Deﬁnition 8.2.7 (∞-codegree). Deﬁne the ∞-codegree of f and g as

∞ codeg (f, g) := min max codeg(fi−1, fi). f=f0,...,fn=g i=1,...,n

Proposition 8.2.10. Let f, g, h : X∗ → Y ∗ and f 0, g0 : Z∗ → X∗ be morphisms of ﬁltered simplicial complexes. Then,

(1) deg(f) = codeg∞(f, f) ≤ codeg∞(f, g) ≤ codeg(f, g).

∞ ∗ ∗+r (2) If codeg (f, g) ≤ r, then f, g induce the same maps from PHk(X ) → PHk(Y ).

(3) (Ultrametricity) codeg∞(f, h) ≤ max codeg∞(f, g), codeg∞(g, h).

(4) codeg∞(f ◦ f 0, g ◦ g0) ≤ codeg∞(f, g) + codeg∞(f 0, g0).

145 ∞ Proof. (1) codeg (f, g) ≤ codeg(f, g) can be seen by taking f = f0, f1 = g. deg(f) ≤

0 1 codeg(f, g) since for any f = f0, . . . , fn = g, deg(f) ≤ codeg(f , f ). This also shows that deg(f) = codeg∞(f, f) ≤ codeg∞(f, g).

(2) There exists f = f0, . . . , fn = g such that codeg(fi−1, fi) ≤ r. By Remark 8.2.9, fi−1, fi

∗ ∗+r induce the same maps from PH(X ) to PH(Y ). Hence f0 = f, g = fn also induce the same maps.

(3) Follows by concatenating sequences of functions.

∞ (4) Let f = f1, . . . , fn = g be the sequence realizing codeg (f, g). Then, for any morphism h whose range is same with the domain of f, g, by Remark 8.2.9 we have

∞ codeg (f ◦ h, g ◦ h) ≤ max codeg(fi−1 ◦ h, fi ◦ h) i

≤ max codeg(fi−1, fi) + deg(h) i = codeg∞(f, g) + deg(h)

Similarly we have

codeg∞(h ◦ f, h ◦ g) ≤ codeg∞(f, g) + deg(h).

Now by using these and part i),iii) above, we get

codeg∞(f ◦ f 0, g ◦ g0) ≤ max(codeg∞(f ◦ f 0, g ◦ f 0), codeg∞(g ◦ f 0, g ◦ g0))

≤ max(codeg∞(f, g) + deg(g0), codeg∞(f 0, g0) + deg(g))

≤ codeg∞(f, g) + codeg∞(f 0, g0).

Deﬁnition 8.2.8 (Interleaving distance between ﬁltered simplicial complexes). For ≥ 0, an -interleaving between X∗ and Y ∗ consists of morphisms f : X∗ → Y ∗, g : Y ∗ → X∗ such

146 that

∞ ∞ deg(f), deg(g) ≤ , and codeg (g ◦ f, idX∗ ), codeg (f ◦ g, idY ∗ ) ≤ 2.

In this case we say that X∗ and Y ∗ are -interleaved. We deﬁne

F ∗ ∗ ∗ ∗ dI (X ,Y ) := inf{ ≥ 0 : X and Y are -interleaved}.

We then have:

F Proposition 8.2.11. dI is a (pseudo-)distance between ﬁltered simplicial complexes.

F ∗ ∗ Proof. Non-negativity and symmetry follow from the deﬁnition. dI (X ,X ) = 0 since idX∗ gives a 0-interleaving. Let us show the triangle inequality. Let f : X∗ → Y ∗, g : Y ∗ → X∗ be an -interleaving between X∗ and Y ∗ and f 0 : Y ∗ → Z∗, g0 : Z∗ → Y ∗ be an 0-interleaving

between Y ∗ and Z∗. Let us show that f 0 ◦ f, g ◦ g0 is an ( + 0)-interleaving between X∗,Z∗.

deg(f 0 ◦ f), deg(g ◦ g0) ≤ + 0,

which follows from the deﬁnition of degree. By Remark 8.2.10 we have

∞ 0 0 ∞ 0 0 ∞ codeg (g ◦ g ◦ f ◦ f, idX∗ ) ≤ max(codeg (g ◦ g ◦ f ◦ f, g ◦ f), codeg (g ◦ f, idX∗ ))

∞ 0 0 ≤ max(deg(g) + codeg (g ◦ f , idY ∗ ) + deg(f), 2)

≤ max( + 20 + , 2) = 2( + 0).

Similarly

∞ 0 0 0 codeg (f ◦ f ◦ g ◦ g , idZ∗ ) ≤ 2( + ).

This completes the proof.

∗ ∗ F ∗ ∗ Deﬁnition 8.2.9. We call X and Y equivalent if dI (X ,Y ) = 0.

147 X0 X1 X2 X3

v2 v2 v3 * X v v v1 v v1 v0 v0 1 0 0

∗ ∗ Figure 8.2: X := ∆3.

F Note that the interleaving distance dI is diﬀerent from the interleaving distance between the corresponding persistence homology modules, as there are simplicial complexes which are not contiguous but have same homology.

Because of Theorem 8.1.1, equivalent filtered simplicial complexes have the same persistent homologies. In the next section we see that weakly isomorphic filtered simplicial complexes (See Definition 8.2.4) are equivalent. For now, let us give an example to show that the converse is not true.

∗ Example 8.2.12 (Equivalence is weaker than weak isomorphism). Deﬁne ∆n as the ﬁltered

simplicial complex with vertex set {0, . . . , n} and size function Dn(α) := max{i : i ∈ α}

∗ ∗ (see Figure 8.2). Note that for any tripod (R, p, q) between ∆n, ∆m we have dis(R) ≥

∗ ∗ |Dn(p(R)) − Dm(q(R))| = |m − n|, hence dGH(∆m, ∆n) ≥ |m − n|/2. We now show that

F ∗ ∗ dI (∆m, ∆n) = 0. The topological basis of this is the fact that any two maps onto a simplex

∗ ∗ are contiguous. Without loss of generality assume that m ≤ n. Let ι : ∆m → ∆n be the

∗ ∗ morphism given by the inclusion of the vertex set and let π : ∆n → ∆m be the map given by k 7→ min(m, k). Since both maps are size non-increasing and size functions are deﬁned by the

maximum, both maps have degree 0. Also note that (1) π ◦ ι = id, and (2) if α ⊆ {0, . . . , n}

has maximal element i, then so does α∪ι◦π(α). Hence, Dn(α) = Dn(α∪ι◦π(α)). This shows

148 F ∗ ∗ that codeg(ι ◦ π, id) = 0. Therefore dI (∆m, ∆n) = 0. In Section 8.6 we give a construction generalizing this one (see Remark 8.6.1).

F 8.2.3 Remarks About the Deﬁnition of dI .

It is possible [32, 26] to define a related but strictly stronger notion of -interleaving between filtered simplicial complexes than the one given in Definition 8.2.8. Given filtered simplicial complexes X∗ and Y ∗, an -strong interleaving between X∗ and Y ∗ is a pair of morphisms f : X∗ → Y ∗ and g : Y ∗ → X∗ such that

deg(f), deg(g) ≤ , and codeg(g ◦ f, idX∗ ), codeg(f ◦ g, idY ∗ ) ≤ 2.

The difference with Definition 8.2.8 is that codeg∞ has been replaced by (the generally larger number) codeg. The problem with this definition is that it does not give a metric as

F ∗ ∗ ∗ ∗ we show next. Deﬁne dcI (X ,Y ) as the inﬁmal ≥ 0 such that X and Y are -strongly interleaved.

Note that the deﬁnition of codeg∞ uses chains of morphisms. Such a sequence of mor-

F phisms used in the proof of Proposition 8.2.11 to show the triangle inequality for dI . The topological basis of the necessity of considering chains is the following: If simplicial maps f, f 0 : S → T are contiguous and g, g0 : T → U are contiguous, it does not necessarily follow that g ◦f, g0 ◦f 0 : S → U are contiguous. Instead, what we have is g ◦f is contiguous to g ◦f 0 which is in turn contiguous to g0 ◦ f 0. Note that contiguity is not an equivalence relation between simplicial morphisms.

F F dcI is not a metric. Let us give a concrete example to show that dcI does not satisfy the

∗ triangle inequality. For a non-negative integer n, let Xn be the ﬁltered simplicial complex with vertex set {v0, . . . , vn} and such that

149 0 (1) the cells of Xn coincide with the set of all edges of the form [vi, vi+1],

t (2) Xn is the full simplex for t ≥ 1,

t (3) Xn = ∅ for t < 0, and

t 0 (4) Xn = Xn for 0 ≤ t < 1.

