\section{Introduction}
\label{sec:1}
We introduce fundamental notation and terminologies on (differentiable) manifolds and (differentiable) maps between them.
For a differentiable manifold $X$, $T_xX$ denotes the tangent space at $x$. 
A point $x \in X$ in the manifold of the domain of a differentiable map $c:X \rightarrow Y$ is said to be {\it singular} if the rank of the differential $dc_x:T_xX \rightarrow T_{c(x)}Y$ is smaller than $\min \{\dim X,\dim Y\}$. 
We define the {\it singular set} of $c$ as the set of all singular points of $c$, denoted by $S(c)$.
Points in the image $c(S(c))$ are called {\it singular values} of $c$.
{\it Regular values} of the map are points in the manifold of the target which are not singular values.

${\mathbb{R}}^k$ denotes the $k$-dimensional Euclidean space for $k \geq 1$, endowed with the natural differentiable structure and the standard metric. 
${\mathbb{R}}^1$ is also denoted by $\mathbb{R}$ as usual. For $x \in {\mathbb{R}}^k$, $||x|| \geq 0$ denotes the distance between the origin $0 \in {\mathbb{R}}^k$ and $x$.
$S^k:=\{x \mid x \in {\mathbb{R}}^{k+1}, ||x||=1.\}$ denotes the $k$-dimensional unit sphere for $k \geq 0$. This is a 
compact smooth submanifold with no boundary (in ${\mathbb{R}}^{k+1}$) and connected for $k>0$. $D^k:=\{x \mid x \in {\mathbb{R}}^{k}, ||x|| \leq 1.\}$ denotes the $k$-dimensional unit disk for $k \geq 1$. This is a connected smooth submanifold in ${\mathbb{R}}^k$. 
\begin{Def}
\label{def:1}
A {\it Morse} function $c:X \rightarrow \mathbb{R}$ is a smooth function on a smooth manifold $X$ satisfying the following two  conditions.
\begin{enumerate}
\item Singular points are always in the interior of the manifold. 
\item At each singular point $x$ of which it has the form $$(x_1,\cdots,x_m) \mapsto {\Sigma}_{j=1}^{m-i(p)} {x_j}^2-{\Sigma}_{j=1}^{i(p)} {x_{m-i(p)+j}}^2+c(x)$$
for an integer $0 \leq i(p) \leq m$, which is shown to be uniquely chosen, and for suitable coordinates.
\end{enumerate}
A {\it Morse-Bott} function $c:X \rightarrow \mathbb{R}$ is a smooth function represented as the composition of a smooth submersion with a Morse function at each singular point.
\end{Def}
\subsection{Reeb spaces and Reeb graphs}
From a differentiable map $c:X \rightarrow Y$, we can define an equivalence relation ${\sim}_c$ on $X$ as follows: $x_1 {\sim}_{c} x_2$ holds if and only if they are in a same connected component of $c^{-1}(y)$ for some point $y$.

\begin{Def}
The quotient space $W_c:=X/{\sim}_c$ is defined as the {\it Reeb space} of $c$.
\end{Def}
For the Reeb space of $c$, $q_c:X \rightarrow W_c$ denotes the quotient space. We can define a map which is denoted by $\bar{c}$ uniquely so that we have the relation $c=\bar{c} \circ q_c$. 

Reeb spaces are in considerable cases graphs such that the vertex sets are the sets of all points $p \in W_c$ giving the preimages ${q_c}^{-1}(p)$ containing at least one singular point of the maps $c$. This is called the {\it Reeb graph} of $c$.
For example, for smooth functions on compact manifolds with finitely many singular values, Reeb spaces are homeomorphic to graphs and if they are on closed manifolds, Reeb graphs are obtained (\cite{saeki}). Note that a graph is naturally topologized and (PL) homeomorphic to a $1$-dimensional polyhedron.
The present paper concentrates on such smooth functions on closed manifolds.

