Stable Bi-Maps on Surfaces and Their Graphs

ABSTRACT

In this paper we study stable bi-maps $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ from a global viewpoint, where M is a smooth closed orientable surface and $f_1:M\to \mathrm{ℝ},f_2:M\to {\mathrm{ℝ}}^2$ are stable maps. We associate a graph to F, so-called ℛℳ-graph and study its properties. The ℛℳ-graph captures more information about the topological structure of M than other graphs that appear in literature. Moreover, some graph realization theorems are obtained.

Keywords:
Stable maps; ℛℳ-graphs; closed surfaces

1 INTRODUCTION

The graph theory has been increasingly used to solve various real-world problems, beyond, of course, the Mathematics problems. The graph theory has applications in Optimization (logistic and transportation problems), Organic Chemistry, Physics (statistical mechanics and solid state Physics), Electrical Engineering (communication network and Coding theory), among others. We can find a wide scope of these applications in the references ^{5), (14}. Some recent applications of graph theory include genome sequencing, starlight interferometer program and DNA sequence. The special class of bipartite graphs that will appear in this paper has many interesting and even surprising applications (see for instance ¹⁶⁾ ).

In Singularity theory, many works resort to graph theory to describe local/global invariants and combinatorial models to investigate the recognition and classification problem involving maps or map germs. The authors themselves of this paper have some works in this direction (see, for instance, ^{3), (8}). It is important to highlight that the Singularity theory has many practical applications in Physics, Engineering, Thermodynamic, among others.

In this work, we resort graphs to study stable bi-maps $F=(f_1,f_2)$ defined on a smooth closed orientable surface $M\subset {\mathrm{ℝ}}^3$. Stable bi-maps means a pair of stable maps, i.e., $f_1:M\to \mathrm{ℝ}$ and $f_2:M\to {\mathrm{ℝ}}^2$ are stable maps. Stable maps have been investigated by several authors and have many interesting applications (see, for instance, ^{3), (4), (8), (11), (13), (18}).

Let us describe the stable bi-maps as above and to show how to associate the graph theory to investigate it.

Denote by $C^{\infty }(M,{\mathrm{ℝ}}^p)$ the set of all C ^∞ maps from M to ℝ^p , p = 1, 2.

Firstly, consider a stable map $f_1:M\to \mathrm{ℝ}$. For this type of map, it is known that the Reeb graph is a global topological invariant associated to f ₁ (cf. ^{5), (15}). The Reeb graph describes the topology of the surface M. Reeb graphs appear with many applications in Computational Geometry, Computer Graphics, Engineering, Applied Mathematics, etc. We will denote the Reeb graph associated to a stable map $f_1:M\to \mathrm{ℝ}$ by ℛ-graph of f ₁. The Figure 1 shows a practical application of ℛ-graphs on the study of human body shape and posture using 3D images (cf. ¹⁷).

Figure 1:
ℛ-graphs associated to human body shapes (for details, see ¹⁷). This figure was extracted from the reference ¹⁷, fig. 2, page 157.

Notice that in Figure 1 (4), we have two distinct body shapes represented by the same ℛ-graph, (c) and (f). Since ℛ-graphs do not distinguish these two body shapes, this kind of problem motivates us to introduce a new graph which encodes information that just the ℛ-graph does not encode.

This new invariant is constructed as follows. We consider a stable map of type $f_2:M\to {\mathrm{ℝ}}^2$. From Whitney theorem (cf. ¹⁸), the singular set of f ₂ (denoted by $\Sigma f_2\subset M$) consists of curves of double points, possibly containing isolated cusp points. The apparent contour of f ₂ (i.e., f ₂ (Σ f ₂) the image of the singular set) consists of a number of immersed curves in ℝ² (possibly with cusps) whose self-intersections are all transverse and disjoint from the cusps (if any). The singular and regular components in M codify relevant information about the stable map f ₂. In fact, in ⁸ graphs with weights on the vertices were introduced as a global topological invariant for stable maps of type $f_2:M\to {\mathrm{ℝ}}^2$. These weighted graphs describe the position of the singular and the regular sets of f ₂ in M. Its edges, vertices and weights corresponding to the singular components, regular components and the genus of the regular components of M, respectively. We will denote the weighted graph associated to $f_2:M\to {\mathrm{ℝ}}^2$ by ℳ-graph of f ₂.

In Figure 1, if we consider f ₂ as the projection map of the human body shapes (4-a) and (4-d) on the (floor) plane, then the ℳ-graphs associated to f ₂ can be used to distinguish them while the ℛ-graphs cannot. To illustrate how the ℳ-graphs associated to f ₂ can distinguish different body shapes, see Figure 2.

Figure 2:
ℳ-graphs related to three distinct human body shapes.

By the other hand, there are situations in the converse sense: we can have the same ℳ-graphs but the ℛ-graphs are different. See, for instance, the Figure 3, (a) and (b).

Figure 3:
Example of stable bi-maps from sphere.

Then, it seems natural to consider the pair ℛ-graph and ℳ-graph, instead of considering just one of them. The pair captures more information about the topological structure of M than only one of them.

In this way, the motivation of this work is to consider stable bi-maps of type $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ and its associated ℛℳ-graphs. The ℛℳ-graph is exactly the pair formed by the ℛ-graph associated to f ₁ and by the ℳ-graph associated to f ₂. Moreover, we can use the ℛℳ-graphs to investigate the topological structure of the surface M and of the singular set Σf ₂. Previously, we showed a possible application of ℛℳ-graphs to study patterns of the human body. This kind of application is very interesting to Computer Vision. There will certainly be many other practical applications of such ℛℳ-graphs.