∗ ∗ Note that Xn is included in Xn+1 via the morphism vi 7→ vi for all i = 0, . . . , n. Also,

∗ ∗ Xn+1 surjects onto Xn via the morphism vi 7→ vi for i = 0, . . . , n and vn+1 7→ vn. By using

F ∗ ∗ F ∗ ∗ these maps, we see that dcI (Xn,Xn+1) is 0. However, for n ≥ 3, dcI (Xn,X0 ) is not 0, as no

0 F constant map from Xn to itself is contiguous to the identity. Therefore, dcI fails to satisfy the triangle inequality, for otherwise one would have

F ∗ ∗ F ∗ ∗ F ∗ ∗ F ∗ ∗ 0 < dcI (X0 ,X3 ) ≤ dcI (X0 ,X1 ) + dcI (X1 ,X2 ) + dcI (X2 ,X3 ) = 0,

a contradiction.

8.2.4 Stability and the Proof of Theorem 8.1.1

In this section we prove Theorem 8.1.1. We ﬁrst need the following

Lemma 8.2.13. Let f : X∗ → Y ∗, g : Y ∗ → X∗ be morphisms. Let R be the correspondence

between the vertex sets of X∗ and Y ∗ containing the graphs of f and g. Then (f, g) is an

dis(R)-interleaving.

∗ ∗ Proof. Let pX (resp. pY ) be the projection map from R to the vertex set of X (resp. Y ).

Let α be a non-empty subset of the vertex set of X∗. Note that

−1 f(α) ⊆ pY (pX (α)).

150 Let := dis(R). We have

−1 DY ∗ (f(α)) ≤ DY ∗ (pY (pX (α)))

−1 ≤ DX∗ (pX (pX (α))) +

= DX∗ (α) + .

Hence deg(f) ≤ . Similarly deg(g) ≤ .

−1 Note that pX (pY (f(α))) contains both α and g ◦ f(α). Hence

−1 DX∗ (g ◦ f(α) ∪ α) ≤ DX∗ (pX (pY (f(α))))

−1 ≤ DY ∗ (pY (pY (f(α)))) +

= DY ∗ (f(α)) +

≤ DX∗ (α) + 2.

This shows that

∞ codeg (g ◦ f, idX∗ ) ≤ codeg(g ◦ f, idX∗ ) ≤ 2.

Similarly,

∞ codeg (f ◦ g, idY ∗ ) ≤ 2.

This completes the proof.

Proof of Theorem 8.1.1. By the deﬁnition of interleavings for ﬁltered simplicial complexes

and Remark 8.2.10, an -interleaving between ﬁltered simplicial complexes induces an - interleaving between their persistence modules. Hence

∗ ∗ F ∗ ∗ dI(PHk(X ), PHk(Y )) ≤ dI (X ,Y ).

Now let R be a correspondence between the vertex sets of X∗ and Y ∗. Then there are

morphism f : X∗ → Y ∗, g : Y ∗ → X∗ such that R contains graphs of f and g. By Lemma

151 8.2.13, X∗ and Y ∗ are dis(R)-interleaved. Since R was an arbitrary correspondence, by

Remark 8.2.3

F ∗ ∗ ∗ ∗ dI (X ,Y ) ≤ 2dGH(X ,Y ).

Remark 8.2.14. Note that the main point of Theorem 8.1.1 is not that the interleaving

distance between persistence homology modules are less than twice the Gromov Hausdorﬀ

distance between ﬁltered simplicial complexes (which is already proven in the case of Vietoris-

Rips complexes, see [22]), but the newly introduced interleaving distance between ﬁltered

simplicial complexes sits in between.

8.3 The Vertex Quasi-distance and Simpliﬁcation

Removing vertices from simplicial complexes while preserving the homotopy type is con-

sidered in the papers [58, 74, 5, 7]. Latschev [54] applied a similar idea to Vietoris-Rips

ﬁltration. Here we generalize it to arbitrary ﬁltered simplicial complexes, while controlling

F the eﬀect of a removal in terms of dI . Let us start with the following deﬁnition.

Deﬁnition 8.3.1 (The vertex quasi-distance). Let X∗ be a ﬁltered simplicial complex. Given

∗ vertices v, w of X , deﬁne the vertex quasi-distance δX (v, w) to be the minimal δ ≥ 0 such

that

DX (α ∪ {v}) + δ ≥ DX (α ∪ {w}), for each non-empty set of vertices α.

Note that taking α as the full vertex set already requires δ ≥ 0, hence we can equivalently deﬁne δX (v, w) by δX (v, w) := maxα DX (α ∪ {w}) − DX (α ∪ {v}) .

152 Although the vertex quasi-distance is not necessarily symmetric (i.e. δX (v, w) may be

diﬀerent from δX (w, v)), the following remark shows that it satisﬁes other properties of a

metric. Such structures are called quasimetric spaces.

Remark 8.3.1 (Quasimetric). For all vertices v, v0, v00,

1. δX (v, v) = 0.

0 0 00 00 2. δX (v, v ) + δX (v , v ) ≥ δX (v, v ).

Deﬁnition 8.3.2 (Codensity function). For each vertex v let

• δX (v) := minw6=v δX (v, w). This is called the codensity of vertex v.

∗ ∗ • δ(X ) := minv δX (v) (minimal codensity of X ).

We introduce this invariant to control the effect of removing a vertex from a filtered simplicial complex on its persistent homology. Before proving Proposition 8.1.2 let us precisely define what we mean by removing a vertex.

Definition 8.3.3 (Filtered subcomplex). A filtered subcomplex of a filtered simplicial complex X∗ is a filtered simplicial complex Y ∗ such that for each t, Y t is a subcomplex of Xt. We call Y ∗ a full filtered subcomplex if each Y i is a full subcomplex of Xt, precisely a simplex of Xt is a simplex of Y t if and only if its vertices are in Y t. Note that a full subcomplex is determined by its vertex set. Therefore, if we take a vertex v from a filtered simplicial complex X∗ with vertex set V , there exists a unique full filtered subcomplex of it such that the vertex set at index t is the vertex set of Xt minus v. We denote this subcomplex by

(X − {v})∗.

153 Remark 8.3.2 (Restriction). The size function of (X − {v})∗ is the restriction of the size function of X∗.

Proof of Proposition 8.1.2. Let w 6= v be a vertex such that δX (v, w) = δX (v). Let f :

X∗ → (X −{v})∗ be the map which is identity on all vertices except v and maps v to w. Let

∗ ∗ ι :(X − {v}) → X be the inclusion map. Let us show that (f, ι) is a δX (v)-interleaving.

We have deg(ι) = 0. Let α be a non-empty subset of the vertex set of X∗. If v∈ / α then f(α) = α. If v ∈ α, then f(α) ⊆ α ∪ {w}, hence

DX (f(α)) ≤ DX (α ∪ {w}) ≤ DX (α ∪ {v}) + δX (v) = DX (α) + δX (v).

∞ Hence deg(f) ≤ δX (v). Since f ◦ ι = id(X−{v})∗ , codeg (f ◦ ι, id(X−v)∗ ) = 0. If v∈ / α, then

α ∪ ι ◦ f(α) = α. If v ∈ α then α ∪ ι ◦ f(α) = α ∪ {w}, hence

DX (α ∪ ι ◦ f(α)) = DX (α ∪ {w}) ≤ DX (α ∪ {v}) + δX (v) = DX (α) + δX (v).

∞ ∗ ∗ Hence codeg (ι ◦ f, idX∗ ) ≤ δX (v). Therefore X , (X − {v}) are δX (v)-interleaved.

Computational consequences. Note that δX (v, w) can only become smaller after we remove a vertex since the maximum in the deﬁnition of vertex quasi-distance is now taken on a smaller set. However, this does not imply that δX (v) also become smaller after removing a vertex, since it is possible that the removed vertex w is the vertex realizing δX (v) = δX (v, w).

