# Persistence Diagram Bundles:
A Multidimensional Generalization of Vineyards

Abigail Hickok

###### Abstract.

I introduce the concept of a persistence diagram (PD) bundle, which is the
space of PDs for a fibered filtration function (a set
$\\{f_{p}:\mathcal{K}^{p}\to\mathbb{R}\\}_{p\in B}$ of filtrations that is
parameterized by a topological space $B$). Special cases include vineyards,
the persistent homology transform, and fibered barcodes for multiparameter
persistence modules. I prove that if $B$ is a smooth compact manifold, then
for a generic fibered filtration function, $B$ is stratified such that within
each stratum $Y\subseteq B$, there is a single PD “template” (a list of
“birth” and “death” simplices) that can be used to obtain the PD for the
filtration $f_{p}$ for any $p\in Y$. If $B$ is compact, then there are
finitely many strata, so the PD bundle for a generic fibered filtration on $B$
is determined by the persistent homology at finitely many points in $B$. I
also show that not every local section can be extended to a global section (a
continuous map $s$ from $B$ to the total space $E$ of PDs such that
$s(p)\in\text{PD}(f_{p})$ for all $p\in B$). Consequently, a PD bundle is not
necessarily the union of “vines” $\gamma:B\to E$; this is unlike a vineyard.
When there is a stratification as described above, I construct a cellular
sheaf that stores sufficient data to construct sections and determine whether
a given local section can be extended to a global section.

##  1\. Introduction

In topological data analysis (TDA), our aim is to understand the global shape
of a data set. Often, the data set takes the form of a collection of points in
$\mathbb{R}^{n}$, called a point cloud, and we hope to analyze the topology of
a lower-dimensional space that the points lie on. TDA has found applications
in a variety of fields, such as biology [29], neuroscience [12], and chemistry
[27].

We use _persistent homology_ (PH), a tool from algebraic topology [21]. The
first step of persistent homology is to construct a filtered complex from our
data; a _filtered complex_ is a nested sequence

(1)
$\mathcal{K}_{r_{0}}\subseteq\mathcal{K}_{r_{1}}\subseteq\cdots\subseteq\mathcal{K}_{r_{n}}\subseteq\cdots$

of simplicial complexes. For example, one of the standard ways to build a
filtered complex from point cloud data is to construct the Vietoris–Rips
filtered complex. At filtration-parameter value $r$, the Vietoris–Rips complex
$\mathcal{K}_{r}$ includes a simplex for every subset of points within $r$ of
each other. In persistent homology, one studies how the topology of
$\mathcal{K}_{r}$ changes as the filtration parameter-value $r$ increases. As
$r$ grows, new homology classes (which represent “holes” in the data) are born
and old homology classes die. One way of summarizing this information is a
_persistence diagram_ : a multiset of points in the extended plane
$\overline{\mathbb{R}}^{2}$. If there is a homology class that is born at
filtration-parameter value $b$ and dies at filtration-parameter value $d$, the
persistence diagram contains the point $(b,d)$.

Developing new methods for analyzing how the topology of a data set changes as
multiple parameters vary is a very active area of research [7]. For example,
if a point cloud evolves over time (i.e., it is a dynamic metric space), then
one maybe interested in using time as a second parameter, in addition to the
filtration parameter $r$. Common examples of time-evolving point clouds
include swarming or flocking animals whose positions and/or velocities are
represented by points ([15, 36, 25]). In such cases, one can obtain a filtered
complex
$\mathcal{K}_{r_{0}}^{t}\subseteq\mathcal{K}_{r_{1}}^{t}\subseteq\cdots\subseteq
K_{r_{n}}^{t}$ at every time $t$ by constructing, e.g., the Vietoris–Rips
filtered complex for the point cloud at time $t$. It is also common to use the
density of the point cloud as a parameter ([30, 10, 6]). Many other parameters
can also vary in the topological analysis of point clouds or other types of
data sets.

One can use a _vineyard_ [14] to study a 1-parameter family of filtrations
$\\{\mathcal{K}_{r_{0}}^{t}\subseteq\mathcal{K}_{r_{1}}^{t}\subseteq\cdots\mathcal{K}_{r_{n}}^{t}\\}_{t\in\mathbb{R}}$
such as that obtained from a time-varying point cloud. At each
$t\in\mathbb{R}$, one can compute the PH of the filtration
$\mathcal{K}_{r_{0}}^{t}\subseteq\mathcal{K}_{r_{1}}^{t}\subseteq\cdots\subseteq\mathcal{K}_{r_{n}}^{t}$
and obtain a persistence diagram PD$(t)$. A vineyard is visualized as the
continuously-varying “stack of PDs”
$\\{\textnormal{PD}(t)\\}_{t\in\mathbb{R}}$. See Figure 1 for an illustration.
As $t\in\mathbb{R}$ varies, the points in the PDs trace out curves (“vines”)
in $\mathbb{R}^{3}$. Each vine corresponds to a homology class (i.e., one of
the holes in the data), and shows how the persistence of that homology class
changes with time (or, more generally, as some other parameter varies).

Figure 1. An example of a vineyard. There is a persistence diagram for each
time $t$. Each curve is a vine in the vineyard. (This figure is a slightly
modified version of a figure that appeared originally in [26].)

However, one cannot use a vineyard for a set of filtrations that is
parameterized by a space that is not a subset of $\mathbb{R}$. For example,
suppose that we have a time-varying point cloud whose dynamics depend on some
system-parameter values $\mu_{1},\ldots,\mu_{m}\in\mathbb{R}$. Many such
systems exist. For example, the D’Orsogna model is a multi-agent dynamical
system that models attractive and repulsive interactions between particles
[11]. Each particle is represented by a point in a point cloud. In certain
parameter regimes, there are interesting topological features, such as mills
or double mills [31]. For each time $t\in\mathbb{R}$ and for each
$\mu_{1},\ldots,\mu_{m}\in\mathbb{R}$, one can obtain a filtered complex

(2)
$\mathcal{K}_{r_{0}}^{t,\mu_{1},\ldots,\mu_{m}}\subseteq\mathcal{K}_{r_{1}}^{t,\mu_{1},\ldots,\mu_{m}}\subseteq\cdots\subseteq\mathcal{K}_{r_{n}}^{t,\mu_{1},\ldots,\mu_{m}}$

by constructing, e.g., the Vietoris–Rips filtered complex for the point cloud
at time $t$ at system-parameter values $\mu_{1},\ldots,\mu_{m}$. A
parameterized set of filtered complexes like the one in (2) cannot be studied
using a vineyard for the simple reason that there are too many parameters.

Such a parameterized set of filtered complexes cannot be studied using
multiparameter PH [8], either. A _multifiltration_ is a set
$\\{\mathcal{K}_{\bm{u}}\\}_{\bm{u}\in\mathbb{R}^{n}}$ of simplicial complexes
such that $\mathcal{K}_{\bm{u}}\subseteq\mathcal{K}_{\bm{v}}$ whenever
$\bm{u}\leq\bm{v}$. _Multiparameter PH_ is the
$\mathbb{F}[x_{1},\ldots,x_{n}]$ module obtained by applying homology (over a
field $\mathbb{F}$) to a multifiltration. (For more details, see reference
[8].) The parameterized set of filtered complexes in (2) is not typically a
multifiltration because it is not necessarily the case that
$\mathcal{K}_{r_{i}}^{t,\mu_{1},\ldots,\mu_{m}}\not\subseteq\mathcal{K}_{r_{i}}^{t^{\prime},\mu_{1}^{\prime},\ldots,\mu_{m}^{\prime}}$
for all values of $t$, $t^{\prime}$, $\\{\mu_{i}\\}$, and
$\\{\mu_{i}^{\prime}\\}$. Not only is there not necessarily an inclusion
$\mathcal{K}_{r_{i}}^{t,\mu_{1},\ldots,\mu_{m}}\xhookrightarrow{}\mathcal{K}_{r_{i}}^{t^{\prime},\mu_{1}^{\prime},\ldots,\mu_{m}^{\prime}}$,
but there is not any given simplicial map
$\mathcal{K}_{r_{i}}^{t,\mu_{1},\ldots,\mu_{m}}\to\mathcal{K}_{r_{i}}^{t^{\prime},\mu_{1}^{\prime},\ldots,\mu_{m}^{\prime}}$.
Therefore, we cannot use multiparameter PH.

In what follows, we will work with a slightly different notion of filtered
complex than that of (1). A _filtration function_ is a function
$f:\mathcal{K}\to\mathbb{R}$, where $\mathcal{K}$ is a simplicial complex,
such that every sublevel set $\mathcal{K}_{r}:=\\{\sigma\in\mathcal{K}\mid
f(\sigma)\leq r\\}$ is a simplicial complex (i.e., $f(\tau)\leq f(\sigma)$ if
$\tau$ is a face of $\sigma$). For every $r\leq s$, we have that
$\mathcal{K}_{r}\subseteq\mathcal{K}_{s}$. A simplex $\sigma\in\mathcal{K}$
appears in the filtration at $r=f(\sigma)$. By setting
$\\{r_{i}\\}=\textnormal{Im}(f)$, where $r_{i}<r_{i+1}$, we obtain a nested
sequence as in (1). Conversely, given a nested sequence of simplicial
complexes, the associated filtration function is
$f(\sigma)=\min\\{r_{i}\mid\sigma\in\mathcal{K}_{r_{i}}\\}$, with
$\mathcal{K}=\bigcup_{i}\mathcal{K}_{r_{i}}$.

###  1.1. Contributions

I introduce the concept of a _persistence diagram (PD) bundle_ , in which PH
varies over an arbitrary “base space” $B$. A PD bundle gives a way of studying
a _fibered filtration function_ , which is a set
$\\{f_{p}:\mathcal{K}^{p}\to\mathbb{F}\\}_{p\in B}$ of functions such that
$f_{p}$ is a filtration of a simplicial complex $\mathcal{K}^{p}$. At each
$p\in B$, the sublevel sets of $f_{p}$ form a filtered complex. For example,
in (2), we have $B=\mathbb{R}^{n+1}$ and we obtain a fibered filtration
function
$\\{f_{t,\mu_{1},\ldots,\mu_{m}}:\mathcal{K}\to\mathbb{R}\\}_{(t,\mu_{1},\ldots,\mu_{m})\in\mathbb{R}^{m+1}}$
by defining $f_{t,\mu_{1},\ldots,\mu_{m}}$ to be the filtration function
associated with the filtered complex in (2). The associated PD bundle is the
space of persistence diagrams PD$(f_{p})$ as they vary with $p\in B$ (see
Definition 3.2). In the special case in which $B$ is an interval in
$\mathbb{R}$, a PD bundle is equivalent to a vineyard.

I prove that for “generic” fibered filtration functions (see Section 4.2), the
base space $B$ can be stratified in a way that makes PD bundles tractable to
compute and analyze. Theorem 4.15 says that for a “generic” fibered filtration
function on a smooth compact manifold $B$, the base space $B$ is stratified
such that within each stratum, there is a single PD “template” that can be
used to obtain $PD(f_{p})$ at any point $p$ in the stratum. Proposition 4.5
shows that _all_ “piecewise-linear” PD bundles (see Definition 3.4) have such
a stratification. The template is a list of (birth, death) simplex pairs, and
the diagram $\text{PD}(f_{p})$ is obtained by evaluating $f_{p}$ on each
simplex. In particular, when $B$ is a smooth compact manifold, the number of
strata is finite, so the PD bundle is determined by the PH at a finite number
of points in the base space.

I show that unlike vineyards, PD bundles do not necessarily decompose into a
union of “vines”. More precisely, there may not exist continuous maps
$\gamma_{1},\ldots,\gamma_{m}:B\to E$ such that

(3) $\text{PD}(f_{p})=\bigcup_{i=1}^{m}\gamma_{i}(p)$

for all $p\in B$. This is a consequence of Proposition 5.3, in which it is
shown that nontrivial global sections are not guaranteed to exist. That is,
given a point $z_{0}\in PD(f_{p_{0}})$ for some $p_{0}\in B$, it may not be
possible to extend $p_{0}\mapsto z_{0}$ to a continuous map $s:B\to
E:=\\{(p,z)\mid z\in PD(f_{p})\\}$ such that $s(p)\in\text{PD}(f_{p})$ for all
$p$. This behavior is a feature that gives PD bundles a richer mathematical
structure than vineyards.

For any fibered filtration with a stratification as described above (see
Theorem 4.15 and Proposition 4.5), I construct a “compatible cellular sheaf”
(see Section 6.2) that stores the data in the PD bundle. Rather than analyzing
the entire PD bundle, which consists of continuously varying PDs over the base
space $B$, we can analyze the cellular sheaf, which is discrete. For example,
in Proposition 6.4, I prove that an extension of $p_{0}\mapsto z_{0}$ to a
global section exists if a certain associated global section of the cellular
sheaf exists. A compatible cellular sheaf stores sufficient data to
reconstruct the associated PD bundle and analyze its sections.

Though not the focus of this paper, I also give a simple example of vineyard
instability in Appendix A.4. It is often quoted in the research literature
that “vineyards are unstable”; however, this “well-known fact” has been shared
only in private correspondence and, to the best of my knowledge, has never
been published. The example of vineyard instability is furnished from an
example in Proposition 5.3.

###  1.2. Related work

PD bundles are a generalization of vineyards, which were introduced in [14].
Two other important special cases of PD bundles are the fibered barcode of a
multiparameter persistence module [4] and the persistent homology transform
($B=S^{n}$) from shape analysis [2, 32]. I discuss the special case of fibered
barcodes in detail in Section 3.2.2; the base space $B$ is a subset of the
space of lines in $\mathbb{R}^{n}$. The persistent homology transform (PHT) is
defined for a constructible set $M\subseteq\mathbb{R}^{n+1}$. For any unit
vector $v\in S^{n}$, one defines the filtration $M_{r}^{v}=\\{x\in M\mid
x\cdot v\leq r\\}$ (i.e., the sublevel filtration of the height function with
respect to the direction $v$). PHT is the map that sends $v\in S^{n}$ to the
persistence diagram for the filtration $\\{M_{r}^{v}\\}_{r\in\mathbb{R}}$. The
significance of PHT is that it is a sufficient statistic for shapes in
$\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ [32]. Applications of PHT are numerous
and include protein docking [34], barley-seed shape analysis [3], and heel-
bone analysis in primates [32].

For PHT, Curry et al. [18] proved that the base space $S^{n}$ is stratified
such that the PHT of a shape $M$ is determined by the PH of
$\\{M_{r}^{v}\\}_{r\in\mathbb{R}}$ for finitely many directions $v\in S^{n}$
(one direction $v$ per stratum). This is related to the stratification given
by Theorem 4.15, in which I show that a “generic” PD bundle whose base space
$B$ is a compact smooth manifold (such as $S^{n}$) is similarly stratified and
thus determined by finitely many points in $B$ (one $p\in B$ per stratum). The
primary difference between the stratifications in [18] and Theorem 4.15 is
that in [18], each stratum is a subset in which the order of the vertices of a
triangulated shape $M$ (as ordered by the height function) is constant,
whereas in Theorem 4.15, each stratum is a subset in which the order of the
simplices (as ordered by the filtration function) is constant. The
stratification result of the present paper (Theorem 4.15) applies to general
PD bundles, while [18] applies only to PHT.

The stratification that we study in the present paper is used in [23] to
develop an algorithm for computing “piecewise-linear” PD bundles (see
Definition 3.4). The algorithm relies on the fact that for any piecewise-
linear PD bundle on a compact triangulated base space $B$, there are a finite
number of strata, so the PD bundle is determined by the PH at a finite number
of points in $B$.

The existence (or nonexistence) of nontrivial global sections in PD bundles is
related to the study of “monodromy” in fibered barcodes of multiparameter
persistence modules [9]. Cerri et al. [9] constructed an example in which
there is a path through the fibered barcode that loops around a “singularity”
(a PD in the fibered barcode for which there is a point in the PD with
multiplicity greater than one) and finishes in a different place than where it
starts.

###  1.3. Organization

This paper proceeds as follows. I review background on persistent homology in
Section 2. In Section 3, I give the definition of a PD bundle, with some
examples, and I compare PD bundles to multiparameter PH. In Section 4, I show
how to stratify the base space $B$ into strata in which the (birth, death)
simplex pairs are constant (see Theorem 4.15 and Proposition 4.5). I discuss
sections of PD bundles and the existence of monodromy in Section 5. I
construct a compatible cellular sheaf in Section 6.2. I conclude and discuss
possible directions for future research in Section 7. In Appendix A.4, I use
the example of monodromy from Section 5 to construct an example of vineyard
instability. In Appendix A.5, I provide technical details that are needed to
prove Theorem 4.15.

##  2\. Background

We begin by reviewing persistent homology (PH) and cellular sheaves. For a
more thorough treatment of PH, see [19, 28], and for more on cellular sheaves,
see [22, 16].

###  2.1. Filtrations

Consider a simplicial complex $\mathcal{K}$. A _filtration function_ on
$\mathcal{K}$ is a real-valued function $f:\mathcal{K}\to\mathbb{R}$ such that
if $\tau\in\mathcal{K}$ is a face of $\sigma\in K$, then $f(\tau)\leq
f(\sigma)$. The _filtration value_ of a simplex $\sigma\in\mathcal{K}$ is
$f(\sigma)$. The $r$-sublevel sets
$\mathcal{K}_{r}:=\\{\sigma\in\mathcal{K}\mid f(\sigma)\leq r\\}$ form a
_filtered complex_. The condition that $f(\tau)\leq f(\sigma)$ if
$\tau\subseteq\sigma$ guarantees that $\mathcal{K}_{r}$ is a simplicial
complex for all $r$. For all $s\leq r$, we have
$\mathcal{K}_{s}\subseteq\mathcal{K}_{r}$. The parameter $r$ is the
_filtration parameter_. For an example, see Figure 2.

(a) $\mathcal{K}_{0}$

(b) $\mathcal{K}_{1}$

(c) $\mathcal{K}_{2}$

(d) $\mathcal{K}_{3}$

(e) $\mathcal{K}_{4}$

Figure 2. An example of a filtration. The simplicial complex $\mathcal{K}_{i}$
has the associated filtration-parameter value $i$. (This figure appeared
originally in [24].)

For example, suppose that $X=\\{x_{i}\\}_{i=1}^{M}$ is a point cloud. Let
$\mathcal{K}$ be the simplicial complex that has a simplex $\sigma$ with
vertices $\\{x_{j}\\}_{j\in J}$ for all $J\subseteq\\{1,\ldots,M\\}$. The
_Vietoris–Rips filtration function_ is $f(\sigma)=\frac{1}{2}\max_{j,k\in
J}\\{\lVert x_{j}-x_{k}\rVert\\}$, where $\\{x_{j}\\}_{j\in J}$ are the
vertices of $\sigma$.

###  2.2. Persistent homology

Let $f:\mathcal{K}\to\mathbb{R}$ be a filtration function on a finite
simplicial complex $\mathcal{K}$, and let $\\{K_{r}\\}_{r\in\mathbb{R}}$ be
the associated filtered complex. Let $r_{1}<\cdots<r_{N}$ be the filtration
values of the simplices of $\mathcal{K}$. These are the critical points at
which simplices are added to the filtration. For all $s\in[r_{i},r_{i+1})$, we
have $\mathcal{K}_{s}=\mathcal{K}_{r_{i}}$.

For all $i\leq j$, the inclusion
$\iota^{i,j}:\mathcal{K}_{r_{i}}\hookrightarrow\mathcal{K}_{r_{j}}$ induces a
map $\iota^{i,j}_{*}:H_{*}(\mathcal{K}_{r_{i}},\mathbb{F})\to
H_{*}(\mathcal{K}_{r_{j}},\mathbb{F})$ on homology, where $\mathbb{F}$ is a
field that we set to $\mathbb{Z}/2\mathbb{Z}$ for the rest of this paper. The
_$q$ th-persistent homology_ (PH) is the pair

$\Big{(}\\{H_{q}(\mathcal{K}_{r_{i}},\mathbb{F})\\}_{1\leq i\leq
N}\,,\\{\iota_{*}^{i,j}\\}_{1\leq i\leq j\leq N}\Big{)}\,.$

The Fundamental Theorem of Persistent Homology yields compatible choices of
bases for the vector spaces $H_{q}(\mathcal{K}_{r_{i}},\mathbb{F})$, which we
use below in our definition of a persistence diagram.