The last part of this paper is dedicated to investigate the graph realization problem. This is a very interesting and difficult problem. Given a graph Γ, in which conditions Γ can be realized as an ℛℳ-graph associated to some bi-stable map $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$? We answer this question in Section 5 for some cases (see Theorems 5.3, 5.4 and 5.5). Recently, many authors have dedicated themselves to the realization problem of ℛ-graphs associated with Morse functions or Morse-Bott functions (see, for instance, the works of L.P. Michalak ¹², I. Gelbukh ⁶, N. Kitazawa ⁹, among others). The second and third named authors of this paper, have also a recent work about the realization problem of the calls MB-Reeb graphs ². The MB-Reeb graphs are a kind of generalization of the R-graphs and they were introduced in ² to investigate the topological classification of circle-valued Morse-Bott functions defined on surfaces.

Some techniques applied here for a ℳ-graph such as surgeries, codimension one transitions, etc., are based on the references ^{8), (13}.

2 STABLE BI-MAPS

Let M be a closed orientable surface and N = ℝ^p (p = 1, 2). Two smooth maps $f,g:M\to N$ are said to be 𝒜-equivalent (or just equivalent) if there are orientation-preserving diffeomorphisms, $k:M\to M$ and $l:N\to N$ such that $g\circ k=l\circ f$. A smooth map $f:M\to N$ is said to be stable if all maps sufficiently closed to f (with respect to Whitney C ^∞-topology) are equivalent to f (see ⁷ for more details).

A smooth map $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ is said to be a stable bi-map if each fi, i = 1, 2, is a stable map.

Considering $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ be a stable bi-map notice that we have the following properties:

• a) Since $f_1:M\to \mathrm{ℝ}$ is stable then f ₁ is Morse with distinct critical values. That is, every critical point of f ₁ is non-degenerate and each critical level curve of f ₁ has a finite number of critical points, all distincts.

• b) Since $f_2:M\to {\mathrm{ℝ}}^2$ is stable then its singular points are only folds and isolated cups. Remind that a point p ∈ M is a regular point of f ₂ if the map f ₂ is a local diffeomorphism around p. Otherwise, the point p is said to be a singular point. According to Whitney theorem (cf. ¹⁸), the singularities of any stable map $f_2:M\to {\mathrm{ℝ}}^2$ are (locally) of fold type $(x,y)\mapsto (x,y^2)$ or of cusp type $(x,y)\mapsto (x^3+yx,y)$.

• c) The set of all singular points of f ₂, denoted by Σf ₂, is called singular set of f ₂. The singular set of f ₂ consists of (finitely many) disjoint embedded closed curves in M. The image of singular set of f ₂, i.e., f ₂(Σf ₂), is called the apparent contour of f ₂. The apparent contour of f ₂ is a finite number of immersed closed plane curves with finite number of cups and finite number of transverse intersections and self-intersections (disjoint from the set of cups). The regular set of f ₂, given by M\Σf ₂, consists of all regular points of f ₂. Since M is a smooth closed orientable surface, the singular set Σf ₂ is a finite collection of closed regular simple curves on M formed by fold points with possible isolated cusp points that divide M in a set of regular regions.

Let $j:M\to {\mathrm{ℝ}}^3$ be an immersion of M in ℝ³ and $\overrightarrow{v}\ne 0$ be any vector in ℝ³. We can decompose ℝ³ in a direct sum R _v ⊕ 𝒫 _v , where R _v is a parallel line to $\overrightarrow{v}$ and 𝒫 _v is a orthogonal plane to $\overrightarrow{v}$. Then we can consider two stable projections as following:

• 1. ${\pi }_v^1:j\left(M\right)\to R_v$ given by the restriction to j(M) of the canonical projection of ℝ³ in R _v .

• 2. ${\pi }_v^2:j\left(M\right)\to {\mathcal{P}}_v$ given by the restriction to j(M) of the canonical projection of ℝ³ in 𝒫 _v .

Let $h:R_v\to \mathrm{ℝ}$ and $g:{\mathcal{P}}_v\to {\mathrm{ℝ}}^2$ be diffeomorphisms and consider $f_1:M\to \mathrm{ℝ}$ and $f_2:M\to {\mathrm{ℝ}}^2$ two stable maps. Then we can define the following stable maps using ${\pi }_v^1$ and ${\pi }_v^2$:

$f_1=h\circ {\pi }_v^1\circ j\mathrm{a}\mathrm{n}\mathrm{d}{f}_2=g\circ {\pi }_v^2\circ j.$

These stable maps f ₁ and f ₂ as above defined are called projection to a parallel line to$\overrightarrow{v}$ and projection to an orthogonal plane to$\overrightarrow{v}$, respectively, associated to the immersion j (see Figure 3).

Observe that when f ₁ is a projection to a parallel line to $\overrightarrow{v}$ and f ₂ is a projection to a orthogonal plane to $\overrightarrow{v}$, then the pair F = (f ₁ , f ₂) is clearly a stable bi-map.

From now one, we will always consider in this paper stable bi-maps F = (f ₁ , f ₂) where f ₁ is a projection to a parallel line $\overrightarrow{v}$, f ₂ is a projection to an orthogonal plane to $\overrightarrow{v}$ and $\overrightarrow{v}\ne 0$ is a fixed vector in ℝ³.

