Notes on Reeb graphs of real algebraic functions which may not be planar

We define the set $S_{D}:=\{(x_{1},\cdots x_{k},y_{1},\cdots y_{l})\in\overline{D}\times{\mathbb{R}}^{m-k}\subset{\mathbb{R}}^{k}\times{\mathbb{R}}^{m+l-k}={\mathbb{R}}^{m+l}\mid 1\leq i\leq l,f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}=0\}$. $y_{i}$ is regarded as a point in ${\mathbb{R}}^{d_{i}}$ here with the condition ${\Sigma}_{j=1}^{l}d_{j}=m+l-k$ and we can choose the dimensions as positive dimensions suitably since $m$ is sufficiently large. We see that this set is an $m$-dimensional non-singular real algebraic manifold and also represented as some disjoint union of connected components of the zero set of the real polynomial map into ${\mathbb{R}}^{l}$ obtained canonically from the $l$ real polynomials $f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}$. A main ingredient is implicit function theorem on this polynomial map. We consider several cases.
 
Case 1 In the case such that for the point $(x_{1},\cdots x_{k},y_{1},\cdots y_{l})\in S_{D}$, $(x_{1},\cdots x_{k})\in D$.

Since $D$ is an NC domain, for each $f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}$, we consider the partial derivative at the point for each variant $y_{j,j^{\prime}}$ where $j^{\prime}$ is an integer $1\leq j^{\prime}\leq d_{j}$. The value is $0$ for $j\neq i$ and it is not zero for some $y_{i,j_{i}}$ with a suitable integer $1\leq j_{i}\leq d_{i}$. The rank of the map into ${\mathbb{R}}^{l}$ defined canonically from the real polynomials $f_{j}(x_{1},\cdots x_{k})-||y_{j}||^{2}$ is $l$ at the point.
 
Case 2 In the case such that for the point $(x_{1},\cdots x_{k},y_{1},\cdots y_{l})\in S_{D}$, $(x_{1},\cdots x_{k})\in S_{A,0}$.

Since $D$ is an NC domain, for each $f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}$ with $i\in{\mathbb{N}}_{l}-A$, we consider the partial derivative at the point for each variant $y_{j,j^{\prime}}$ where $j^{\prime}$ is an integer $1\leq j^{\prime}\leq d_{j}$. The value is $0$ for $j\neq i$ and it is not zero for some $y_{i,j_{i}}$.

For each $f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}$ with $i\in A$, $y_{i}$ is the origin. We consider the partial derivative at the point for each variant $y_{j,j^{\prime}}$ where $j^{\prime}$ is an integer $1\leq j^{\prime}\leq d_{j}$. The value is always $0$. We consider the map into ${\mathbb{R}}^{|A|}$ defined canonically from the $|A|$ polynomials $f_{i}(x_{1},\cdots x_{k})-||y_{i}||^{2}$. We consider the partial derivatives by each variant $x_{j}$ for $1\leq j\leq k$. We have a matrix of the form $|A|\times k$ consisting of the values of the partial derivatives at the point. By the assumption on the transversality on the intersections of the hypersurfaces $S_{j}$, the rank is $|A|$.

According to this argument, the rank of the map into ${\mathbb{R}}^{l}$ defined canonically from the real polynomials $f_{j}(x_{1},\cdots x_{k})-||y_{j}||^{2}$ is $l$ at the point.
 