The _$q$ th persistence diagram_ $PD_{q}(f)$ is a multiset of points in the
extended plane $\overline{\mathbb{R}}^{2}$ that summarizes the $q$th-
persistent homology. The PD contains the diagonal as well as a point for every
generator. We say that a generator $\gamma\in
H_{q}(\mathcal{K}_{r_{i}},\mathbb{F})$ is _born_ at $r_{i}$ if it is not in
the image of $\iota^{i,i-1}_{*}$. The homology class $\gamma$ subsequently
_dies_ at $r_{j}>r_{i}$ if $\iota_{*}^{i,j}(\gamma)=0$ and
$\iota_{*}^{i,j-1}(\gamma)\neq 0$. If $\iota_{*}^{i,j}(\gamma)\neq 0$ for all
$j>i$, then $\gamma$ never dies. For every generator, the PD contains the
point $(r_{i},r_{j})$ if the generator is born at $r_{i}$ and dies at $r_{j}$,
or else contains the point $(r_{i},\infty)$ if the generator is born at
$r_{i}$ and never dies.

###  2.3. Birth and death simplex pairs

Computing persistent homology can be reduced to computing the set of “birth”
and “death” simplices for the generating homology classes. Informally, a
_birth simplex_ $\sigma_{b}$ is a $q$-simplex that creates a new
$q$-dimensional homology class when it is added to the filtration and a _death
simplex_ is a $(q+1)$-simplex that destroys a $q$-dimensional homology class
when it is added to the filtration. For example, in Figure 2, the 1D PH has
one generator. Its birth simplex is the $1$-simplex $(0,3)$ and its death
simplex is the $2$-simplex $(0,2,3)$. For every pair $(\sigma_{b},\sigma_{d})$
of (birth, death) simplices, the persistence diagram contains the point
$(f(\sigma_{b}),f(\sigma_{d}))$. For every unpaired birth simplex
$\sigma_{b}$, the persistence diagram contains the point
$(f(\sigma_{b}),\infty)$.

In [20], Edelsbrunner and Harer presented an algorithm for computing the
(birth, death) simplex pairs of a filtration $f:\mathcal{K}\to\mathbb{R}$. Let
$\sigma_{1},\ldots,\sigma_{N}$ be the simplices of $\mathcal{K}$, indexed such
that $i<j$ if $\sigma_{i}$ is a proper face of $\sigma_{j}$.

######  Definition 2.1.

The _simplex order_ induced by $f$ is the strict partial order $\prec_{f}$ on
$\mathcal{K}$ such that $\sigma_{i}\prec_{f}\sigma_{j}$ if and only if
$f(\sigma_{i})<f(\sigma_{j})$.

If the simplex orders $\prec_{f_{1}},\prec_{f_{2}}$ induced by filtrations
$f_{1},f_{2}$ (respectively) are the same, then
$f_{1}(\sigma_{i})<f_{1}(\sigma_{j})$ if and only if
$f_{2}(\sigma_{i})<f_{2}(\sigma_{j})$ and
$f_{1}(\sigma_{i})=f_{1}(\sigma_{j})$ if and only if
$f_{2}(\sigma_{i})=f_{2}(\sigma_{j})$.

The algorithm of [20] requires a _compatible simplex indexing_.

######  Definition 2.2.

A _compatible simplex indexing_ is a function
$\text{idx}:\mathcal{K}\to\\{1,\ldots,N\\}$ such that
$\text{idx}(\sigma_{i})<\text{idx}(\sigma_{j})$ if
$\sigma_{i}\prec_{f}\sigma_{j}$ or $\sigma_{i}$ is a proper face of
$\sigma_{j}$. Because a compatible simplex indexing may not be unique, we fix
the _simplex indexing induced by $f$_ to be the unique function
$\text{idx}_{f}:\mathcal{K}\to\\{1,\ldots,N\\}$ such that
$\text{idx}_{f}(\sigma_{i})<\text{idx}_{f}(\sigma_{j})$ if either
$\sigma_{i}\prec_{f}\sigma_{j}$ or if $f(\sigma_{i})=f(\sigma_{j})$ and $i<j$.

The function $\text{idx}_{f}$ is a compatible simplex indexing because if
$\sigma_{i}$ is a proper face of $\sigma_{j}$, then $i<j$ and
$f(\sigma_{i})\leq f(\sigma_{j})$. The sequence
$\text{idx}_{f}^{-1}(1),\ldots,\text{idx}_{f}^{-1}(N)$ of simplices is ordered
by the value of $f$ on each simplex, with ties broken by the order of the
simplices in the sequence $\sigma_{1},\ldots,\sigma_{N}$. The indexing
$\text{idx}_{f}$ is defined such that if we define
$\mathcal{K}^{\prime}_{j}:=\\{\sigma\in\mathcal{K}\mid\text{idx}_{f}(\sigma)\leq
j\\}$, then

$\mathcal{K}_{1}^{\prime}\subseteq\mathcal{K}_{2}^{\prime}\subseteq\cdots\subseteq\mathcal{K}^{\prime}_{N}$

is a nested sequence of simplicial complexes and if
$r_{i}=f(\sigma_{j_{1}})=\cdots=f(\sigma_{j_{k}})$, where $j_{1}<\cdots<j_{k}$
and $\\{r_{i}\\}=\text{Im}(f)$, with $r_{i}<r_{i+1}$, then

$\mathcal{K}_{r_{i}}=\mathcal{K}^{\prime}_{j_{1}}\subset\mathcal{K}^{\prime}_{j_{2}}\subset\cdots\subset\mathcal{K}_{j_{k}}^{\prime}\subset\mathcal{K}_{r_{i+1}}\,.$

In other words, $\\{\mathcal{K}^{\prime}_{j}\\}$ is a refinement of
$\\{\mathcal{K}_{r_{i}}\\}$.

The following lemma is a straightforward corollary of the work in [19], and we
will rely on it repeatedly in the present paper.

######  Lemma 2.3 ([19]).

If $f_{0},f_{1}:\mathcal{K}\to\mathbb{R}$ are two filtration functions such
that $\prec_{f_{0}}$ is the same as $\prec_{f_{1}}$, then
$\text{idx}_{f_{0}}=\text{idx}_{f_{1}}$ and $f_{1}$ and $f_{2}$ both induce
the same set of (birth, death) simplex pairs.

###  2.4. Updating persistent homology when the simplex indexing is updated

One of the main contributions of [14], in which vineyards were introduced, is
an algorithm for updating the (birth, death) simplex pairs when the simplex
indexing changes. We review the relevant details in this subsection.

Suppose that
$\text{idx}_{f_{0}},\text{idx}_{f_{1}}:\mathcal{K}\to\\{1,\ldots,N\\}$ are the
simplex indexings that are induced by filtrations
$f_{0},f_{1}:\mathcal{K}\to\mathbb{R}$ (respectively), and suppose that
$\text{idx}_{f_{0}}$ and $\text{idx}_{f_{1}}$ differ only by a transposition
of a pair $(\sigma,\tau)$ of consecutive simplices. That is,
$\text{idx}_{f_{0}}(\tau)=\text{idx}_{f_{0}}(\sigma)+1$,
$\text{idx}_{f_{1}}(\tau)=\text{idx}_{f_{0}}(\sigma)$, and
$\text{idx}_{f_{1}}(\sigma)=\text{idx}_{f_{0}}(\tau)$. Let
$S_{\text{idx}_{f_{0}}}$ and $S_{\text{idx}_{f_{1}}}$ be the sets of (birth,
death) simplex pairs for $f_{0}$ and $f_{1}$, respectively.111Recall that, by
Lemma 2.3, the pairs depend only on the simplex orders $\text{idx}_{f_{0}}$,
$\text{idx}_{f_{1}}$, which is why we label the sets by their associated
simplex indexings. The update rule of [14] gives us bijection
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}:S_{\text{idx}_{f_{0}}}\to
S_{\text{idx}_{f_{1}}}$.

We review the key properties of the bijection
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}$. We write

$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}=(\phi_{b}^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}},\phi_{d}^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}})\,,$

where
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}_{b}:S_{\text{idx}_{f_{0}}}\to\mathcal{K}$
maps a simplex pair $(\sigma_{b},\sigma_{d})\in S_{\text{idx}_{f_{0}}}$ to the
birth simplex of
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}((\sigma_{b},\sigma_{d}))$ and
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}_{d}:S_{\text{idx}_{f_{0}}}\to\mathcal{K}$
maps $(\sigma_{b},\sigma_{d})\in S_{\text{idx}_{f_{0}}}$ to the death simplex
of $\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}((\sigma_{b},\sigma_{d}))$.
If $(\sigma_{b},\sigma_{d})\in S_{\text{idx}_{f_{0}}}$ is a pair such that
$\sigma_{b},\sigma_{d}\not\in\\{\sigma,\tau\\}$, then
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}((\sigma_{b},\sigma_{d}))=(\sigma_{b},\sigma_{d})$.
If $(\sigma_{b}^{1},\sigma_{d}^{1})\in S_{\text{idx}_{f_{0}}}$ is the pair
that contains $\sigma$, then let $\lambda\in\\{b,d\\}$ be the index such that
$\sigma^{1}_{\lambda}=\sigma$. Similarly, if
$(\sigma_{b}^{2},\sigma_{d}^{2})\in S_{\text{idx}_{f_{0}}}$ is the pair that
contains $\tau$, then let $\mu\in\\{b,d\\}$ be the index such that
$\sigma^{1}_{\mu}=\tau$. The key fact about the update rule of [14] is that
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}$ is defined such that either

$\displaystyle\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}((\sigma^{1}_{b},\sigma^{2}_{d}))$
$\displaystyle=(\sigma^{1}_{b},\sigma^{1}_{d})\,,$
$\displaystyle\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}((\sigma^{2}_{b},\sigma^{2}_{d}))$
$\displaystyle=(\sigma^{2}_{b},\sigma^{2}_{d})\,,$

or

$\displaystyle\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}_{\lambda}((\sigma^{1}_{b},\sigma^{1}_{d}))$
$\displaystyle=\tau\,,\qquad\phi^{\text{idx}_{f_{0}},\text{idx}_{f_{1}}}_{\lambda^{c}}((\sigma^{1}_{b},\sigma^{1}_{d}))=\sigma^{1}_{\lambda^{c}}\,,$
$\displaystyle\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}_{\mu}((\sigma^{2}_{b},\sigma^{2}_{d}))$
$\displaystyle=\sigma\,,\qquad\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}_{\mu^{c}}((\sigma^{2}_{b},\sigma^{2}_{d}))=\sigma^{1}_{\mu^{c}}\,,$

where

$\lambda^{c}:=\begin{cases}b\,,&\lambda=d\\\
d\,,&\lambda=b\end{cases}\qquad\mu^{c}:=\begin{cases}b\,,&\mu=d\\\
d\,,&\mu=b\,.\end{cases}$

In other words, either $\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}$ is the
identity map or $\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}$ swaps
$\sigma$ and $\tau$ in the pairs that contain them. The particular case
depends on the order of
$f_{0}(\sigma_{b}^{1}),f_{0}(\sigma_{d}^{1}),f_{0}(\sigma_{b}^{2}),f_{0}(\sigma_{d}^{2})$
(see [14] for details; they are not relevant to the present paper).

More generally, suppose that $\text{idx}_{f_{0}}$, $\text{idx}_{f_{1}}$ are
the simplex indexings induced by any two filtrations $f_{0},f_{1}$, where
$\text{idx}_{f_{0}}$ and $\text{idx}_{f_{1}}$ are no longer required to differ
only by the transposition of two consecutive simplices. Let
$S_{\text{idx}_{f_{0}}}$ and $S_{\text{idx}_{f_{1}}}$ be the sets of (birth,
death) simplex pairs for $f_{0}$ and $f_{1}$, respectively. The update rule of
Cohen-Steiner et al. [14] defines a bijection
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}:S_{\text{idx}_{f_{0}}}\to
S_{\text{idx}_{f_{1}}}$ as follows. Every permutation can be decomposed into a
sequence of transpositions that transpose consecutive elements, so there is a
sequence $\zeta_{0},\ldots,\zeta_{m}$ of simplex indexings such that
$\zeta_{0}=\text{idx}_{f_{0}}$, $\zeta_{m}=\text{idx}_{f_{1}}$, and
$\zeta_{i}$, $\zeta_{i+1}$ differ only by the transposition of two consecutive
simplices. Cohen-Steiner et al. [14] defined

(4)
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}:=\phi^{\zeta_{m-1},\,\zeta_{m}}\circ\cdots\circ\phi^{\zeta_{0},\,\zeta_{1}}\,.$

######  Remark 2.4.

If $\text{idx}_{f_{0}}$, $\text{idx}_{f_{1}}$ do not differ by only the
transposition of two consecutive simplices, then the sequence
$\zeta_{0},\ldots,\zeta_{m}$ is not unique. Unfortunately, the definition of
$\phi^{\text{idx}_{f_{0}},\,\text{idx}_{f_{1}}}$ does depend on the sequence
$\zeta_{0},\ldots,\zeta_{m}$ in its definition. This is implicitly shown in
Proposition 5.3.

###  2.5. Cellular Sheaves

A _cell complex_ is a topological space $Y$ with a partition into a set
$\\{Y_{\alpha}\\}_{\alpha\in P_{Y}}$ of subspaces (the _cells_ of the cell
complex) that satisfy the following conditions:

  1. (1)

Every cell $Y_{\alpha}$ is homeomorphic to $\mathbb{R}^{k_{\alpha}}$ for some
$k_{\alpha}\geq 0$. The cell $Y_{\alpha}$ is a _$k_{\alpha}$ -cell_.

  2. (2)

For every cell $Y_{\alpha}$, there is a homeomorphism
$\phi_{\alpha}:B^{k_{\alpha}}\to\overline{X_{\alpha}}$, where $B^{k_{\alpha}}$
is the closed $k_{\alpha}$-dimensional ball, such that
$\phi_{\alpha}(\text{int}(B^{k_{\alpha}}))=X_{\alpha}$.

  3. (3)

Axiom of the Frontier: If the intersection $\overline{Y_{\beta}}\cap
Y_{\alpha}$ is nonempty, then $Y_{\alpha}\subseteq\overline{Y_{\beta}}$. We
say that $Y_{\alpha}$ is a _face_ of $Y_{\beta}$.

  4. (4)

Locally finite: Every $x\in X$ has an open neighborhood $U$ such that $U$
intersects finitely many cells.

For example, a polyhedron is a cell complex whose $k$-cells are the
$k$-dimensional faces of the polyhedron. A graph is a another example of a
cell complex; the $0$-cells are the vertices and the $1$-cells are the edges.

We will first review the most general definition of cellular sheaves, which
uses category theory, and then we will specialize to the case of interest for
the present paper, which does not require category theory. Let $Y$ be a cell
complex with cells $\\{Y_{\alpha}\\}_{\alpha\in P_{Y}}$, and let $\mathcal{D}$
be a category. The set $P_{Y}$ is a poset with the relation $\alpha\leq\beta$
if $Y_{\alpha}\subseteq\overline{Y_{\beta}}$.

A _$\mathcal{D}$ -valued cellular sheaf on $Y$_ consists of

  1. (1)

An assignment of an object $\mathcal{F}(Y_{\alpha})\in\mathcal{D}$ (the
_stalk_ of $\mathcal{F}$ at $Y_{\alpha}$) for every cell $Y_{\alpha}$ in $Y$,
and

  2. (2)

A morphism $\mathcal{F}_{\alpha\leq\beta}:F(Y_{\alpha})\to F(Y_{\beta})$ (a
_restriction map_) whenever $Y_{\alpha}$ is a face of $Y_{\beta}$. The
morphisms must satisfy the _composition condition_ :

(5)
$\mathcal{F}_{\beta\leq\gamma}\circ\mathcal{F}_{\alpha\leq\beta}=\mathcal{F}_{\alpha\leq\gamma}$

whenever $\alpha\leq\beta\leq\gamma$.

Equivalently, a $\mathcal{D}$-valued cellular sheaf on $Y$ is a functor
$\mathcal{F}:P_{Y}\to\mathcal{D}$, where $P_{Y}$ is considered as a category.

A _global section_ of a cellular sheaf $\mathcal{F}$ is a function

$s:\\{Y_{\alpha}\\}_{\alpha\in
P_{Y}}\to\bigcup_{\alpha}\mathcal{F}(Y_{\alpha})$

such that

  1. (1)

$s(Y_{\alpha})\in\mathcal{F}(Y_{\alpha})$ (i.e., $s(Y_{\alpha})$ is a choice
of element in the stalk at $Y_{\alpha}$), and

  2. (2)

if $\alpha\leq\beta$, then
$s(Y_{\beta})=\mathcal{F}_{\alpha\leq\beta}(s(Y_{\alpha}))$ .

In what follows, we will consider a Set-valued cellular sheaf. The objects of
the category Set are sets, and the morphisms between sets $A$ and $B$ are the
functions from $A$ to $B$. A Set-valued cellular sheaf on a cell complex $Y$
consists of

  1. (1)

A set $\mathcal{F}(Y_{\alpha})$ for every cell $Y_{\alpha}$ in $Y$, and

  2. (2)

A function
$\mathcal{F}_{\alpha\leq\beta}:\mathcal{F}(Y_{\alpha})\to\mathcal{F}(Y_{\beta})$
whenever $Y_{\alpha}$ is a face of $Y_{\beta}$. The functions must satisfy the
condition:

$\mathcal{F}_{\beta\leq\gamma}\circ\mathcal{F}_{\alpha\leq\beta}=\mathcal{F}_{\alpha\leq\gamma}$

whenever $\alpha\leq\beta\leq\gamma$.

##  3\. Definition of a Persistence Diagram Bundle

A vineyard is a $1$-parameter set of persistence diagrams that is computed
from a $1$-parameter set of filtration functions on a simplicial complex
$\mathcal{K}$. We generalize a vineyard to a “persistence diagram bundle” as
follows.

######  Definition 3.1 (Fibered filtration function).

A _fibered filtration function_ is a set
$\\{f_{p}:\mathcal{K}^{p}\to\mathbb{R}\\}_{p\in B}$, where $B$ is a
topological space, $\\{\mathcal{K}^{p}\\}_{p\in B}$ is a set of simplicial
complexes parameterized by $B$, and $f_{p}$ is a filtration function on
$\mathcal{K}^{p}$.

When $\mathcal{K}^{p}\equiv\mathcal{K}$ for all $p\in B$, we define
$f:\mathcal{K}\times B\to\mathbb{R}$ to be the function
$f(\sigma,p):=f_{p}(\sigma)$. In a slight abuse of notation, we refer to
$f:\mathcal{K}\times B\to\mathbb{R}$, rather than to $\\{f_{p}:\mathcal{K}\to
B\\}_{p\in B}$, as the fibered filtration function. For all $p\in B$, the
function $f(\cdot,p):\mathcal{K}\to\mathbb{R}$ is a filtration of
$\mathcal{K}$. For several examples with $\mathcal{K}^{p}\equiv\mathcal{K}$,
see Section 3.1.

######  Definition 3.2 (Persistence diagram bundle).

Let $\\{f_{p}:\mathcal{K}^{p}\to\mathbb{R}\\}_{p\in B}$ be a fibered
filtration function. The _base_ of the bundle is $B$. The _$q$ th total space_
of the bundle is $E:=\\{(p,z)\\}\mid p\in B,z\in PD_{q}(f_{p})\\}$, with the
subspace topology inherited from the inclusion $E\hookrightarrow
B\times\overline{\mathbb{R}}^{2}$.222Technically, $E$ is a multiset because
persistence diagrams are multisets. However, when considering $E$ as a
topological space (which we do in Section 5 to study continuous paths in $E$
and sections of the PD bundle), we consider $E$ as a set. The _$q$ th
persistence diagram bundle_ is the triple $(E,B,\pi)$, where $\pi:E\to B$ is
the projection $(p,z)\mapsto p$.

