Characterizing $3$-dimensional manifolds represented as connected sums of Lens spaces, $S^{2}\times S^{1}$, and torus bundles over the circle by certain Morse-Bott functions

We refer to several figures from [11] where we do not assume essential arguments first presented in [11]. Some arguments here are presented in [11] and several existing studies.
 
STEP 1 The manifold $M$ of the domain of the Morse-Bott function $f:M\rightarrow\mathbb{R}$ must be of the presented class of $3$-dimensional manifolds.

STEP 1-1 The preimage ${q_{f}}^{-1}(N(v))$ of a small regular neighborhood $N(v)$ of a vertex $v$ where $f$ has a local extremum.
We investigate the preimage ${q_{f}}^{-1}(N(v))$ of a small regular neighborhood $N(v)$ of a vertex $v$ where $f$ has a local extremum. We respect the four cases from (2).
 
CASE 1-1-1
In the first case (2a), the preimage is diffeomorphic to the disk $D^{3}$ and we have a Morse function having exactly one singular point in the interior. It is regarded as a so-called height function of the disk $D^{3}$.
 
CASE 1-1-2
In the second case (2b), the preimage is diffeomorphic to the product $S^{1}\times D^{2}$ and we have a Morse-Bott function the set of all singular points of which is seen as the set $S^{1}\times\{0\}\subset S^{1}\times{\rm Int}\ D^{2}$. This can be deformed into a simple STF Morse function with exactly two singular points by a suitable small homotopy, thanks to fundamental singularity theory and elementary topological arguments.
 
CASE 1-1-3
In the third case (2c), the preimage is diffeomorphic to the product $S^{1}\times S^{1}\times D^{1}$ or $S^{2}\times D^{1}$ and we have a Morse-Bott function the set of all singular points of which is seen as the set $S^{1}\times S^{1}\times\{0\}\subset S^{1}\times S^{1}\times{\rm Int}\ D^{1}$ or $S^{2}\times\{0\}\subset S^{2}\times{\rm Int}\ D^{1}$. The latter case of these two functions can be deformed into a simple STF Morse function with exactly two singular points by a suitable small homotopy, thanks to fundamental singularity theory and elementary topological arguments.

Note that the former function here is closely related to a key ingredient in STEP 1-2 and our new arguments.
 
CASE 1-1-4
In the fourth case (2d), the preimage is shown to be diffeomorphic to the complementary set of the interior of a smoothly embedded disk $D^{3}$ in the $3$-dimensional real projective space ${\mathbb{R}P}^{3}$ by a fundamental argument on $3$-dimensional manifolds. Note that the $3$-dimensional real projective space is a circle bundle over $S^{2}$, for example. We have a Morse-Bott function the set of all singular points of which is seen as a copy of the real projective plane ${\mathbb{R}P}^{2}$ embedded in a certain canonical way in the $3$-dimensional real projective space ${\mathbb{R}P}^{3}$.
 
STEP 1-2 Deforming the Morse-Bott function $f$ locally around the preimages ${q_{f}}^{-1}(N(v))$ of small regular neighborhoods $N(v)$ of vertices $v$ where $f$ has no local extremum.
 
By a suitable small homotopy, we can deform the Morse-Bott function around the preimages ${q_{f}}^{-1}(N(v))$ of small regular neighborhoods $N(v)$ of vertices $v$ where $f$ has no local extremum preserving the function on the complementary set of the interior of the disjoint union of these local preimages ${q_{f}}^{-1}(N(v))$. The local functions are originally Morse by (3), and by (1), STF Morse functions. By [18, Lemma 6.6], these local functions are changed into simple STF Morse functions. Such an argument is also presented in [11] and we present again.

We also present additional arguments and figures from [11]. FIGURE 1 shows local forms of simple STF Morse functions by Reeb digraphs with information on preimages.

Blue (red) colored edges show edges the preimages of single points in the interior of which are diffeomorphic to $S^{2}$ (resp. $S^{1}\times S^{1}$). Black dots are for vertices. We adopt this rule for our figures. We can consider the case of $-c$ for the function $c$ and we omit the case. We also respect this fundamental rule, hereafter.

FIGURE 2 shows local changes. Each homotopy is realized by a suitable small homotopy. This is from fundamental singularity theory together with the fundamental fact on the 3-dimensional manifold theory: the preimage of the presented local graph (local complex) is diffeomorphic to a manifold obtained by removing the interiors of two disjointly and smoothly embedded copies of the disk $D^{3}$ in the interior of $S^{1}\times D^{2}$.