We consider a point $p_{1}\in{\mathbb{R}}^{k}$ such that $p_{1}$ is sufficiently close to $D$ and not in the closure $\overline{D}$. More precisely, we can formulate this by $p_{1}\in U_{D}-\overline{D}$ for example. Then according to the assumption, $f_{i}(p_{1,1},\cdots p_{1,k})<0$ for some $i$, the natural set $S_{D,p_{1}}:=\{(p_{1},p_{2})\in{\mathbb{R}}^{k}\times{\mathbb{R}}^{m+l-k}={\mathbb{R}}^{m+l}\mid 1\leq i\leq l,f_{i}(p_{1,1},\cdots p_{1,k})-||p_{2}||^{2}=0\}$ is empty. From this with the arguments on Case 1 and Case 2, $S_{D}$ is regarded as a smooth compact submanifold of dimension $m$ with no boundary in ${\mathbb{R}}^{m+l}$ and some disjoint union of connected components of the zero set of the real polynomial map into ${\mathbb{R}}^{l}$ defined canonically from the $l$ polynomials. $f_{D}$ is defined as the natural projection of $M:=S_{D}\subset{\mathbb{R}}^{m+l}$ to ${\mathbb{R}}^{k}$. We have (2) and (3) mainly from the arguments on the ranks of the differentials and implicit function theorem before.

We can see (1), (4), (5) and (6) from the construction easily.

This completes the proof. ∎

We define a new open set in ${\mathbb{R}}^{k}\times{\mathbb{R}}={\mathbb{R}^{k+1}}$. More precisely, we redefine $D$ as a suitable open set there. We abuse the notation in Definition 4 for example.

Consider ${\pi}_{{\mathbb{R}}^{k+1},{\mathbb{N}}_{\leq k+1},{\mathbb{N}}_{\leq k}}$ and the preimages of $D$ and the hypersurfaces $S_{j}$ for this projection. Consider ${\pi}_{{\mathbb{R}}^{k+1},{\mathbb{N}}_{\leq k+1},\{1,k+1\}}$ and the preimages of an $(i,t_{1},t_{2})$-type domain with circles and the circles surrounding the domain for this projection. We can redefine our new open set $D$ as the intersection of the two preimages of the two NC domains. We can also redefine our new hypersurfaces surrounding the new domain as the preimages of the original hypersurfaces. Let the new hypersurface denote $S_{j}$ for the preimage of $S_{j}$ in the original situation and $S_{l+j}$ for the preimage of the $j$-th circle in the family of the $i$-circles surrounding the $(i,t_{1},t_{2})$-type domain with circles. The outermost sufficiently large circle is denoted by $S_{l+i+1}$.

We explain about the fact that our new set $D$ is an NC domain. We can easily check the conditions on the real polynomials and the zero sets and the fact that the hypersurfaces are non-singular, by our construction. The preimages are regarded as the natural cylinders for the original hypersurfaces. We check the condition on the transversality of the intersections of hypersurfaces.

$S_{j_{1}}$ and $S_{j_{2}}$ do not intersect for $j_{1}<j_{2}\leq l$ or $l+1\leq j_{1}<j_{2}\leq l+(i+1)=l+i+1$.

We consider the remaining case. Suppose that $S_{j_{1}}\bigcap S_{j_{2}}$ is not empty for some $j_{1}<j_{2}$ and that $p$ is a point in this intersection. Let $p_{1}$ denote the value obtained by mapping this by the projection to the first component. We can consider the following exactly three cases by the assumption or the condition C4.
 
Case A The preimage ${g_{K_{D}}}^{-1}(p_{1})$ for the original function $g_{K_{D}}$ contains no vertices and $p_{1}\neq t_{j}$ for $j=1,2$.
A normal vector at $p\in S_{j_{1}}$ must be parallel to a vector of the form ${\Sigma}_{j=1}^{k}t_{j}e_{j}$ for some $j\neq 1$ and $t_{j}\neq 0$. A normal vector at $p\in S_{j_{2}}$ must be parallel to a vector of the form $t_{1}e_{1}+t_{k+1}e_{k+1}$ with $t_{k+1}\neq 0$. Remember that the outermost circle in the $(i,t_{1},t_{2})$-type domain with circles is chosen sufficiently large and that $S_{l+i+1}$ is also a sufficiently large cylinder of the circle.
 