For example, when $B$ is an interval in $\mathbb{R}$ and
$\mathcal{K}^{p}\equiv\mathcal{K}$, Definition 3.2 reduces to that of a
vineyard: a $1$-parameter set of PDs for a $1$-parameter set of filtrations of
$\mathcal{K}$. As discussed in Section 1, PHT is a special case with
$B=S^{d}$. The fibered barcode of a multiparameter persistence module is
another special case; we will discuss it in Section 3.2.2.

######  Remark 3.3.

In Definition 3.2, we are suggestively using the language of fiber bundles.
However, it is important to note that a PD bundle is not guaranteed to be a
true fiber bundle. The fibers need not be homeomorphic to each other for all
$p\in B$. At “singularities” (points $p_{*}\in B$ at which
$\text{PD}(f_{p_{*}})$ has an off-diagonal point with multiplicity), points in
$\text{PD}(f_{p})$ for nearby $p$ may merge into each other, changing the
homotopy type of the fiber. However, if $f:\mathcal{K}\times B\to\mathbb{R}$
is continuous and $p\in B$ is not a singularity, then there is a neighborhood
$U\subseteq B$ and a homeomorphism $\phi:\pi^{-1}(U)\to
U\times\text{PD}(f_{p_{*}})$ that preserves fibers (i.e., a local
trivialization).

As a special case of fibered filtration functions, we define _piecewise-linear
fibered filtration functions_ , which are simpler to analyze.

######  Definition 3.4 (Piecewise-linear fibered filtration function).

Let $\\{f_{p}:\mathcal{K}^{p}\to\mathbb{R}\\}_{p\in B}$ be a fibered
filtration function such that $\mathcal{K}^{p}\equiv\mathcal{K}$. We define
$f(\sigma,p):=f_{p}(\sigma)$ for all $\sigma\in\mathcal{K}$ and $p\in B$. If
$B$ is a simplicial complex and $f(\sigma,\cdot)$ is linear on each simplex of
$B$ for all simplices $\sigma\in\mathcal{K}$, then $f$ is a _piecewise-linear
fibered filtration function._ The resulting PD bundle is a _piecewise-linear
PD bundle_.

For instance, the fibered filtration function in Example 3.6, below, is
piecewise linear.

###  3.1. Examples

The following are concrete examples of PD bundles. We begin with the example
that motivated PD bundles in Section 1.

######  Example 3.5.

Suppose that $X(t,\bm{\mu})=\\{x_{1}(t,\bm{\mu}),\ldots,x_{k}(t,\bm{\mu})\\}$
is a point cloud that varies continuously with time $t\in\mathbb{R}$ and
system-parameter values $\mu_{1},\ldots,\mu_{m}\in\mathbb{R}$. We obtain a
fibered filtration function $f:\mathcal{K}\times\mathbb{R}^{m+1}\to\mathbb{R}$
by defining $f(\cdot,(t,\bm{\mu})):\mathcal{K}\to\mathbb{R}$ to be the
Vietoris–Rips filtration function for the point cloud $X(t,\bm{\mu})$ at all
$(t,\bm{\mu})\in\mathbb{R}^{m+1}$ (or any other filtration for the point cloud
at each $(t,\bm{\mu})$). The simplicial complex $\mathcal{K}$ is the
simplicial complex that has a simplex for every subset of points in the point
cloud.

######  Example 3.6.

Consider a color image. Enumerate the pixels and let $r(i)$, $g(i)$, and
$b(i)$ denote the red, green, and blue values of the $i$th pixel. Triangulate
each pixel to obtain a simplicial complex $\mathcal{K}$. (Every pixel is split
into two triangles.) Let $B=\\{(w_{1},w_{2})\in[0,1]^{2}\mid 0\leq
w_{1}+w_{2}\leq 1\\}$. For all $(w_{1},w_{2})\in B$, define
$p(i,(w_{1},w_{2}))=w_{1}r(i)+w_{2}g(i)+(1-w_{1}-w_{2})b(i)$. The function
$p(i,(w_{1},w_{2}))$ is a weighted average of the red, green, and blue values
of the $i$th pixel. Define a piecewise-linear fibered filtration function
$f:\mathcal{K}\times B\to\mathbb{R}$ as follows. For a $2$-simplex $\sigma$,
define $f(\sigma,{\bf w})=p(i(\sigma),{\bf w})$, where $i(\sigma)$ is the
pixel containing $\sigma$. For any other simplex $\sigma\in\mathcal{K}$,
define $f(\sigma,{\bf w})=\min\\{f(\tau,{\bf
w})\mid\sigma\subseteq\tau,\,\dim(\tau)=2\\}$. At ${\bf w}=(1,0)$, ${\bf
w}=(0,1)$, and ${\bf w}=(1,1)$, the filtration function $f(\cdot,{\bf w})$ is
the sublevel filtration by red, green, and blue pixel values, respectively. At
all other ${\bf w}\in B$, the filtration function $f(\cdot,{\bf w})$ is the
sublevel filtration by a weighted average of the red, green, and blue pixel
values.

######  Example 3.7.

Let $\mu_{1},\ldots,\mu_{m}\in\mathbb{R}$ denote the system-parameter values
of some discrete dynamical system. For given system-parameter values
$\bm{\mu}\in\mathbb{R}^{m}$, let $x_{i}^{\bm{\mu}}\in\mathbb{R}^{n}$ be the
solution at the $i$th time step and let
$X(\bm{\mu})=\\{x_{0}^{\bm{\mu}},\ldots,x_{k}^{\bm{\mu}}\\}$ be the set of
points obtained after the first $k$ time steps. For example, persistent
homology has been used to study orbits of the linked twist map (a discrete
dynamical system) [1]. We obtain a fibered filtration function
$f:\mathcal{K}\times\mathbb{R}^{m}\to\mathbb{R}$ by defining
$f(\cdot,\bm{\mu}):\mathcal{K}\to\mathbb{R}$ to be the Vietoris–Rips
filtration function for the point cloud $X(\bm{\mu})$ (or any other filtration
for the point cloud at each $\bm{\mu}$). The simplicial complex $\mathcal{K}$
has a simplex for every subset of points in the point cloud.

######  Example 3.8.

Suppose that $X(t)=\\{x_{1}(t),\ldots,x_{k}(t)\\}$ is a time-varying point
cloud in a compact triangulable subset $S\subseteq\mathbb{R}^{n}$. Let
$\rho_{h}(\cdot,t)$ be a kernel density estimator at time $t$, with bandwidth
parameter $h>0$. For fixed $h$ and $t$, we define a filtered complex by
considering sublevel sets of $\rho_{h}$ as follows. Let $\mathcal{K}$ be a
triangulation of $S\subseteq\mathbb{R}^{n}$. A vertex $v$ of $\mathcal{K}$ is
included in the simplicial complex $\mathcal{K}_{r}^{(t,h)}$ if
$\rho_{h}(v,t)\leq r$, and a simplex of $\mathcal{K}$ is included in
$\mathcal{K}_{r}^{t,h}$ if all of its vertices are in
$\mathcal{K}_{r}^{(t,h)}$. For each $t$ and $h$, the set
$\\{\mathcal{K}^{(t,h)}_{r}\\}_{r\in\mathbb{R}}$ is a filtered complex. We
obtain a fibered filtration function
$f:\mathcal{K}\times\mathbb{R}_{+}^{2}\to\mathbb{R}$ by defining
$f(\cdot,(t,h))$ to be the filtration function associated with the filtered
complex $\\{\mathcal{K}_{r}^{(t,h)}\\}_{r\in\mathbb{R}}$.

Density sublevels of time-varying point clouds were also considered by
Corcoran et al. [15], who studied a school of fish swimming in a shallow pool
that was modeled as a subset of $\mathbb{R}^{2}$. However, Corcoran et al.
[15] fixed a bandwidth parameter $h$ and a sublevel $r$, and only studied how
the PH changed with time (by using zigzag PH).

###  3.2. Comparison to multiparameter PH

Multiparameter PH was introduced in [8]; see [7] for a review. Typically, a
fibered filtration function does not induce a multifiltration, but the fibered
barcode of a multiparameter peristence module is an example of a PD bundle.

####  3.2.1. Multifiltrations

We review the definition of a multifiltration and compare it to the definition
of a fibered filtration.

######  Definition 3.9.

A _multifiltration_ is a set
$\\{\mathcal{K}_{\bm{u}}\\}_{\bm{u}\in\mathbb{Z}^{n}}$ of simplicial complexes
such that if $\bm{u}\leq\bm{v}$, then
$\mathcal{K}_{\bm{u}}\subseteq\mathcal{K}_{\bm{v}}$.

The inclusion
$\iota^{\bm{u},\bm{v}}:\mathcal{K}_{\bm{u}}\hookrightarrow\mathcal{K}_{\bm{v}}$
induces a map
$\iota_{*}^{\bm{u},\bm{v}}:H_{q}(\mathcal{K}_{\bm{u}},\mathbb{F})\to
H_{q}(\mathcal{K}_{\bm{v}},\mathbb{F})$ from the $q$th homology of
$\mathcal{K}_{\bm{u}}$ to the $q$th homology of $\mathcal{K}_{\bm{v}}$ over a
field $\mathbb{F}$. Given a multifiltration
$\\{\mathcal{K}_{\bm{u}}\\}_{\bm{u}\in\mathbb{Z}^{n}}$, the _multiparameter
persistence module_ is the graded $\mathbb{F}[x_{1},\ldots,x_{n}]$-module
$\bigoplus_{\bm{u}\in\mathbb{Z}^{n}}H_{q}(\mathcal{K}_{\bm{u}},\mathbb{F})$.
The action of $x_{i}$ on a homogeneous element $\gamma\in
H_{q}(\mathcal{K}_{\bm{u}},\mathbb{F})$ is given by
$x_{i}\gamma=\iota_{\bm{u},\bm{v}}^{*}(\gamma)$, where
$v_{j}=u_{j}+\delta_{ij}$.

######  Remark 3.10.

Some researchers define multifiltrations more generally as functors
$\mathcal{F}:\mathcal{P}\to\bf{Simp}$, where $\mathcal{P}$ is any poset and
$\bf{Simp}$ is the category of simplicial complexes, with simplicial maps as
morphisms. Definition 3.9 is the specific case in which
$\mathcal{P}=\mathbb{Z}^{n}$ and
$\mathcal{F}_{\bm{u}\leq\bm{v}}:\mathcal{K}_{\bm{u}}\to\mathcal{K}_{\bm{v}}$
is an inclusion map.

To see why a fibered filtration function does not typically induce a
multifiltration, consider a fibered filtration function
$\\{f_{p}:\mathcal{K}^{p}\to\mathbb{R}\\}_{p\in B}$ with $B=\mathbb{R}^{n}$.
Let $\mathcal{K}_{r}^{p}:=\\{\sigma\in\mathcal{K}^{p}\mid f_{p}(\sigma)\leq
r\\}$ denote the $r$-sublevel set of $f_{p}$. It is not necessarily the case
that $\mathcal{K}_{s}^{p_{1}}\subseteq\mathcal{K}_{r}^{p_{2}}$ whenever $r\leq
s$ and $p_{1}\leq p_{2}$. Moreover, there are no canonical simplicial maps
$\mathcal{K}_{s}^{p_{1}}\to\mathcal{K}_{r}^{p_{2}}$, so it is not guaranteed
that $\\{\mathcal{K}_{r}^{p}\\}_{(p,r)\in\mathbb{R}^{n}\times\mathbb{R}}$ is a
multifiltration even in the general sense of Remark 3.10. Therefore, such a
set of filtered complexes cannot be analyzed using multiparameter persistent
homology.

####  3.2.2. Fibered barcodes

Consider a multifiltration
$\\{\mathcal{K}_{\bm{u}}\\}_{\bm{u}\in\mathbb{R}^{n}}$. Let
$\overline{\mathcal{L}}$ denote the space of lines in $\mathbb{R}^{n}$ with a
parameterization of the form

$\displaystyle L:\mathbb{R}\to\mathbb{R}^{n}\,,$ $\displaystyle
L(r)=r\bm{v}+\bm{b}\,,\qquad\bm{v}\in[0,\infty)^{n},\,\lVert\bm{v}\rVert=1,\,\bm{b}\in\mathbb{R}^{n}.$

For example, when $n=2$, the space $\overline{\mathcal{L}}$ is the space of
lines in $\mathbb{R}^{2}$ with non-negative slope, including vertical lines.
For each line $L\in\overline{\mathcal{L}}$, we define
$\mathcal{K}^{L}_{r}:=\mathcal{K}_{L(r)}$. That is,
$\\{\mathcal{K}^{L}_{r}\\}_{r\in\mathbb{R}}$ is the filtered complex obtained
by restricting the multifiltration
$\\{\mathcal{K}_{\bm{u}}\\}_{\bm{u}\in\mathbb{R}^{n}}$ to the line
$L\subseteq\mathbb{R}^{n}$. The set
$\\{\mathcal{K}^{L}_{r}\\}_{r\in\mathbb{R}}$ is a filtered complex because
$L(r)_{i}\leq L(s)_{i}$ for all $r\leq s$ and $i\in\\{1,\ldots,n\\}$. The
_fibered barcode_ [4] is the map that sends $L\in\overline{\mathcal{L}}$ to
the barcode for the persistent homology of
$\\{\mathcal{K}^{L}_{r}\\}_{r\in\mathbb{R}}$.

A fibered barcode is a PD bundle whose base space is
$B=\overline{\mathcal{L}}$. For $L\in\overline{\mathcal{L}}$, the filtration
function is

$\displaystyle f_{L}:\mathcal{K}^{L}\to\mathbb{R}\,,$ $\displaystyle
f_{L}(\sigma)=\min\\{r\mid\sigma\in\mathcal{K}_{L(r)}\\}\,,$

where $\mathcal{K}^{L}:=\bigcup_{r\in\mathbb{R}}\mathcal{K}_{L(r)}$. Unlike
the other examples in Section 3.1, the simplicial complex $\mathcal{K}^{L}$ is
not independent of $L\in\overline{\mathcal{L}}$.

##  4\. A Stratification of the Base Space

There are many different notions of a stratified space [35]. In the present
paper, what we mean by a stratification is the following definition.

######  Definition 4.1.

A _stratification_ of a topological space $B$ is a nested sequence

$B^{0}\subseteq B^{1}\subseteq B^{2}\subseteq\cdots\subseteq B^{n}=B$

of closed subsets $B^{m}$ such that the following hold:

  1. (1)

For all $m$, the space $B^{m}\setminus B^{m-1}$ is either empty or a smooth
$m$-dimensional submanifold of $B$ (where we set $B^{-1}:=\emptyset$). The
_$m$ -dimensional strata_ are the connected components of $B^{m}\setminus
B^{m-1}$. We denote the set of strata by $\mathcal{Y}$.

  2. (2)

The set $\mathcal{Y}$ of strata is _locally finite_ : every $p\in B$ has an
open neighborhood $U$ such that $U$ intersects finitely many elements of
$\mathcal{Y}$.

  3. (3)

The set $\mathcal{Y}$ of strata satisfy the _Axiom of the Frontier_ : If
$Y_{\alpha}$, $Y_{\beta}\in\mathcal{Y}$ are strata such that
$Y_{\beta}\cap\overline{Y_{\alpha}}$, then
$Y_{\beta}\subseteq\overline{Y_{\alpha}}$. We write that $Y_{\beta}$ is a
_face_ of $Y_{\alpha}$.

In the present paper, $B$ is the base space of a fibered filtration function.

Theorem 4.15 says that for any “generic” smooth fibered filtration function
(see Section 4.2), the base space $B$ can be stratified so that in each
stratum $Y\subseteq B$, the set of (birth, death) simplex pairs is constant
and can be used to obtain $\text{PD}(f_{p})$ for any $p\in Y$.

###  4.1. Piecewise-linear fibered filtrations

As a warm-up, we first consider piecewise-linear fibered filtration functions,
which will provide intuition for the general case. However, note that
Proposition 4.5 below is not simply a special case of Theorem 4.15, in which
we consider generic smooth fibered filtrations on smooth compact manifolds
(see Section 4.2). Here, we consider _all_ piecewise-linear fibered
filtrations, rather than only generic piecewise-linear fibered filtrations.

First, we establish some notation and definitions.

######  Definition 4.2.

An _open half-space_ of an affine space $A$ is one of the two connected
components of $A\setminus H$ for some hyperplane $H$.

For example, an open half-space of $\mathbb{R}^{n}$ is a set of the form
$\\{\bm{x}\in\mathbb{R}^{n}\mid A\bm{x}>\bm{b}\\}$ for some $n\times n$ matrix
$A$ and some vector $\bm{b}\in\mathbb{R}^{n}$.

######  Definition 4.3.

An _open polyhedron_ is the intersection of open half-spaces.

For example, an open polygon $P$ (a polygon without its faces) in
$\mathbb{R}^{2}$ is an open 2D polyhedron because it is the intersection of
half-spaces of $\mathbb{R}^{2}$. The 1D faces (i.e., edges) of $P$ are 1D
polyhedra because an edge is a subset of a line $L\subseteq\mathbb{R}^{2}$ and
the edge is the intersection of two half-spaces of $L$. The 0D faces (i.e.,
vertices) of $P$ are 0D polyhedra.

We fix a simplicial complex $\mathcal{K}$ for the remainder of this section.
For each pair $(\sigma,\tau)$ of simplices in $\mathcal{K}$, we define

(6) $I(\sigma,\tau):=\\{p\in B\mid f(\sigma,p)=f(\tau,p)\\}\,.$

######  Lemma 4.4.

Suppose that $f:\mathcal{K}\times B\to\mathbb{R}$ is a continuous fibered
filtration function (i.e., $f(\sigma,\cdot):B\to\mathbb{R}$ is continuous for
all simplices $\sigma\in\mathcal{K}$) and that $Y$ is a path-connected subset
of $B$. If all pairs $(\sigma,\tau)$ of simplices satisfy either
$I(\sigma,\tau)\cap Y=\emptyset$ or $I(\sigma,\tau)\cap Y=Y$, then the simplex
order is constant in $Y$. That is, there is a strict partial order $\prec_{Y}$
on $\mathcal{K}$ such that for all $y\in Y$, we have that $\prec_{f(\cdot,y)}$
is the same as $\prec_{Y}$.

###### Proof.

Let $(\sigma,\tau)$ be a pair of simplices. If $I(\sigma,\tau)\cap Y=Y$, then
$f(\sigma,p)=f(\tau,p)$ for all $p\in Y$, so
$\sigma\not\prec_{f(\cdot,p)}\tau$ and $\tau\not\prec_{f(\cdot,p)}\sigma$ for
all $p\in Y$. If $I(\sigma,\tau)\cap Y=\emptyset$, then $f(\sigma,p)\neq
f(\tau,p)$ for all $p\in Y$. Let $p_{0}$ be a point in $Y$. Without loss of
generality, $\tau\prec_{f(\cdot,p_{0})}\sigma$. Therefore,
$f(\sigma,p_{0})>f(\tau,p_{0})$. To obtain a contradiction, suppose that
$f(\sigma,p_{1})<f(\tau,p_{1})$ for some $p_{1}\in Y$. Let $\gamma:[0,1]\to Y$
be a continuous path from $p_{0}$ to $p_{1}$, and let
$g(s)=f(\sigma,\gamma(s))-f(\tau,\gamma(s))$ for $s\in[0,1]$. By the
Intermediate Value Theorem, there is an $s_{*}\in[0,1]$ such that
$f(\sigma,\gamma(s_{*}))=f(\tau,\gamma(s_{*}))$, but this is a contradiction.
Therefore, $f(\sigma,p)>f(\tau,p)$ for all $p\in Y$, which implies that
$\tau\prec_{f(\cdot,p)}\sigma$ for all $p\in Y$. ∎

######  Proposition 4.5.

Let $B$ be a simplicial complex. If $f:\mathcal{K}\times B\to\mathbb{R}$ is a
piecewise-linear fibered filtration function, then $B$ can be partitioned into
disjoint polyhedra $P$ on which the simplex order induced by $f$ is constant.
That is, there is a strict partial order $\prec_{P}$ on $\mathcal{K}$ such
that $\prec_{f(\cdot,p)}$ is the same as $\prec_{P}$ for all $p\in P$.
Consequently, the set $\\{(\sigma_{b},\sigma_{d})\\}$ of (birth, death)
simplex pairs for $f$ is constant in each $P$ and for any $p\in P$, the
persistence diagram $\text{PD}(f_{p})$ consists of the diagonal and the
multiset $\\{(f(\sigma_{b}),f(\sigma_{d})\\}$.

