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for an integer 0 ≤ i(p) ≤ m, which is shown to be uniquely chosen, and for suitable coordinates. A Morse-Bott function is a smooth function represented as the composition of a smooth submersion with a Morse function at each singular point. 1.1. Reeb spaces and Reeb graphs. From a differentiable map c : X → Y , we can define an equivalence relation ∼c on X as follows: x1∼cx2 holds if and only if they are in a same connected component of c−1(y) for some point y.

Definition 1. The quotient space Wc := X/∼c is defined as the Reeb space of c.

For the Reeb space of c, qc : X → Wc denotes the quotient space. We can define a map which is denoted byc ¯ uniquely so that we have the relation c =¯c ◦ qc. Reeb spaces are in considerable cases graphs such that the vertex sets are the sets −1 of all points p ∈ Wc giving the preimages qc (p) containing at least one singular point of the maps c. This is called the of c. For example, Reeb spaces are Reeb graphs for smooth functions on compact manifolds with finitely many singular values ([20]). 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 [5]–[10] and [21], for example: so-called fold maps such that preimages of regular values are disjoint unions of spheres are studied for ex- ample. We do not concentrate on fold maps in the present study. Fold maps are, in short, smooth maps from smooth manifolds with no boundaries into manifolds with no boundaries represented as products maps of Morse functions and identity maps on open disks 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 [23] for general related theory and see also [14] for cases where the manifolds of the targets are surfaces, for example. For Reeb graphs and Reeb spaces, there exist various papers. [19] seems to be one of pioneering papers. Other papers will be presented in the present paper. 1.2. Some notions on graphs which are topologized canonically. A graph is naturally (PL) homeomorphic to a 1-dimensional polyhedron. An isomorphism between two graphs K1 and K2 is a (PL) homeomorphism from K1 to K2 mapping the vertex set of K1 onto the vertex set of K2. Definition 2. A continuous real-valued function g on a graph K is said to be a good function if it is injective on each edge. In the present paper, we only consider finite (and connected) graphs. It imme- diately follows that a finite graph has such a function if and only if it has no loops as edges. 1.3. Main problem and Main Theorems. The present paper studies another important problem as follows. Main Problem. 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 REALIZING GRAPHS AS REEB GRAPHS OF MORSE FUNCTIONS AND PREIMAGES 3 graph isomorphic to the graph and preimages of regular values for which are as prescribed? More explicitly, can we construct functions as Morse functions?

This is so-called realization problems of graphs as Reeb graphs of smooth func- tions of suitable classes. [22] is a pioneering paper on this. Other related papers will be presented in the present paper. In the present paper, we first show the following results as Main Theorems. 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. The equivalence class which a smooth manifold belongs to is said to be the diffeomorphism type for a manifold. A handlebody is a smooth manifold of a suitable class of compact, connected and smooth manifolds whose dimensions are greater than 1. This notion will be presented in the next section.

Main Theorem 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 rK on the edge set E satisfying either of the following three for each edge e. Let m> 1 be an integer. (1) At each edge of K containing a vertex where g has a local extremum, the m−1 value of rK is the diffeomorphism type for S . (2) Each vertex where g has a local extremum is of degree 1. (3) The values of rK are always diffeomorphism types for closed and connected manifolds diffeomorphic to boundaries of most fundamental handlebodies. Then there exist an m-dimensional closed, connected and orientable manifold M and a Morse function f on M satisfying the following three properties.

(1) The Reeb graph Wf of f is isomorphic to K: we can take a suitable iso- morphism φ : Wf → K compatible with the remaining properties. (2) If we consider the natural quotient map qf : M → Wf and for each point φ(p) ∈ K (p ∈ Wf ) that is not a vertex and that is in an edge e, then the −1 diffeomorphism type for the preimage qf (p) is rK (e). (3) For each point p ∈ M mapped by qf to a vertex vp := qf (p) ∈ Wf , f(p)= g ◦ φ(vp).

Main Theorem 2. 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 rK on the edge set E satisfying either of the following three for each edge e. (1) At each edge of K containing a vertex where g has a local extremum, the m−1 value of rK is the diffeomorphism type for S . (2) Each vertex where g has a local extremum is of degree 1. (3) The values of rK are always diffeomorphism types for 3-dimensional closed, connected, and orientable manifolds. Then there exist a 4-dimensional closed, connected and orientable manifold M and a Morse function f on M satisfying the following three properties.

(1) The Reeb graph Wf of f is isomorphic to K: we can take a suitable iso- morphism φ : Wf → K compatible with the remaining properties. 4 NAOKI KITAZAWA

(2) If we consider the natural quotient map qf : M → Wf and for each point φ(p) ∈ K (p ∈ Wf ) that is not a vertex and that is in an edge e, then the −1 diffeomorphism type for the preimage qf (p) is rK (e). (3) For each point p ∈ M mapped by qf to a vertex vp := qf (p) ∈ Wf , f(p)= g ◦ φ(vp). The following theorem is previously obtained and motivates us to study the present problems. Main Theorem 1 can be regarded as an extension of Theorem 1. Theorem 1 ([11]). 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 rK on the edge set E. These two functions satisfy the following conditions. (1) At each edge of K containing a vertex where g has a local extremum, the value of rK is 0. (2) Each vertex where g has a local extremum is of degree 1. Then there exist a 3-dimensional closed, connected and orientable manifold M and a Morse function f on M satisfying the following three properties.

