---
abstract: |
    The [*Reeb space*]{} of a function or a map on a manifold is defined as the space of all the connected components of inverse images.

    A Reeb space represents the manifold compactly. In fact, such stuffs are fundamental and useful tools in geometric theory of Morse functions and more general maps not so ill: in other words, a branch of the global singularity theory. The author has been interested in the following problem first established and explicitly solved by Sharko: can we construct an explicit good function inducing a given graph as the Reeb space ([*Reeb graph*]{})? Such problems have been explicitly solved by several researchers and after that the author has set problems of new types and solved them.

    A [*pseudo quotient map*]{} on a differentiable manifold is a surjective continuous map onto a lower dimensional polyhedron. More precisely, a [*pseudo quotient map*]{} is defined as a map locally regarded as the natural quotient map onto the Reeb space defined from a differentiable map of a given class. They were first defined by Kobayashi and Saeki in 1996 as useful objects in the theory of global singularity related to generic maps into the plane and later the author has used these objects in new explicit situations starting from redefining.

    In this paper, we consider classes of continuous or differentiable maps on manifolds, introduce the [*Reeb graphs*]{} of the maps and redefine pseudo quotient maps onto graphs in a bit new way. Last, we attack and solve a new problem on construction before.
address: 'Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku Fukuoka 819-0395, Japan'
author:
- Naoki Kitazawa
title: Maps on manifolds onto graphs locally regarded as a quotient map onto a Reeb space and construction problem
---

Introduction {#sec:1}
============

Reeb spaces and graphs and differentiable functions realizing given graphs as Reeb graphs
-----------------------------------------------------------------------------------------

The [*Reeb space*]{} of a continuous map of a suitable class on a topological space is the space of all the connected components of inverse images. For a differentiable function, consider the set of all the points in the Reeb space coinciding with the set of all the connected components of inverse images including [*singular points*]{}: a [*singular point*]{} of a smooth map is a point at which the rank of the differential drops. For Morse functions, functions with finitely many singular points on closed manifolds and functions of several suitable classes, the spaces are graphs such that the vertex sets are the sets defined before. They are called the [*Reeb graphs*]{} of the maps. They seem to have been first defined in [@reeb].

Reeeb graphs and spaces are fundamental and important in the algebraic and differential topological theory of Morse functions and their generalizations, or in other words, the theory of global singularity.

We introduce several terminologies and a problem on construction of good functions inducing Reeb graphs isomorphic to given graphs.

The [*singular set*]{} of a differentiable map is defined as the set of all the singular points. A [*singular value*]{} is a point in the target manifold such that the inverse image contains a singular point and a [*regular value*]{} is a point in the target manifold which is not a singular value. The [*singular value set*]{} is the image of the singular set.

\[prob:1\] Can we construct a differentiable function with good geometric properties inducing a given graph as the Reeb graph? We do not fix a manifold on which we construct a desired function.

A problem of this type was first considered and explicitly solved by Sharko ([@sharko]). [@batistacostamezasarmiento], [@martinezalfaromezasarmientooliveira], [@masumotosaeki] and [@michalak] are important studies related to this. Later the author has set and solved explicit cases in [@kitazawa4] and [@kitazawa5].

We comment on studies of the author. Different from the other studies, conditions on inverse images of regular values are posed and manifolds appearing there may not be spheres, for example.

Pseudo quotient maps
--------------------

A [*pseudo quotient map*]{} on a differentiable manifold is a surjective continuous map onto a lower dimensional polyhedron. and defined as a map locally regarded as the natural quotient map onto the Reeb space of a differentiable map of a suitable class. They were first defined by Kobayashi and Saeki in 1996 ([@kobayashisaeki]) as useful objects in the theory of global singularity related to generic maps of dimensions larger than $2$ into the plane. Later the author has used these objects in new explicit situations starting from redefining in [@kitazawa2] and [@kitazawa3] for example.

The content of the present paper
--------------------------------

In this paper, we perform the following.

-   We consider classes of continuous or differentiable maps on differentiable manifolds for maps of which we introduce the [*Reeb graphs*]{} and we introduce the Reeb graphs of these maps. This is a refinement of the definition of a Reeb graph which has not appeared.

-   We redefine pseudo quotient maps on differentiable manifolds of a class before.

-   We set a construction problem and give an answer (Theorem \[thm:1\]) with several terminologies needed. These are a problem of a new type and a result of a new type.

The author is a member of the project Grant-in-Aid for Scientific Research (S)

(17H06128 Principal Investigator: Osamu Saeki) “Innovative research of geometric topology and singularities of differentiable mappings”

(https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17H06128/ ) and supported by this project.

Classes of continuous or differentiable maps on differentiable manifolds and the Reeb graphs of the maps of these classes
=========================================================================================================================

Let $(r,s)$ be a pair of non-negative integers satisfying $r>s$ or a pair such that $r=\infty$ and that $s$ is a non-negative integer.