###### Proof.

Let $\Delta$ be an $n$-dimensional simplex of the simplicial complex $B$ and
let $\sigma$ and $\tau$ be distinct simplices of $\mathcal{K}$. Because
$f(\sigma,\cdot)|_{\Delta}$ and $f(\tau,\cdot)|_{\Delta}$ are linear, the set
$I(\sigma,\tau)\cap\Delta$ is one of the following:

  1. (1)

the intersection of an $(n-1)$-dimensional hyperplane with $\Delta$ ;

  2. (2)

$\emptyset$ ;

  3. (3)

$\Delta$ ;

  4. (4)

a vertex of $\Delta\,$.

Therefore, the set
$\partial\Delta\bigcup\\{I(\sigma,\tau)\cap\Delta\mid\emptyset\subset(I(\sigma,\tau)\cap\Delta)\subset\Delta\\}_{\sigma,\tau\in\mathcal{K}}$
partitions $\Delta$ into polyhedra. By Lemma 4.4, the simplex order induced by
$f$ is constant on each polyhedron. The last statement of Proposition 4.5
follows from Lemma 2.3. ∎

For example, if $B$ is a triangulated surface, then the set

(7) $L:=\bigcup_{\Delta\in
B}\partial\Delta\cup\\{I(\sigma,\tau)\cap\Delta\mid\emptyset\subset(I(\sigma,\tau)\cap\Delta)\subset\Delta\\}_{\sigma,\tau\in\mathcal{K}}$

partitions $\Delta$ into polyhedra such that the simplex order is constant on
each polyhedron, including the 1D polyhedra (i.e., edges) and the 0D polyhedra
(i.e., vertices). The polygonal subdivision induced by $L$ is called a _line
arrangement_ $\mathcal{A}(L)$. For an example of such a line arrangement, see
Figure 3.

Figure 3. A line arrangement that represents the partition of a triangulated
surface $B$ (the base space) into polyhedra on which the simplex order is
constant.

###  4.2. Generic smooth fibered filtrations

We now consider generic smooth fibered filtration functions. Throughout
Section 4.2, we consider a smooth fibered filtration function of the form
$f:\mathcal{K}\times B\to\mathbb{R}$ for some $n$-dimensional smooth compact
manifold $B$ and some simplicial complex $\mathcal{K}$. (A fibered filtration
$f$ is _smooth_ if $f(\sigma,\cdot):B\to\mathbb{R}$ is smooth for all
$\sigma\in\mathcal{K}$.) To make precise the notion of a “generic” fibered
filtration function, we consider perturbations of $f$ of a certain form.
Because the filtration value of a simplex $\sigma$ must be at least as large
as the filtration value of any face $\tau$ at all $p\in B$, we consider only
perturbations $f_{\bm{a}}:\mathcal{K}\times B\to\mathbb{R}$ of the form

$f_{\bm{a}}(\sigma_{i},p):=f(\sigma_{i},p)+a_{i}\,,$

where $\bm{a}$ is an element of the set

(8) $A:=\\{\bm{a}\in\mathbb{R}^{N}\mid a_{i}\leq a_{j}\text{ for all }i\leq
j\\}\ $

and $\sigma_{1},\ldots,\sigma_{N}$ are the simplices of $\mathcal{K}$, indexed
such that $i<j$ if $\sigma_{i}$ is a proper face of $\sigma_{j}$. By
construction, $f_{\bm{a}}$ is a fibered filtration function for all $\bm{a}\in
A$.

For each simplex $\sigma_{k}$ in $\mathcal{K}$, we define the manifold

$M_{k}:=\\{(p,f(\sigma_{k},p))\mid p\in B\\}\subseteq B\times\mathbb{R}\ $

and for each $\bm{a}\in A$, we define the manifold

(9) $M_{\bm{a},k}:=\\{(p,f_{\bm{a}}(\sigma_{k},p))\mid p\in B\\}\subseteq
B\times\mathbb{R}\,.$

For each pair $(\sigma_{i},\sigma_{j})$ of simplices in $\mathcal{K}$, we
define $I(\sigma_{i},\sigma_{j})$ as in (6). The set
$I(\sigma_{i},\sigma_{j})$ is the projection of $M_{i}\cap M_{j}\subseteq
B\times\mathbb{R}$ to a subset of $B$. For each $\bm{a}\in A$, we define the
set

$I_{\bm{a}}(\sigma_{i},\sigma_{j}):=\\{p\in B\mid
f_{\bm{a}}(\sigma_{i},p)=f_{\bm{a}}(\sigma_{j},p)\\}\,.$

We also define

$\displaystyle E^{m}$
$\displaystyle:=\\{I(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I(\sigma_{i_{m}},\sigma_{j_{m}})\\}\,,$

which is the set of all $m$-way intersections of sets
$I(\sigma_{i},\sigma_{j})$. For all $\bm{a}\in A$, we define

(10)
$E^{m}_{\bm{a}}:=\\{I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})\\}\,.$

Lastly, we define

(11)
$E^{m,k}_{\bm{a}}:=\\{I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})\mid i_{r},j_{r}\leq k\text{ for all
}r\\}\,,$

which is the set of $m$-way intersections that only involve the simplices
$\sigma_{1},\ldots,\sigma_{k}$.

######  Remark 4.6.

There are several facts to keep in mind. First, it is not guaranteed that
$I_{\bm{a}}(\sigma_{i},\sigma_{j})$ is homeomorphic to
$I(\sigma_{i},\sigma_{j})$ even for arbitrarily small $\bm{a}$. Additionally,
the sets $I_{\bm{a}}(\sigma_{i},\sigma_{j})$ are not “independent” of each
other; a perturbation of $f(\sigma,\cdot)$ for a single simplex $\sigma$
causes a perturbation of $I(\sigma,\tau)$ for all $\tau\in\mathcal{K}$.
Furthermore, not every element of $E^{m}$ is an $(n-m)$-dimensional
submanifold, even generically. For example, if
$I(\sigma_{i_{2}},\sigma_{i_{1}})$ and $I(\sigma_{i_{3}},\sigma_{i_{2}})$ are
$(n-1)$-dimensional submanifolds that intersect transversely, then
$I(\sigma_{i_{3}},\sigma_{i_{1}})\cap I(\sigma_{i_{3}},\sigma_{i_{2}})\cap
I(\sigma_{i_{2}},\sigma_{i_{1}})=I(\sigma_{i_{3}},\sigma_{i_{2}})\cap
I(\sigma_{i_{2}},\sigma_{i_{1}})$ is an $(n-2)$-dimensional submanifold,
rather than an $(n-3)$-dimensional submanifold. Finally,
$\bigcap_{r=1}^{m}I(\sigma_{i_{r}},\sigma_{j_{r}})$ is not necessarily equal
to the projection of $\bigcap_{r=1}^{m}(M_{i_{r}}\cap M_{j_{r}})$ to $B$. In
other words, not every intersection in $B$ lifts to an intersection of the
manifolds $\\{M_{k}\\}_{k=1}^{N}\subseteq B\times\mathbb{R}$. These are the
main subtleties in the proof of Theorem 4.15.

######  Definition 4.7.

Let $S$ be an element of $E^{m}_{\bm{a}}$, where $E^{m}_{\bm{a}}$ is defined
as in (10). The set $S$ is _$m$ -reduced_ if it equals a set of the form
$I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})$, where $i_{1}>i_{2}>\cdots>i_{m}$
and $i_{r}>j_{r}$ for all $r$.

For example, if $\sigma_{i_{1}}$, $\sigma_{i_{2}}$, and $\sigma_{i_{3}}$ are
distinct simplices, then $I_{\bm{a}}(\sigma_{i_{3}},\sigma_{i_{2}})\cap
I_{\bm{a}}(\sigma_{i_{2}},\sigma_{i_{1}})$ is $2$-reduced, but
$I_{\bm{a}}(\sigma_{i_{3}},\sigma_{i_{1}})\cap
I_{\bm{a}}(\sigma_{i_{3}},\sigma_{i_{2}})\cap
I_{\bm{a}}(\sigma_{i_{2}},\sigma_{i_{1}})$ is not $3$-reduced. We define

(12) $\displaystyle\overline{E^{m}_{\bm{a}}}$ $\displaystyle:=\\{S\in
E^{m}_{\bm{a}}\mid S\text{ is }m\text{-reduced}\\}\,,$ (13)
$\displaystyle\overline{E^{m,k}_{\bm{a}}}$ $\displaystyle:=\\{S\in
E^{m,k}_{\bm{a}}\mid S\text{ is }m\text{-reduced}\\}\,,$

where $E^{m}_{\bm{a}}$ is defined as in (10) and $E^{m,k}_{\bm{a}}$ is defined
as in (11).

######  Lemma 4.8.

For all $m\geq 1$, all $k$, and all $\bm{a}\in A$, where $A$ is defined as in
(8), every $S\in E^{m,k}_{\bm{a}}$ belongs to
$\overline{E^{m^{\prime},k}_{\bm{a}}}$ for some $m^{\prime}\leq m$, where
$E^{m,k}_{\bm{a}}$ and $\overline{E^{m^{\prime},k}_{\bm{a}}}$ are defined as
in (11) and (13), respectively.

###### Proof.

We prove the lemma by induction on $m$. For all $k$, every $S\in
E^{1,k}_{\bm{a}}$ is $1$-reduced by definition. Assume that Lemma 4.8 is true
for $m-1\geq 1$, and let $S$ be an element of $E^{m,k}_{\bm{a}}$. The set $S$
is equal to a set of the form

$I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})\,,$

where $i_{r}>j_{r}$ for all $r$ and $k\geq i_{1}\geq i_{2}\geq\cdots\geq
i_{m}$ without loss of generality. By the induction hypothesis,

$I_{\bm{a}}(\sigma_{i_{2}},\sigma_{j_{2}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})=I_{\bm{a}}(\sigma_{i_{2}^{\prime}},\sigma_{j_{2}^{\prime}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{\ell}^{\prime}},\sigma_{j_{\ell}^{\prime}})$

for some $\ell\leq m$, where $i^{\prime}_{r}>j^{\prime}_{r}$ for all $r$ and
$k\geq i_{1}\geq i_{2}\geq
i^{\prime}_{2}>i^{\prime}_{3}>\cdots>i^{\prime}_{\ell}$. If
$i_{1}>i^{\prime}_{2}$, then $S$ is an element of $E^{\ell,k}$ and we are
done. Otherwise,

$I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap
I_{\bm{a}}(\sigma_{i_{2}^{\prime}},\sigma_{j_{2}^{\prime}})=I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap
I_{\bm{a}}(\sigma_{j_{1}},\sigma_{j_{2}^{\prime}})$

because $i_{1}=i_{2}^{\prime}$. If $j_{1}=j_{2}^{\prime}$, then

$S=I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap
I_{\bm{a}}(\sigma_{i_{3}^{\prime}},\sigma_{j_{3}^{\prime}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{\ell}^{\prime}},\sigma_{j_{\ell}^{\prime}})\,,$

so $S$ is $(\ell-1)$-reduced. Otherwise,

$S=I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap
I_{\bm{a}}(\sigma_{j_{1}},\sigma_{j_{2}^{\prime}})\cap
I_{\bm{a}}(\sigma_{i_{3}^{\prime}},\sigma_{j_{3}^{\prime}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{\ell}^{\prime}},\sigma_{j_{\ell}^{\prime}})\,,$

where $k\geq i_{1}>j_{1},j_{2}^{\prime}$ and $i_{1}>i_{r}^{\prime}$ for all
$r\geq 3$. By the induction hypothesis, the set
$I_{\bm{a}}(\sigma_{j_{1}},\sigma_{j_{2}^{\prime}})\cap
I_{\bm{a}}(\sigma_{i_{3}^{\prime}},\sigma_{j_{3}^{\prime}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{\ell}^{\prime}},\sigma_{j_{\ell}^{\prime}})$ belongs to
$\overline{E^{\ell^{\prime},k-1}_{\bm{a}}}$ for some
$\ell^{\prime}\leq\ell-1$, so $S$ belongs to
$\overline{E^{\ell^{\prime}+1,k}_{\bm{a}}}$, where
$\ell^{\prime}+1\leq\ell\leq m$. ∎

######  Lemma 4.9.

For almost every $\bm{a}\in A$ (where $A$ is defined as in (8)), we have that
$M_{\bm{a},i}$ intersects $M_{\bm{a},j}$ transversely333When manifolds $M_{1}$
and $M_{2}$ intersect transversely, we will use the notation $M_{1}\pitchfork
M_{2}$. for $i\neq j$ and every $S\in\overline{E^{m}_{\bm{a}}}$ is either
$\emptyset$ or an $(n-m)$-dimensional submanifold of $B$ for all
$m\in\\{1,\ldots,n\\}$, where $M_{\bm{a},i}$ is defined as in (9) and
$\overline{E^{m}_{\bm{a}}}$ is defined as in (12).

###### Proof.

Define $g^{ij}(p):=f(\sigma_{i},p)-f(\sigma_{j},p)$ for all $i\neq j$. For
almost every $\bm{a}\in A$, the quantity $a_{j}-a_{i}$ is a regular value of
$g^{ij}$ by Sard’s Theorem. The set of regular values is open for all $i\neq
j$ because $g^{ij}$ is smooth and $B$ is compact. Therefore, there is an
$\epsilon^{*}$ such that for all $i\neq j$, every
$y\in(a_{j}-a_{i}-2\epsilon^{*},a_{j}-a_{i}+2\epsilon^{*})$ is a regular value
of $g^{ij}$.

Given an $\bm{a}$ and $\epsilon^{*}$ as above, it suffices to show that for
almost every $\bm{\epsilon}\in\mathbb{R}^{N}$ with
$|\epsilon_{i}|\leq\epsilon^{*}$, we have that every
$S\in\overline{E^{m}_{\bm{a+\epsilon}}}$ is an $(n-m)$-dimensional submanifold
of $B$ for all $m$. For $m=1$, every element of
$\overline{E^{1}_{\bm{a+\epsilon}}}$ is of the form
$I_{\bm{a+\epsilon}}(\sigma_{i},\sigma_{j})$ for some $i\neq j$. The set
$I_{\bm{a+\epsilon}}(\sigma_{i},\sigma_{j})$ is the
$(a_{j}-a_{i}+\epsilon_{j}-\epsilon_{i})$-level set of $g^{ij}$. Because
$(a_{j}-a_{i}+\epsilon_{j}-\epsilon_{i})$ is a regular value of $g^{ij}$, the
set $I_{\bm{a+\epsilon}}(\sigma_{i},\sigma_{j})$ is an $(n-1)$-dimensional
submanifold of $B$ and we must have $M_{\bm{a},i}\pitchfork M_{\bm{a},j}$.

For $m\geq 2$, observe that

$\overline{E^{m}_{\bm{a}}}=\overline{E^{m,2}_{\bm{a}}}\cup\Big{(}\bigcup_{k=3}^{N}\overline{E^{m,k}_{\bm{a}}}\setminus\overline{E^{m,k-1}_{\bm{a}}}\Big{)}\,,$

where $\overline{E^{m,k}_{\bm{a}}}$ is defined as in (13). We induct on
$k\in\\{2,\ldots,N\\}$, where $N$ is the number of simplices in $\mathcal{K}$.
When $k=2$, we have

$\overline{E^{m,2}_{\bm{a+\epsilon}}}=\begin{cases}\\{I_{\bm{a+\epsilon}}(\sigma_{1},\sigma_{2})\\}\,,&m=1\\\
\emptyset\,,&m\geq 2\,,\end{cases}$

so every $S\in\overline{E^{m,2}_{\bm{a+\epsilon}}}$ is either $\emptyset$ or
an $(n-m)$-dimensional submanifold of $B$. Now suppose that $k>2$ and that
every element of $S\in\overline{E^{m,k-1}_{\bm{a+\epsilon}}}$ is either
$\emptyset$ or an $(n-m)$-dimensional submanifold for all $m$. Every element
$S$ in
$\overline{E^{m,k}_{\bm{a+\epsilon}}}\setminus\overline{E^{m,k-1}_{\bm{a+\epsilon}}}$
is equal to a set of the form

$S=I_{\bm{a+\epsilon}}(\sigma_{k},\sigma_{\ell})\cap S^{\prime}\,,$

where $\ell\leq k-1$ and
$S^{\prime}\in\overline{E^{m-1,k-1}_{\bm{a+\epsilon}}}$. We define the vectors
$\bm{\epsilon}^{j}:=(0,\ldots,0,\epsilon_{j},0,\ldots,0)$ and
$\bm{b}^{j}:=\bm{a+\epsilon}-\bm{\epsilon}^{j}$. Note that
$I_{\bm{a}+\bm{\epsilon}}(\sigma_{k},\sigma_{\ell})=I_{\bm{b}+\bm{\epsilon}^{k}}(\sigma_{k},\sigma_{\ell})$
because $b_{i}^{k}+\epsilon_{i}^{k}=a_{i}+\epsilon_{i}$ for all $i$, and
$\overline{E^{m-1,k-1}_{\bm{a+\epsilon}}}=\overline{E^{m-1,k-1}_{\bm{b}^{k}}}$
because $a_{i}+\epsilon_{i}=b_{i}$ for all $i\leq k-1$. Therefore, every
$S\in\overline{E^{m,k}_{\bm{a+\epsilon}}}\setminus\overline{E^{m,k-1}_{\bm{a+\epsilon}}}$
is equal to a set of the form

$S=I_{\bm{b}^{k}+\bm{\epsilon}^{k}}(\sigma_{k},\sigma_{\ell})\cap S^{\prime}$

for some $S^{\prime}\in\overline{E^{m-1,k-1}_{\bm{b}^{k}}}$ and $\ell\leq
k-1$. Because $g^{\ell k}$ has no critical values between
$b_{\ell}-b_{k}-\epsilon_{k}$ and $b_{\ell}-b_{k}$, we have that
$I_{\bm{b}^{k}+\bm{\epsilon}^{k}}(\sigma_{k},\sigma_{\ell})$ is diffeomorphic
to $I_{\bm{b}^{k}}(\sigma_{k},\sigma_{\ell})$ for all
$\epsilon_{k}\in(-\epsilon^{*},\epsilon^{*})$. In other words,
$I_{\bm{b}^{k}+\bm{\epsilon}^{k}}(\sigma_{k},\sigma_{\ell})$ is a perturbation
of $I_{\bm{b}^{k}}(\sigma_{k},\sigma_{\ell})$ for all
$\epsilon_{k}\in(-\epsilon^{*},\epsilon^{*})$. By Thom’s Transversality
Theorem, $I_{\bm{b}^{k}+\bm{\epsilon}^{k}}(\sigma_{k},\sigma_{\ell})$
intersects every $S^{\prime}\in E_{\bm{b}^{k}}^{m-1,k-1}$ transversely for
almost every $\epsilon_{k}\in(-\epsilon^{*},\epsilon^{*})$. This shows that
$S$ is either $\emptyset$ or an $(n-m)$-dimensional submanifold of $B$ for
almost every $\epsilon_{k}\in(\epsilon^{*},\epsilon^{*})$. Because there are
finitely many elements in $\overline{E^{m,k}_{\bm{a}}}$, we must have that
every $S\in\overline{E^{m,k}_{\bm{a}}}$ is either $\emptyset$ or an
$(n-m)$-dimensional submanifold of $B$ for almost every
$\epsilon_{k}\in(\epsilon^{*},\epsilon^{*})$. Induction on $k$ concludes the
proof. ∎

######  Lemma 4.10.

For all $\bm{a}\in A$, with $A$ defined as in (8), define

(14) $B^{n}_{\bm{a}}:=B\,\qquad B^{m}_{\bm{a}}:=\bigcup_{\ell\leq
m}\bigcup_{S\in\overline{E_{\bm{a}}^{n-\ell}}}S\qquad\text{ for }m<n\,.$

If $\bm{a}\in A$ is such that every $S\in\overline{E_{\bm{a}}^{n-\ell}}$ is
either $\emptyset$ or an $\ell$-dimensional smooth submanifold for every
$\ell\in\\{1,\ldots,n\\}$, where $\overline{E_{\bm{a}}^{n-\ell}}$ is defined
as in (12), then $B^{m}_{\bm{a}}\setminus B^{m-1}_{\bm{a}}$ is the disjoint
union of smooth $m$-dimensional manifolds.