Reeb spaces represent the manifolds of the domains compactly. In specific cases, they inherit topological information such as homology groups and cohomology rings much, shown explicitly in \cite{kitazawa0.1}--\cite{kitazawa0.6} and \cite{saekisuzuoka}, for example: so-called {\it fold} maps such that preimages of regular values are disjoint unions of spheres are studied for example.

We do not concentrate on {\it fold} maps in the present study. {\it Fold} maps are, in short, smooth maps from smooth manifolds with no boundaries into manifolds with no boundaries represented as the product maps of Morse functions and identity maps on the interiors of copies of a unit disk locally (for suitable coordinates). The Reeb spaces of fold maps are shown to be polyhedra whose dimensions are same as those of the manifolds of the targets: see \cite{shiota} for general related theory and see also \cite{kobayashisaeki} for cases where the manifolds of the targets are surfaces, for example.

For Reeb graphs and Reeb spaces, there exist various papers. \cite{reeb} seems to be one of pioneering papers. Other papers will be presented in the present paper.
\subsection{Some notions on graphs which are topologized canonically.}

\begin{Def}
An {\it isomorphism} between two graphs $K_1$ and $K_2$ is a (PL) homeomorphism from $K_1$ to $K_2$ mapping the vertex set of $K_1$ onto the vertex set of $K_2$ where they are topologized naturally.
\end{Def}
\begin{Def}
A continuous real-valued function $g$ on a graph $K$ is said to be a {\it good} function if it is injective on each edge.
\end{Def}

In the present paper, we only consider finite (and connected) graphs. It immediately follows that a finite graph has such a function if and only if it has no loops as edges.

\subsection{Main problem and Main Theorems.}
The present paper studies another important problem as follows.
\begin{MainProb}
For a finite and connected graph with at least one edge which may be a (so-called) multigraph, can we construct a smooth function on a (compact and connected) manifold (satisfying some good conditions) which induces the Reeb graph isomorphic to the graph and preimages of regular values for which are as prescribed? More explicitly, can we construct functions as Morse functions?
\end{MainProb}
This is so-called realization problems of graphs as Reeb graphs of smooth functions of suitable classes. \cite{sharko} is a pioneering paper on this. This concerns smooth functions on closed surfaces whose Reeb graphs are isomorphic to given graphs. Other related papers will be presented in the present paper.


In the present paper, we show the following results as the main theorem.

For the family of (all) smooth manifolds, we can induce the following equivalence relation: two smooth manifolds are equivalent if and only if they are diffeomorphic each other. 
\begin{Def}
The equivalence class which a smooth manifold belongs to is said to be the {\it diffeomorphism type} for the manifold.
\end{Def}
A ({\it most fundamental}) {\it handlebody} is a smooth manifold of a suitable class of compact, connected and smooth manifolds whose dimensions are greater than $1$. The notion of a most fundamental handlebody will be presented in the next section.

\begin{MainThm}
\label{mthm:1}
Let $K$ be a finite and connected graph which has at least one edge and no loops. Let there exist a good function $g$ on $K$. Suppose that a diffeomorphism type for some manifold is assigned to each edge by a map $r_K$ on the edge set $E$. Let $m>1$ be an integer. Suppose the following conditions.
\begin{itemize}
\item At each edge of $K$ containing a vertex where $g$ has a local extremum, the value of $r_K$ is the diffeomorphism type for $S^{m-1}$.
\item Each vertex where $g$ has a local extremum is of degree $1$.
\end{itemize}
Then we have the following two.
\begin{enumerate}
\item \label{mthm:1.1} Assume also that the values of $r_K$ are always diffeomorphism types for closed and connected manifolds diffeomorphic to the boundaries of some $m$-dimensional most fundamental handlebodies. Then there exists an $m$-dimensional closed, connected and orientable manifold $M$ and a Morse function $f$ on $M$ satisfying the following three properties.
\begin{enumerate}
\item \label{mthm:1.1.1} The Reeb graph $W_f$ of $f$ is isomorphic to $K${\rm :} we can take a suitable isomorphism $\phi:W_f \rightarrow K$ compatible with the remaining properties of these three properties.
\item \label{mthm:1.1.2} If we consider the natural quotient map $q_f:M \rightarrow W_f$ and for each point $\phi(p) \in K$ {\rm (}$p \in W_f${\rm )} that is not a vertex and that is in an edge $e$, then the diffeomorphism type for
 the preimage ${q_f}^{-1}(p)$ is $r_K(e)$.
\item \label{mthm:1.1.3} For each point $p \in M$ mapped by $q_f$ to a vertex $v_p:=q_f(p) \in W_f$, $f(p)=g \circ \phi(v_p)$.
\end{enumerate}
\item \label{mthm:1.2}
Assume that $m=4$. Then there exists a $4$-dimensional closed, connected and orientable manifold $M$ and a Morse function $f$ on $M$ satisfying the three properties {\rm (\ref{mthm:1.1.1})}--{\rm (\ref{mthm:1.1.3})} in {\rm (\ref{mthm:1.1})}.
\end{enumerate}
\end{MainThm}