Still, the observation of the monotonicity of δX (v, w) gives us a method to simplify a ﬁltered simplicial complex while bounding the approximation error in the persistent homology. Let

∗ ∗ us enumerate the vertex set of X as (v1, . . . , vn) and let Q(X ) be the n × n matrix given by [δX (vi, vj)]i,j. This method is streamlined in Listing 8.1. Note that when the procedure

F ∗ ∗ terminates, the interleaving distance dI (X ,Y ) ≤ errorBound. In the following subsections

154 Listing 8.1: Simpliﬁcation Method. Note: at the end of the execution Y* is full subcomplex of X* with vertex set W. Note: the procedure ComputeCodensityMatrix() is discussed in the next section. INPUT: X∗ , N: number of vertices to be removed OUTPUT: Y∗ , a full subcomplex and errorBound SET Q=ComputeCondensityMatrix(X∗), errorBound=0, W=Vertex set of X∗ . for k from 1 to N (i,j)=index of the minimal nondiagonal element of Q errorBound=errorBound + Q(i , j) remove W(i) from W remove i−th row and column from Q endfor

we discuss how to decrease the time complexity and/or error bound obtained from this method, if we are only interested in certain degrees of homology.

8.3.1 Computing δX(v, w): The Procedure ComputeCodensityMatrix()

The simpliﬁcation method given in Listing 8.1 calls the procedure ComputeCodensityMatrix()

which calculates the matrix [δX (vi, vj)]i,j. In this section we explain the mathematical ideas behind it. We will not provide pseudo-code as the procedure will be made evident.

Normally, the definition of δX (v, w) (Definition 8.3.1) requires checking all non-empty subsets of the vertex set V of X∗, which in total gives us a complexity of O(2n). However, we can achieve a better complexity if our filtered simplicial complex has some structure.

Proposition 8.3.3 below shows that that if the ﬁltered simplicial complex is clique, then we only need to check singletons in order to calculate δX (v, w). Recall that a simplicial complex is called clique if a simplex is included in it whenever its 1-skeleton is included. A

ﬁltered simplicial complex X∗ is called clique if each Xt is clique.

155 Proposition 8.3.3 (The case of clique ﬁltered simplicial complexes). Let X∗ be a clique

ﬁltered simplicial complex with vertex set V and size function DX . Then

0 0 δX (v, v ) = max DX ({v , w}) − DX ({v, w}) . w∈V

We have the following lemma whose proof we omit:

∗ Lemma 8.3.4. Let X be a clique ﬁltered simplicial complex with the size function DX .

Then,

DX (α) = max DX ({v, w}). v,w∈α

0 Proof of Proposition 8.3.3. Let := maxw∈V DX ({v , w}) − DX ({v, w}) . Recall that

0 0 δX (v, v ) = max DX (α ∪ {v }) − DX (α ∪ {v}), α⊆V,α6=∅

0 0 hence δX (v, v ) ≥ . Let us show that δX (v, v ) ≤ . Let α be a non-empty subset of V . Then by Lemma 8.3.4 we have

0 0 0 DX (α ∪ {v }) = max( max DX ({w, w }), max DX ({v , w})) w,w0∈α w∈α

0 ≤ max( max DX ({w, w }), max DX ({v, w}) + ) w,w0∈α w∈α

0 ≤ max( max DX ({w, w }), max DX ({v, w})) + w,w0∈α w∈α

= DX (α ∪ {v}) + .

0 Since α was arbitrary, δX (v, v ) ≤ .

Deﬁnition 8.3.4. As a generalization of the concept of a clique complex, let us call a

simplicial complex k-clique if a simplex is contained in it if and only if its k-skeleton is

contained in it. A clique complex is 1-clique with respect to this deﬁnition.

By a proof similar to that of Proposition 8.3.3, we can obtain the following generalization:

156 Proposition 8.3.5. The case of k-clique ﬁltered simplicial complexes Let k be a positive

integer and let X∗ be a ﬁltered simplicial complex such that for each t Xt is k-clique. Then

for all vertices v and v0,

0 0 δX (v, v ) = max DX (α ∪ {v }) − DX (α ∪ {v}) . α,0<|α|≤k

Remark 8.3.6. By Proposition 8.3.5, Lemma 7 of the paper [71] is equivalent in our termi-

nology to saying δ(p, πα(p)) = 0, hence the codensity δ(p) = 0. The homology isomorphism

in Lemma 8 of [71] then follows from the fact that removing a codensity zero point does not

aﬀect homology.

Now we use Proposition 8.3.5 to show we can turn a given ﬁltered simplicial complex

into one satisfying the assumptions in Proposition 8.3.5 without losing persistent homology

information in degrees less than k.

Proposition 8.3.7. Let X∗ be a ﬁltered simplicial complex. Let Y ∗ be the ﬁltered simplicial

complex with the same vertex set as X∗ such that for each t, a simplex is in Y t if and only

if its k-skeleton is in Xt. Note that Y ∗ is well deﬁned and Xt ⊆ Y t for each t. We have:

(1) Y t is k-clique for all t.

∗ ∼ ∗ (2) PH

(3) DY (α) = maxβ⊆α,0<|β|≤k+1 DX (β).

Proof. (1) Assume the k-skeleton αk of α is contained in Y i. Then the k-skeleton of αk,

which is αk itself is contained in Xi. Therefore α is contained in Y i.

(2) Note that if α is a simplex of dimension less than or equal to k, then its k-skeleton is

itself, hence it is contained in Xi if and only if it is contained in Y i. Therefore the inclusion

157 X∗ → Y ∗ is identity in the level of k-skeleton. Therefore, it induces an isomorphism between

homology groups of degree less than k.

∗ ∗ (3) Let r := maxβ⊆α,0<|β|≤k+1 DX (β). Since the k-skeletons of X and Y are the same, for

|β| ≤ k + 1 we have DX (β) = DY (β). Therefore

DY (α) ≥ max DY (β) = max DX (β) = r. β⊆α,0<|β|≤k+1 β⊆α,0<|β|≤k+1

r Now let us show that DY (α) ≤ r. We need to show that α ∈ Y . By the deﬁnition of r, the k-th skeleton of α is contained in Xr. This implies that α is in Y r.

8.3.2 Specializing δX(v, w) According to Homology Degree

In this section we reﬁne our ideas so that given a ﬁltered simplicial complex X∗ and

k ∈ N, the bound given in Proposition 8.1.2 is better adapted to scenarios when one only

∗ wishes to compute persistent homologies PHj(X ) for j ≥ k.

∗ ∗ Given a ﬁltered simplicial complex and k ∈ N, let Y = Tk(X ) be the ﬁltered simplicial complex with the same vertex such that a simplex is in Y t if it is in Xt and each simplex in

its k-skeleton is contained in a k-simplex of Xt. In other words, we remove simplices from

Xt which have dimension less than k and are not contained in any k-simplex of Xt. Note that Y ∗ is well deﬁned and Y t ⊆ Xt for each t.

∗ Proposition 8.3.8. Denote the vertex quasi-distance for X by δX , and by δY denote the

∗ ∗ vertex quasidistance of Y = Tk(X ). Then, we have:

∗ ∗ (1) PH≥k(Y ) = PH≥k(X ).

(2) DY (α) = minβ⊇α,|β|≥k+1 DX (β).

(3) δY (v, w) ≤ δX (v, w).

158 (4) Let m ≥ k be a non-negative integer. If X∗ satisﬁes the property that for each t, α ∈ Xt

if and only if the m-skeleton of α is in Xt, then Y ∗ satisﬁes this property too.

Proof. (1) Note that for k0 ≥ k, the k0-simplices of Xi is same with the k0-simplices of Y i.

Since the homology of degree k0 is determined by such cells, the inclusion Y ∗ ⊆ X∗ induces

∗ ∗ the isomorphism PH≥k(Y ) → PH≥k(X ).

(2) By the identity mentioned above, for β with |β| ≥ k + 1 we have DY (β) = DX (β). Let

us denote r := minβ⊇α,|β|≥k+1 DX (β). Then we have

DY (α) ≤ min DY (β) = min DX (β) = r. β⊇α,|β|≥k+1 β⊇α,|β|≥k+1

Let us show that DY (α) ≥ r. Let s < r. Let us show that DY (α) > s. If the dimension

of α is greater than or equal to k, then DY (α) = DX (α) = r > s. Now assume that α has

dimension less than k. By deﬁnition of r, every k-cell containing α has size strictly greater

s than s, therefore α is not contained in X . Hence, we have DY (α) > s. Since s < r was arbitrary DY (α) ≥ r.

(3) Let α be a simplex and β be the simplex with dimension greater than or equal to k containing α ∪ {v} such that DY (α ∪ {v}) = DX (β). Then we have

DY (α ∪ {w}) − DY (α ∪ {v}) ≤ DY (β ∪ {w}) − DY (α ∪ {v})

= DX (β ∪ {w}) − DX (β)

= DX (β ∪ {w}) − DX (β ∪ {v})

≤ δX (v, w).