The Figure 3 illustrates three different stable bi-maps from sphere 𝕊². Notice that in (a) and (b), the projections to the orthogonal plane 𝒫 _v have the same number of connected components of the singular set, while the projections to R _v have distinct number of singular points. Already in (b) and (c), the projections to R _v have the same number of singular points while the projections to 𝒫 _v have distint numbers of singular curves. In this way, the classsical invariant related to stable projections to the orthogonal plane 𝒫 _v do not distinguish (a) and (b) but they distinguish (b) and (c). By other hand, the classical invariant associated to the projections to the parallel line R _v can distinguish (a) and (b) but not (b) and (c).

In fact, the classical invariant associated to the projection to R _v (i.e., the stable map f ₁) was codified in the literature by the Reeb graph associated to f ₁. The Reeb graphs were introduced by Reeb in ¹⁵. Here the Reeb graphs will be denoted by ℛ-graphs. The next Subsection 3.1 will be dedicated to explain the ℛ-graphs.

By other hand, the classical invariant associated to the projection to 𝒫 _v (i.e., the stable map f ₂) was codified in the literature by Hacon-Mendes-Romero in ⁸, by a graph with weights in the vertices. Here these graphs will be denoted by ℳ-graphs or Mendes-graphs. The next Subsection 3.2 will be dedicated to explain the ℳ-graphs.

The Figure 4 illustrates the respectives ℛ-graphs and ℳ-graphs in the three previous examples given in Figure 3. Notice that in (a) and (b) the respective ℛ-graphs are not isomorphic while in (b) and (c) only the ℳ-graphs are not isomorphic. The Figure 5 shows one more example of two stable maps from sphere 𝕊² and their respective ℛ-graphs and ℳ-graphs.

Figure 4:
ℛ-graphs and ℳ-graphs corresponding to the Figure 3.

Figure 5:
Examples of ℛ-graphs and ℳ-graphs associated to stable maps on sphere.

This suggests that these two graphs togheter may give more information to differ two immersions of M in ℝ³ that just one of them. This is our motivation to introduce, in the next Section, the called ℛℳ-graph which is a pair of graphs.

3 ℛℳ-GRAPHS ASSOCIATED TO BI-STABLE MAPS

Given a smooth closed orientable surface M, let $j:M\to {\mathrm{ℝ}}^3$ be an embedding and as said before, let us consider bi-stable maps of type $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, where f ₁ is a projection in a line and f ₂ is a projection on a plane. By simplicity, if we consider R _v = ℝ and 𝒫 _v = ℝ², we can take

$f_1={\pi }_1\circ j\mathrm{a}\mathrm{n}\mathrm{d}{f}_2={\pi }_2\circ j,$

where π _i denotes the canonical projection in ℝⁱ , i = 1, 2.

Definition 3.1. If 𝒢 ¹ is the ℛ-graph associated to $f_1:M\to \mathrm{ℝ}$ and 𝒢 ² is the ℳ-graph associated to $f_2:M\to {\mathrm{ℝ}}^2$, then we say that the pair of graphs (𝒢 ¹ , 𝒢 ² ) is the ℛℳ-graph associated to the stable bi-map $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$.

In the next pictures, the ℛℳ-graphs will be illustrated always in this order (𝒢 ¹ , 𝒢 ² ). Moreover, the notation 𝒢 ⁱ(V ⁱ , E ⁱ ) will indicate that the graph 𝒢 ⁱ is a graph with V ⁱ vertices and E ⁱ edges, i = 1, 2.

In an ℛℳ-graph, the ℛ-graph contributes to determine the position of the maximum and minimum points (local and global) of f ₁ while the ℳ-graph contributes to determine the position of the regular regions and singular curves of f ₂ in M.

3.1 ℛ-graphs

Given a stable map $f_1:M\to \mathrm{ℝ}$ we consider the following equivalence relation on M:

$x~y\iff f_1\left(x\right)=f_1\left(y\right)$ and x and y are in the same connected component of $f_1^{-1}\left(f_1\right(x\left)\right)$.

Then the quotient space M/∼ admits the structure of a connected graph where the vertices are the connected components of critical level curves $f_1^{-1}\left(v\right)$ which contains critical points of f ₁ and each edge is formed by points that correspond to connected components of level curves $f_1^{-1}\left(w\right)$, where w ∈ ℝ is a regular value and v ∈ ℝ is a critical value of f ₁.

Each vertex of the graph can be of three types, depending on the critical points in the connected component as shown in Figure 6.

Figure 6:
Incidence rules for the vertices in a Reeb graph.

Let v ₁ , . . . , v _r ∈ ℝ be the critical values of f ₁. We choose a base point v ₀ ∈ ℝ and an orientation. We can reorder the critical values such that v ₀ ≤ v ₁ < · · · < v _r and we label each vertex with the index i ∈ {1,..., r}, if it corresponds to the critical value v _i .

Definition 3.2. The graph given by M/∼ together with the labels of the vertices, as previously defined, is said to be the Reeb graph (or ℛ-graph) associated to $f_1:M\to \mathrm{ℝ}$.

It is known that the Reeb graph is a complete topological invariant for stable functions (cf. ¹). The example in Figure 7 shows that the ℛ-graph is an invariant that distinguishes what the image set of the critical points cannot tell us.

Figure 7:
Examples of ℛ-graphs from sphere.

Follows from the possible incidence rules of edges and vertices in an ℛ-graph that all its vertices have degree 1 or 3. Graphs with this property will be called here 1-trivalent graphs.