The following theorem is previously obtained and motivates us to study the present problems. (\ref{mthm:1.1}) of Main Theorem can be regarded as an extension of Theorem \ref{thm:1} except the last part. (\ref{mthm:1.2}) of Main Theorem is an application of (\ref{mthm:1.1}) with a few additional arguments in the original proof of Theorem \ref{thm:1}.
\begin{Thm}[\cite{kitazawa}]
\label{thm:1}
Let $K$ be a finite and connected graph which has at least one edge and no loops. Let there exist a good function $g$ on $K$. Suppose that a non-negative integer is assigned to each edge by a map $r_K$ on the edge set $E$. These two functions satisfy the following conditions.
\begin{itemize}
\item At each edge of $K$ containing a vertex where $g$ has a local extremum, the value of $r_K$ is $0$.
\item Each vertex where $g$ has a local extremum is of degree $1$.
\end{itemize}
Then there exist a $3$-dimensional closed, connected and orientable manifold $M$ and a Morse function $f$ on $M$ satisfying the following three properties.
\begin{enumerate}
\item The Reeb graph $W_f$ of $f$ is isomorphic to $K$ and we can take a suitable isomorphism $\phi:W_f \rightarrow K$ compatible with the remaining properties of these three properties.
\item For each point $\phi(p) \in K$ {\rm (}$p \in W_f${\rm )} in the interior of an edge $e$, the preimage ${q_f}^{-1}(p)$ is a closed, connected, and orientable surface of genus $r_K(e)$.
\item For each point $p \in M$ mapped by $q_f$ to a vertex $v_p:=q_f(p) \in W_f$, $f(p)=g \circ \phi(v_p)$.
\end{enumerate}
Furthermore, if we drop the two conditions on $g$, $r_K$ and the graph $K$, then there exist a $3$-dimensional closed, connected and orientable manifold $M$ and a smooth function $f$ on $M$ satisfying the three properties just before and the following two properties hold.
\begin{enumerate}
\item At each singular point where $f$ does not have a local extremum, it is a Morse function.
\item At each singular point where $f$ has a local extremum, except finitely many such singular points, it is a Morse-Bott function. 
\end{enumerate}
\end{Thm}

We prove Main Theorem in the next section. Roughly speaking, methods are similar to the methods in the proof of Theorem \ref{thm:1} or the main theorem of \cite{kitazawa} (and methods in the proofs of main Theorems of  \cite{kitazawa2}, and \cite{kitazawa3}). However, related differential topological arguments on manifolds belonging to certain classes we need are mutually different. Proofs of Main Theorem needs related new arguments.
\section{Proofs of Main Theorems and remarks.}
We review several notions and some technique from differential topology of smooth manifolds.

A {\it k-handle} is a smooth manifold diffeomorphic to a smooth manifold of the form $D^k \times D^{m-k}$ ($0 \leq k \leq m$) as a cornered manifold where $D^0$ denotes the one-point set.

A {\it linear} bundle is a bundle whose fiber is diffeomorphic to a Euclidean space, a unit sphere or a unit disk and whose structure group is linear and acts in a canonical way on the fiber.
As a fundamental fact on differential topology of manifolds, it is well-known that we can always eliminate corners for smooth manifolds in certain canonical ways and obtain smooth manifolds with no corners. The resulting diffeomorphism types are unique.