Since α was arbitrary, δY (v, w) ≤ δX (v, w).

159 (4) Let α be a simplex whose m-skeleton αm is in Y t. Let us show that α ∈ Y t. Note that

α is in Xt. If the dimension of α is greater or equal than k, then α is in Y t. Now assume that the dimension of α is less than k. Then we have that αm = α is in Y t.

Summary. If we are only interested in degree k persistent homology, then we can ﬁrst apply the clique-ﬁcation process described in Proposition 8.3.7 for k + 1 so that the calculation of

∗ k n each entry δX (v, w) of the matrix Q(X ) becomes a O(n ) task instead of O(2 ), where n is the number of vertices. Then we can apply the process described in Proposition 8.3.8 so that we lower the values of δX (v, w) and get a better error bound for the simpliﬁcation process.

Then we can start our simpliﬁcation process. After removing a vertex we have two options, we can either keep working with the original codensity matrix to get the upper bound on the change in persistent homology, or we may want to compute the codensity matrix again, since its elements may decrease after removal. Note that if we remove a vertex from a k-clique

ﬁltered simplicial complex, it will still be k-clique. Hence calculating the codensity matrix does not become more costly after removing a vertex.

8.4 An Application to the Vietoris-Rips Filtration of Finite Met- ric Spaces and Graphs

In this section, we apply the ideas we developed in previous sections to the Vietoris-Rips

ﬁltration of ﬁnite metric spaces and graphs.

8.4.1 Finite Metric Spaces

We show can we get a much smaller upper bound (compared to the Gromov-Hausdorﬀ bound) on the eﬀect of the removal of a vertex on the persistent homology through codensity considerations, if we restrict our attention to persistent homology of degree ≥ 1.

160 ∗ Remark 8.4.1. Let X be the Vietoris-Rips complex of a ﬁnite metric space (M, dM ). Then for each x and y in M we have

δX (x, y) = dM (x, y).

In particular, this implies that the codensity of a point is the distance to the nearest neighbor.

Proof. For the Vietoris-Rips ﬁltration,the size function is the diameter. Note that for α =

{x}, diamM (α∪{y})−diamM (α∪{x}) = dM (x, y)−0 = dM (x, y), hence δX (x, y) ≥ dM (x, y).

Note that, by triangle inequality for any z we have dM (y, z) ≤ dM (x, z) + dM (x, y) and this implies that for any subset α we have diamM (α ∪ {y}) ≤ diamM (α ∪ {x}) + dM (x, y) and this implies that δX (x, y) ≤ dM (x, y). Hence δX (x, y) = dM (x, y).

Remark 8.4.1 shows that the codensity bound on the effect of removing a vertex directly from the Vietoris-Rips filtration is not better than the Gromov-Hausdorff bound. However, let us see how we can get much better bounds by applying methods mentioned Section 8.3.2

∗ if we are only interested in PHk≥1(VR (X)).

Let us denote the modiﬁed size function described in Proposition 8.3.8 by diamM,k. More precisely, for α ⊆ M, we have

diamM,k(α) := min diamM (β). β⊇α,|β|≥k+1

Note that diamM,0 = diamM . Let us denote the corresponding ﬁltered simplicial complex

∗ ∗ by Xk . By Proposition 8.3.8, degree ≥ k persistent homology of X = VR(M) is the same

∗ as that of Xk . Therefore, if we are interested in persistence homology of degree at least 1,

∗ then instead of working with the Vietoris-Rips complex, we can work with X1 which has the advantage of having a smaller codensity function.

161 ∗ Proposition 8.4.2. Let M be a metric space, X1 be the modiﬁed Vietoris-Rips complex

∗ described as above and δ1 be the codensity function of X1 . Let x be a point in M and y be the closest point to x. Then

δ1(x, y) = max 0, max dM (y, p) − dM (x, p) . p6=x,y

∗ Proof. Since the Vietoris-Rips ﬁltration is clique, by Proposition 8.3.8 X1 is also clique. By Proposition 8.3.3, we have δ1(x, y) = maxp∈M diam1({y, p}) − diam1({x, p}) . Hence,

δ1(x, y) = max diamM,1({p, y}) − diamM,1({p, x}) p∈M = max dM (x, y) − diamM,1({x}), diamM,1({y}) − dM (x, y), max dM (p, y) − dM (x, y) p6=x,y = max 0, max dM (y, p) − dM (x, p) . p6=x,y

The following is an example where the modiﬁed codensity described in Proposition 8.4.2

is signiﬁcantly smaller than the original codensity, which is equal to the distance to the

nearest neighbor for the Vietoris-Rips ﬁltration by Remark 8.4.1.

Example 8.4.3 (Circle with ﬂares (see Figure 8.1)). Let M be a ﬁnite metric space described

as follows: It is a ﬁnite set of points selected from a circle and some ﬂares attached to it, see

Figure 8.1. Let us show that for an endpoint x of a ﬂare in M, δ1(x) = 0. Note that this

implies that our method (see Listing 8.1) will inductively remove all points in ﬂares without

∗ any cost on PH≥1(VR (M)) until only the points on the circle are left. Note that this is

signiﬁcantly less than both the Gromov-Hausdorﬀ distance between the original space M

and the ﬁnal space M 0, and the sum of Gromov-Hausdorﬀ costs of successively removing

single points.

162 Let y be the closest to point to x in M. Since x is a endpoint in a ﬂare, for each p 6= x

we have dM (x, p) = dM (x, y) + dM (y, p), in particular dM (y, p) ≤ dM (x, p). Therefore, by

Proposition 8.4.2 we have δ1(x) ≤ δ1(x, y) = 0.

8.4.2 Application to Metric Graphs

In this section we use our results to provide some characterization results related to the

Vietoris-Rips persistent homology of metric graphs along the lines of [3].

Recall that a metric graph is a topological graph with a length structure. Up to isometry,

a metric graph is determined by the lengths of its edges. Here we only consider ﬁnite graphs,

i.e. ones having ﬁnitely many edges.

Let us call a metric graph simple if it can be constructed by inductively wedging circles and closed intervals (see Figure 8.3). Recall that the wedge sum of two pointed topological spaces (X, p), (Y, q) is deﬁned by

a X ∨ Y := (X, p) (Y, q)/{p, q}.

It is a pointed topological space where the chosen point is the point representing the identiﬁed

points p and q. Furthermore, if (X, dX ), (Y, dY ) are metric spaces, then the wedge sum carries

the metric d which reduces to dX and dY on the copies of X and Y , and for each x ∈ X and

y ∈ Y , d(x, y) := dX (x, p) + dY (y, q).

The main proposition we prove in this section is the following. A similar result can be

found in [3, Proposition 13]. Here we do not put any restriction on edge lengths.

n Proposition 8.4.4. Let G be a simple metric graph obtained by using circles (Ci)i=1 and some intervals. Then we have

n ∗ M ∗ PH≥1(VR (G)) = PH≥1(VR (Ci)). i=1 163 In particular, since a metric tree is a simple graph without circles, a metric tree has trivial

PH≥1.

We use the following lemma in the proof of Proposition 8.4.4. This lemma is in itself an application of Theorem 8.1.1 and Proposition 8.1.2. A similar result is stated in [3] which requires elements from the theory of folds, elementary simplicial collapses, and LC reductions. Our proof is self contained.

∗ Lemma 8.4.5. Let I be a closed interval in R. Then PH≥1(VR (I)) = 0.

In the proof of Lemma 8.4.5, we use the following statement.

Proposition 8.4.6. Let X be a metric space, k be a non-negative integer and r be a non- negative real number. Let R be a ring, J be the poset of ﬁnite subsets of X and F : J →

r R − Mod be the functor A 7→ Hk(VR (A)) where the coeﬃcients of homology is taken in the

r ring R. Then colimF = Hk(VR (X)).

The fact that the Vietoris-Rips complex of a metric space is the colimit of the Vietoris-

Rips complexes of its ﬁnite subspaces is mentioned in [1, Section 7]. One may be able to prove Proposition 8.4.6 starting from this fact but we contribute a diﬀerent more direct proof.

Proof of Proposition 8.4.6. Let us start by ﬁxing some notation. Let A be a ﬁnite subset of

r X. We denote the homology maps induced by the inclusion A,→ X by ιA :Hk(VR (A)) →

r Hk(VR (X)). If B is another ﬁnite subset of X such that A ⊆ B, we denote the homology

r r r map induced by this inclusion by ιA,B :Hk(VR (A)) → Hk(VR (B)). If z is chain in VR (X) such that the vertices of z is contained in A, we denote the homology class it represents in

r Hk(VR (A)) by [z]A.