###### Proof.

We have that

$B_{\bm{a}}^{m}\setminus
B_{\bm{a}}^{m-1}=\bigcup_{S\in\overline{E^{n-m}_{\bm{a}}}}\Big{(}S\setminus\bigcup_{\begin{subarray}{c}S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}\\\
\ell\leq m-1\end{subarray}}S^{\prime}\Big{)}\,.$

If $S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}$ is a subset of
$S\in\overline{E^{n-m}_{\bm{a}}}$, then $S^{\prime}$ is a closed subset of
$S$. Therefore, the set
$S\setminus\Big{(}\bigcup_{\begin{subarray}{c}S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}\\\
\ell\leq m-1\end{subarray}}S^{\prime}\Big{)}$ is an open subset of the smooth
manifold $S$, which implies that
$S\setminus\Big{(}\bigcup_{\begin{subarray}{c}S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}\\\
\ell\leq m-1\end{subarray}}S^{\prime}\Big{)}$ is a smooth manifold. If $S_{1}$
and $S_{2}$ are distinct elements of $\overline{E^{n-m}_{\bm{a}}}$, then

$\Big{(}S_{1}\setminus\bigcup_{\begin{subarray}{c}S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}\\\
\ell\leq
m-1\end{subarray}}S^{\prime}\Big{)}\cap\Big{(}S_{2}\setminus\bigcup_{\begin{subarray}{c}S^{\prime}\in\overline{E^{n-\ell}_{\bm{a}}}\\\
\ell\leq m-1\end{subarray}}S^{\prime}\Big{)}=\emptyset\,,$

which completes the proof. ∎

For the remainder of Section 4.2, let $\\{B^{m}_{\bm{a}}\\}_{m=0}^{n}$ be
defined as in (14), and define

(15) $\mathcal{Y}_{\bm{a}}=\bigcup_{m=0}^{n}\mathcal{Y}^{m}_{\bm{a}}\,,$

where $\mathcal{Y}^{m}_{\bm{a}}$ is the set of connected components of
$B^{m}_{\bm{a}}\setminus B^{m-1}_{\bm{a}}$ (with
$B^{-1}_{\bm{a}}:=\emptyset$).

######  Lemma 4.11.

Let $A$ be defined as in (8). If $\bm{a}\in A$ is such that each
$Y\in\mathcal{Y}_{\bm{a}}$ is a manifold, then the simplex order induced by
$f$ is constant in each $Y$. That is, there is a strict partial order
$\prec_{Y}$ on $\mathcal{K}$ such that $\prec_{f_{\bm{a}}(\cdot,y)}$ is the
same as $\prec_{Y}$ for all $y\in Y$.

###### Proof.

Let $Y\in\mathcal{Y}_{\bm{a}}$. The set $Y$ is connected by definition.
Because $Y$ is a manifold, it is also path-connected. For each pair
$(\sigma,\tau)$ of simplices, we have by construction that $Y\cap
I_{\bm{a}}(\sigma,\tau)$ equals either $\emptyset$ or $Y$. (In fact, this
statement holds for all $\bm{a}\in A$ and does not require $Y$ to be a
manifold.) By Lemma 4.4, the simplex order is constant in $Y$. ∎

######  Lemma 4.12.

For almost every $\bm{a}\in A$ (where $A$ is defined as in (8)), we have that
$\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a submanifold
of $B$ and

(16)
$T_{p}\Big{(}\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=\bigcap_{r=1}^{m}T_{p}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$

for all points
$p\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ and all sets
$\\{(i_{r},j_{r})\\}_{r=1}^{m}$ of index pairs such that
$\\{i_{r},j_{r}\\}\neq\\{i_{s},j_{s}\\}$ if $r\neq s$.

###### Proof.

Because there are finitely many sets of index pairs, it suffices to fix a set
$\\{(i_{r},j_{r})\\}_{r=1}^{m}$ of index pairs and show that (16) holds for
all $y\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ for
almost every $\bm{a}\in A$. By Lemmas 4.8 and 4.9, the set
$\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a manifold for
almost every $\bm{a}\in A$. By Lemma A.2, there is a finite open cover
$\\{U_{k}\\}_{k=1}^{K}$ of $B$ such that for each $k$, there is a disjoint
partition $\bigcup_{\ell}J_{\ell,k}=\\{1,\ldots,m\\}$ such that
$\\{i_{r},j_{r}\mid r\in J_{\ell_{1},k}\\}\cap\\{i_{r},j_{r}\mid r\in
J_{\ell_{2},k}\\}=\emptyset$ if $\ell_{1}\neq\ell_{2}$ and

$\pi\Big{(}\bigcap_{r\in J_{\ell,k}}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}\cap U_{k}=\bigcap_{r\in
J_{\ell,k}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U_{k}$

for all $\ell$, where $\pi$ is the projection $\pi:B\times\mathbb{R}\to
B$.444Recall that an arbitrary intersection
$\bigcap_{r}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\subseteq B$ does not
necessarily lift to an intersection $\bigcap_{r=1}^{m}(M_{\bm{a},i}\cap
M_{\bm{a},j})\subseteq B\times\mathbb{R}$. Lemma A.2 says that in local
neighborhoods $U\subseteq B$, we can partition the set $\\{1,\ldots,m\\}$ into
subsets $J_{\ell}$ such that the intersection $\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ does lift to a subset of
the intersection $\bigcap_{r\in J_{\ell}}(M_{\bm{a},i}\cap M_{\bm{a},j})$.
Because the number $K$ of open sets is finite, it suffices to fix $U_{k}$ and
show that (16) holds for all
$p\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U_{k}$ for
almost every $\bm{a}\in A$.

By Lemma A.4, we have

$T_{p}\Big{(}\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=T_{p}\Big{(}\bigcap_{\ell}\bigcap_{r\in
J_{\ell,k}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=\bigcap_{\ell}T_{p}\Big{(}\bigcap_{r\in
J_{\ell,k}}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}\Big{)}$

for all $p\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U_{k}$ for almost every $\bm{a}\in A$. By Lemma A.5, we have

$\bigcap_{\ell}T_{p}\Big{(}\bigcap_{r\in
J_{\ell,k}}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}\Big{)}=\bigcap_{\ell}\bigcap_{r\in
J_{\ell,k}}T_{p}(I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$

for all $p\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U_{k}$ for almost every $\bm{a}\in A$. ∎

For any strict partial order $\prec$ on $\mathcal{K}$, we define

$\displaystyle Z^{\prec}_{\bm{a}}:=$ $\displaystyle\\{p\in B\mid
f_{\bm{a}}(\sigma,p)<f_{\bm{a}}(\tau,p)\text{ if }\sigma\prec\tau$ (17)
$\displaystyle\qquad\text{ and }f_{\bm{a}}(\sigma,p)=f_{\bm{a}}(\tau,p)\text{
if }\sigma\not\prec\tau\text{ and }\tau\not\prec\sigma\\}\,.$

That is, $Z_{\bm{a}}^{\prec}$ is the subset of $B$ such that for all $z$ in
$Z_{\bm{a}}^{\prec}$, the strict partial order $\prec_{f_{\bm{a}}(\cdot,z)}$
is the same as $\prec$.

######  Lemma 4.13.

Let $A$ be defined as in (8). If $\bm{a}\in A$ is such that

  1. (1)

every $Y\in\mathcal{Y}_{\bm{a}}$ is a manifold, where $\mathcal{Y}_{\bm{a}}$
is defined as in (15),

  2. (2)

$M_{\bm{a},i}\pitchfork M_{\bm{a},j}$ for all $i\neq j$, where $M_{\bm{a},i}$
is defined as in (9),

  3. (3)

$\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a manifold for
all sets $\\{(i_{r},j_{r}\\}_{r=1}^{m}$ of index pairs, and

  4. (4)

$T_{p}\Big{(}\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=\bigcap_{r=1}^{m}T_{p}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$
for all sets $\\{(i_{r},j_{r}\\}_{r=1}^{m}$ of index pairs and all
$p\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$,

then $\mathcal{Y}_{\bm{a}}$ is locally finite.

###### Proof.

Let $p$ be a point in $B$. There are finitely many strict partial orders
$\prec_{1},\ldots,\prec_{i}$ on $\mathcal{K}$. By Lemma A.6, we have that for
each $j\in\\{1,\ldots,i\\}$, there is a subset
$\mathcal{Y}^{\prec_{j}}_{\bm{a}}\subseteq\mathcal{Y}_{\bm{a}}$ such that
$Z^{\prec_{j}}_{\bm{a}}=\bigcup_{Y\in\mathcal{Y}^{\prec_{j}}_{\bm{a}}}Y$. For
each $j$, the point $p$ has a neighborhood $U_{j}$ that intersects at most one
$Y\in\mathcal{Y}^{\prec_{j}}_{\bm{a}}$ by Lemma A.7. Therefore
$\bigcap_{j=1}^{i}U_{j}$ is a neighborhood of $p$ that intersects at most $i$
elements of $\mathcal{Y}_{\bm{a}}$. ∎

######  Lemma 4.14.

Let $A$ be defined as in (8). If $\bm{a}\in A$ is such that

  1. (1)

every $S\in\overline{E_{\bm{a}}^{n-\ell}}$ is an $\ell$-dimensional smooth
submanifold for every $\ell\in\\{1,\ldots,n\\}$, where
$\overline{E_{\bm{a}}^{n-\ell}}$ is defined as in (12),

  2. (2)

$M_{\bm{a},i}\pitchfork M_{\bm{a},j}$ for all $i\neq j$, where $M_{\bm{a},i}$
is defined as in (9),

  3. (3)

$\bigcap_{i=r}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a manifold for
all sets $\\{(i_{r},j_{r}\\}_{r=1}^{m}$ of index pairs, and

  4. (4)

$T_{p}\Big{(}\bigcap_{i=r}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=\bigcap_{i=1}^{m}T_{p}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$
for all sets $\\{(i_{r},j_{r}\\}_{r=1}^{m}$ of index pairs and all
$p\in\bigcap_{i=r}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$,

then $\mathcal{Y}_{\bm{a}}$ satisfies the Axiom of the Frontier in Definition
4.1, where $\mathcal{Y}_{\bm{a}}$ is defined as in (15)

###### Proof.

By Lemma 4.10, each $Y\in\mathcal{Y}_{\bm{a}}$ is a manifold. Let $Y_{\alpha}$
be an element of $\mathcal{Y}_{\bm{a}}$. It suffices to show that if
$Y_{\beta}\neq Y_{\alpha}$ is another element of $\mathcal{Y}_{\bm{a}}$ and
$Y_{\beta}\cap\partial Y_{\alpha}\neq\emptyset$, where $\partial Y_{\alpha}$
denotes the boundary of the manifold $Y_{\alpha}$, then
$Y_{\beta}\subseteq\partial Y_{\alpha}$.

By Lemma 4.11, the simplex order induced by $f$ is constant on each $Y$, so
there is a strict partial order $\prec_{\alpha}$ on $\mathcal{K}$ such that
$\prec_{f_{\bm{a}}(\cdot,y)}$ is the same as $\prec_{\alpha}$ for all $y\in
Y_{\alpha}$. Let $Z^{\prec_{\alpha}}_{\bm{a}}$ be defined as in (4.2). By
Lemma A.6, there is a subset
$\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}\subseteq\mathcal{Y}_{\bm{a}}$ such that
$Y_{\alpha}\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}$ and
$Z^{\prec_{\alpha}}_{\bm{a}}=\bigcup_{Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}}Y$.
We have $\partial
Z^{\prec_{\alpha}}_{\bm{a}}=\bigcup_{Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}}\partial
Y$ because $\mathcal{Y}_{\bm{a}}$ is locally finite by Lemma 4.13. Therefore,

(18) $Y_{\beta}\cap\partial
Z^{\prec_{\alpha}}_{\bm{a}}=\bigcup_{Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}}(Y_{\beta}\cap\partial
Y)\,.$

By Lemmas A.6 and A.8, we have that if $Y\in\mathcal{Y}_{\bm{a}}$ intersects
$\partial Z^{\prec_{\alpha}}_{\bm{a}}$, then $Y\subseteq\partial
Z^{\prec_{\alpha}}_{\bm{a}}$. Therefore $Y_{\beta}\subseteq\partial
Z^{\prec_{\alpha}}_{\bm{a}}$ because $Y_{\beta}\cap\partial
Z^{\prec_{\alpha}}_{\bm{a}}$ contains $Y_{\beta}\cap\partial
Y_{\alpha}\neq\emptyset$. Together with (18), this shows that

(19)
$Y_{\beta}=\bigcup_{Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}}(Y_{\beta}\cap\partial
Y)\,.$

By Lemma A.7, every point in $B$ has a neighborhood that intersects at most
one $Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}$, so $\partial
Y^{\prime}\cap\partial Y=\emptyset$ for all
$Y,Y^{\prime}\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}$ such that $Y\neq
Y^{\prime}$. Because $Y_{\beta}$ is connected (by definition) and
$Y_{\beta}\cap\partial Y_{\alpha}\neq\emptyset$, we must have that
$Y_{\beta}\cap\partial Y=\emptyset$ for all
$Y\in\mathcal{Y}^{\prec_{\alpha}}_{\bm{a}}$ such that $Y\neq Y_{\alpha}$. By
(19),

$Y_{\beta}=Y_{\beta}\cap\partial Y_{\alpha}\subseteq\partial Y_{\alpha}\,.$

∎

######  Theorem 4.15.

Let $B$ be a smooth compact $n$-dimensional manifold. For every $\bm{a}\in A$,
define $\\{B^{m}_{\bm{a}}\\}_{m=0}^{n}$ as in (14), with $A$ defined as in
(8). For almost every $\bm{a}\in A$, we have that
$\\{B^{m}_{\bm{a}}\\}_{m=0}^{n}$ is a stratification of $B$. In each stratum
$Y$, the simplex order induced by $f_{\bm{a}}$ is constant. (In other words,
there is a strict partial order $\prec_{Y}$ on $\mathcal{K}$ such that
$\prec_{f_{\bm{a}}(\cdot,y)}$ is the same as $\prec_{Y}$ for all $y\in Y$.)
Consequently, the set $\\{(\sigma_{b},\sigma_{d})\\}$ of (birth, death)
simplex pairs is constant in each stratum $Y$ and for any $p\in Y$, the
persistence diagram $PD(f_{p})$ consists of the diagonal (with infinite
multiplicity) and the multiset $\\{(f(\sigma_{b},p),f(\sigma_{d},p))\\}$.

###### Proof.

By Lemmas 4.8, 4.9, 4.10, 4.12, 4.13, and 4.14,
$\\{B_{\bm{a}}^{m}\\}_{m=0}^{n}$ is a stratification of $B$ for almost every
$\bm{a}\in A$. By Lemma 4.11, the simplex order induced by $f_{\bm{a}}$ is
constant in each stratum $Y\in\mathcal{Y}_{\bm{a}}$ whenever
$\\{B_{\bm{a}}^{m}\\}_{m=0}^{n}$ is a stratification of $B$. The last
statement of Theorem 4.15 follows from Lemma 2.3. ∎

##  5\. Monodromy in PD Bundles

######  Definition 5.1 (Local section).

Let $(E,B,\pi)$ be a PD bundle. A _local section_ is a continuous map $s:U\to
E$, where $U$ is an open set in $B$ and $\pi\circ s(p)=p$ for all $p\in U$.

For example, consider a vineyard, in which $B$ is an interval $I$ in
$\mathbb{R}$. Let $(t_{0},T)$ be an open interval in $I$. A local section in
the vineyard is a map $s:(t_{0},T)\to E$ that parameterizes an open subset of
one of the vines (a curve in $\mathbb{R}^{3}$).

######  Definition 5.2 (Global section).

Let $(E,B,\pi)$ be a PD bundle. A _global section_ is a continuous map $s:B\to
E$ with $\pi\circ s(p)=p$ for all $p\in B$. In particular, a _nontrivial
global section_ is a global section $s:B\to E$ such that there exists a
$p_{*}\in B$ for which $s(p_{*})$ is not on the diagonal of $PD(f_{p_{*}})$.

In a vineyard, every local section can be extended to a global section. In
other words, we can trace out how the persistence of a single homology class
changes over $B=[t_{0},t_{1}]\subseteq\mathbb{R}$, so there are individual
“vines” in the vineyard. We will show that local sections of a PD bundle
cannot necessarily be extended to global sections. Consequently, a PD bundle
does not necessarily have a decomposition of the form (3); if it does, then
each $\gamma$ is a global section.

######  Proposition 5.3.

There is a PD bundle $(E,B,\pi)$ for which no nontrivial global sections
exist.

###### Proof.

The proof is constructive. Let $\mathcal{K}$ be the simplicial complex in
Figure 4a, which has vertices $0$, $1$, $2$, and $3$. Let $a$ be the edge with
vertices $(0,1)$, let $b$ be the edge with vertices $(0,2)$, let $c$ be the
triangle with vertices $(0,1,2)$, and let $d$ be the triangle with vertices
$(0,2,3)$.

Let $f:\mathcal{K}\times\mathbb{R}^{2}\to\mathbb{R}$ be a continuous fibered
filtration function that satisfies the following conditions:

$\displaystyle f(c,(x,y)),f(d,(x,y))$
$\displaystyle>f(a,(x,y)),f(b,(x,y))>0\qquad\text{for all
}(x,y)\in\mathbb{R}^{2}\,,$ $\displaystyle f(a,(x,y))$
$\displaystyle>f(b,(x,y))\,,\qquad y>0\,,$ $\displaystyle f(a,(x,y))$
$\displaystyle<f(b,(x,y))\,,\qquad y<0\,,$ $\displaystyle f(c,(x,y))$
$\displaystyle>f(d,(x,y))\,,\qquad x>0\,,$ $\displaystyle f(c,(x,y))$
$\displaystyle<f(d,(x,y))\,,\qquad x<0\,,$ $\displaystyle f(\sigma,(x,y))$
$\displaystyle=0\,,\qquad\text{for all other }\sigma,\text{ for all }x,y\,.$

The conditions on the fibered filtration function $f$ are illustrated in
Figure 4b.

(a)

(b)

(c)

Figure 4. (A) The simplicial complex $\mathcal{K}$ that is defined in the
proof of Proposition 5.3. (B) The conditions on the fibered filtration
$f:\mathcal{K}\times\mathbb{R}^{2}\to\mathbb{R}$ that is defined in the proof
of Proposition 5.3. (C) The (birth, death) simplex pairs in each quadrant for
the 1D PH.

These conditions imply that simplices $a$ and $b$ swap their order along the
$x$-axis and the simplices $c$ and $d$ swap their order along the $y$-axis.

In Figure 4c, we list the (birth, death) simplex pairs for the 1D PH in each
quadrant. In quadrants $1$, $2$, and $4$, the simplex pairs are $(a,c)$ and
$(b,d)$. In quadrant $3$, the simplex pairs are $(a,d)$ and $(b,c)$.