Hence, the ℛ-graph associated to a stable map f ₁ is 1-trivalent. Moreover, the Euler characteristic of M is given by $\chi \left(M\right)=2(V^1-E^1)$. In other words, the topology of surface M can be determined by the ℛ-graph associated to f ₁.

Follows from the Poincaré-Hopf theorem that every ℛ-graph associated to a stable map $f_1:{\mathbb{S}}^2\to \mathrm{ℝ}$ is a 1-trivalent tree.

Theorem 3.1.Any 1-trivalent graph 𝒢¹ (V ¹ , E ¹) is a ℛ-graph associated to a stable map$f_1:M\to \mathrm{ℝ}$where the Euler characteristic of M is$\chi \left(M\right)=2(V^1-E^1)$.

Proof. Given a 1-trivalent graph 𝒢 ¹ (V ¹ , E ¹), we can construct a surface M such that 𝒢 ¹ (V ¹ , E ¹) is the ℛ-graph associated to the height function $f:M\to \mathrm{ℝ}$. In fact, consider an embedding of the graph 𝒢 ¹ (V ¹ , E ¹) in ℝ³. Let M be the boundary of a thin tubular neighborhood of 𝒢 ¹ (V ¹ , E ¹). Then M is diffeomorphic to a surface with $\chi \left(M\right)=2(V^1-E^1)$. Moreover, the restriction of the height function on M, $f:M\to \mathrm{ℝ}$, is a stable map whose ℛ-graph is equivalent to 𝒢 ¹ (V ¹ , E ¹). □

3.2 ℳ-graphs

Given a stable map $f_2:M\to {\mathrm{ℝ}}^2$, its topological information may be conveniently encoded in a determined weighted graph. In fact, given the pair (M, Σf ₂), we may reconstruct (up to diffeomorphism) a weighted graph associated to f ₂ (cf. ⁸), in the following sense: the edges and vertices of this weighted graph correspond to the singular curves and the connected components of the regular set, respectively. An edge is incident to a vertex if and only if the corresponding singular curve to the edge lies in the boundary of the regular region corresponding to the vertex. The weight of a vertex is defined as the genus of the corresponding region. This graph is called ℳ-graph (or Mendes-graph) associated to f ₂.

It is known that the ℳ-graph is a complete topological invariant for stable maps from M to the plane (cf. ⁸).

Let 𝒢 ² (V ² , E ² ,W ²) be the ℳ-graph associated to f ₂, where V ², E ² and W ² correspond to the number of vertices, edges and the total weight of the graph, respectively. Then E ² represents the number of connected components of Σf ₂; V ² the number of connected components of M\Σf ₂ and W the total sum of genus of the components of M\Σf ₂.

If W ² = 0, that is, if 𝒢 ² is a graph without weight, we will denote by simplicity 𝒢 ² (V ² , E ²) omitting W ² in the graph notation.

The Figure 8 illustrates two stable maps $f_2,g_2:{\mathbb{S}}^2\to {\mathrm{ℝ}}^2$ whose ℳ-graphs are non equivalent, where j _i indicates the respective embedding from 𝕊² in ℝ³ and π _i the canonical projection, i = 1, 2. This example shows that the ℳ-graph is an invariant that distinguishes what the apparent contour sets of f ₂ and g ₂ can not distinguish.

Figure 8:
Examples of ℳ-graphs associated to stable maps from sphere.

Given a stable map $f_2:M\to {\mathrm{ℝ}}^2$ always it is possible to obtain its ℳ-graph associated. Moreover, the Euler characteristic of M is given by $\chi \left(M\right)=2(V^2-E^2-W^2)$. In other words, the topology of surface M can be determined by ℳ-graph 𝒢 ² (V ² , E ² ,W ²) associated to f ².

Given the orientations of M and ℝ², a region of M is said to be positive (resp. negative) if the map f ₂ preserves (resp. reverses) orientation. Since each component of Σf ₂ is the boundary of a positive and a negative region, the signs of the vertices are assigned alternately, that is, the ℳ-graph associated to stable map $f_2:M\to {\mathrm{ℝ}}^2$ is bipartite.

To be a bipartite graph is a necessary condition for a graph to be a ℳ-graph for some stable map $f_2:M\to {\mathrm{ℝ}}^2$.

Theorem 3.2. (⁸) Any bipartite connected graph (with arbitrarily weighted vertices) is the ℳ-graph of a stable map$f_2:M\to {\mathrm{ℝ}}^2$.

The proof of Theorem 3.2 is based in a convenient manipulation of codimension one transitions (lips and beaks) in the space C ^∞(M, ℝ²) and convenient surgeries of stable maps (see ⁸ for more details). Transitions and surgeries involving stable maps will be treated with more details in the next Section.

As a consequence of Theorem 3.2, any tree with W ² = 0 may be realized as the ℳ-graph of some stable map $f_2:{\mathbb{S}}^2\to {\mathrm{ℝ}}^2$.

From now on, we focus on ℳ-graph without weighted vertices (that is, W ² = 0). The case W ² > 0 will be studied in a future work in progress.

4 TOOLS USED TO CONSTRUCT STABLE BI-MAPS

Let ℰ ^∞(M, ℝ^p ) be the set of all stable maps in C ^∞(M, ℝ^p ), p = 1, 2.