164 r r r The homology maps ιA :Hk(VR (A)) → Hk(VR (X)) shows that Hk(VR (X)) is a cocone for J. Let us show that it is universal. Let M be another cocone for J with morphisms

r r φA :Hk(VR (A)) → M. Deﬁne a map u :Hk(VR (X)) → M as follows. Given a homology

r class c in Hk(VR (X)), let z be a cycle representing it and let A be a ﬁnite subset containing

0 the vertices of z. Deﬁne u(c) := φA([z]A). Let us show that this map is well deﬁned. Let z

be another cycle representing c and A0 be a ﬁnite subset containing the vertices of z0. Note

that z − z0 is a boundary in VRr(X), let w be a chain in VRr(X) so that ∂w = z − z0. Let

0 0 B a ﬁnite subspace of X containing A, A and vertices of w. Note that [z]B = [z ]B as w is

contained in B. We have

0 0 0 φA([z]A) = φB ◦ ιA,B([z]A) = φB([z]B) = φB([z ]B) = φB ◦ ιA0,B([z ]B) = φA0 ([z ]A0 ).

r This shows that u :Hk(VR (X)) → M is well deﬁned. It is an R-module homomorphism

0 r 0 since if c, c are homology classes in Hk(VR (X)) represented by cycles z, z whose vertices

are contained in a ﬁnite subspace A, then for any λ ∈ R we have

0 0 0 0 u(c + λ c ) = φA([z]A + λ [z ]A) = φA([z]A) + λ φA([z ]A) = u(c) + λ u(c ).

r Given a homology class c in Hk(VR (A)) which is represented by a cycle z, we have

u ◦ ιA(c) = φA([z]A) = φA(c),

hence u commutes with structure maps. The uniqueness of such u follows from the fact

r that for every homology class c in Hk(VR (X)) there exists a ﬁnite subset A such that c is

r r contained in the image of ιA :Hk(VR (A)) → Hk(VR (X)).

Proof of Lemma 8.4.5. By Proposition 8.4.6, it is enough to show that for any ﬁnite subset

r A of I, k ≥ 1 and r ≥ 0, we have Hk(VR (A)) = 0. Fix A, r as above. We proceed by

165 Figure 8.3: A simple metric graph.

induction. Order A as x1 < ··· < xn. The case n = 1 is obvious. Let n > 1. By Proposition

0 8.4.2 δ1(xn, xn−1) = 0, where δ1 is deﬁned as in Section 8.4.1. Hence, if we let A = A−{xn}, then by Theorem 8.1.1, Proposition 8.1.2, and Proposition 8.3.8 there is a 0-interleaving

∗ ∗ 0 between PH≥1(VR (A)) and PH≥1(VR (A )). This completes the proof.

The other main ingredient of the proof of Proposition 8.4.4 is the following:

Proposition 8.4.7. Let X and Y be metric spaces. Then VR∗(X ∨ Y ) has the same persistent homology with the wedge sum VR∗(X) ∨ VR∗(Y ).

Proposition 8.4.7 can be found in [3, Proposition 4]. We give an alternative proof via barycentric subdivision which is more concise.

Proof of Proposition 8.4.7. Let p ∈ X, q ∈ Y be the chosen points for the wedge sum. Note that VRr(X) ∨ VRr(Y ) is contained in VRr(X ∨ Y ) for each r ≥ 0. Let us show that this inclusion induces an isomorphism between homology groups. Note that this is enough for our proof as these inclusions commutes with the structure maps of both ﬁltered simplicial complexes.

Order the disjoint union of X ∨ Y and denote the smallest element of a ﬁnite subset σ by min(σ). It is known that for a simplicial complex S with ordered vertices, the map B(S) → S

166 from the barycentric subdivision B(S) of S to S deﬁned by σ → min(σ) is simplicial and

induces an isomorphism between homology groups [55, p. 166,167].

Consider the map f : BVRr(X ∨ Y ) → VRr(X) ∨ VRr(Y ) deﬁned as follows: f(σ) =

min(σ) if σ is contained in X or Y , f(σ) = p = q else. Let us see that this map is simplicial.

r Take a simplex S = (σ1 ⊂ σ2 ⊂ · · · ⊂ σn) in B VR (X ∨ Y ) . Without loss of generality,

assume that k is the maximal integer such that for i ≤ k, σi ⊆ X and furthermore assume

that for i > k, σi is neither contained in X nor in Y . If k = n, then f(S) ⊆ σn, hence it

r r is a simplex in VR (X) ∨ VR (Y ). If k 6= n, then σn contains elements from both X and

r Y , hence σn ∪ {p} is a simplex in in VR (X ∨ Y ), which in turn implies that σk ∪ {p} is a

r r r simplex in VR (X). Hence, f(S) ⊆ σk ∪ {p} is a simplex in VR (X) ∨ VR (Y ). Therefore, f is simplicial.

Consider the following (non-commutative) diagram:

VRr(X) ∨ VRr(Y ) i VRr(X ∨ Y )

ϕ ψ f j BVRr(X) ∨ BVRr(Y ) BVRr(X ∨ Y ).

Here, i, j are inclusions and ϕ, ψ are maps deﬁned by σ 7→ min(σ). This diagram is non-

commutative only because i ◦ f is not equal to ψ. All other commutativity relations hold.

Let us show that i ◦ f is contiguous to ψ, hence the non-commutativity disappears when we

pass to homology.

r As above, take a simplex S = (σ1 ⊂ σ2 ⊂ · · · ⊂ σn) in B VR (X ∨ Y ) . Without loss of

generality, assume that k is the maximal integer such that for i ≤ k, σi ⊆ X and furthermore

assume that for i > k, σi is neither contained in X nor in Y . If k = n, then S is in the image

of j, therefore i ◦ f(S) = ψ(S). If k < n, then we have

r i ◦ f(S) ∪ ψ(S) ⊆ σn ∪ {p} ∈ VR (X ∨ Y ).

167 This shows the contiguity. Now we have the following commutative diagram:

r r i∗ r H∗ VR (X) ∨ VR (Y ) H∗ VR (X ∨ Y )

ϕ∗ ψ∗ f∗ r r j∗ r H∗ B VR (X) ∨ B VR (Y ) H∗ B VR (X ∨ Y ) .

Now, the induced map f∗ is surjective since ϕ∗ is surjective and f∗ is injective since ψ∗ is

−1 injective. Hence f∗ is an isomorphism, which implies that i∗ = ψ∗ ◦ f∗ is an isomorphism.

Now, we can start the proof of Proposition 8.4.4.

Proof of Proposition 8.4.4. Recall that given two simplicial complexes S,T ,H≥1(S ∨ T ) is L naturally isomorphic to H≥1(S) H≥1(T ) (see [45, Corollary 2.25]). The result then follows

from Lemma 8.4.5 and Proposition 8.4.7.

F 8.5 Classiﬁcation of Filtered Simplicial Complexes via dI

The idea of removing vertices from a simplicial complex one by one without changing its homotopy type to get a minimal core determined up to isomorphism is used in [58, 74, 5, 7].

Here we show that a similar result holds for ﬁltered complexes and in this way we prove

Theorem 8.1.3.

Definition 8.5.1 (Simple filtered simplicial complex). A filtered simplicial complex X∗ is

called simple if δ(X∗) > 0.

Lemma 8.5.1 (Non-identity morphisms). Every non-identity morphism f : X∗ → X∗ has

∞ ∗ codeg (f, idX∗ ) ≥ δ(X ).

168 ∞ Proof. Let idX∗ = f0, f1, . . . , fn = f be a family of morphisms realizing δ := codeg (f, idX∗ ).

Without loss of generality, we can assume that f1 is non-identity. Note that codeg(f1, idX∗ ) ≤

δ. Let v be a vertex such that w := f1(v) diﬀerent from v. Now, we have

DX (α ∪ {w}) ≤ DX (α ∪ {v}) ∪ (f1(α) ∪ {w})

≤ DX (α ∪ {v}) + δ.

Since α was arbitrary,

∞ ∗ codeg (f, idX∗ ) = δ ≥ δX (v, w) ≥ δ(X ).

Proposition 8.5.2. Let X∗ and Y ∗ be simple ﬁltered simplicial complexes such that for some r ≥ 0,

min δ(X∗), δ(Y ∗) > r.