We can consider a suitable iteration of local changes of functions presented in FIGURE 2 to have another new Morse-Bott function $f_{0}:M\rightarrow\mathbb{R}$ and a new Reeb digraph such that the union of all edges the preimages of single points in the interiors of which are diffeomorphic to $S^{1}\times S^{1}$ consists of connected components represented as either of the following. If we need, then we change the local functions for CASE 1-1-2 into simple STF Morse functions locally.

Note that for Reeb spaces and Reeb graphs, preimages are considered for the map $q_{c}:W_{c}\rightarrow Y$ for a given map $c:X\rightarrow Y$. We also remember the map $\bar{c}$ with the relation $c=\bar{c}\circ q_{c}$.

We note that, hereafter, we present some new arguments, which do not appear in [11].

• •

  A closed arc $I$ satisfying the following: vertices of the boundary points, points of $\partial I$, are of degree $2$ and ones where the function $\bar{f_{0}}:W_{f_{0}}\rightarrow\mathbb{R}$ does not have a local maximum, and vertices in the interior are of degree $2$ and ones where the function $\bar{f_{0}}:W_{f_{0}}\rightarrow\mathbb{R}$ has a local extremum. We can also check that the preimage ${q_{f}}^{-1}(N(I))$ of a small regular neighborhood $N(I)$ of $I$ is diffeomorphic to a manifold obtained by removing the interiors of two smoothly and disjointly embedded copies of $D^{3}$ from a Lens space, $S^{2}\times S^{1}$, or $S^{3}$.

• •

  A connected component $C$ homeomorphic to the circle $S^{1}$ satisfying the following: at each vertex $v\in C$ there where the restriction ${\bar{f_{0}}}{\mid}_{C}$ to $C$ of the function $\bar{f_{0}}:W_{f_{0}}\rightarrow\mathbb{R}$ has a local extremum, the original function $\bar{f_{0}}$ also has a local extremum at the vertex $v$ and $v$ is also of degree $2$ in the graph $W_{f_{0}}$. We can also check that the preimage ${q_{f}}^{-1}(N(C))$ of a small regular neighborhood $N(C)$ of $C$ is diffeomorphic to a manifold obtained by removing the interiors of finitely many smoothly and disjointly embedded copies of $D^{3}$ from a torus bundle over the circle.

Remembering STEP 1-1 with CASE 1-1-1–CASE 1-1-4 and Theorem 1 for example, we can see that the closed, connected and orientable manifold $M$ of the domain is diffeomorphic to a manifold obtained in the following way.

• •

  We prepare a suitable finite family of torus bundles over $S^{1}$, Lens spaces, copies of $S^{2}\times S^{1}$, and copies of $S^{3}$.

• •

  We remove the interiors of finitely many smoothly and disjointly embedded copies of $D^{3}$ from the prepared manifolds suitably.

• •

  We glue the resulting manifolds suitably to have our desired closed, connected and orientable manifold along the boundaries, each of which is diffeomorphic to $S^{2}$. The resulting manifold is diffeomorphic to $M$.

Our manifold $M$ is also of the class of $3$-dimensional closed, connected and orientable manifolds presented in the statement of Theorem 2.

This completes STEP 1.
 
STEP 2 Construct a desired Morse-Bott function on a $3$-dimensional manifold $M$ of the presented class, conversely.

On a sphere $S^{3}$, we have a Morse function with exactly two singular points. It is also a specific case of simple STF Morse functions.

We can reconstruct a desired simple STF Morse function on each lens space $M_{j}$, $S^{1}\times S^{2}$ and $S^{3}$ from the colored digraph in FIGURE 3. Rules for colors are for preimages as presented: blue (red) colored edges are for preimages of single points diffeomorphic to $S^{2}$ (resp. $S^{1}\times S^{1}$). Blue colored vertices are for removal of (the interiors of) small regular neighborhoods of the vertices and the preimages: preimages here are diffeomorphic to $D^{3}$ (${\rm Int}\ D^{3}$).

In addition, we can also reconstruct a desired Morse-Bott function $f_{j}:M_{j}\rightarrow\mathbb{R}$ on each torus bundle $M_{j}$ over the circle $S^{1}$ from the colored digraph in FIGURE 4. Rules for colors and blue vertices are same as those in the previous situation.

On a general $M$, represented as a connected sum of the manifolds $M_{j}$, by gluing these local Morse-Bott functions suitably, we have a desired Morse-Bott function $f:M\rightarrow\mathbb{R}$.

For our reconstruction here, the author has discussed more general cases in [8, 9] for example. However we do not understand these studies of course. Our construction in the present proof is very fundamental, natural and specific.