In this work we are considering stable bi-maps of type $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ which can be decomposed (locally) as $f_i={\pi }_i\circ j,i=1,2$, where j is an embedding from M in ℝ³ and π _i are the canonical projections from j(M) to ℝ and ℝ², respectively, i = 1, 2. Replacing the embedding j by another embedding from M in ℝ³, we can obtain new stable bi-maps. This procedure can be done by taking small perturbations of the embedding j, so that they may alter or not the images of the projections π ₁ and π ₂. The new stable bi-maps obtained in this procedure have associated new ℛℳ-graphs. Then, it is natural to ask whether these changes modify the new ℛℳ-graphs or not.

The alterations used here to obtain new stable bi-maps correspond to transitions and surgeries and they are described with more details in the next Subsections.

4.1 Elementary Morse transitions

In this Subsection let us consider stable maps in ℰ ^∞(M, ℝ).

A Morse transition corresponds to an isotopy from a given stable map to another in a different path component of ℰ ^∞(M, ℝ). Thus, a Morse transition allows to transform a stable map $f_1:M\to \mathrm{ℝ}$ in another ${\tilde{f}}_1:M\to \mathrm{ℝ}$ in such a way that their respective ℛ-graphs have a different number of vertices or the same number of vertices with noncompatible labels.

A Morse transition T is called elementary if the isotopy T transforms f ₁ in ${\tilde{f}}_1$ through one of the following ways:

• [C ] The isotopy T creates a new edge in ℛ-graph of f ₁. That is, if T (0) = f ₁ and its ℛ-graph has s saddles and m max/min points then $T\left(1\right)={\tilde{f}}_1$ and the ℛ-graph of ${\tilde{f}}_1$ has s + 1 saddles and m + 1 max/min points, with the new saddle and max/min point being connected by a new edge.

• [−C ] It is the inverse transition of C. That is, when the isotopy collapses an edge of ℛ-graph of f ₁, with the vertices that were removed being previously connected by an edge. In this case, the ℛ-graph of ${\tilde{f}}_1$ has s− 1 saddles and m− 1 max/min points.

The Figure 9 indicates examples of elementary Morse transitions.

Figure 9:
The effect of the elementary Morse transitions in the ℛ-graphs.

For more details about the consequences of Morse transitions in a Reeb graph see ¹⁰.

Since elementary Morse transitions do not generate any new critical curve related to projection π ₂ (in the plane), the ℳ-graph has no change after C or − C transitions. In other words, elementary Morse transitions alter the ℛℳ-graph associated to original stable bi-map $F=(f_1,f_2)=({\pi }_1\circ j,{\pi }_2\circ j)$, changing only its ℛ-graph.

Definition 4.3.Given a ℛ-graph 𝒢¹, we say that a C transition is a 1-extension over the graph 𝒢 ¹ .

A sequence of n 1-extensions over a ℛ-graph increases n vertices of degree 1 (max/min points) and n vertices of degree 3 (saddles) over the original ℛ-graph.

Any 1-trivalent tree 𝒢 ¹ (V ¹ ,V ¹ − 1) can be obtained as a ℛ-graph of a stable map from M to R by applying a sequence with (V ¹ − 2)/2 1-extensions over the ℛ-graph of canonical height function from 𝕊² to ℝ.

Proposition 4.1.All pair of trees 𝒢¹ (V ¹ ,V ¹ − 1), 𝒢 ² (2, 1)) is a ℛℳ-graph of some stable bi-map$F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, where 𝒢 ¹ (V ¹ ,V ¹ − 1) is a 1-trivalent tree.

Proof. Since 𝒢 ¹ (V ¹ ,V ¹ − 1) is an 1-trivalent tree, then V ¹ is even and (V ¹ − 2)/2 is a integer number. Consider the standard stable bi-map $G=(g_1,g_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ where $g_i={\pi }_i\circ j$ and $j:{\mathbb{S}}^2\to {\mathrm{ℝ}}^3$ is the inclusion.

Let (𝒢 ¹ (2, 1), 𝒢 ² (2, 1)) be the ℛℳ-graph associated to G. After a sequence with (V ¹ − 2)/2 1-extensions over the ℛℳ-graph of G without altering the singular set of g ₂, we obtain a new stable bi-map $F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ which realizes the graph 𝒢 ¹ (V ¹ ,V ¹ − 1), 𝒢 ² (2, 1)). In fact, each 1-extension increases two edges and two vertices in the ℛ-graph and do not alter the ℳ-graph. □

4.2 Lips, beaks and swallowtail transitions

In this Subsection we will consider transitions that alter only the ℳ-graph in a ℛℳ-graph associated to a stable bi-map $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$. They are the same transitions that change the regular and singular sets of f ₂, namely the lips transitions, denoted by L and beaks transitions, denoted by B (see Figure 10). We denote by − B and − L, respectively, the inverse transitions of B and L. These transitions also change the number of cusps by ±2 and they are sufficient to show that any tree with weight equal zero can be realized as a graph of a stable map from 𝕊² to ℝ² (see ⁸).

Figure 10:
Lips and beaks transitions.

Let 𝒢 ²(V ² , E ²) be the ℳ-graph (without weight) associated to a stable map $f_2:M\to {\mathrm{ℝ}}^2$. Then, the lips transition increases by 1 the number of regions in M (i.e., vertices in V ²) and the number of singular curves in M (i.e., edges in E ²). The beaks transition can be classified in four different cases (see Figure 12):

Figure 11:
Beaks transition and C-transition on the torus.

Figure 12:
Decomposition of beaks transition.