F ∗ ∗ ∗ ∗ F ∗ ∗ If dI (X ,Y ) ≤ r/2, then 2 dGH(X ,Y ) = dI (X ,Y ). Furthermore, in this case there exists an invertible morphism f : X∗ → Y ∗ with inverse g : Y ∗ → X∗ such that the value above is equal to max(deg(f), deg(g)).

∗ ∗ F ∗ ∗ Proof. By Theorem 8.1.1, we already know that 2dGH(X ,Y ) ≥ dI (X ,Y ). Let us show

∗ ∗ F ∗ ∗ that 2dGH(X ,Y ) ≤ dI (X ,Y ). Let f : X∗ → Y ∗, g : Y ∗ → X∗ be morphisms realizing the interleaving distance

F ∗ ∗ := dI (X ,Y ). Then,

∞ ∗ codeg (g ◦ f, idX∗ ) ≤ 2 ≤ r < δ(X ),

hence by Lemma 8.5.1 g ◦ f = idX∗ . Similarly f ◦ g = idY ∗ . Note that this implies that

= max(deg(f), deg(g)). If we deﬁne R as the graph of f, then R is a correspondence between the vertex sets of X∗ and Y ∗. It is enough to show that dis(R) ≤ .

169 Let β be a non-empty subset of R. Let us denote the projection maps from R to the

∗ ∗ vertex sets of X and Y by pX and pY respectively. Let α := pX (β). Since R is the graph

of f, pY (β) = f(α). Now we have,

DY ∗ (pY (β)) − DX∗ (pX (β)) = DY ∗ (f(α)) − DX∗ (α)

≤ deg(f) ≤ , and

DX∗ (pX (β)) − DY ∗ (pY (β)) = DX∗ (α) − DY ∗ (f(α))

= DX∗ (g(f(α))) − DY ∗ (f(α))

≤ deg(g) ≤ ,

hence

|DX∗ (pX (β)) − DY ∗ (pY (β))| ≤ .

Since β was arbitrary, we have

∗ ∗ F ∗ ∗ 2dGH(X ,Y ) ≤ dis(R) ≤ = dI (X ,Y ).

Proof of Theorem 8.1.3. Existence: By Proposition 8.1.2, by removing v such that δX (v) = 0 one by one, we get a simple ﬁltered simplicial complex C∗ such that X∗ is equivalent to C∗,

F ∗ ∗ ∗ i.e. dI (X ,C ) = 0. Note that for a ﬁltered simplicial complex P with a single vertex,

∗ ∗ ∗ δX (P ) = ∞ hence it is simple. Since C is obtained from X by removing vertices, it is a full subcomplex.

Uniqueness: Assume C∗,T ∗ are simple ﬁltered simplicial complexes equivalent to X∗. Then,

F by the triangle inequality for dI they are equivalent to each other. Hence by Proposition

170 8.5.2, taking r = 0, we see that C∗,T ∗ are isomorphic, since the map f becomes a size preserving bijection as both f and its inverse has degree 0.

Remark 8.5.3 (Cores and isomorphism). As it is explained in the proof above, we obtain the

∗ core of X by removing vertices v with δX (v) = 0 one by one. Since the core is determined

up to isomorphism, the order in which we remove the points does not matter, in any case we

remove the same number of points and although we may reach diﬀerent subcomplexes, they

will be necessarily isomorphic.

∗ F ∗ ∗ ∗ Theorem 8.1.3 implies the following. Let C(X) = {C | dI (X ,C ) = 0}, that is, C(X ) consists of all the ﬁltered simplicial complexes equivalent to X∗. Let m(X∗) be the minimal

possible cardinality over all vertex sets of elements in C(X∗).

Corollary 8.5.4 (The core is minimal). The vertex set of the core C(X∗) has minimal

cardinality m(X∗).

Proof. Let C∗ ∈ C(X∗) be such that its vertex sets has minimal cardinality m(X∗). It follows

that C∗ is simple for otherwise, according to Proposition 8.1.2, we would be able to reduce

F its size. Then, by the triangle inequality for dI (Proposition 8.2.11) and Theorem 8.1.3, the distance between C(X∗) and C∗ is also zero. But since both C∗ and C(X∗) are simple,

Theorem 8.1.3 implies that they have to be isomorphic. In particular, their vertex sets ought

to have the same cardinality.

F 8.6 An Example where dI dGH

F We now give a generalization of Example 8.2.12 which shows that dI can be much smaller than dGH.

171 X0 X1 X2 X* w w wv v

0 1 2 X w,1 X w,1 X w,1 * v X W,1 v0 0

w w v w v

F Figure 8.4: These two ﬁltered simplicial complexes are at 0 dI -distance while they are at 1 Gromov-Hausdorﬀ distance at least 2 .

∗ Let X be a ﬁltered simplicial complex with the size function DX . Given a vertex w and

∗ a real number r ≥ 0, we deﬁne a single vertex extension Xw,r as follows. The underlying

∗ ˜ vertex set is the vertex set of X plus a new vertex v0. We deﬁne a size function D on this ˜ new vertex set as follows. We set D({v0}) := DX ({w}) + r and for a nonempty subset α of the vertex set of X∗, we set

˜ ˜ D(α) := DX (α), D(α ∪ {v0}) := DX (α ∪ {w}) + r

(see Figure 8.4). Let us show that D˜ is monotonic with respect to inclusion. Let α be

∗ ˜ ˜ non-empty subset of the vertex set of Xr,w. If v0 ∈ α, then D(α ∪ {v0}) = D(α). If v0 ∈/ α, then ˜ ˜ D(α ∪ {v0}) = DX (α ∪ {w}) + r ≥ DX (α) = D(α).

˜ ˜ Hence, in any case D(α ∪ {v0}) ≥ D(α). Now let α ⊆ β. If v0 ∈/ β, then

˜ ˜ D(β) = DX (β) ≥ DX (α) = D(α).

If v0 ∈ β, then

˜ ˜ ˜ D(β) = DX (β ∪ {w} − {v0}) + r ≥ DX (α ∪ {w} − {v0}) + r = D(α ∪ {v0}) ≥ D(α).

172 ˜ ∗ ∗ Hence D is a size funtion and Xw,r is a ﬁltered simplicial complex. Note that X is a full

∗ subcomplex of Xw,r obtained by removing the vertex v0. Let us show that δXw,r (v0) = 0. For

∗ any non-empty subset α of the vertex set of Xw,r, we have

˜ ˜ ˜ D(α ∪ {w}) ≤ D(α ∪ {v0, w}) = DX (α − {v0} ∪ {w}) + r = D(α ∪ {v0})

F ∗ ∗ Hence δXw,r (v0, w) = 0, so δw,r(v0) = 0. By Proposition 8.1.2, dI (Xw,r,X ) = 0.

∗ ∗ Now let us show that dGH(Xw,r,X ) ≥ r/2. Let (Z, p, p˜) be any tripod between the

˜ ∗ ∗ vertex sets V, V of X ,Xw,r. Then

˜ ˜ ˜ dis(Z) ≥ D(˜p(Z)) − DX (p(Z)) = D(V ) − DX (V ) = r.

∗ ∗ Since Z was arbitrary, dGH(Xw,r,X ) ≥ r/2. Therefore, if we take r 0, then

F ∗ ∗ ∗ ∗ dI (Xw,r,X ) = 0 r/2 ≤ dGH(Xw,r,X ).

∗ Remark 8.6.1. Recall ∆n from Example 8.2.12, with the vertex set {0, . . . , n} and the

∗ ∗ size function given by maximum. Note that ∆n+1 = (∆n)w=n,r=1. This also shows that

F ∗ ∗ dI (∆m, ∆n) = 0.

8.7 Chain Construction

r By a simplicial family we mean a collection (S )r≥0 of ﬁnite simplicial complexes with

r s simplicial maps fr,s : S → S for 0 ≤ r ≤ s with the regular commutativity conditions.

Here, we only consider constructible simplicial families (i.e. changes happen at finitely many indices). Note that such families also induce persistence modules by applying homology. Here we introduce a construction which we call the chain construction which takes a simplicial family as an input and returns a filtered simplicial complex. The advantage of such a construction can be explained as follows. There are efficient tools for calculating persistent

173 homology of ﬁltered simplicial complexes which does not work for simplicial families who

are not ﬁltrations. Hence, our construction enables one to use the persistence calculation

tools of simplicial ﬁltrations for arbitrary simplicial families. Another advantage is that since

ﬁltrations are metric objects (recall the size function), metric methods, including the ones

we cover in this chapter, can be used to analyze arbitrary simplicial families.