Let $(E,\mathbb{R}^{2},\pi)$ be the corresponding PD bundle, where
$E=\\{((x,y),z)\mid(x,y)\in\mathbb{R}^{2},z\in PD_{1}(f(\cdot,(x,y)))\\}$ is
the total space and $\pi$ is the projection to $\mathbb{R}^{2}$. We will show
that $(E,\mathbb{R}^{2},\pi)$ has no nontrivial global sections.

If $s:B\to E$ is a global section and $s(p_{*})$ is on the diagonal of
$\text{PD}(f(\cdot,p_{*}))$ for some $p_{*}\in B$, then $s$ is a trivial
section because $f(\sigma_{b},p)\neq f(\sigma_{d},p)$ for all $p$ for any
(birth, death) simplex pairs $(\sigma_{b},\sigma_{d})$ at $p$. Therefore, if
$s:B\to E$ is a nontrivial global section, $s(p)$ is not on the diagonal of
$\text{PD}(f(\cdot,p))$ for any $p\in B$.

Suppose that $\gamma:[0,1]\to E$ is a continuous path such that $\gamma(u)$ is
not on the diagonal of $\text{PD}(f(\cdot,(x,y)))$ for any
$(x,y)\in\mathbb{R}^{2}$ and such that

(20) $\pi\circ\gamma(u)=\theta(u):=(\cos(2\pi u+\pi/4),\sin(2\pi u+\pi/4))\in
S^{1}\,.$

That is, $\pi\circ\gamma$ is a parameterization of $S^{1}$ that starts in the
first quadrant of $\mathbb{R}^{2}$ at $p_{0}=(\sqrt{2}/2,\sqrt{2}/2)$. The
path $\gamma$ is determined uniquely by its initial condition $\gamma(0)$. The
simplex pairs in the first quadrant are $(a,c)$ and $(b,d)$, so $\gamma(0)$
equals either $(p_{0},(f(a,p_{0}),f(c,p_{0})))$ or
$(p_{0},(f(b,p_{0}),f(d,p_{0})))$. In Figure 5, we illustrate the two
possibilities for the path $\gamma$. If
$\gamma(0)=(p_{0},(f(a,p_{0}),f(c,p_{0})))$, then
$\gamma(1)=(p_{0},(f(b,p_{0}),f(d,p_{0})))$; if
$\gamma(0)=(p_{0},(f(b,p_{0}),f(d,p_{0})))$, then
$\gamma(1)=(p_{0},(f(a,p_{0}),f(c,p_{0})))$. In either case,
$\gamma(0)\neq\gamma(1)$.

Figure 5. A visualization of the two choices for the path $\gamma:[0,1]\to E$
in the proof of Proposition 5.3, where $E$ is the total space of the PD
bundle. We show $10$ fibers of the PD bundle for various points
$p\in\mathbb{R}^{2}$. The first nine PDs (labeled $0$ through $8$) are PDs for
points $p\in S^{1}$; the $k$th PD is the PD at $t_{k}=\theta(u_{k})$, where
$u_{k}=k/8$ and $\theta(u)$ is the parameterization of $S^{1}$ given by (20).
Note that $\theta(0)=(\sqrt{2}/2,\sqrt{2}/2)\in S^{1}$. The two choices for
the path $\gamma(u)$, which depend only on the choice of $\gamma(u_{0})$, are
shown in red and blue, respectively. For each $k$, the red (respectively,
blue) dot in the $k$th PD is equal to $\gamma(u_{k})$ when $\gamma(u_{0})$ is
the red (respectively, blue) point in the $0$th PD. Observe that
$\gamma(u_{0})\neq\gamma(u_{8})$ even though $p_{0}=p_{8}$. The unlabeled PD
at the origin is the PD for the origin in $\mathbb{R}^{2}$, at which there is
a “singularity.”

To obtain a contradiction, suppose that there were a nontrivial global section
$s:\mathbb{R}^{2}\to E$. Let $\gamma:[0,1]\to E$ be the path
$\gamma=s\circ\theta$, where $\theta$ is the parameterization of $S^{1}$
defined in (20). Then $\gamma(0)\neq\gamma(1)$ because $\gamma$ is a path
satisfying (20), but $\gamma(0)=s(p_{0})=\gamma(1)$. ∎

Note that we will use the fibered filtration
$f:\mathcal{K}\times\mathbb{R}^{2}\to\mathbb{R}$ that was constructed in
Proposition 5.3 as a running example throughout Section 6.2.

######  Remark 5.4.

Even when $\dim(B)=1$, it is not guaranteed that a nontrivial global section
exists. To see this, consider the 1D PH of the fibered filtration function
above restricted to $S^{1}\subseteq\mathbb{R}^{2}$. In this example,
$\dim(B)=1$ and a nontrivial global section does not exist.

######  Remark 5.5.

In the example of Proposition 5.3, it was the “singularity” (the point
$(0,0)\in\mathbb{R}^{2}$ at which the PD had a point of multiplicity greater
than one) that prevented the existence of a nontrivial global section.
Restricting the PD bundle to $B^{\prime}:=\mathbb{R}^{2}\setminus\\{(0,0)\\}$
yields a true fiber bundle; each fiber is homeomorphic to the disjoint union
of a line (the diagonal) and two points (the off-diagonal points). It is well
known that fiber bundles over contractible spaces are trivial (i.e., the total
space is homeomorphic to the product of the base and a fiber.) However,
$B^{\prime}$ is not contractible, so our PD bundle restricted to $B^{\prime}$
is not guaranteed to be trivial. Indeed, what we showed in Proposition 5.3 is
that it is not. By comparison to a vineyard,

  1. (1)

Singularities do not occur for generic fibered filtrations
$f:\mathcal{K}\times\mathbb{R}\to\mathbb{R}$. A singularity occurs at
$p_{*}\in B$ when there are two (birth, death) simplex pairs
$(\sigma_{b}^{1},\sigma_{d}^{1})$, $(\sigma_{b}^{2},\sigma_{d}^{2})$ at
$p_{*}$ such that $p_{*}\in I(\sigma_{b}^{1},\sigma_{b}^{2})\cap
I(\sigma_{d}^{1},\sigma_{d}^{2})$. When $\dim(B)=1$, the intersection
$I(\sigma_{b}^{1},\sigma_{b}^{2})\cap I(\sigma_{d}^{1},\sigma_{d}^{2})$ is
empty in the generic case, so singularities do not typically exist when
$\dim(B)=1$.

  2. (2)

Even when singularities do occur in a vineyard, there should not be monodromy
in the vineyard. As in the example above, we can remove the singularities from
$\mathbb{R}$ to obtain a disjoint union of intervals $B_{1},\ldots,B_{m}$ such
that when we restrict the vineyard to a $B_{i}$, we have a fiber bundle.
Intervals in $\mathbb{R}$ are contractible, so these fiber bundles must be
trivial. By continuity, we can glue together the fiber bundles over each
$B_{i}$ to see that our PD bundle cannot have monodromy.

##  6\. A Compatible Cellular Sheaf

For a given fibered filtration function that induces a stratification of $B$
as in Theorem 4.15, we construct a compatible cellular sheaf. We discuss a
motivating example in Section 6.1, and give the definition in Section 6.2.

###  6.1. A motivating example

Again consider the example in the proof of Proposition 5.3, and also again
consider the path $\gamma:[0,1]\to E$ that is determined uniquely by the
choice of

$\gamma(0)\in\\{(p_{0},(f(a,p_{0}),f(c,p_{0}))),(p_{0},(f(b,p_{0}),f(d,p_{0})))\\}\,,$

where $p_{0}=(\sqrt{2}/2,\sqrt{2}/2)$. The two possibilities for the path
$\gamma$ are illustrated in Figure 5. For example, if
$\gamma(0)=(p_{0},(f(a,p_{0}),f(c,p_{0})))$, then

$\gamma(u)=\begin{cases}\Big{(}\theta(u),(f(a,\theta(u)),f(c,\theta(u)))\Big{)}\,,&u\in[0,1/8]\\\
\Big{(}\theta(u),(f(a,\theta(u)),f(c,\theta(u)))\Big{)}\,,&u\in[1/8,3/8]\\\
\Big{(}\theta(u),(f(b,\theta(u)),f(c,\theta(u)))\Big{)}\,,&u\in[3/8,5/8]\\\
\Big{(}\theta(u),(f(b,\theta(u)),f(d,\theta(u)))\Big{)}\,,&u\in[5/8,7/8]\\\
\Big{(}\theta(u),(f(b,\theta(u)),f(d,\theta(u)))\Big{)}\,,&u\in[7/8,1]\,,\\\
\end{cases}$

where $\theta(u)$ is the parameterization of $S^{1}$ given by (20). As we move
through the quadrants of $\mathbb{R}^{2}$, the point in the PD that represents
the pair $(a,c)$ in the first quadrant becomes the point that represents the
pair $(a,c)$ in the second quadrant, which becomes the point that represents
the pair $(b,c)$ in the third quadrant, which becomes the point that
represents the pair $(b,d)$ in the fourth quadrant, which becomes the point
that represents the pair $(b,d)$ in the first quadrant. One can do a similar
analysis for the case in which $\gamma(0)=(p_{0},(f(b,p_{0}),f(d,p_{0})))$.

This analysis yields a bijection of the (birth, death) simplex pairs for any
pair of adjacent quadrants. We illustrate the bijections in Figure 6. The
bijection between the simplex pairs in a given quadrant and one of its
adjacent quadrants is the same as the bijection defined by the update rule of
Cohen-Steiner et al. [14] for updating the simplex pairs in a vineyard. A
combinatorial perspective on Proposition 5.3 is that there is no consistent
way of choosing a simplex pair in each quadrant such that if
$(\sigma_{b},\sigma_{d})$ is the (birth, death) simplex pair chosen for a
given quadrant and $(\tau_{b},\tau_{d})$ is the (birth, death) simplex pair
chosen for an adjacent quadrant, then $(\sigma_{b},\sigma_{d})$ and
$(\tau_{b},\tau_{d})$ are matched in the bijection between the two quadrants.
This is because if we choose an initial simplex pair in one of the quadrants
and then walk in a circle through the other quadrants, then the simplex pair
at which we finish is different from the initial simplex pair. For example, if
we start at $(a,c)$ in the first quadrant, then we finish at $(b,d)$ when we
return to the first quadrant, and vice versa. This is a discrete way of
illustrating the non-existence of a nontrivial global section.

Figure 6. The (birth, death) simplex pairs in each quadrant for the 1D PH of
the fibered filtration function in the proof of Proposition 5.3 (see also
Figure 4). For each pair of adjacent quadrants, there is a bijection between
their sets of simplex pairs; this bijection is equal to the bijection given by
the update rule of Cohen-Steiner et al. [14]. The red lines connect simplex
pairs that are in bijection with each other.

###  6.2. Definition of a compatible cellular sheaf

I generalize the discussion in Section 6.1 to fibered filtration functions of
the form $f:\mathcal{K}\times B\to\mathbb{R}$ that have a stratification (see
Definition 4.1) of $B$ such that in each stratum $Y$, the simplex order that
is induced by $f$ is constant. (In other words, there is a strict partial
order $\prec_{Y}$ on $\mathcal{K}$ such that $\prec_{f(\cdot,y)}$ is the same
as $\prec_{Y}$ for all $y\in Y$.) Theorem 4.15 guarantees that such a
stratification exists for generic fibered filtration functions, and
Proposition 4.5 guarantees that such a stratification exists for all
piecewise-linear fibered filtration functions. We denote the set of strata by
$\mathcal{Y}=\\{Y_{\alpha}\\}_{\alpha\in J}$ for some index set $J$.

######  Definition 6.1.

Suppose that $\mathcal{F}$ is a Set-valued cellular sheaf whose cell complex,
stalks, and morphisms are of the following form:

  1. (1)

The cell complex: The cell complex on which $\mathcal{F}$ is constructed is
the graph $G$ such that there is a vertex $v_{\alpha}$ for each stratum
$Y_{\alpha}\in\mathcal{Y}$ and an edge
$e_{\beta,\alpha}=(v_{\beta},v_{\alpha})$ if $Y_{\beta}\in\mathcal{Y}$ is a
face of $Y_{\alpha}$. The $0$-cells of the cell complex are the vertices of
$G$ and the $1$-cells are the edges of $G$.

  2. (2)

The stalks: Let $S_{\alpha}$ denote the set of (birth, death) simplex pairs
for a stratum $Y_{\alpha}$. The stalk at a $0$-cell $v_{\alpha}\in G$ is
$\mathcal{F}(v_{\alpha}):=S_{\alpha}$. For a $1$-cell $e_{\beta,\alpha}\in G$,
where $Y_{\beta}$ is a face of $Y_{\alpha}$, the stalk at $e_{\beta,\alpha}$
is $\mathcal{F}(e_{\beta,\alpha}):=S_{\alpha}$.

  3. (3)

The morphisms: If $Y_{\beta}\in\mathcal{Y}$ is a face of
$Y_{\alpha}\in\mathcal{Y}$, then the morphism $\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}:\mathcal{F}(v_{\alpha})\to\mathcal{F}(e_{\beta,\alpha})$ is
the identity map and the morphism $\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}:\mathcal{F}(v_{\beta})\to\mathcal{F}(e_{\beta,\alpha})$ is

$\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}:=\phi^{\text{idx}_{\beta},\,\text{idx}_{\alpha}}\,,$

where $\phi^{\text{idx}_{\beta},\,\text{idx}_{\alpha}}$ is of the form in (4)
and $\text{idx}_{\alpha}:\mathcal{K}\to\\{1,\ldots,N\\}$ and
$\text{idx}_{\beta}:\mathcal{K}\to\\{1,\ldots,N\\}$ are the simplex indexings
(recall Definition 2.2) on $Y_{\alpha}$ and $Y_{\beta}$, respectively. (Recall
that by Lemma 4.11, the simplex order induced by $f$ is constant within
$Y_{\alpha}$ and within $Y_{\beta}$.)

Then the cellular sheaf $\mathcal{F}$ is a _compatible cellular sheaf_ for the
fibered filtration function $f:\mathcal{K}\times B\to\mathbb{R}$.

It is not guaranteed that there is a _unique_ compatible cellular sheaf for a
given fibered filtration function $f$. Although the cell complex (the graph
$G$) is determined uniquely by $f$, the stalks and morphisms are not. Recall
from Definition 2.2 that the simplex indexing that is induced by $f$ may
depend on an intrinsic indexing $\sigma_{1},\ldots,\sigma_{N}$ of the
simplices in $\mathcal{K}$. (The intrinsic indexing breaks ties when two
simplices have the same filtration value.) For a stratum $Y_{\alpha}$ such
that $f(\sigma,y)=f(\tau,y)$ for all $y\in Y_{\alpha}$ for some pair
$(\sigma,\tau)$ of simplices, the simplex indexing
$\text{idx}_{f(\cdot,Y_{\alpha})}$ depends on the intrinsic indexing, so
$S_{\alpha}$ may not be determined uniquely by $f$. If $S_{\alpha}$ is not
determined uniquely by $f$, then for any face $Y_{\beta}$ of $Y_{\alpha}$, the
stalks $\mathcal{F}(v_{\alpha})$ and $\mathcal{F}(e_{\beta,\alpha})$ are not
determined uniquely by $f$. As discussed in Remark 2.4, a bijection
$\phi^{\text{idx}_{\beta},\text{idx}_{\alpha}}$ of the form in (4) is not
determined uniquely by $f$ if $\text{idx}_{\beta}$ and $\text{idx}_{\alpha}$
differ by more than the transposition of two consecutive simplices. Therefore,
the morphism $\mathcal{F}_{v_{\beta}\leq e_{\beta,\alpha}}$ is not necessarily
determined uniquely by $f$.

However, many aspects of the stalks and morphisms _are_ determined uniquely by
$f$. Suppose that $Y_{\beta}\in\mathcal{Y}$ is a face of
$Y_{\alpha}\in\mathcal{Y}$. If $f(\sigma,y)\neq f(\tau,y)$ for all $y$ in
$Y_{\alpha}$ and all simplices $\sigma\neq\tau$, then the simplex indexing
$\text{idx}_{f(\cdot,Y_{\alpha})}$ is determined uniquely by $f$, so the
stalks $\mathcal{F}(v_{\alpha})$ and $\mathcal{F}(e_{\beta,\alpha})$ are
determined uniquely by $f$. Theorem 4.15 guarantees that this is the generic
case when $Y_{\alpha}$ is an $n$-dimensional stratum (where $n=\dim(B)$).
There are also conditions under which a morphism is determined uniquely by
$f$. The morphism $\mathcal{F}_{v_{\alpha}\leq e_{\beta,\alpha}}:S_{\alpha}\to
S_{\alpha}$ must be the identity map. The morphism
$\mathcal{F}_{v_{\alpha}\leq
e_{\beta,\alpha}}:=\phi^{\text{idx}_{\beta},\text{idx}_{\alpha}}$ is
determined uniquely by $f$ when $\text{idx}_{\beta}$ and $\text{idx}_{\alpha}$
differ by the transposition of two consecutive simplices. Theorem 4.15
guarantees that this is the generic case when $Y_{\beta}$ is a “top-
dimensional” face of $Y_{\alpha}$ (i.e., when
$\dim(Y_{\beta})=\dim(Y_{\alpha})-1$).

######  Example 6.2.

Again consider a fibered filtration function
$f:\mathcal{K}\times\mathbb{R}^{2}\to\mathbb{R}$ of the form defined in
Proposition 5.3, with $\mathcal{K}$ defined as in Figure 4a with $N=11$
simplices. We construct a compatible cellular sheaf $\mathcal{F}$ as follows.

  1. (1)

The cell complex: The strata are the open quadrants $Q_{1},\ldots,Q_{4}$, the
open half-axes $A_{12},A_{23},A_{34},A_{14}$ with $A_{ij}=(\partial
Q_{i}\cap\partial Q_{j})\setminus\\{\bm{0}\\}$, and the point
$\bm{0}\in\mathbb{R}^{2}$. The associated graph $G$ (the cell complex for
$\mathcal{F}$) has a vertex $v_{Q_{i}}$ for the $i$th quadrant, a vertex
$v_{A_{ij}}$ for the $(i,j)$th half-axis, and a vertex $v_{\bm{0}}$ for the
point $\bm{0}$. The graph $G$ has edges $(v_{A_{ij}},v_{Q_{i}})$ and
$(v_{A_{ij}},v_{Q_{j}})$ for each half-axis $A_{ij}$, and it has an edge
$(v_{\bm{0}},v)$ for every vertex $v\in G$ such that $v\neq v_{\bm{0}}$.

  2. (2)

The stalks:  We index the simplices of $\mathcal{K}$ such that $\sigma_{8}=a$,
$\sigma_{9}=b$, $\sigma_{10}=c$, and $\sigma_{11}=d$, where $a$, $b$, $c$, $d$
are the simplices defined in Figure 4a. The stalk at $v_{Q_{1}}$ is
$S_{Q_{1}}=\\{(a,c),(b,d)\\}$. The vertices $v_{Q_{2}}$ and $v_{A_{12}}$ have
the same stalk $\\{(a,c),(b,d)\\}$; the vertices $v_{Q_{3}}$, $v_{A_{23}}$,
and $v_{34}$ have the same stalk $\\{(a,d),(b,c)\\}$; and the vertices
$v_{Q_{4}}$ and $v_{A_{14}}$ have the same stalk $\\{(b,d),(a,c)\\}$. The
stalks at the edges of $G$ are determined by the stalks at the vertices. In
this example, the stalks at the vertices or edges that correspond to 2D strata
are determined uniquely by $f$, but the stalks at the vertices and edges that
correspond to 0D or 1D strata depend on our choice of intrinsic indexing.