• ${\mathbf{\text{B}}}_v^{+}$: beaks transition increases by 1 the number of regular regions, i.e., it adds 1 vertex and 1 edge on the ℳ-graph;

• ${\mathbf{\text{B}}}_v^{-}$: beaks transition decreases by 1 the number of regular regions, therefore it removes 1 vertex and 1 edge on the ℳ-graph (see Figure 13);

• ${\mathbf{\text{B}}}_w^{+}$: beaks transition increases by 1 the weight, maintains the number of regular regions (vertices) but decreases by 1 the number of edges (see Figure 11);

• ${\mathbf{\text{B}}}_w^{-}$: beaks transition decreases by 1 the weight, maintains the number of regular regions (vertices) but increases by 1 the number of edges.

Figure 13:
Lips and beaks transitions and 2-extensions.

The four types of beaks transition are illustrated (locally) in Figure 12, where in the picture X, X ₁ , Y, Z, Z ₁ and Z ₂ denote (locally) the regular regions where the transitions hold and the numbers 1 and 2 represent the number of singular curves.

Definition 4.4. Given a ℳ-graph 𝒢 ² (V ² , E ²) (without weight), we say that a composition of a lips transition with a beaks transition (in this order) is a 2-extension over a ℳ-graph when:

1. a lips transition L creates a singular curve α with 2 cusps and 1 new regular region D;

2. a beaks transition $-{\mathbf{\text{B}}}_v^{-}$ eliminates the 2 cusps, dividing α into two new singular curves and border of the new region D′, as illustrate Figure 13.

The lips transition creates a new region D inside of region A, decomponding A into B and D, where B is homeomorphic to a cilinder and D is homeomorphic to a disk. The beaks transition transformes the regions B and D into C and D′, where D′ is homeomorphic to a cilinder and C is homeomorphic to a disk.

Lips and beaks transitions can modify the singular set of a stable map from M to the plane, and do not alter the singular set of the height function. That is, the ℛ-graph are preserved by 2-extensions while the ℳ-graph always add two vertices (linked by one edge) with degree two on the graph. The Figure 13 illustrates a sequence of transitions that alter the ℳ-graph but they do not the ℛ-graph.

Definition 4.5. We call line graph, and denoted it by ℒ ²(k), a graph with k vertices with degree 2 and k − 1 edges.

In the Figure 13 we have three examples of ℳ-graph which are line graphs: (a) ℒ ²(2); (b) ℒ ²(3) and (c) ℒ ²(4). Moreover, the lips transition L holds in a positive region of 𝕊² creating a new singular curve with 2 cusps (see (b)). The beaks transition $-{\mathbf{\text{B}}}_v^{-}$ in the negative new region eliminates 2 cusps and adding a new singular curve (see (c)).

Applying 2-extensions we can show that all line graph ℒ ²(k) is a ℳ-graph of some stable map $f_2:{\mathbb{S}}^2\to {\mathrm{ℝ}}^2$. This is a consequence of Theorem 3.2.

4.3 Surgeries of stable bi-maps

Are considered two types of surgeries of stable bi-maps: horizontal and vertical surgeries. We are interested to know the effects of these surgeries over the ℛℳ-graphs. This study is based in the concepts introduced in ⁸.

Let $F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ be a bi-stable map, P and Q be two any regions of surface M, where M may or may not be connected.

Horizontal surgery𝒮^h: Let p ∈ P, q ∈ Q be two singular points of f ₂. A bridge is an embedded arc β in ℝ², which connects the set of singular values of f ₂ (or apparent contour) in its two end points (and nowhere else). A new stable map f _2h can be constructed as follows: the bridge links the apparent contour of f ₂ in its points, f ₂ (p) and f ₂ (q). Choose small disks Dp, Dq in M centered at p, q ∈ Σf ₂, respectively. We can choose small enough disks such that they do not contain any critical point of f ₁. Replace the interiors of Dp and Dq by a tube (i.e., an annulus) connecting these two small disks obtaining a new connected surface N. Then f ₁ and f ₂ may be extended over the tube to give new stable maps f ₁ h and f ₂ h, in such way that f ₁ has only 2 saddle points in this tube, as shown (locally) in Figure 14.

Figure 14:
Horizontal surgery example.

The horizontal surgery 𝒮 ^h connects the two regular regions of f ₁ and adds 2 saddle points, while in f ₂ the horizontal surgery 𝒮 ^h effects are:

i) it links the 2 singular curves if p and q are in two disjoint singular curves (see the graphs in Figure 16 (a)-(b));

ii) it separates a singular curve into two curves, if p and q are in the same singular curve (see the graphs in Figure 16 (c)).

Figure 15:
Vertical surgery of bistable maps.

Figure 16:
Examples of surgeries: (a) vertical and (b) horizontal.

Vertical surgery𝒮^h: Let p ∈ P, q ∈ Q be two singular points of f ₂. We take a surgery between P e Q by identifying two small topological disks Dp and Dq, one positive and one negative, such that each one of them contains only 1 critical point of f ₁ in their interior being 1 maximum and 1 minimum, and its boundary is a connected component of a regular level curve of f ₁. The disks are replaced by a tube which is mapped into ℝ, with no singular points in the interior of the tube. The stable map f ₁ may be extended over the tube to a new stable map f _1v from a new connected surface N to ℝ. Also, the stable map f ₂ can be extended to a new stable map $f_{2v}:N\to {\mathrm{ℝ}}^2$ with one fold curve in the tube (see Figure 15).