(1) The Reeb graph Wf of f is isomorphic to K and we can take a suitable isomorphism φ : Wf → K compatible with the remaining properties. (2) For each point φ(p) ∈ K (p ∈ Wf ) in the interior of an edge e, the preimage −1 qf (p) is a closed, connected, and orientable surface of genus rK (e). (3) For each point p ∈ M mapped by qf to a vertex vp := qf (p) ∈ Wf , f(p)= g ◦ φ(vp).

Furthermore, if we drop the two conditions on g and rK , then at each singular point where the function has a local extremum, except finitely many singular points, the functions are Morse-Bott functions. We prove Main Theorems in the next section. Roughly speaking, the methods are similar to the method of the proof of Theorem 1 or main theorem of [11] (and the methods of proofs of main Theorems of [12], and [13]). However, related differential topological arguments on manifolds belonging to some classes we need are mutually different. Main Theorems need such new arguments.

2. Proofs of Main Theorems and remarks. We review several notions and some technique from differential of smooth manifolds. A k-handle is a smooth manifold diffeomorphic to a smooth manifold of the form Dk ×Dm−k (0 ≤ k ≤ m) as a cornered manifold. Note that D0 denotes a one-point set. A 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 ways and obtain smooth manifolds with no corners. The resulting diffeomorphism types are unique. We introduce a procedure of constructing compact(, connected) and smooth manifolds of good classes. (1) Take an integer m> 1. REALIZING GRAPHS AS REEB GRAPHS OF MORSE FUNCTIONS AND PREIMAGES 5

m (2) Set H0 as a non-empty space and an m-dimensional compact and smooth manifold. m (3) Set k = 0 and take a point in ∂H0 , or set a positive integer k

A proof of Main Theorem 1. As the original proof of Theorem 1, we construct local Morse functions around each vertex, respecting the three properties and glue them together. We explain the construction of the local functions to complete the proof. Around each vertex where g has a local extremum, we construct a Morse function on a copy of the unit disk obtained by considering the natural height. Around each vertex v where g does not have a local extremum, we will construt a local Morse function. This is a new ingredient. m Set H0 as a copy of the m-dimensional unit disk in the situation before. We attach l ≥ 0 (m − 1)-handles and suitable handles to obtain an arbitrary most fundamental handlebody whose boundary consists of exactly l + 1 connected com- ponents which are also the boundaries of most fundamental handlebodies by the definition. We remove a smoothly embedded copy of the m-dimensional unit disk m from the submanifold H0 in such a most fundamental handlebody. We prepare two compact and connected manifolds obtained in such ways and glue them via a diffeomorphism between the boundary components originally the copies of the m- dimensional unit disk are attached to. ∂H1 and ∂H2 denote the original boundaries of the original most fundamental handlebodies. By the exposition on the relation- ship between singularities of Morse functions and handles, we easily have a Morse function such that the preimage of the minimum is ∂H1, that of the maximum is ∂H2, and that the singular value is g(v) and in the interior of the image, which is a closed interval. The three conditions on rK imply that this completes the proof with the argu- ments in the original proof of Theorem 1. 

Remark 1. For theory on diffeomorphism types for handlebodies and attachments of handles, see papers by Wall ([24]–[29]) and see also [18] for example. We prove Main Theorem 2. For systematic topological theory on 3-dimensional manifolds including so-called (integral) Dehn surgeries, see [4] for example.

Proof. By the theory of (integral) Dehn-surgeries on 3-dimensional manifolds, every 3-dimensional closed, connected and orientable manifolds can be regarded as the boundaries of 4-dimensional most fundamental handlebodies which are orientable. Main Theorem 1 and an argument on the orientability of the manifold M of the domain in the last of the original proof of Theorem 1 completes the proof. 

Remark 2. We close the present paper by introducing related studies among the others with results of the author. [15] 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. [16] studies deformations of Morse functions and their Reeb graphs. They motivated the author to give a new answer as [11] or Theorem 1. [20] is regarded as a paper motivated by [11]. This studies very general cases where diffeomorphism types for closed (compact), connected and smooth manifolds assigned to edges are very general satisfying only conditions on so-called cobordism relations on closed (or compact) smooth manifolds. On the other hand, this does REALIZING GRAPHS AS REEB GRAPHS OF MORSE FUNCTIONS AND PREIMAGES 7 not study functions with very explicit singularities such as Morse functions, Morse- Bott functions, and so on. [1] and [2] 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, [12] and [13] are also studies on Main Problem. They study cases where preimages may not be compact and non-orientable cases for Theorem 1 respectively. Proofs of results there are also as presented in the last of the first section,

3. Acknowledgement, grants and data. The author is a member of the project JSPS KAKENHI Grant Number JP17H06128 ”Innovative research of geometric topology and singularities of differentiable map- pings” (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 [20] with [11]. 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.

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