Deﬁnition 8.7.1 (Chains of vertices). Let S∗ be a simplicial family. A chain of vertices is a family

(vt)t∈I⊆R

r such that for each r and s in I with t ≤ s, vr is a vertex of S and fr,s(vr) = vs.

A chain (vt)t∈I is called a subchain of a chain (wt)t∈J if I ⊆ J and for r ∈ I, vr = wr.A

maximal chain of vertices in S∗ is a chain which is not a strict subchain of any other chain.

∗ We denote the set of all maximal chains of S by VS.

Remark 8.7.1. Since S∗ is assumed to be constructible, every maximal chain in it has a

minimal element and hence has the form (vt)t≥a.

Remark 8.7.2. If S∗ is a ﬁltered chain complex with the vertex set V , then V can be

naturally identiﬁed with VS by sending a chain to its minimal element.

Deﬁnition 8.7.2 (Chain construction). Let S∗ be a simplicial family. Deﬁne Chain(S∗) to

be the simplicial ﬁltration with set of vertices is VS and its size function is deﬁned as follows:

∗ ∗ r r r D(v0, . . . , vn) := inf{r :[v0, . . . , vn] ∈ S }

A morphism of simplicial families S∗, T ∗ is a collection of simplicial maps f r : Sr → T r

commuting with the structure maps. The main theorem of this section is the following.

174 Theorem 8.7.3. Let S∗ be a simplicial family. There is a morphism from Chain(S∗) to S∗ which is a homotopy equivalence at each given index. In particular, S∗ and Chain(S∗) have isometric persistent homology.

s ∗ s s Proof. Given s > 0 Deﬁne ψ : Chain(S ) → S by (v∗) 7→ vs. The fact that this is a simplicial map follows from the deﬁnition of the size function of the chain construction.

∗ ψ is a morphism of simplicial families since given a maximal chain v∗, then vs = fr,s(vr) whenever the both sides are well deﬁned. Let us show that φs is a homotopy equivalence.

Let v1, . . . , vn be the vertex set of Ss. By the constructibility of S∗, we can choose

1 n i i i i maximal chains w∗, . . . , w∗ such that ws = v for each i = 1, . . . , n. The function v 7→ w∗

s ∗ s s induces a simplicial map ϕ : S → Chain(S ) . Note that ψ ◦ ϕ = idSs . It is enough to

s 1 k show that ϕ ◦ ψ is contiguous to the identity map. Let σ = (v∗, . . . , v∗ ) be a simplex in

∗ s j α(j) Chain(S ) . Let α : {1, . . . , k} → {1, . . . , n} be the function such that vs = v . Note that we have

s 1 k α(1) α(k) ϕ ◦ ψ (σ) ∪ σ = (v∗, . . . , v∗ , w∗ , . . . , w∗ ).

This simplex has size less than or equal to s since

1 k α(1) α(k) 1 k (vs , . . . , vs , ws , . . . , ws ) = (vs , . . . , vs ) which is a simplex in Ss. This completes the proof.

175 Chapter 9: Future work

The work I have done in metric graph approximations and persistence has led me to consider the following problems/perspectives:

• I believe that metric graph embeddings of ﬁnite metric spaces can be studied with the tools and the approach of our paper [61]. I am currently exploring this possibility.

• Another question I am planning to study is ﬁnding higher dimensional analogues of the graph approximation results in our paper [61]. This question leads us to search of the following families of spaces and constructions: What should a geodesic space be replaced with for analogous results to hold? What should metric graphs be replaced with as approximating objects? What type of modiﬁcation of the Reeb construction should be used?

• Whereas the Reeb graph with respect to a distance function to a reference point helps a great deal in understanding graph approximations, it is difficult to guarantee its existence as a metric graph in the general setting of geodesic spaces. Thus, I would like to find local metric and topological conditions on a geodesic space X to guarantee the existence of Xp as a finite metric graph.

• There is another way of looking at the graph approximation problem, initiated by

Gromov [44], which considers surjections onto graphs and minimizes the maximal diameter of the ﬁbers of this map. How would our results in [61] change under these approach?

176 • In [46], Haussmann shows that given a compact Riemannian manifold X, for small enough, the Vietoris-Rips complex VR(M) is homotopy equivalent to M. I would like to

ﬁnd necessary conditions for a compact geodesic space to satisfy a similar statement.

177 Bibliography

[1] Micha l Adamaszek and Henry Adams. The vietoris–rips complexes of a circle. Paciﬁc Journal of Mathematics, 290(1):1–40, 2017.

[2] Micha lAdamaszek, Henry Adams, and Florian Frick. Metric reconstruction via optimal transport. SIAM Journal on Applied Algebra and Geometry, 2(4):597–619, 2018.

[3] Michal Adamaszek, Henry Adams, Ellen Gasparovic, Maria Gommel, Emilie Purvine, Radmila Sazdanovic, Bei Wang, Yusu Wang, and Lori Ziegelmeier. Vietoris-rips and cech complexes of metric gluings. arXiv preprint arXiv:1712.06224, 2017.

[4] Henry Adams and Joshua Mirth. Metric thickenings of euclidean submanifolds. Topology and its Applications, 254:69–84, 2019.

[5] Dominique Attali, Andr´eLieutier, and David Salinas. Eﬃcient data structure for representing and simplifying simplicial complexes in high dimensions. International Journal of Computational Geometry & Applications, 22(04):279–303, 2012.

[6] Mich`eleAudin and Mihai Damian. Morse theory and Floer homology. Springer, 2014.

[7] Jonathan Ariel Barmak and Elias Gabriel Minian. Strong homotopy types, nerves and collapses. Discrete & Computational Geometry, 47(2):301–328, 2012.

[8] Ulrich Bauer, Xiaoyin Ge, and Yusu Wang. Measuring distance between reeb graphs. In Proceedings of the thirtieth annual symposium on Computational geometry, page 464. ACM, 2014.

[9] Ulrich Bauer, Claudia Landi, and Facundo M´emoli. The reeb graph edit distance is universal. arXiv preprint arXiv:1801.01866, 2018.

[10] Ulrich Bauer and Michael Lesnick. Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the thirtieth annual symposium on Computational geometry, page 355. ACM, 2014.

[11] Ulrich Bauer, Elizabeth Munch, and Yusu Wang. Strong equivalence of the interleaving and functional distortion metrics for reeb graphs. arXiv preprint arXiv:1412.6646, 2014.

178 [12] Andrew J Blumberg and Michael Lesnick. Universality of the homotopy interleaving distance. arXiv preprint arXiv:1705.01690, 2017.

[13] Andrew J Blumberg and Michael A Mandell. Quantitative homotopy theory in topological data analysis. Foundations of Computational Mathematics, 13(6):885–911, 2013.

[14] Magnus Bakke Botnan and Gard Spreemann. Approximating persistent homology in euclidean space through collapses. Applicable Algebra in Engineering, Communication and Computing, 26(1-2):73–101, 2015.

[15] Brian H Bowditch. Notes on gromov’s hyperbolicity criterion for path-metric spaces. In Group theory from a geometrical viewpoint. Institut des Hautes Etudes Scientiﬁques [IHES], 1991.

[16] Glen E Bredon. Topology and geometry, volume 139. Springer Science & Business Media, 2013.

[17] Peter Bubenik and Jonathan A Scott. Categoriﬁcation of persistent homology. Discrete & Computational Geometry, 51(3):600–627, 2014.

[18] Dmitri Burago, Yuri Burago, and Sergei Ivanov. A course in metric geometry, volume 33. American Mathematical Soc., 2001.

[19] Gunnar Carlsson. Topology and data. Bulletin of the American Mathematical Society, 46(2):255–308, 2009.

[20] Hamish Carr, Jack Snoeyink, and Ulrike Axen. Computing contour trees in all dimensions. Computational Geometry, 24(2):75–94, 2003.

[21] Nicholas J Cavanna, Mahmoodreza Jahanseir, and Donald R Sheehy. A geometric perspective on sparse ﬁltrations. arXiv preprint arXiv:1506.03797, 2015.

[22] Frédéric Chazal, David Cohen-Steiner, Leonidas J Guibas, Facundo Mémoli, and Steve Y Oudot. Gromov-hausdorff stable signatures for shapes using persistence. In Computer Graphics Forum, volume 28-5, pages 1393–1403. Wiley Online Library, 2009.

[23] Fr´ed´ericChazal, Vin De Silva, Marc Glisse, and Steve Oudot. The structure and stability of persistence modules. Springer, 2016.

[24] Fr´ed´ericChazal, Vin De Silva, and Steve Oudot. Persistence stability for geometric complexes. Geometriae Dedicata, 173(1):193–214, 2014.