The vertical surgery 𝒮 ^h identifies a neighborhood of a maximum point of f ₁ as a neighborhood of a minimum point of f ₁, eliminating two singular points. In f ₂, the surgery 𝒮 ^h always creates a new singular curve. In other words, the vertical surgery 𝒮 ^h takes away two critical points of the singular set Σf ₁ while it adds one critical curve to Σf ₂ (see Figure 15 and 16-(b)).

Notice that the regions P and Q can be in a same connected component of M or in two different connected components. If P and Q are regions of the same connected component of M the singular points p, q ∈ Σf ₂ may or may not be in the same singular curve.

We denote by ${\mathcal{S}}^h(f_1,f_2)=(f_{1h},f_{2h}):N\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ the new stable bi-map resulting after the horizontal surgery 𝒮 ^h , and by 𝒮 ^h (𝒢) the effect in the graph 𝒢 after applying a horizontal surgery 𝒮 ^h . Analogously we denote by ${\mathcal{S}}^v(f_1,f_2)=(f_{1v},f_{2v}):N\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ the new stable bi-map resulting after the vertical surgery 𝒮 ^v and by 𝒮 ^v (𝒢) the effect in the graph 𝒢 after applying a vertical surgery 𝒮 ^v .

Let P and Q be two regions of M that are in two connected component of M, then the surgeries 𝒮 ^h and 𝒮 ^v over P and Q are called connected horizontal sum (see Figure 15-(a)) and connected vertical sum (see Figure 15-(a)).

Definition 4.6. We denote by 𝒮 ^h1 (resp. 𝒮 ^v1 ) the horizontal surgery (resp. vertical surgery) of stable maps from M to the plane that connects two regions of a connected surface M. We denote by 𝒮 ^h0 (resp. 𝒮 ^v0 ) the horizontal surgery (resp. vertical surgery) that connects two regular regions of two different connected components of M.

This local process of f ₂, for 𝒮 ^ηα (η = h, v and α = 0, 1) is illustrated in Figures 14 and 15, which shows the effect of these surgeries on the graphs. It induces the following surgeries on the ℛ-graph and ℳ-graph (see Figure 15):

A horizontal surgery on graphs which identifies two edges ur and vs, identifying the vertices u with v and r with s, will be denoted by:

(a) 𝒮 ^h0: when ru and sv are in different connected graphs, joining these two graphs (Figure 14-(a)).

(b) 𝒮 ^h1: when ru and sv are in the same connected component of a graph, creating a new cycle in the graph (see Figure 14-(b) and (c)).

A vertical surgery on graphs which connects two vertices, r positive and v negative, by an edge rv, will be denoted by:

(c) 𝒮 ^v0: when r and v are in different connected graphs, joining these two graphs (see Figure 15-(a)).

(d) 𝒮 ^v1: when r and v are in the same connected graph, creates a new cycle in the graph (see Figure 15-(b) and (c)).

The next result is a consequence from horizontal surgery and vertical surgeries.

Proposition 4.2. Let $({\mathcal{G}}_1^1,{\mathcal{G}}_1^2)$ and $({\mathcal{G}}_2^1,{\mathcal{G}}_2^2)$ be the ℛℳ-graphs associated to two stable bi-maps $F=(f_1,f_2):P⟶\mathrm{ℝ}\times {\mathrm{ℝ}}^2$ and $G=(g_1,g_2):Q⟶\mathrm{ℝ}\times {\mathrm{ℝ}}^2$ , respectively. Then $\left({\mathcal{S}}^{\alpha \eta }\right({\mathcal{G}}_1^1,{\mathcal{G}}_2^1),{\mathcal{S}}^{\alpha \eta }({\mathcal{G}}_1^2,{\mathcal{G}}_2^2\left)\right)$ is the ℛℳ-graph associated to the stable bi-map

$\left({\mathcal{S}}^{\alpha \eta }\right(f_1,g_1),{\mathcal{S}}^{\alpha \eta }(f_2,g_2\left)\right):{\mathcal{S}}^{\alpha \eta }(P,Q)\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2,\alpha =h,v\mathit{\text{and}}\eta =0,1.$

The Figure 16 illustrates examples of surgeries of stable maps from sphere: (a) horizontal surgery 𝒮 ^h1 and (b) vertical surgery 𝒮 ^v1 .

5 REALIZATION OF GRAPHS (𝒢 ¹ , 𝒢 ² )

Lemma 5.1.All pair of trees (𝒢 ¹(2, 1), 𝒢 ²(V ² ,V ² − 1)) is a ℛℳ -graph of some stable bi-map$F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$.

Proof. Consider the pair of canonical maps (given by height function) $G=(g_1,g_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, such that the ℛℳ-graph associated to G is (𝒢 ¹(2, 1), 𝒢 ²(2, 1)). Since 𝒢 ²(V ² ,V ² − 1) is a tree, let ℒ ²(k + 1) be the biggest line subgraph of 𝒢 ²(V ² ,V ² − 1) which connects two peripheral vertices of 𝒢 ²(V ² ,V ² − 1), where k +1 ≤V ². Then, the pair (𝒢 ¹(2, 1), ℒ ²(k +1)) can be realized as the following:

• i) If k is odd, k− 1 is even. Passing through a sequence with (k − 1)/2 of 2-extensions (without altering the singular set of g ₁), we obtain a stable bi-map $H=(h_1,h_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ which realizes the graph (𝒢 ¹(2, 1), ℒ ²(k + 1)), because each 2-extension increases two edges and two vertices in the ℳ-graph and does not change the ℛ-graph. After this, we can obtain a stable bi-map $F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, as required, realizing the ℛℳ-graph (𝒢 ¹ (2, 1), 𝒢 ² (V ² ,V ² − 1)), taking V ² −k lips transitions over H = (h ₁ , h ₂), in convenient regions.