We introduce an operation of constructing compact smooth manifolds of good classes.
Take an integer $m>1$.
\begin{enumerate}
\item Set ${H_0}^m$ as a non-empty space which is also an $m$-dimensional compact and smooth manifold.
\item We do either of the following two.
\begin{enumerate}
\item Set $k=0$ and take two distinct points in $\partial {H_0}^m$. We choose a small closed tubular neighborhood $N_0$ for the discrete set consisting of the two points. This is always seen as a normal bundle which is trivial and diffeomorphic to the disjoint union $D^{m-1} \sqcup D^{m-1}$.
\item Set a positive integer $k<m-1$ and take a smooth submanifold diffeomorphic to $S^k$ whose suitable small closed tubular neighborhood $N_0$ is regarded as a trivial linear bundle over $S^k$ seen as a normal bundle.
\end{enumerate}
\item Take a ($k+1$)-handle, identified with $D^{k+1} \times D^{m-k-1}$ via a diffeomorphism between the cornered manifolds.  
\item Attach the $k$-handle $D^{k+1} \times D^{m-k-1}$ just before to $\partial {H_0}^m$ via a bundle isomorphism between the trivial linear bundle $\partial D^{k+1} \times D^{m-k-1} \subset D^{k+1} \times D^{m-k-1}$ whose fiber is diffeomorphic to $D^{m-k-1}$ and the small closed tubular neighborhood $N_0$.
\item We eliminate the corner of the resulting manifold in the canonical way and set the resulting manifold as ${H_1}^m$.
\item We do a finite iteration of this procedure and obtain $m$-dimensional smooth manifolds with no corners ${H_{l_0}}^m$ for $l_0 \geq 1$ where $l_0$ is an integer: at each step we first choose an $m$-dimensional compact and smooth manifold ${H_{l_0-1}}^m$ instead of "${H_0}^m$" and the closed tubular neighborhood $N_{l_0-1}$ instead of "$N_0$".
\end{enumerate}
\begin{Def}
In this operation, if ${H_0}^m$ is diffeomorphic to the unit disk, then we call the manifold ${H_{l_0}}^m$ a {\it handlebody} for $l_0 \geq 1$. If ${H_0}^m$ is diffeomorphic to the unit disk and each smooth submanifold diffeomorphic to a unit sphere and the small closed tubular neighborhood $N_{l_0-1}$ are always in the boundary $\partial {H_0}^m \subset {H_0}^m$ and mutually disjoint, then we call the $m$-dimensional manifold ${H_{l_0}}^m$ a {\it most fundamental} handlebody.
\end{Def}

\begin{Ex}
A copy of the $m$-dimensional unit disk $D^m$ is a most fundamental handlebody for $m \geq 2$. We attach two handles to the boundary of a copy ${H_0}^m$ of the $m$-dimensional unit disk simultaneously to obtain this. More precisely, we attach a $1$-handle and an ($m-1$)-handle to obtain the desired handlebody by defining the closed tubular neighborhoods $N_0$ and $N_1$ before satisfying the following two conditions.
\begin{enumerate}
\item $N_0$ is the union of a suitable two copies of $D^{m-1}$ smoothly and disjointly embedded in $\partial {H_0}^m$. 
\item $N_1$ is the closed tubular neighborhood of a copy of $S^{m-2}$ smoothly embedded in $\partial {H_0}^m$ and two connected components of $N_0$ are in distinct connected components of $ \partial {H_0}^m-N_1$.
\end{enumerate}
${H_2}^m$ is diffeomorphic to $D^m$.
\end{Ex}

Any compact, connected and smooth manifold always has a Morse function such that at distinct singular points, the values are distinct.

Choose an arbitrary disjoint ordered pair of non-empty subsets represented as disjoint unions of connected components of the boundary such that their union is the original boundary. We have such a Morse function such that the preimage of the minimum and that of the maximum coincide with the former submanifold and the latter submanifold, respectively.