[25] FrédéricChazal, Ruqi Huang, and Jian Sun. Gromov–hausdorff approximation of filamentary structures using reeb-type graphs. Discrete & Computational Geometry, 53(3):621–649, 2015.

179 [26] Samir Chowdhury and Facundo M´emoli.Persistent homology of asymmetric networks: An approach based on dowker ﬁltrations. arXiv preprint arXiv:1608.05432, 2016.

[27] Samir Chowdhury, Facundo M´emoli,and Zane T Smith. Improved error bounds for tree representations of metric spaces. In Advances in Neural Information Processing Systems, pages 2838–2846, 2016.

[28] Michel Coonaert, Thomas Delzant, and Athanase Papadopoulos. Géométrieet théorie des groupes: les groupes hyperboliques de Gromov. Springer, 1990.

[29] William Crawley-Boevey. Decomposition of pointwise ﬁnite-dimensional persistence modules. Journal of Algebra and its Applications, 14(05):1550066, 2015.

[30] Vin De Silva, Elizabeth Munch, and Amit Patel. Categoriﬁed reeb graphs. Discrete & Computational Geometry, 55(4):854–906, 2016.

[31] Tamal K Dey, Fengtao Fan, and Yusu Wang. Computing topological persistence for simplicial maps. In Proceedings of the thirtieth annual symposium on Computational geometry, page 345. ACM, 2014.

[32] Tamal K Dey, Facundo M´emoli,and Yusu Wang. Multiscale mapper: Topological summarization via codomain covers. In Proceedings of the twenty-seventh annual acm- siam symposium on discrete algorithms, pages 997–1013. SIAM, 2016.

[33] Tamal K Dey, Dayu Shi, and Yusu Wang. Simba: An eﬃcient tool for approximating rips-ﬁltration persistence via simplicial batch collapse. Journal of Experimental Algo- rithmics (JEA), 24(1):1–5, 2019.

[34] Andreas Dress, Katharina T Huber, Jacobus Koolen, Vincent Moulton, and Andreas Spillner. Basic phylogenetic combinatorics. Cambridge University Press, 2012.

[35] Andreas WM Dress. Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Ad- vances in Mathematics, 53(3):321–402, 1984.

[36] Richard O Duda, Peter E Hart, and David G Stork. Pattern classiﬁcation. John Wiley & Sons, 2012.

[37] Herbert Edelsbrunner and John Harer. Computational topology: an introduction. Amer- ican Mathematical Soc., 2010.

[38] Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simpliﬁcation. In Proceedings 41st Annual Symposium on Foundations of Computer Science, pages 454–463. IEEE, 2000.

180 [39] Herbert Edelsbrunner and Dmitriy Morozov. Persistent homology: theory and practice. Technical report, Lawrence Berkeley National Lab.(LBNL), Berkeley, CA (United States), 2012.

[40] Herbert Federer. Geometric measure theory. Springer, 2014.

[41] Robert E Greene and Hung-Hsi Wu. c∞ approximations of convex, subharmonic, and plurisubharmonic functions. In Annales Scientiﬁques de l’Ecole´ Normale Sup´erieure, volume 12-1, pages 47–84, 1979.

[42] Mikhael Gromov. Hyperbolic groups. In Essays in group theory, pages 75–263. Springer, 1987.

[43] Mikhael Gromov et al. Filling riemannian manifolds. Journal of Diﬀerential Geometry, 18(1):1–147, 1983.

[44] Mikhail Gromov. Width and related invariants of riemannian manifolds. Ast´erisque, 163(164):93–109, 1988.

[45] Allen Hatcher. Algebraic topology. Cambridge University Press, 2005.

[46] Jean-Claude Hausmann et al. On the vietoris-rips complexes and a cohomology theory for metric spaces. Ann. Math. Studies, 138:175–188, 1995.

[47] Nicholas Jardine and Robin Sibson. Mathematical taxonomy. London etc.: John Wiley, 1971.

[48] Mikhail Katz. Diameter-extremal subsets of spheres. Discrete & Computational Geom- etry, 4(2):117–137, 1989.

[49] Mikhail Katz. On neighborhoods of the kuratowski imbedding beyond the ﬁrst ex- tremum of the diameter functional. Fundamenta Mathematicae, 137(3):161–175, 1991.

[50] Mikhail Katz. The rational ﬁlling radius of complex projective space. Topology and its Applications, 42(3):201–215, 1991.

[51] Mikhail Katz et al. The ﬁlling radius of two-point homogeneous spaces. Journal of Diﬀerential Geometry, 18(3):505–511, 1983.

[52] Michael Kerber and R Sharathkumar. Approximate ˇcech complex in low and high dimensions. In International Symposium on Algorithms and Computation, pages 666– 676. Springer, 2013.

[53] Urs Lang. Injective hulls of certain discrete metric spaces and groups. Journal of Topology and Analysis, 5(03):297–331, 2013.

181 [54] Janko Latschev. Vietoris-rips complexes of metric spaces near a closed riemannian manifold. Archiv der Mathematik, 77(6):522–528, 2001.

[55] Solomon Lefschetz. Algebraic topology, volume 27. American Mathematical Soc., 1942.

[56] Michael Lesnick. The theory of the interleaving distance on multidimensional persistence modules. Foundations of Computational Mathematics, 15(3):613–650, 2015.

[57] Luofei Liu. The mapping properties of ﬁlling radius and packing radius and their applications. Diﬀerential Geometry and its Applications, 22(1):69–79, 2005.

[58] Jiˇr´ıMatouˇsek. Lc reductions yield isomorphic simplicial complexes. Contributions to Discrete Mathematics, 3(2), 2008.

[59] Facundo M´emoli. A distance between ﬁltered spaces via tripods. arXiv preprint arXiv:1704.03965, 2017.

[60] Facundo M´emoliand Osman Berat Okutan. The distortion of the reeb quotient map on riemannian manifolds. arXiv preprint arXiv:1801.01562, 2018.

[61] Facundo M´emoliand Osman Berat Okutan. Metric graph approximations of geodesic spaces. arXiv preprint arXiv:1809.05566, 2018.

[62] Facundo Mémoliand Osman Berat Okutan. Quantitative simplification of filtered simplicial complexes. arXiv preprint arXiv:1801.02812, 2018.

[63] Facundo M´emoliand Osman Berat Okutan. Reeb posets and tree approximations. arXiv preprint arXiv:1801.01555, 2018.

[64] Facundo M´emoli,Anastasios Sidiropoulos, and Kritika Singhal. Sketching and clustering metric measure spaces. arXiv preprint arXiv:1801.00551, 2018.

[65] John Milnor and James D Stasheﬀ. Characteristic Classes.(AM-76), volume 76. Prince- ton university press, 2016.

[66] Dmitriy Morozov, Kenes Beketayev, and Gunther Weber. Interleaving distance between merge trees. Discrete and Computational Geometry, 49(22-45):52, 2013.

[67] Jos´e Perea. Birs 2018 lecture. http://www.birs.ca/events/2018/5-day- workshops/18w5140/videos/watch/201808081206-Perea.html, 2018.

[68] Georges Reeb. Sur les points singuliers d’une forme de pfaﬀ completement integrable ou d’une fonction numerique [on the singular points of a completely integrable pfaﬀ form or of a numerical function]. Comptes Rendus Acad. Sciences Paris, 222:847–849, 1946.

182 [69] Vanessa Robins. Towards computing homology from ﬁnite approximations. In Topology proceedings, volume 24-1, pages 503–532, 1999.

[70] Charles Semple, Mike A Steel, Richard A Caplan, Mike Steel, et al. Phylogenetics, volume 24. Oxford University Press on Demand, 2003.

[71] Donald R Sheehy. Linear-size approximations to the vietoris–rips ﬁltration. Discrete & Computational Geometry, 49(4):778–796, 2013.

[72] Zigaˇ Virk. 1-dimensional intrinsic persistence of geodesic spaces. Journal of Topology and Analysis, pages 1–39, 2018.

[73] Nikita A Zinov’ev. On graph approximations of surfaces with small area. St Petersburg Mathematical Journal, 17(3):435–442, 2006.

[74] Afra Zomorodian. The tidy set: a minimal simplicial set for computing homology of clique complexes. In Proceedings of the twenty-sixth annual symposium on Computa- tional geometry, pages 257–266. ACM, 2010.

[75] Afra Zomorodian and Gunnar Carlsson. Computing persistent homology. Discrete & Computational Geometry, 33(2):249–274, 2005.

183