• ii) If k is even, we can first obtain a stable bi-map H = (h ₁ , h ₂) which realizes the graph (𝒢 ¹ (2, 1), ℒ ²(k + 1)) as done in item i). Then, we can obtain a stable bi-map $F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, as required, realizing the ℛℳ-graph (𝒢 ¹ (2, 1), 𝒢 ² (V ² ,V ² − 1), taking V ² −k + 1 lips transitions over H = (h ₁ , h ₂), in convenient regions. □

Proposition 5.3. If 𝒢¹ is a 1-trivalent tree and 𝒢² is a tree whithout weights then the graph (𝒢¹, 𝒢²) is a ℛℳ-graph of some stable bi-map $F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$.

Proof. Let 𝒢¹(V ¹,V ¹ − 1) be a 1-trivalent tree and 𝒢²(V ²,V ² − 1) be a tree with W ² = 0. Consider ℒ²(k + 1) the biggest line subgraph of 𝒢² (V ²,V ² − 1). Then by Lemma 5.1, the graph (𝒢¹(2, 1), ℒ²(k + 1)) can be realized by some stable bi-map $G=(g_1,g_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$. Since V ¹ is even and each 1-extension increases 2 vertices and 1 edge to the ℛ-graph, then passing through a sequence with (V ¹ − 2)/2 1-extensions over G = (g ₁, g ₂) we obtain a stable bi-map $F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$ which realizes the graph (𝒢¹(V ¹,V ¹ − 1), 𝒢²(V ²,V ² − 1)), as required. □

From Poincaré-Hopf theorem, Theorem 3.2 and Proposition 5.3, we obtain the following result:

Theorem 5.3.A graph (𝒢 ¹ , 𝒢 ² ) is a ℛℳ-graph for a stable bi-map$F=(f_1,f_2):{\mathbb{S}}^2\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$if and only if 𝒢 ¹ is a tree 1-trivalent and 𝒢 ² is a tree without weights (i.e., with W ² = 0).

Theorem 5.4.Let 𝒢¹be a 1-trivalent graph and 𝒢²be a bipartite graph (with W² = 0). Suppose χ(𝒢 ¹) = χ(𝒢 ²). Then the graph (𝒢 ¹ , 𝒢 ²) is a ℛℳ-graph associated to a stable bi-map$F=(f_1,f_2):M\to \mathrm{ℝ}\times {\mathrm{ℝ}}^2$, where$\chi \left(M\right)=2\chi \left({\mathcal{G}}^1\right)=2\chi \left({\mathcal{G}}^2\right)$.

Proof. Since χ(𝒢 ¹) = χ(𝒢 ²), both graphs have the same number m of cycles. Consider a pair of support trees (𝒯 ¹ , 𝒯 ²) defined in the following way:

• i) 𝒯 ¹ is obtained by choosing one edge in each one of the m cycles of 𝒢 ¹ and subdividing each chosen edge obtaining m new vertices of degree 2. Then we cleave each new vertex v _i (i = 1, · · · , m) in to two new vertices v _i1 and v _i2 , obtaining 2m new vertices of degree 1.

• ii) 𝒯 ² is a spanning tree of 𝒢 ², that is, a tree that contains all the vertices of 𝒢 ², obtained by removing one edge of each cycle of 𝒢 ².

By Theorem 5.3, (𝒯 ¹ , 𝒯 ²) is a ℛℳ-graph of a stable bi-map $G=(g_1,g_2):{\mathbb{S}}^2⟶\mathrm{ℝ}\times {\mathrm{ℝ}}^2$. Consider the regions U _i1 and U _i2 of 𝕊² corresponding to the neighborhoods of the points associated with a pair of vertices v _i1 and v _i2 obtained in (i), respectively. For each pair of vertices v _i1 and v _i2 , we may realize a vertical surgery over U _i1 and U _i2 obtaining a new stable bi-map $F=(f_1,f_2):M⟶\mathrm{ℝ}\times {\mathrm{ℝ}}^2$ (where M is a surface with genus m) which realizes the ℛℳ-graph (𝒢 ¹ , 𝒢 ²). □

The next result is a consequence of the previous results and remarks.

Theorem 5.5.Consider a pair of graphs (𝒢 ¹ , 𝒢 ²), with χ(𝒢 ¹) = χ(𝒢 ²). Then (𝒢 ¹ , 𝒢 ²) is a ℛℳ-graph for some stable bi-map$F=(f_1,f_2):M⟶\mathrm{ℝ}\times {\mathrm{ℝ}}^2$, where M is a closed oriented surface with$\chi \left(M\right)=2\chi \left({\mathcal{G}}^1\right)=2\chi \left({\mathcal{G}}^2\right)$, if and only if 𝒢 ¹ is a 1-trivalent graph and 𝒢 ² is a bipartite graph without weights (i.e. W ² = 0).

Acknowledgments

The third named author has been partially supported by grants 2018/25157-3 and 2019/21181-0 São Paulo Research Foundation (FAPESP).

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Publication Dates

• Publication in this collection
  29 May 2023
• Date of issue
  Apr-Jun 2023

History

• Received
  20 Dec 2021
• Accepted
  04 Oct 2022