  3. (3)

The morphisms: There are only three distinct nonidentity morphisms. The first
two are

$\displaystyle\mathcal{F}_{v_{A_{23}}\leq
e_{(A_{23},Q_{2})}},\,\mathcal{F}_{v_{\bm{0}}\leq
e_{(\bm{0},Q_{2})}}:\\{(a,c),(b,d)\\}$ $\displaystyle\to\\{(a,d),(b,c)\\}$
$\displaystyle(a,c)$ $\displaystyle\mapsto(b,c)$ $\displaystyle(b,d)$
$\displaystyle\mapsto(a,d)\,,$ $\displaystyle\mathcal{F}_{v_{A_{34}}\leq
e_{(A_{34},Q_{4})}},\,\mathcal{F}_{v_{\bm{0}}\leq
e_{(\bm{0},Q_{4})}}:\\{(a,d),(b,c)\\}$ $\displaystyle\to\\{(a,c),(b,d)\\}$
$\displaystyle(a,d)$ $\displaystyle\mapsto(a,c)$ $\displaystyle(b,c)$
$\displaystyle\mapsto(b,d)\,.$

The third distinct nonidentity morphism is a map

$\mathcal{F}_{v_{\bm{0}}\leq
e_{(\bm{0},Q_{1})}}:\\{(a,d),(b,c)\\}\to\\{(a,c),(b,d)\\}\,.$

As we move from $\bm{0}$ to $Q_{1}$, we swap the simplex indices of $a$ and
$b$ and we also swap the simplex indices of $c$ and $d$ (in the simplex
indexing induced by $f$). The morphism is not canonical because the bijection
$\phi^{\text{idx}_{\bm{0}},\text{idx}_{Q_{1}}}$ depends on whether one first
swaps $a$ and $b$ or one first swaps $c$ and $d$. Therefore, we may define
either

$\displaystyle\mathcal{F}_{v_{\bm{0}}\leq e_{(\bm{0},Q_{1})}}:(a,d)$
$\displaystyle\mapsto(a,c)\,,$ $\displaystyle(b,c)$
$\displaystyle\mapsto(b,d)$

or

$\displaystyle\mathcal{F}_{v_{\bm{0}}\leq e_{(\bm{0},Q_{1})}}:(a,d)$
$\displaystyle\mapsto(b,d)\,,$ $\displaystyle(b,c)$
$\displaystyle\mapsto(a,c)\,.$

Both choices results in a compatible cellular sheaf.

###  6.3. Sections of the cellular sheaf

Let $\mathcal{F}$ be any compatible cellular sheaf for a fibered filtration
$f:\mathcal{K}\times B\to\mathbb{R}$. We write

(21) $\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}=\Big{(}\mathcal{F}^{b}_{v_{\beta}\leq
e_{\beta,\alpha}},\,\mathcal{F}^{d}_{v_{\beta}\leq
e_{\beta,\alpha}}\Big{)}\,,$

where $\mathcal{F}^{b}_{v_{\beta}\leq e_{\beta,\alpha}}:S_{\beta}\to
S_{\alpha}$ maps a pair $(\sigma_{b},\sigma_{d})\in S_{\beta}$ to the birth
simplex of $\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}((\sigma_{b},\sigma_{d}))$ and
$\mathcal{F}^{d}_{v_{\beta}\leq e_{\beta,\alpha}}:S_{\beta}\to S_{\alpha}$
maps $(\sigma_{b},\sigma_{d})\in S_{\beta}$ to the death simplex of
$\mathcal{F}_{v_{\beta}\leq e_{\beta,\alpha}}((\sigma_{b},\sigma_{d}))$.
(Recall that $S_{\alpha}$, $S_{\beta}$ are the stalks at the vertices
$v_{\alpha}$, $v_{\beta}$ that are associated with the strata $Y_{\alpha}$,
$Y_{\beta}$.)

In this subsection, we show that one can view sections of $\mathcal{F}$ as
sections of the associated PD bundle.

######  Lemma 6.3.

Let $Y_{\beta}$ be a face of $Y_{\alpha}$. Assume that $f:\mathcal{K}\times
B\to\mathbb{R}$ is continuous (i.e., $f(\sigma,\cdot)$ is continuous for all
simplices $\sigma$ in $\mathcal{K}$). Then for any point $p\in Y_{\beta}$ and
any pair $(\sigma_{b},\sigma_{d})$ in $\mathcal{F}(v_{\beta})$, we have

(22)
$\big{(}f(\sigma_{b},p),f(\sigma_{d},p)\Big{)}=\Big{(}f(\mathcal{F}^{b}_{v_{\beta}\leq
e_{\beta,\alpha}}((\sigma_{b},\sigma_{d})),p)\,,f(\mathcal{F}^{d}_{v_{\beta}\leq
e_{\beta,\alpha}}((\sigma_{b},\sigma_{d})),p)\Big{)}\,,$

where $\mathcal{F}^{b}$ and $\mathcal{F}^{d}$ are defined as in (21).

###### Proof.

If the simplex orders in $Y_{\alpha}$ and $Y_{\beta}$ differ only by a
transposition of simplices $(\sigma,\tau)$ with consecutive indices in the
orderings, then we must have $f(\sigma,p)=f(\tau,p)$ for all $p\in Y_{\beta}$
because $f$ is continuous and $Y_{\beta}\subseteq\overline{Y_{\alpha}}$. By
definition, $\mathcal{F}_{v_{\beta}\leq e_{\beta,\alpha}}$ is either the
identity map or the map that swaps $\sigma$ and $\tau$ in the pairs that
contain them. In either case, (22) holds because $f(\sigma,p)=f(\tau,p)$ for
all $p\in Y_{\beta}$. Equation (22) holds in general because
$\mathcal{F}_{v_{\beta}\leq e_{\beta,\alpha}}$ is defined as the composition
of such maps. ∎

The following proposition says that a global section of a compatible cellular
sheaf $\mathcal{F}$ corresponds to a global section of the PD bundle.

######  Proposition 6.4.

Let $z_{0}$ be a non-diagonal point in $PD_{q}(f_{p_{0}})$ for some $p_{0}\in
B$, let $(\sigma_{b},\sigma_{d})$ be the (birth, death) simplex pair such that
$z_{0}=(f(\sigma_{b},p_{0}),f(\sigma_{d},p_{0}))$, and let $Y_{0}$ be the
stratum that contains $p_{0}$. Suppose that $\mathcal{F}$ is a compatible
cellular sheaf, and let $v_{0}$ be the vertex in the graph $G$ (see Definition
6.1) that is associated with $Y_{0}$. If there is a global section
$\overline{s}$ of the cellular sheaf $\mathcal{F}$ such that
$\overline{s}(v_{0})=(\sigma_{b},\sigma_{d})$, then there is a global section
$s$ of the PD bundle such that $s(p_{0})=z_{0}$.

###### Proof.

Let $\overline{s}$ be a global section of the cellular sheaf $\mathcal{F}$
such that $\overline{s}(v_{0})=(\sigma_{b},\sigma_{d})$. For every stratum
$Y_{\alpha}$, we write

$\overline{s}(v_{\alpha})=(\overline{s}_{b}(v_{\alpha}),\overline{s}_{d}(v_{\alpha}))\,,$

where $\overline{s}_{b}(v_{\alpha})$ is the birth simplex of
$\overline{s}(v_{\alpha})$ and $\overline{s}_{d}(v_{\alpha})$ is the death
simplex of $\overline{s}(v_{\alpha})$. Let $Y:B\to\\{Y_{\alpha}\\}$ be the
function that maps $p\in B$ to the unique stratum $Y_{\alpha}$ that contains
it.

We define $s:B\to E$ to be the function

$s(p):=\Big{(}p,f(\overline{s}_{b}(v_{Y(p)}),p),f(\overline{s}_{d}(v_{Y(p)}),p)\Big{)}\,.$

To show that $s:B\to E$ is a global section of the PD bundle, it remains to
show that it is continuous. The function $s|_{Y_{\alpha}}$ is continuous for
all strata $Y_{\alpha}$ because $f(\sigma,\cdot)$ is continuous for all
simplices $\sigma\in\mathcal{K}$. Therefore, it suffices to show that
$s|_{\overline{Y_{\alpha}}}$ is continuous on each face $Y_{\beta}$ of
$Y_{\alpha}$. Because $\overline{s}$ is a section of the cellular sheaf,

$\overline{s}(v_{\alpha})=\mathcal{F}_{v_{\beta}\leq
e_{\beta,\alpha}}(\overline{s}(v_{\beta}))\,.$

By Lemma 6.3,

$\Big{(}f(\overline{s}_{b}(v_{\beta}),p),f(\overline{s}_{d}(v_{\beta}),p)\Big{)}=\Big{(}f(\mathcal{F}^{b}_{v_{\beta}\leq
e_{\beta,\alpha}}(\overline{s}(v_{\beta})),p),f(\mathcal{F}^{d}_{v_{\beta}\leq
e_{\beta,\alpha}}(\overline{s}(v_{\beta})),p)\Big{)}$

for all points $p\in Y_{\beta}$. Therefore,

$\Big{(}f(\overline{s}_{b}(v_{\beta}),p),f(\overline{s}_{d}(v_{\beta}),p)\Big{)}=\Big{(}f(\overline{s}_{b}(v_{\alpha}),p),f(\overline{s}_{d}(v_{\alpha}),p)\Big{)}$

for all $p\in Y_{\beta}$, which completes the proof. ∎

##  7\. Conclusions

###  7.1. Summary

In this paper, I introduced the concept of a persistence diagram (PD) bundle,
a framework that can be used to study the persistent homology of a fibered
filtration function (i.e., set of filtrations parameterized by an arbitrary
“base space” $B$). Special cases of PD bundles include vineyards [14], the
persistent homology transform (PHT) [32], the fibered barcode of a
multiparameter persistence module [4], and the barcode-decorated merge tree
[17].

In Theorem 4.15, I proved that if $B$ is a smooth compact manifold, then for
generic fibered filtrations, $B$ is stratified so that the simplex order is
constant within each stratum. When such a stratification exists, the PD bundle
is determined by the PDs at a locally finite (or finite, if $B$ is compact)
subset of points in $B$. In Proposition 4.5, I showed that every piecewise-
linear PD bundle has such a stratification into polyhedra. This polyhedral
stratification is utilized in [23] in an algorithm for computing piecewise-
linear PD bundles.

I showed that, unlike vineyards, which PD bundles generalize, not every local
section of a PD bundle can be extended to a global section (see Proposition
5.3). The implication is that PD bundles do not necessarily decompose into
“vines” in the way that vineyards do (see (3)).

Lastly, I introduced a cellular sheaf that is compatible with a given PD
bundle. In Proposition 6.4, I proved that one can determine whether a local
section can be extended to a global section by determining whether or not
there is an associated global section of a compatible cellular sheaf. A
compatible cellular sheaf is a discrete mathematical data structure for
summarizing the data in a PD bundle.

###  7.2. Discussion

For a given fibered filtration function $f$ with a stratification as in
Theorem 4.15, I defined a compatible cellular sheaf $\mathcal{F}$ over a graph
$G$. It is tempting to instead define an associated cellular sheaf directly on
the stratification of $B$. In particular, when $f$ is piecewise linear, the
strata are polyhedra, so the stratification is guaranteed to be a cellular
decomposition. One could certainly define stalks $\mathcal{F}(Y_{\alpha})$ and
functions $\mathcal{F}(Y_{\beta})\to\mathcal{F}(Y_{\beta})$ in the same way as
in Definition 6.1. The problem is that $\mathcal{F}$ would not necessarily
satisfy the composition condition (see (5)). For instance, this issue occurs
in Example 6.2 for the same reason that the morphism
$\mathcal{F}_{v_{\bm{0}}\leq e_{(\bm{0},Q_{1})}}$ in the example is not
canonical (see the discussion in Example 6.2).

Additionally, I note that one could have defined a compatible cellular cosheaf
rather than a sheaf.

###  7.3. Future research

I conclude with some questions and proposals for future work:

  * •

What are the conditions under which a PD bundle must have a decomposition of
the form (3)?

I conjecture that if $B\setminus B^{*}$ is contractible, where $B^{*}$ is the
set of singularities (i.e., points $p_{*}\in B$ at which there is a point in
PD$(f_{p_{*}})$ with multiplicity greater than one), then there is a
decomposition of the form (3). See the discussion in Remark 5.5.

  * •

What algebraic or computational methods can we use to analyze global sections
and to compute obstructions to the existence of global sections?

It may help to consider the cellular-sheaf perspective from Section 6.2, which
turns the question into a discrete problem that one can study computationally.
One can also generalize Turner’s vineyard-module perspective [33].

  * •

PHT is a PD bundle over the base space $B=S^{n}$. Are there constructible sets
$M\subseteq\mathbb{R}^{n+1}$ for which the associated PHT exhibits monodromy?
What is the geometric interpretation (in terms of $M$)?

  * •

Arya et al. [5] showed that the PHT of a constructible set $M$ can be
calculated by “gluing together” the PHT of smaller, simpler subsets of $M$.
Can one generalize these results to all PD bundles?

  * •

When are PD bundles “stable”?

PD bundles are “fiberwise stable” in the sense that if
$f_{1},f_{2}:\mathcal{K}\times B\to\mathbb{R}$ are two fibered filtrations,
then the bottleneck distance between $\text{PD}(f_{1}(\cdot,p))$ and
$\text{PD}(f_{2}(\cdot,p))$ is bounded above by $\lVert
f_{1}-f_{2}\rVert_{\infty}$ for all $p\in B$ [13]. However, this does not
guarantee that the global structure of a PD bundle is stable. For example, it
is well known that the structure of a vineyard is not stable (see Appendix A.4
for an example). However, vineyards are stable for generic $1$-parameter
filtrations; if none of the vines intersect (which is the generic case), then
sufficiently small perturbations of the filtration result in small
perturbations of each vine in the vine decomposition (see (3)). I expect that
an analogous result holds for generic fibered filtration functions over any
base space $B$.

  * •

It will also be interesting to study real-world applications of PD bundles,
such as the examples that were mentioned in Section 3.1.

## Acknowledgements

I am very grateful for discussions with Andrew Blumberg, which led to the
investigation of monodromy in PD bundles. I also thank Ryan Grady and Karthik
Viswanathan for helpful discussions.

## Appendix

###  A.4. Vineyard instability

For any $\epsilon>0$, we construct two $1$-parameter filtration functions
$f^{+}_{\epsilon}$, $f^{-}_{\epsilon}$ that are $\epsilon$-perturbations of
each other (that is,
$|f^{+}_{\epsilon}(\sigma,p)-f^{-}_{\epsilon}(\sigma,p)|<\epsilon$ for all
simplices $\sigma\in\mathcal{K}$ and all points $p\in B$) but such that for
any bijection between the vines in the respective vineyards, not all of the
matched vines are close to each other. In fact, we can define
$f^{+}_{\epsilon}$ and $f^{-}_{\epsilon}$ so that their vines are arbitrarily
far apart.

We construct our example by restricting the filtration function from Section 5
to certain paths through $\mathbb{R}^{2}$. Let $\mathcal{K}$ and
$f:\mathcal{K}\times\mathbb{R}^{2}\to\mathbb{R}$ be defined as in the proof of
Proposition 5.3. (See Figure 4b.) Because $f$ is continuous, we have that for
any $\epsilon>0$, there is a $\delta>0$ such that
$|f(\sigma,p)-f(\sigma,\bm{0})|<\epsilon/2$ when $\lVert
p\rVert<\sqrt{2}\delta$ and $\sigma$ is any simplex in $\mathcal{K}$. We
define the paths

$\displaystyle\gamma^{+}_{\epsilon}(t)$
$\displaystyle:=\begin{cases}(t,t)\,,&|t|\geq\delta\\\
(-\delta,\delta+2t)\,,&-\delta<t<0\\\ \ (-\delta+2t,\delta)\,,&0\leq
t<\delta\,,\end{cases}$ $\displaystyle\gamma_{\epsilon}^{-}(t)$
$\displaystyle:=\begin{cases}(t,t)\,,&|t|\geq\delta\\\
(\delta+2t,-\delta)\,,&-\delta<t<0\\\ (\delta,-\delta+2t)\,,&0\leq
t<\delta\,.\end{cases}$

See Figure 7 for a plot of the paths $\gamma_{\epsilon}^{\pm}(t)$.

Figure 7. The paths $\gamma_{\epsilon}^{+}(t)$ and $\gamma_{\epsilon}^{-}(t)$.

Let $f^{\pm}_{\epsilon}:\mathcal{K}\times\mathbb{R}\to\mathbb{R}$ be the
$1$-parameter filtration functions defined by
$f^{\pm}(\sigma,t):=f(\sigma,\gamma_{\epsilon}^{\pm}(t))$. By construction,
the filtrations $f^{+}_{\epsilon}$ and $f^{-}_{\epsilon}$ are
$\epsilon$-perturbations of each other.

Let $V^{+}$ and $V^{-}$ be the vineyards for $f^{+}_{\epsilon}$ and
$f^{-}_{\epsilon}$, respectively, for the $1$st degree PH. The vineyards
$V^{\pm}$ each have two vines $v_{1}^{\pm}$, $v_{2}^{\pm}$, which are paths
$v_{i}^{\pm}:\mathbb{R}\to\mathbb{R}^{3}$. The vines are

$\displaystyle v^{+}_{1}(t)$
$\displaystyle=\begin{cases}(f(a,\gamma^{+}(t)),f(d,\gamma^{+}(t)))\,,&t\leq-\delta/2\\\
f(b,\gamma^{+}(t)),f(d,\gamma^{+}(t)))\,,&t>-\delta/2\,,\end{cases}$
$\displaystyle v^{+}_{2}(t)$
$\displaystyle=\begin{cases}(f(b,\gamma^{+}(t)),f(c,\gamma^{+}(t)))\,,&t\leq-\delta/2\\\
f(a,\gamma^{+}(t)),f(c,\gamma^{+}(t)))\,,&t>-\delta/2\,,\end{cases}$
$\displaystyle v^{-}_{1}(t)$
$\displaystyle=\begin{cases}(f(a,\gamma^{-}(t)),f(d,\gamma^{-}(t)))\,,&t\leq-\delta/2\\\
f(a,\gamma^{-}(t)),f(c,\gamma^{-}(t)))\,,&t>-\delta/2\,,\end{cases}$
$\displaystyle v^{-}_{2}(t)$
$\displaystyle=\begin{cases}(f(b,\gamma^{-}(t)),f(c,\gamma^{-}(t)))\,,&t\leq-\delta/2\\\
f(b,\gamma^{-}(t)),f(d,\gamma^{-}(t)))\,,&t>-\delta/2\,.\end{cases}$

There is no bijection $\phi:\\{1,2\\}\to\\{1,2\\}$ such that $v_{1}^{+}$ and
$v_{2}^{+}$ are close to $v_{\phi(1)}^{-}$ and $v_{\phi(2)}^{-}$,
respectively. This is because

$\displaystyle\lVert v_{1}^{+}(t)-v_{1}^{-}(t)\rVert^{2}$
$\displaystyle=|f(b,(t,t))-f(a,(t,t))|^{2}+|f(d,(t,t)))-f(c,(t,t))|^{2}\,,\qquad
t>-\delta/2\,,$ $\displaystyle\lVert v_{1}^{+}(t)-v_{2}^{-}(t)\rVert^{2}$
$\displaystyle=|f(b,(t,t))-f(a,(t,t))|^{2}+|f(d,(t,t)))-f(c,(t,t))|^{2}\,,\qquad
t\leq-\delta/2\,,$ $\displaystyle\lVert v_{2}^{+}(t)-v_{2}^{-}(t)\rVert^{2}$
$\displaystyle=|f(b,(t,t))-f(a,(t,t))|^{2}+|f(d,(t,t)))-f(c,(t,t))|^{2}\,,\qquad
t>-\delta/2\,,$ $\displaystyle\lVert v_{2}^{+}(t)-v_{1}^{-}(t)\rVert^{2}$
$\displaystyle=|f(b,(t,t))-f(a,(t,t))|^{2}+|f(d,(t,t)))-f(c,(t,t))|^{2}\,,\qquad
t\leq-\delta/2$

and we can define $f$ so that $|f(b,(t,t))-f(a,(t,t))|$ and
$|f(d,(t,t))-f(c,(t,t))|$ are arbitrarily large for $t\neq 0$.

###  A.5. Technical Details of Section 4

All notation is defined as in Section 4.