Let $X$ be a $C^r$ manifold of dimension $m>1$ and $Y$ be a $C^r$ manifold of dimension $1$. Let $A \subset X$ be a measure zero set. A map $f:(X,A) \rightarrow N$ between the $C^r$ manifolds is said to be a [*$(C^r,C^s)$*]{} map if $f$ is of class $C^r$ at any point in $X-A$ and of class $C^s$ at any point in $A$. $A$ is called the [*measure zero set*]{} of $f$

Consider the Reeb space $W_f$ of the map $f$. Let $V$ be the set of all the points including singular points or points in $A$. If we can regard $W_f$ as a graph whose vertex set is $V$, then we call the graph the [*Reeb graph*]{} of $f$.

A pseudo quotient map of a class of maps
========================================

Let $(r,s)$ be a pair of non-negative integers satisfying $r>s$ or a pair such that $r=\infty$ and that $s$ is a non-negative integer. Two $C^r$ maps $c_1:X_1 \rightarrow Y_1$ and $c_2:X_2 \rightarrow Y_2$ are said to be [*$C^s$ equivalent*]{} if a pair $({\phi}_X,{\phi}_Y)$ of $C^s$ diffeomorphisms satisfying ${\phi}_Y \circ c_1=c_2 \circ {\phi}_X$ exists.

Let $X_1$ and $X_2$ be differentiable manifolds of dimension $m>1$ and $Y$ be a graph. Let $r$ be a positive integer or $\infty$. Two continuous maps $c_1:X_1 \rightarrow Y$ and $c_2:X_2 \rightarrow Y$ are said to be [*$C^r$-PL equivalent*]{} if there exists a $C^r$ diffeomorphism $\phi$ satisfying $c_1=c_2 \circ {\phi}_X$ exists.

Let $m>2$ be a positive integer. Let $\mathcal{C}$ be a class of $(C^r,C^s)$ maps from $m$-dimensional differentiable manifolds into $1$-dimensional ones whose Reeb spaces are regarded as Reeb graphs. A continuous map $q$ on an $m$-dimensional differentiable manifold onto a graph is said to be a [*pseudo quotient map*]{} of the class $\mathcal{C}$ if the following hold.

1.  At each point $p$ in the interior of an edge, consider a small closed interval $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a $C^r$ trivial bundle, appearing locally around a closed interval in the interior of an edge in the Reeb graph of a map of the class $\mathcal{C}$.

2.  At each vertex $p$, consider a small regular neighborhood $C_p$ including the point, $q {\mid}_{q^{-1}(C_p)}:q^{-1}(C_p) \rightarrow C_p$ is $C^r$-PL equivalent to a PL map $q_{C_p}$ over a small regular neighborhood of a vertex in the Reeb graph of a map $f_p$ of $\mathcal{C}$ such that the domain includes no point in the measure zero set of the map $f_p$, or $C^s$-PL equivalent to a PL map over a small regular neighborhood of a vertex in the Reeb graph of a map of $\mathcal{C}$.

A pseudo quotient map of a class $\mathcal{C}$ is said to be [*realized*]{} as a quotient map of the class $\mathcal{C}$ if by identifying the target graphs suitably, we can regard the original map as $C^r$-PL equivalent to the quotient map onto the Reeb graph of a map of the class.

Examples will be presented in the next section.

The main theorem and its proof
==============================

We introduce the main theorem, related to Problem \[prob:1\],

A [*fold*]{} map is a $C^{\infty}$ map at each singular point which is represented as a Morse function and an identity map on a $C^{\infty}$ manifold. A [*special generic*]{} map is a fold map at each singular point which is represented as a natural height function of an unit open ball and an identity map on a $C^{\infty}$ manifold.

Precise explanations on these maps are in [@golubitskyguillemin], [@saeki] and [@saeki2], for example.

For an integer $n>1$, let $L_n$ be a $1$-dimensional polyhedron represented as $${\bigcup}_{k=0}^{n-1} \{(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n}) \mid 0 <r \leq 1 \} \bigcup \{(0,0)\} \subset {\mathbb{R}}^2.$$

![$L_n$.[]{data-label="fig:1"}](ln.eps){width="30mm"}

Let $q$ be a continuous map from a $C^s$ manifold of dimension $m>1$ onto $L_n \subset {\mathbb{R}^2}$. Suppose that for a transformation ${\phi}_{n}$ on $L_n$ defined as $${\phi}_{n}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{2(k+1)\pi}{n}, r \sin \frac{2(k+1)\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{n}$ satisfying ${\phi}_n \circ q=q \circ {\Phi}_n$ and that for one ${\phi}_{r}$ on $L_n$ defined as $${\phi}_{r}(r \cos \frac{2k\pi}{n}, r \sin \frac{2k\pi}{n})=(r \cos \frac{-2k\pi}{n}, r \sin \frac{-2k\pi}{n})$$ there exists a $C^s$ diffeomorphism ${\Phi}_{r}$ satisfying ${\phi}_r \circ q=q \circ {\Phi}_r$. Then $q$ is said to be [*$D_n$-symmetric*]{}.

\[thm:1\] There exist a class $\mathcal{C}$ of $(C^r,C^0)$ maps whose Reeb spaces are regarded as Reeb graphs and a class ${\mathcal{Q}}_{\mathcal{C}}$ of pseudo quotient maps of the class satisfying the following.