Note also that we can have a resulting Reeb digraph as a so-called cactus graph: a cactus graph is a graph such that each edge is contained in at most one circle (simple cycle).
 
This completes the proof. ∎

Here, we refer to several published articles [8, 9] and arguments. We also refer to some preprints of the author [10, 11].

We can have the following subgraph $K^{\prime}$ of $K$ in the unique way.

• •

  The subgraph $K^{\prime}$ contains no edge in cycles $C_{j}$.

• •

  The subgraph $K^{\prime}$ is maximal among all subgraphs $K^{\prime\prime}$ satisfying the previous condition.

We do the following operation for each vertex $v_{C}$ of $K^{\prime}$ originally contained in some cycle $C_{j}$.

We add two oriented edges $e_{v_{C,K^{\prime}},{\rm l}}$ and $e_{v_{C,K^{\prime}},{\rm h}}$. We also add the first edge $e_{v_{C,K^{\prime}},{\rm l}}$ as an edge departing from a new vertex $v_{C,K^{\prime},{\rm l}}$ and entering $v_{C}$. We also add the second edge $e_{v_{C,K^{\prime}},{\rm h}}$ as an edge departing from the vertex $v_{C}$ and entering a new vertex $v_{C,K^{\prime},{\rm h}}$.

Let ${K_{0}}^{\prime}$ denote the resulting digraph. We extend the map $l_{K_{g}}$ as a map such that at each new edge the value is $0$. Let the map denoted by ${l_{K_{g},0}}^{\prime}$.

We also do the following operation for each vertex $v_{C}$ of each cycle $C_{j}$. The notation ”$v_{C}$ here” is same as the vertex $v_{C}$ before.

We add another two oriented edges $e_{v_{C},{\rm l}}$ and $e_{v_{C},{\rm h}}$. We also add the first edge $e_{v_{C},{\rm l}}$ as an edge departing from a new vertex $v_{C,{\rm l}}$ and entering $v_{C}$. We also add the second edge $e_{v_{C},{\rm h}}$ as an edge departing from the vertex $v_{C}$ and entering a new vertex $v_{C,{\rm h}}$. Let $C_{0,j}$ denote the resulting digraph, obtained as a result of the change from $C_{j}$.

We further extend the maps $l_{K_{g}}$ and ${l_{K_{g},0}}^{\prime}$ as a map such that at each of additional new edges the value is $0$. Let the map denoted by $l_{K_{g},0}$.

For ${K_{0}}^{\prime}$, we can have a Morse-Bott function $f^{\prime}:M^{\prime}\rightarrow\mathbb{R}$ on a suitably chosen manifold $M^{\prime}$ as in the statement of Theorem 2: for the preimages we respect the map ${l_{K_{g},0}}^{\prime}$ (and the map $l_{K_{g},0}$). We can construct the function in such a way that the set of all singular points of the function is a disjoint union of copies of $S^{2}$, $S^{1}$, $S^{1}\times S^{1}$ and one-point sets: we consider CASE 1-1-1, CASE 1-1-2 and CASE 1-1-3 in the proof of Theorem 2. For the class of manifolds $M^{\prime}$ belongs to, we need additional consideration on local construction around each vertex $v$ where for the digraph ${K_{0}}^{\prime}$ both an edge entering $v$ and an edge departing from $v$ exist. We construct locally as in [11, (3) in STEP 2 in the proof of Theorem 2]. Or we also construct as presented in the articles [8, 9]: for example [8, Proof of Theorem 1.2]. In this construction, it is most important to choose the local function so that we can deform the local function by a suitable homotopy into a simple STF Morse function having the preimage of some single value diffeomorphic to $S^{2}$ and containing no singular point of the simple STF Morse function.