If either of the subsets is empty instead here, then we have a Morse function with similar properties such that either the preimage of the minimum or the maximum coincides with the boundary.


It is also well-known that Morse functions satisfying the presented properties exist densely in the space of all smooth functions topologized suitably. Related to this, for fundamental theory on singularity theory and differential topological properties of Morse functions (and fold maps), see \cite{golubitskyguillemin} for example.

Let us review a correspondence between Morse functions and handles. This plays essential roles in our construction of functions including our proof of Main Theorem.

From a Morse function as presented here on a compact manifold $E$ whose boundary $\partial E$ is non-empty and represented as the disjoint union $F_1 \sqcup F_2$ where $F_j$ is not empty and the disjoint union of finitely many connected components of $\partial E$. We have an explicit decomposition of $E$ into ($k$-)handles in the construction of the $m$-dimensional manifold ${H_{l_0}}^m$ before. We can set $F_1$, $F_2$ and ${H_0}^m$ so that the following conditions hold.
\begin{itemize}
\item ${H_0}^m$ is a (small) collar neighborhood $F_1 \times [-1,0]$ of $F_1$, identified with $F_1 \times \{-1\} \subset F_1 \times [-1,0]$.
\item $F_1 \times \{-1\}$ is the preimage of the minimum and that $F_2$ is the preimage of the maximum.
\end{itemize}
In this situation, each $k$-handle naturally corresponds to a singular point $p$ such that $i(p)=k$ in Definition \ref{def:1} and for the $j_1$-th handle and the $j_2$-th handle with $j_1<j_2$, the corresponding value at the singular point corresponding to the $j_2$-th singular point is greater than the corresponding value at the singular point corresponding to the $j_1$-th singular point.

Conversely, from a manifold ${H_0}^m$ diffeomorphic to $F_1 \times [-1,0]$ and such a system of handles for an $m$-dimensional compact and smooth manifold $E:={H_{l_0}}^m$ such that $\partial E=(F_1 \times \{-1\}) \sqcup F_2$ we have a Morse function satisfying the properties where $F_1$ and $F_2$ are non-empty and closed manifolds which may not be connected.

As a specific case, we consider a compact manifold $E$ such that the following three hold where we abuse notation in the construction of handlebodies and more general compact manifolds.
\begin{itemize}
\item The boundary $\partial E$ is represented as the disjoint union $F_1 \sqcup F_2$ where $F_j$ is not empty and the disjoint union of finitely many connected components of $\partial E$. 
\item Small closed tubular neighborhoods $N_j$ ($0 \leq j \leq l_0-1$) are always in the boundary $\partial {H_0}^m$ of ${H_0}^m \subset E$ and mutually disjoint as in the definition of a most fundamental handlebody.
\item ${H_0}^m$ is a (small) collar neighborhood $F_1 \times [-1,0]$ of $F_1$, identified with $F_1 \times \{-1\}$.
\end{itemize}
In this case we have a Morse function with exactly one singular value instead where there may exist singular points at least $2$. The singular set is a discrete subset of course and a finite subset. Conversely, from such a Morse function, we have a system of handles as presented here.

Related theory is explained in \cite{milnor} for example. This is also a key ingredient in \cite{michalak}. Earlier versions of \cite{kitazawa} adopt expositions via this theory. 

We go to our proof of Main Theorem. We also abuse our notation in the construction of handlebodies and more general compact manifolds before in the proof. 

\begin{proof}[A proof of Main Theorem.]
We prove (\ref{mthm:1.1}). As the original proof of Theorem \ref{thm:1}, we construct a local Morse function around each vertex, respecting the three properties (\ref{mthm:1.1.1})--(\ref{mthm:1.1.3}) and glue them together.
We explain the construction of the local functions to complete the proof.

Around each vertex $v$ where $g$ has a local extremum, we construct a Morse function on a copy of the unit disk obtained by considering the natural height. This is a so-called {\it height} function. This function $f_{v,m}:D^{m} \rightarrow \mathbb{R}$ is defined by $f_{v,m}(x):=\pm||x||^2+g(v)$ and $0$ is mapped to $v$ by the composition of $q_{f_{v,m}}$ with a suitable PL homeomorphism ${\phi}_v$ from the Reeb space onto a small regular neighborhood of $v$ in $K$.