1.  For maps of the class $\mathcal{C}$, inverse images of regular values are disjoint unions of standard spheres.

2.  For maps of the class ${\mathcal{Q}}_{\mathcal{C}}$, around each vertex of degree $n>1$, the map is regarded as a $D_n$-symmetric map onto $L_n$ by applying a suitable identification.

3.  For any finite connected graph which is not a single point, we can construct a map of the class ${\mathcal{Q}}_{\mathcal{C}}$ onto the graph.

4.  For a map of the class ${\mathcal{Q}}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then the map is realized as a map into $S^1$ or $\mathbb{R}$ of the class $\mathcal{C}$.

In the proof, we first perform local construction around each vertex (Step 1–3) and in Step 4, we complete the construction by constructing remaining parts. Last, we give the strict definitions of $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. We will see that this completes the proof except for the fourth condition. Last we discuss the fourth condition.

Step 1 Around a vertex of degree $2$.\
We consider a trivial $C^r$ bundle over $[-1,1]$ whose fiber is a standard sphere. We compose a surjective function over $[-1,1]$ defined by $t \mapsto t^3$. $0 \in [-1,1]$ is the vertex of degree $2$. The other points are not in the vertex set. The map is regarded as a $D_2$-symmetric map onto $L_2$.\
 \
Step 2 Around a vertex of degree $n \geq 3$.\
We consider a $C^{\infty}$ map on a $C^{\infty}$ manifold of dimension $m>1$ into the plane whose image is the bounded domain surrounded by the five arcs as FIGURE \[fig:2\]. We construct the map as a special generic map such that the restriction to the singular set is embedding and the singular value set is the disjoint union of two thick arcs. Note that inverse images of regular values are standard spheres ($m>2$) or two point sets ($m=2$). For more precise facts on special generic maps into the plane, see [@saeki2] for example.

![A special generic map into the plane.[]{data-label="fig:2"}](zahyoone.eps){width="30mm"}

Let $f$ be a $C^{\infty}$ function whose value is $1$ on the interval $x \leq 0$ and which is strictly increasing on the interval $x \geq 0$. We can do in the step before so that by composing a $C^{\infty}$ map $T$ defined as $$T(x,y):=
\begin{cases}
(x,y) & (y \leq 0) \\
(f(y)x,y) & (y \geq 0) 
\end{cases}$$ the resulting image is as FIGURE \[fig:3\]: the arc connecting two thick arcs is defined as a part of a quadratic curve of a form $x^2-y^2=a$. We can determine $c(t)$ and we can naturally determine a $C^{\infty}$ function $c$ on $[0,t]$ based on quadratic curves of this form. We can extend this to the bounded domain except two arcs in the bottom as a $C^{\infty}$ map. However, we can extend to the arcs as a continuous map: we define the map so that the values on these two arcs are $(0,-1)$. Thus we can define a map $C$ on the bounded domain, Through the steps, we can define a $(C^r,C^0)$ function on the given $m$-dimensional manifold to a closed interval $[-1,c(t)]$ where we identify $x$ with $(0,x)$ for $-1 \leq x \leq c(t)$.

![A deformation of the special generic map into the plane of FIGURE \[fig:2\].[]{data-label="fig:3"}](zahyotwo.eps){width="30mm"}

We consider the map on an $m$-dimensional manifold into the plane obtained in the explanation of FIGURE \[fig:2\] and $n$-copies of such a map. We deform these maps by scaling suitably and attach the maps as FIGURE \[fig:4\] on the arcs corresponding to ones including $(0,-1)$ in the original image and the inverse images. $D_k$ are the images of the maps: maps are suitably scaled so that $(0,-1)$ goes to $(0,0)$ and that the angle formed by the pairs of the arcs including $(0,0)$ are equal, for example. We can obtain a local map around the vertex satisfying the first two conditions. Note that the measure zero set of the local function is obtained by identifying the equations of $n$-copies of $S^{m-1}$.