The first series of lemmas is used to prove Lemma 4.12, which shows that for
almost every $\bm{a}\in A$, the tangent space of the intersection of sets
$I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is equal to the intersection of
their tangent spaces.

######  Lemma A.1.

For almost every $\bm{a}\in A$, we have

$T_{p}\Big{(}\bigcap_{j\in J}M_{\bm{a},j}\Big{)}=\bigcap_{j\in
J}T_{p}(M_{\bm{a},j})$

for all $J\subseteq\\{1,\ldots,N\\}$ and all $p\in\bigcap_{j\in
J}M_{\bm{a},j}$.

###### Proof.

Because there are finitely many subsets of $\\{1,\ldots,N\\}$, it suffices to
show that for a given $J\subseteq\\{1,\ldots,N\\}$, we have
$T_{p}\Big{(}\bigcap_{j\in J}M_{\bm{a},j}\Big{)}=\bigcap_{j\in
J}T_{p}(M_{\bm{a},j})$ for all $p\in\bigcap_{j\in J}M_{\bm{a},j}$ for almost
every $\bm{a}\in A$. Let $\\{j_{i}\\}_{i=1}^{k}$ be the elements of $J$, where
$j_{i}<j_{i+1}$ for all $i$. Because transverse intersections are generic, we
have $M_{\bm{a},j_{i}}\pitchfork(M_{\bm{a},j_{1}}\cap\cdots\cap
M_{\bm{a},j_{i-1}})$ for every $i$ for almost every $\bm{a}\in A$. For such an
$\bm{a}\in A$, we have

$T_{p}\Big{(}\bigcap_{j\in J}M_{\bm{a},j}\Big{)}=T_{p}(M_{\bm{a},j_{k}})\cap
T_{p}\Big{(}\bigcap_{i=1}^{k-1}M_{\bm{a},j_{i}}\Big{)}$

because $M_{\bm{a},j_{k}}\pitchfork(M_{\bm{a},j_{1}}\cap\cdots\cap
M_{\bm{a},j_{k-1}})$. Therefore,

$T_{p}\Big{(}\bigcap_{j\in
J}M_{\bm{a},j}\Big{)}=\bigcap_{i=1}^{k}T_{p}(M_{\bm{a},j_{i}})$

by induction on $i$. ∎

######  Lemma A.2.

Let $\bm{a}\in A$, and let $\\{(i_{r},j_{r})\\}_{r=1}^{m}$ be a set of index
pairs such that $\\{i_{r},j_{r}\\}\neq\\{i_{s},j_{s}\\}$ if $r\neq s$. If $B$
is a compact manifold, then there is a finite open cover
$\\{U_{k}\\}_{k=1}^{K}$ and a disjoint partition
$\bigcup_{\ell}J_{\ell,k}=\\{1,\ldots,m\\}$ for each $k$ such that

$\\{i_{r},j_{r}\mid r\in J_{\ell_{1},k}\\}\cap\\{i_{r},j_{r}\mid r\in
J_{\ell_{2},k}\\}=\emptyset$

if $\ell_{1}\neq\ell_{2}$ and

$\pi\Big{(}\bigcap_{r\in J_{\ell,k}}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}\cap U_{k}=\bigcap_{r\in
J_{\ell,k}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U_{k}$

for all $\ell$, where $\pi$ is the projection $\pi:B\times\mathbb{R}\to B$.

###### Proof.

Suppose that $y\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$.
Let $J^{0}_{\ell}:=\\{\ell\\}$ be an initial disjoint partition of
$\\{1,\ldots,m\\}$. By definition, $\pi(M_{\bm{a},i_{\ell}}\cap
M_{\bm{a},j_{\ell}})=I_{\bm{a}}(\sigma_{i_{\ell}},\sigma_{j_{\ell}})$. If
$i_{\ell_{1}}=i_{\ell_{2}}$ for some $\ell_{1}\neq\ell_{2}$, then
$f(\sigma_{j_{\ell_{1}}},y)=f(\sigma_{i_{\ell_{1}}},y)=f(\sigma_{j_{\ell_{2}}},y)$,
so $y\in\pi(M_{{\bm{a}},i_{\ell_{1}}}\cap M_{\bm{a},j_{\ell_{1}}}\cap
M_{\bm{a},i_{\ell_{2}}}\cap M_{\bm{a},j_{\ell_{2}}})$. We combine
$J^{0}_{\ell_{1}}$ and $J^{0}_{\ell_{2}}$ into a single subset of the
partition, and we iterate until we obtain a disjoint partition
$\\{J_{\ell,y}\\}_{\ell}$ of $\\{1,\ldots,m\\}$ such that

$\\{i_{r},j_{r}\mid r\in J_{\ell_{1},y}\\}\cap\\{i_{r},j_{r}\mid r\in
J_{\ell_{2},y}\\}=\emptyset$

if $\ell_{1}\neq\ell_{2}$ and

$y\in\pi\Big{(}\bigcap_{r\in J_{\ell,y}}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}$

for all $\ell$. Therefore, for each $\ell$, there is a neighborhood
$U_{\ell,y}$ such that

$\pi\Big{(}\bigcap_{r\in J_{\ell,y}}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}\cap U_{\ell,y}=\bigcap_{r\in
J_{\ell,y}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U_{\ell,k}\,.$

Set $U_{y}:=\bigcap_{\ell}U_{\ell,y}$. Because $B$ is compact, there is a
finite open cover $\\{U_{k}\\}_{k=1}^{K}$, which has the desired properties by
construction. ∎

The following lemma will be repeatedly used in Lemma A.4.

######  Lemma A.3.

Suppose that $g:B\to\mathbb{R}$ is a smooth map and $y\in\mathbb{R}$ is a
regular value with preimage $Z_{y}$. If $Z\subseteq B$ is a submanifold such
that $Z\pitchfork Z_{y}$, then $y$ is a regular value of
$g|_{Z}:Z\to\mathbb{R}$.

###### Proof.

At any $z\in Z_{y}$, we have $\ker(dg_{z})=T_{z}Z_{y}$. Therefore, if $z\in
Z\cap Z_{y}$, then $T_{z}Z\subseteq\ker(dg_{z})$ only if $T_{z}Z\subseteq
T_{z}Z_{y}$. Because $y$ is a regular value of $g:B\to\mathbb{R}$, we have

$\dim(T_{z}(Z_{y}))=\dim B-1\,,$

so $T_{z}(Z_{y})$ is a strict subset of $T_{z}B$. Because $Z\pitchfork Z_{y}$,
we have

$T_{z}Z+T_{z}Z_{y}=T_{z}B\,,$

so $T_{z}Z$ cannot be a subset of $T_{z}Z_{y}$. Therefore, $T_{z}Z$ is not a
subset of $\ker(dg_{z})$. This implies that $dg_{z}|_{T_{z}Z}$ is a surjection
because $\dim\mathbb{R}=1$. Therefore, $y$ is a regular value of $g|_{Z}$. ∎

######  Lemma A.4.

Let $\\{(i_{r},j_{r})\\}_{r=1}^{m}$ be a set of index pairs such that
$\\{i_{r},j_{r}\\}\neq\\{i_{s},j_{s}\\}$ if $r\neq s$. For almost every
$\bm{a}\in A$, we have that if

  1. (1)

$\bigcup_{\ell}J_{\ell}=\\{1,\ldots,m\\}$ is a disjoint partition with

$\\{i_{r},j_{r}\mid r\in J_{\ell_{1}}\\}\cap\\{i_{r},j_{r}\mid r\in
J_{\ell_{2}}\\}=\emptyset$

for $\ell_{1}\neq\ell_{2}$ and

  2. (2)

$U$ is an open set in $B$ such that

(23) $\pi\Big{(}\bigcap_{r\in J_{\ell}}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}\cap U=\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U\,,$

for all $\ell$, where $\pi$ is the projection $\pi:B\times\mathbb{R}\to B$,

then

  1. (1)

the set $\bigcap_{r\in J^{\prime}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$
is a manifold for every $J^{\prime}\subseteq\\{1,\ldots,m\\}$ and

  2. (2)

we have

(24) $T_{y}\Big{(}\bigcap_{\ell}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}=\bigcap_{\ell}T_{y}\Big{(}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$

for every $y\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U$.

###### Proof.

It suffices to show that

(25) $\Big{(}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U\Big{)}\pitchfork\Big{(}\bigcap_{\ell^{\prime}<\ell}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U\Big{)}$

for all $\ell$ and almost every $\bm{a}\in A$. Informally, what we show first
is that at almost every $\bm{a}\in A$, perturbations of $\bm{a}$ produce
perturbations of $\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$ for each $\ell$.
Then at the end of the proof, we apply the fact that transverse intersections
are generic.

By Lemmas 4.8 and 4.9, we may assume that $\bigcap_{r\in
J^{\prime}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a manifold for every
$J^{\prime}\subseteq\\{1,\ldots,m\\}$. By (23), we may assume without loss of
generality that there is a sequence $k_{1}<\cdots<k_{c}$ such that
$j_{1}=k_{1}$ and $i_{r}=j_{r-1}$ for all $r$ and $j_{r+1}=i_{r}$ for all $r$.
In other words, we may assume that $\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is of the form

$I_{\bm{a}}(\sigma_{k_{c}},\sigma_{k_{c-1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{k_{3}},\sigma_{k_{2}})\cap
I_{\bm{a}}(\sigma_{k_{2}},\sigma_{k_{1}})$

for all $\bm{a}$. The idea is that because the intersection lifts to an
intersection of the corresponding manifolds (see (23)), we can pair up the
indices however we like.

Define the function $g_{i}:B\to\mathbb{R}$ by

$g_{i}(p):=f(\sigma_{k_{i}},p)-f(\sigma_{k_{i-1}},p)\,.$

For almost every $\bm{a}\in A$, the quantity $a_{k_{i}}-a_{k_{i-1}}$ is a
regular value of $g_{i}$ for all $i$, and the set of regular values is open.
By the same argument as in the proof of Lemma 4.9, we have

$\Big{(}I_{\bm{a}}(\sigma_{k_{c}},\sigma_{k_{c-1}})\cap
U\Big{)}\pitchfork\Big{(}\bigcap_{i=2}^{c-1}I_{\bm{a}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\cap
U\Big{)}$

for almost every $\bm{a}\in A$. Therefore, $(a_{k_{c}}-a_{k_{c-1}})$ is a
regular value of
$g_{k_{c}}|_{\bigcap_{i=2}^{c-1}I_{\bm{a}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\cap
U}$ by Lemma A.3. Additionally, for $\bm{\epsilon}\in\mathbb{R}^{N}$ such that
$\epsilon_{k_{c}}$ and $\epsilon_{k_{c-1}}$ are sufficiently small, there are
no critical values between $(a_{k_{c}}-a_{k_{c-1}})$ and
$(a_{k_{c}}-a_{k_{c-1}}+\epsilon_{k_{c}}-\epsilon_{k_{c-1}})$. Because there
are no critical values, the set $\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$ (which is the
$(a_{k_{c}}-a_{k_{c-1}})$-level set of
$g_{k}|_{\bigcap_{i=2}^{c-1}I_{\bm{a}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\cap
U}$) is a submanifold of $B$ that is diffeomorphic to
$I_{\bm{a}+\bm{\epsilon}}(\sigma_{k_{c}},\sigma_{k_{c-1}})\cap\Big{(}\bigcap_{i=2}^{c-1}I_{\bm{a}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U$ (which is the
$(a_{k_{c}}-a_{k_{c-1}}+\epsilon_{k_{c}}-\epsilon_{k_{c-1}})$-level set of
$g_{k}|_{\bigcap_{i=2}^{c-1}I_{\bm{a}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\cap
U}$), and these submanifolds are smoothly parameterized by
$\epsilon_{k_{c}},\epsilon_{k_{c-1}}$.

Now consider any $i_{*}\in\\{2,\ldots,c-1\\}$. By induction on $i_{*}$, we
will show that there is a set $A^{\prime}\subseteq A$ such that $A\setminus
A^{\prime}$ has measure zero and such that for all $\bm{a}\in A^{\prime}$, we
have that

  1. (1)

the set $\bigcap_{r\in
J_{\ell}}I_{\bm{a}+\bm{\epsilon}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$ is a
submanifold of $B$ that is diffeomorphic to

$\bigcap_{r\in J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$

for sufficiently small $\bm{\epsilon}\in\mathbb{R}^{N}$, and

  2. (2)

these submanifolds are smoothly parameterized by $\bm{\epsilon}$.

Because $(a_{k_{i_{*}}}-a_{k_{i_{*}-1}})$ is a regular value of $g_{i_{*}}$
and the set of regular values is open, there are no critical values between
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}})$ and
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}-\epsilon_{k_{i_{*}-1}})$ for sufficiently
small $\epsilon_{k_{i_{*}-1}}$. Therefore, for sufficiently small
$\epsilon_{k_{i_{*}-1}}$, the
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}-\epsilon_{k_{i_{*}-1}})$-level set of
$g_{i_{*}}$ is a submanifold of $B$ that is diffeomorphic to the
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}})$-level set, and these submanifolds are
smoothly parameterized by (sufficiently small) $\epsilon_{k_{i_{*}-1}}$.
Because transverse intersections are generic,

$\Big{(}\bigcap_{i=i_{*}+1}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}-1}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U$

is transverse to the
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}-\epsilon_{k_{i_{*}-1}})$-level set of
$g_{i_{*}}$ for almost every (sufficiently small) $\epsilon_{k_{i_{*}-1}}$.
Additionally, if the intersection is transverse, it is transverse for an open
neighborhood of $\epsilon_{k_{i_{*}-1}}$. Therefore, we can assume without
loss of generality that this intersection is transverse at
$\epsilon_{k_{i_{*}-1}}=0$ (if not, we can perturb $a_{k_{i_{*}-1}}$ so that
it is) and for all sufficiently small $\epsilon_{k_{i_{*}-1}}$. This implies
that $(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}-\epsilon_{k_{i_{*}-1}})$ is also a
regular value of $g_{i_{*}}$ restricted to
$\Big{(}\bigcap_{i=i_{*}+1}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}-1}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U$, by Lemma A.3. For sufficiently small $\epsilon_{k_{i_{*}}}$, there are no
critical values between
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}-\epsilon_{k_{i_{*}-1}})$ and
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}+\epsilon_{k_{i_{*}}}-\epsilon_{k_{i_{*}-1}})$.
Therefore, for sufficiently small $\epsilon_{k_{i_{*}}}$,
$\epsilon_{k_{i_{*}-1}}$, we have that

$\Big{(}\bigcap_{i=i_{*}+1}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U\,,$

which is the $(a_{k_{i_{*}}}-a_{k_{i_{*}-1}})$-level set of $g_{i_{*}}$
restricted to

$\Big{(}\bigcap_{i=i_{*}+1}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}-1}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U\,,$

is a submanifold of $B$ that is diffeomorphic to

$\Big{(}\bigcap_{i=i_{*}}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}-1}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U\,,$

which is the
$(a_{k_{i_{*}}}-a_{k_{i_{*}-1}}+\epsilon_{k_{i_{*}}}-\epsilon_{k_{i_{*}-1}})$-level
set of $g_{i_{*}}$ restricted to

$\Big{(}\bigcap_{i=i_{*}+1}^{c}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap\Big{(}\bigcap_{i=2}^{i_{*}-1}I_{\bm{a+\epsilon}}(\sigma_{k_{i}},\sigma_{k_{i-1}})\Big{)}\cap
U\,.$

These submanifolds are smoothly parameterized by $\epsilon_{k_{i_{*}}}$ and
$\epsilon_{k_{i_{*}+1}}$. This concludes the inductive step.

Let $\bm{a}\in A^{\prime}$, where $A^{\prime}$ is defined as it was earlier in
the proof. We showed above that for sufficiently small
$\bm{\epsilon}\in\mathbb{R}^{N}$, the set of manifolds
$\bigcap_{r\in\ell}I_{\bm{a+\epsilon}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$
(parameterized by $\bm{\epsilon}$) is a smoothly parameterized family of
embeddings of $\bigcap_{r\in\ell}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U$ into $U\subseteq B$. Varying $\\{\epsilon_{r}\mid r\in J_{\ell}\\}$ (while
holding $\epsilon_{r}$ constant for $r\not\in J_{\ell}$) produces a smoothly
parameterized family of embeddings of
$\bigcap_{r\in\ell}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$ while
holding $\Big{(}\bigcap_{\ell^{\prime}<\ell}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U\Big{)}$ constant.
Therefore, because transverse intersections are generic,

$\big{(}\bigcap_{r\in J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U\Big{)}\pitchfork\Big{(}\bigcap_{\ell^{\prime}<\ell}\bigcap_{r\in
J_{\ell}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U\Big{)}$

for all $\ell$ for almost every $\bm{\epsilon}$ in a neighborhood of
$\bm{0}\in\mathbb{R}^{N}$. This proves (25), which completes the proof. ∎

######  Lemma A.5.

Let $\\{(i_{r},j_{r})\\}_{r=1}^{m}$ be a set of index pairs such that
$\\{i_{r},j_{r}\\}\neq\\{i_{s},j_{s}\\}$ if $r\neq s$. For almost every
$\bm{a}\in A$, we have that if $U$ is an open set in $B$ such that

(26) $\pi\Big{(}\bigcap_{r=1}^{m}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\Big{)}\cap
U=\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U\,,$

where $\pi$ is the projection $\pi:B\times\mathbb{R}\to B$, then
$\bigcap_{r\in J^{\prime}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a
manifold for every $J^{\prime}\subseteq\\{1,\ldots,m\\}$ and

(27) $T_{y}\Big{(}I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})\Big{)}=\bigcap_{r=1}^{m}T_{y}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$

for all $y\in\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap
U$.

###### Proof.

By Lemmas 4.8 and 4.9, $\bigcap_{r\in
J^{\prime}}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})$ is a manifold for every
$J^{\prime}\subseteq\\{1,\ldots,m\\}$ for almost every $\bm{a}\in A$. We have

$T_{y}\Big{(}I_{\bm{a}}(\sigma_{i_{1}},\sigma_{j_{1}})\cap\cdots\cap
I_{\bm{a}}(\sigma_{i_{m}},\sigma_{j_{m}})\Big{)}\subseteq\bigcap_{r=1}^{m}T_{y}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$

because $\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\subseteq
I_{\bm{a}}(\sigma_{i_{s}},\sigma_{j_{s}})$ for all $s$.

Let
$\bm{v}\in\bigcap_{r=1}^{m}T_{y}\Big{(}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\Big{)}$.
Define $\pi_{[m]}:=\pi|_{\bigcap_{r=1}^{m}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\cap\pi^{-1}(U)}$, and define
$\pi_{r}:=\pi|_{M_{\bm{a},i_{r}}\cap M_{\bm{a},j_{r}}\cap\pi^{-1}(U)}$ for
each $r$. Each $\pi_{r}$ is a diffeomorphism from $M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}}\cap\pi^{-1}(U)$ to
$I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$, and $\pi_{[m]}$ is a
diffeomorphism from $\bigcap_{r=1}^{m}(M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}})\cap\pi^{-1}(U)$ to
$\bigcap_{r=1}^{m}I_{\bm{a}}(\sigma_{i_{r}},\sigma_{j_{r}})\cap U$. Let
$\tilde{y}:=\pi^{-1}_{[m]}$ (which exists because $\pi^{-1}_{[m]}$ is a
diffeomorphism), and let $\tilde{v}:=d\pi^{-1}_{[m]}(v)$ (which exists because
$d\pi_{[m]}$ is an isomorphism). For all $r$, we have
$\tilde{y}=\pi^{-1}_{r}(y)$ and $\tilde{v}:=d\pi^{-1}_{r}(v)$. Therefore,

$\tilde{v}\in\bigcap_{r=1}^{m}T_{\tilde{y}}\Big{(}M_{\bm{a},i_{r}}\cap
M_{\bm{a},j_{r}}\Big{)}\,.$

By Lemma A.1, we have
$T_{\tilde{y}}\Big{(}\bigcap_{r=1}^{m}(M_{\bm{a},{i_{r}}}\cap