Around each vertex $v$ where $g$ does not have a local extremum, we will construct a local Morse function. This is a new ingredient.

Set ${H_0}^m$ as a copy of the $m$-dimensional unit disk in the construction of a handlebody. We can attach $l \geq 0$ ($m-1$)-handles and suitable handles simultaneously as explained in the definition of a most fundamental handlebody to obtain a most fundamental handlebody whose boundary consists of exactly $l+1$ connected components so that the two conditions hold.
\begin{itemize}
\item We attach at least one handle here.
\item The boundary of the union of ${H_0}^m$ and the first $l$ ($m-1$) handles is the disjoint union of $l+1$ copies of $S^{m-1}$ (after the corner is eliminated in the canonical way).
\item Each of the remaining handles is attached to the intersection of $\partial {H_0}^m$ and one of the $l+1$ copies of $S^{m-1}$ before.  
\end{itemize}
The $l+1$ connected components of the resulting boundary are the boundaries of some most fundamental handlebodies by the definition. 

For any ($m-1$)-dimensional manifold $X^{m-1}$ diffeomorphic to the disjoint union of $l+1$ connected manifolds which are all boundaries of some most fundamental handlebodies, we can attach the handles to $\partial {H_0}^m \subset {H_0}^m$ so that the boundary of the resulting most fundamental handlebody is diffeomorophic to $X^{m-1}$ before. We can know this immediately from the construction.

We can remove a smoothly embedded copy of the $m$-dimensional unit disk in the interior of the submanifold ${H_0}^m$ in a most fundamental handlebody obtained in such a way. In other words we can change the manifold ${H_0}^m$ to a one diffeomorphic to $S^{m-1} \times [-1,0]$. 

We prepare two compact and connected manifolds obtained in such ways. Let $E_1$ and $E_2$ denote them. Let ${\partial}_0 {H_1} \subset \partial E_1$ and ${\partial}_0 {H_2} \subset \partial E_2$ denote the connected components of the boundaries originally copies of the unit disks are attached to. Let ${X_1}^{m-1} \subset \partial E_1$ and ${X_2}^{m-1} \subset \partial E_2$ denote the disjoint unions of the remaining connected components of the boundaries. By the exposition on the relationship between singularities of Morse functions and handles, we easily have a Morse function with exactly one singular value such that the preimage of the minimum is ${\partial}_0 H_1$, that the preimage of the maximum is ${X_{1}}^{m-1}$ and that the singular value is $g(v)$. By the same reason, we also have another Morse function with exactly one singular value such that the preimage of the maximum is ${\partial}_0 H_2$, that the preimage of the minimum is ${X_{2}}^{m-1}$ and that the singular value is $g(v)$. We can obtain these functions so that the images are $[m_v,M_v]$ and contain the unique singular values $g(v)$ in the interior $(m_v,M_v)$. We can choose a trivial linear bundle over $[m_v,M_v]$ whose fiber is diffeomorphic to $D^{m-1}$ apart from the singular set and remove the interior for each Morse function. After that we can glue the resulting functions on the resulting manifolds preserving the values at each point in the manifolds of the domains. In this way, we can obtain a local Morse function $f_{v}$ satisfying the following properties.
\begin{itemize}
\item The manifold of the domain is compact and connected and its boundary is diffeomorphic to ${X_1}^{m-1} \sqcup {X_2}^{m-1}$.
\item The image is $[m_v,M_v]$.
\item The preimage of the maximum is diffeomorphic to ${X_1}^{m-1}$. The preimage of the minimun is diffeomorphic to ${X_2}^{m-2}$.
\item There exists exactly one singular value and the singular value is $g(v)$.
\item The Reeb space is homeomorphic to a graph: this is also a specific case of a main theorem of \cite{saeki}.
\item The preimage of the singular value is connected ($q_{f_{v}}(p)$ in the next property is uniquely determined). 
\item There exists a suitable PL homeomorphism ${\phi}_v$ from the Reeb space onto a small regular neighborhood of $v$ in $K$ such that the composition of $q_{f_{v}}$ with ${\phi}_v$ maps $q_{f_{v}}(p)$ to $v$ for any singular point $p$.
\end{itemize}.