Our construction also yields a Morse-Bott function ${f_{0}}^{\prime}:M^{\prime}\rightarrow\mathbb{R}$ on $M^{\prime}$ which is, around each point where the function does not have a local extremum, represented as a simple STF Morse function. In addition, the function ${f_{0}}^{\prime}$ satisfies the following. The union of all edges the preimages of single points in the interiors of which are diffeomorphic to the torus $S^{1}\times S^{1}$ is a disjoint union of intervals: as a small remark, we deform the local function around a vertex for CASE 1-2 in the proof of Theorem 2 into a local simple STF Morse function, if we need. For this, note that for any simple cycle ${C_{{K_{0}}^{\prime}}}^{\prime}$ in the graph ${K_{0}}^{\prime}$, at some vertex $v\in{C_{{K_{0}}^{\prime}}}^{\prime}$, either $g{\mid}_{{C_{{K_{0}}^{\prime}}}^{\prime}}$ does not have a local extremum or $g$ does not have a local extremum on the original graph $K$. This is thanks to the assumption that if the graph $K$ is homeomorphic to the circle $S^{1}$, then at some vertex $v\in K$, $g$ does not have a local extremum. Each connected component of the manifold $M^{\prime}$ belongs to the class of ”the $3$-dimensional manifolds $M$ of Theorem 1”. Remember also that such an argument is presented in STEP 1-2 in the proof of Theorem 2. By the structure of the Morse-Bott function and fundamental arguments on Morse functions, we have a copy ${D^{3}}_{v_{C}}$ of the disk $D^{3}$ small and smoothly embedded into $M^{\prime}$ and mapped onto the union of the two edges $e_{v_{C,K^{\prime}},{\rm l}}$ and $e_{v_{C,K^{\prime}},{\rm h}}$ for each vertex $v_{C}$ before. We can also choose the disk ${D^{3}}_{v_{C}}$ as a space containing exactly two singular points of the function ${f_{0}}^{\prime}$. For related arguments, for our previous result, see also [10, A proof of Main Theorem 1] for example.

For each simple cycle $C_{j}$, we remove all vertices $v_{C_{j}}$ where the function does not have local extrema from $C_{j}$ and have a new simple cycle ${C_{j}}^{\prime}$. We can construct a Morse-Bott function on $f_{j}:M_{j}\rightarrow\mathbb{R}$ and the Reeb digraph isomorphic to ${C_{j}}^{\prime}$ and the preimage of a point in $C^{\prime}$ is diffeomorphic to the torus $S^{1}\times S^{1}$: the point may be a point in the interior of an edge or a vertex. We can also reconstruct a Morse-Bott function $f_{0,j}:M_{j}\rightarrow\mathbb{R}$ which is, around each vertex where the function does not have a local extremum, represented as an STF Morse function not being simple, whose Reeb digraph is isomorphic to $C_{0,j}$ and which respects the map $l_{K_{g},0}$ for the preimages. We can reconstruct the functions so that $f_{j}$ is deformed into $f_{0,j}$ by a suitable homotopy, thanks to fundamental arguments on Morse functions, singular points of them and (generating and canceling) handles. We also have a copy ${D^{3}}_{v_{C_{j}}}$ of the disk $D^{3}$ small and smoothly embedded into $M_{j}$ and mapped onto a closed interval embedded in the interior of the union of two adjacent edges $e_{v_{C_{j}},{\rm l}}$ and $e_{v_{C_{j}},{\rm h}}$ for each vertex $v_{C_{j}}$ before. Remember that for these edges, the notation of the form ”$e_{v_{C},{\rm l}}$, $e_{v_{C},{\rm h}}$, and $v_{C}$” is used before. We have no problem on the usage. We can also choose the disk ${D^{3}}_{v_{C_{j}}}$ as a space containing no singular points of the function $f_{0,j}$. For related arguments, for our previous result, see also [10, A proof of Main Theorem 1] again, for example.

Respecting these arguments and information, we construct a new $3$-dimensional manifold $M$. First we remove the interiors of disks ${D^{3}}_{v_{C}}$ and ${D^{3}}_{v_{C_{j}}}$ from $M_{j}$ and $M^{\prime}$. We glue the resulting manifolds along connected components of the boundaries one after another suitably to have a closed, connected and orientable manifold $M$. More explicitly, we glue connected components originally corresponding to a same vertex $v_{C}=v_{C_{j}}\in K$ where abusing the notation in such a way has no problem. We also glue the functions $f^{\prime}$ and $f_{0,j}$ (, restricted to the remaining manifolds).

We can have a new Morse-Bott function $f^{\prime}:M\rightarrow\mathbb{R}$ on the new manifold $M$ whose Reeb digraph is isomorphic to a digraph obtained by identifying $e_{v_{C,K^{\prime}},{\rm l}}$ with $e_{v_{C},{\rm l}}$, and $e_{v_{C,K^{\prime}},{\rm h}}$ with $e_{v_{C},{\rm h}}$ from ${K_{0}}^{\prime}$ and $C_{0,j}$, canonically. By the structure of the Morse-Bott function and fundamental arguments on Morse functions, especially on singular points of the functions, handles and so-called canceling pairs of singular points of the functions or handles, we have a desired function $f:M\rightarrow\mathbb{R}$. Last, we can also see that $M$ is represented as in the statement.

This completes the proof.

∎