![Attacing copies of a map in FIGURE \[fig:2\].[]{data-label="fig:4"}](zahyothree.eps){width="30mm"}

 \
Step 3 Around a vertex of degree $1$.\
We consider a natural height function on a unit disc of dimension $m>1$ whose image is $[0,1]$ and $0$ or $1$ is the only one vertex. The function is also a Morse function with just $1$ singular point in the interior.\
 \
Step 4 Completing the construction.\
For the interior of each edge, we construct a trivial $C^r$ bundle whose fiber is a standard sphere. Last we glue all the constructed local maps together to obtain a global map.\
 \
Step 5 Define $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$.\
Through Steps 1–4, we construct a map for arbitrary finite graphs which are not single points. Last, we explain about the classes $\mathcal{C}$ and $\mathcal{Q}_{\mathcal{C}}$. $\mathcal{Q}_{\mathcal{C}}$ are naturally defined as a class locally of a form of the obtained forms. We define $\mathcal{C}$ as a map such that around each singular value corresponding to a vertex of degree not $1$, the map is represented as the composition of a presented local map onto a regular neighborhood of a graph, regarded as a map into the plane, a homeomorphism on the plane satisfying the following (we define such a map as an [*almost smooth generalized rotation with reflection*]{}) and a projection onto a straight line $\mathbb{R}$ including $(0,0)$.

1.  The homeomorphism is $C^{\infty}$ on ${\mathbb{R}}^2-\{(0,0)\}$.

2.  For each point not $(0,0)$, the homeomorphism preserves the distance between the point and $(0,0)$.

3.  The homeomorphism maps each straight line originating from $(0,0)$ to another straight line originating from $(0,0)$.

For around each singular value corresponding to a vertex of degree $0$, we define the class so that the local form is as in Step 3: a natural height function on a unit disc. This completes the proof except the fourth condition.

For each finite graph which is not a single point or which has no vertex of degree larger than $3$, then we can orient the graph so that we can construct a continuous map into $S^1$ satisfying the following.

1.  On each edge the map is injective.

2.  The orientation of each edge canonically induced from a canonical orientation of $S^1$ is compatible with the defined orientation.

If the graph has no loop, then we can replace $S^1$ by $\mathbb{R}$.

For a map of the class $\mathcal{Q}_{\mathcal{C}}$, if for the target graph, for each vertex, the degree is at most $3$, then we can orient the graph as this and we can construct a local function compatible with the definition of the class $\mathcal{C}$ and the orientation. This is oweing to the definitions of a $D_2$-symmetric and a $D_3$-symmetric map and an almost smooth generalized rotation with reflection. We can consider a transformation by an almost smooth generalized rotation with reflection to construct a local function compatible with the desired orientation. See FIGURE \[fig:4.5\] for the case of a vertex of degree $2$.

![Almost smooth generalized rotations with reflections and projections (around a vertex of degree $2$: arrows indicate local orientations of the graphs induced naturally from local functions).[]{data-label="fig:4.5"}](rot.eps){width="30mm"}

This completes the proof.

Last we present another example of classes of maps and pseudo quotient maps of the class.

A [*standard-spherical*]{} Morse function is a Morse function such that the following hold ([@kitazawa]).

1.  At distinct singular points the values are distinct.

2.  Inverse images of regular values are disjoint unions of standard spheres.

3.  A vertex of the Reeb graph such that the inverse image includes a singular point not giving a local extremum is a vertex of degree $3$.

We consider pseudo quotient maps of the class of such functions. We regard these functions as $C^{\infty}$ functions whose measure zero sets are empty.

We present local forms of these pseudo quotient maps with several inverse images in FIGURE \[fig:5\].

![Local forms of the pseudo quotient maps with several inverse images.[]{data-label="fig:5"}](sphelocal.eps){width="30mm"}

We investigate the local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image. Consider a small regular neighborhood of the just one non-manifold point in the inverse image of the vertex. See FIGURE \[fig:6\].

![The local form around a vertex of degree $3$ of a pseudo quotient map of the class of the functions with the inverse image (a regular neighborhood of the just one non-manifold point in the inverse image of the vertex).[]{data-label="fig:6"}](localspheind.eps){width="15mm"}

If we remove the inverse image of the vertex, then the resulting space has $4$ connected components: two of them are in the upper part and the others are in the lower part. Moreover, the former two components are mapped onto an interval and the latter two ones are mapped onto the disjoint union of two intervals in the graph.

This leads us to the fact that a pseudo quotient map whose target is as FIGURE \[fig:7\] cannot be realized as a quotient map of the class. Arrows indicate natural local orientations induced from canonically obtained local functions. Note that around a vertex of degree $3$, the local map cannot be regared as a $D_3$-symmetric map.

![The graph represents the target of a pseudo quotient map of the class of standard-spherical functions: arrows indicate local orientations induced canonically from local functions.[]{data-label="fig:7"}](sphenewex.eps){width="15mm"}

For local forms of such functions, see also [@saeki3] for example.

Last, compare this case with Theorem \[thm:1\].

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