Note that the pair $({X_1}^{m-1},{X_2}^{m-1})$ can be also seen as an arbitrary pair of two manifolds represented as the disjoint unions of ($m-1$)-dimensional closed and connected manifolds diffeomorphic to the boundaries of some $m$-dimensional most fundamental handlebodies. The conditions on $g$, $r_K$ and the graph $K$ imply that this completes the proof of (\ref{mthm:1.1}) with the arguments in the original proof of Theorem \ref{thm:1}.
We explain about this more precisely. Around each connected component $R \subset K$ of the complementary set of the interior of the small regular neighborhood of the vertex set $V$ of $K$ (obtained as the disjoint union of all the small regular neighborhoods of the vertices chosen before), we can construct a local function $f_R$ as a trivial smooth bundle. 
$R$ is in the interior of an edge $e_R \subset K$ and PL homeomorphic to a closed interval. We can easily have a PL homeomorphism ${\phi}_R$ from the Reeb space of the function to $R$.
 
We have a desired function $f$ and a desired PL homeomorphism $\phi$ by gluing local smooth functions and local PL homeomorphisms together.

We prove (\ref{mthm:1.2}). We need topological theory on $3$-dimensional manifolds such as so-called ({\it integral}) {\it Dehn surgeries}. For systematic explanations on the theory, see \cite{hempel} for example.

By the theory of (integral) Dehn-surgeries on $3$-dimensional manifolds, every $3$-dimensional closed, connected and orientable manifold can be regarded as the boundary of some $4$-dimensional most fundamental handlebody which is also orientable. (Our proof of) (\ref{mthm:1.1}) and an argument on the orientability of the manifold $M$ of the domain in the last of the original proof of Theorem \ref{thm:1} completes the proof.  
\end{proof}
\begin{Rem}
For theory on diffeomorphism types for handlebodies and attachments of handles, see papers by Wall (\cite{wall}--\cite{wall6}) and see also \cite{ranicki} for example.
\end{Rem}
\begin{Rem}
We close the present paper by introducing related studies among the others with results of the author.

\cite{michalak} has given an answer to Main Problem by constructing a Morse functions such that preimages of regular values are disjoint unions of copies of a unit sphere. \cite{michalak2} studies deformations of Morse functions and their Reeb graphs. They motivated the author to give a new answer as \cite{kitazawa} or Theorem \ref{thm:1}.

\cite{saeki} is regarded as a paper motivated by \cite{kitazawa}. This also studies more general cases where diffeomorphism types for closed (compact), connected and smooth manifolds assigned to edges are general satisfying only conditions on so-called cobordism relations on closed (or compact) smooth manifolds. On the other hand, this does not study functions with very explicit singularities such as Morse(-Bott) functions.

\cite{gelbukh} and \cite{gelbukh2} are related studies where Morse-Bott functions on closed surfaces and general closed manifolds are considered. Studies respecting preimages of regular values are new important problems and regarded as variants of Main Problem.

Last, \cite{kitazawa2} and \cite{kitazawa3} are also studies on Main Problem. The former studies cases where preimages may not be compact. The latter is on non-orientable cases for Theorem \ref{thm:1}. Proofs of results there are also as presented in the last of the first section.
\end{Rem}
\section{Acknowledgement, grants and data.}
\label{sec:3}
\thanks{The author is a member of the project JSPS KAKENHI Grant Number JP17H06128 "Innovative research of geometric topology and singularities of differentiable mappings" (Principal Investigator: Osamu Saeki). This work is also supported by the project. 
The author would like to thank Osamu Saeki again for private discussions on \cite{saeki} with \cite{kitazawa}. These discussions continue to motivate the author to study the present study and related studies further. 
All data supporting the present study are in the present paper.}