Case B The preimage ${g_{K_{D}}}^{-1}(p_{1})$ for the original function $g_{K_{D}}$ contains no vertices and $p_{1}=t_{j}$ for either $j=1$ or $j=2$.
A normal vector at $p\in S_{j_{1}}$ must be parallel to a vector of the form ${\Sigma}_{j=1}^{k}t_{j}e_{j}$ for some $j\neq 1$ and $t_{j}\neq 0$. A normal vector at $p\in S_{j_{2}}$ must be parallel to a vector of the form $t_{1}e_{1}$ for some $t_{1}\neq 0$.
 
Case C The preimage ${g_{K_{D}}}^{-1}(p_{1})$ for the original function $g_{K_{D}}$ contains some vertices and $p_{1}\neq t_{j}$ for $j=1,2$.
In the case $p$ is not a singular point of the function of the restriction of the projection of ${\mathbb{R}}^{k}$ to the first component to ”the original set $\overline{D}-D$”, a normal vector at $p\in S_{j_{1}}$ must be parallel to a vector of the form ${\Sigma}_{j=1}^{k}t_{j}e_{j}$ for some $j\neq 1$ and $t_{j}\neq 0$. $p$ may be a singular point of the function of the restriction of the projection of ${\mathbb{R}}^{k}$ to the first component to ”the original set $\overline{D}-D$” and in this case, a normal vector at $p\in S_{j_{1}}$ must be parallel to a vector of the form $t_{1}e_{1}$ with $t_{1}\neq 0$. A normal vector at $p\in S_{j_{2}}$ must be parallel to a vector of the form $t_{1}e_{1}+t_{k+1}e_{k+1}$ with $t_{k+1}\neq 0$. Remember again that the outermost circle in the $(i,t_{1},t_{2})$-type domain with circles is chosen sufficiently large and that $S_{l+i+1}$ is also a sufficiently large cylinder of the circle.

 
We can see the transversality easily from this. We also see that our new $D$ is an NC domain. $U_{D}$ may be large. However it is regarded as a desired open neighborhood of our new $D$.

We apply Proposition 1 to have a new map $f_{D}:M_{D}\rightarrow{\mathbb{R}}^{k+1}$ whose image is the closure $\overline{D}$ of the new set $D$. This also gives a part of our proof of Main Theorem 2 and a part of our proof of Main Theorem 3 essentially.

We consider the composition of $f_{D}$ with the projection to the first component and its Reeb graph. We have a smooth real algebraic function, denoted by $f_{i}:M_{i}\rightarrow{\mathbb{R}}^{k+1}$ with $\dim M_{i}=m_{i}$. Let us see that $f_{i}$ is our desired function. First, we have the following by fundamental arguments.

• •

  Remember that $\bar{f_{i}}:W_{f_{i}}\rightarrow\mathbb{R}$ enjoys $f_{i}=\bar{f_{i}}\circ q_{f_{i}}$. $\bar{f_{i}}$ is represented as the composition of a suitable piecewise smooth map ${\phi}_{f_{i}}$ onto $K_{D}$ with the function $g_{K_{D}}$.

• •

  The restriction of ${\phi}_{f_{i}}$ to the preimage ${{\phi}_{f_{i}}}^{-1}({g_{K_{D}}}^{-1}(\mathbb{R}-(t_{1},t_{2})))$ of the complementary set of the open interval $(t_{1},t_{2})$ in $\mathbb{R}$ is regarded to give an isomorphism of the two graphs.

• •

  The restriction of ${\phi}_{f_{i}}$ to the preimage ${{\phi}_{f_{i}}}^{-1}({g_{K_{D}}}^{-1}((t_{1},t_{2})))$ of the open interval $(t_{1},t_{2})$ is regarded as an ($i+1$)-fold covering.

• •

  Consider the set of all vertices of the graph $W_{f_{i}}$ in the preimage ${{\phi}_{f_{i}}}^{-1}({g_{K_{D}}}^{-1}((t_{1},t_{2})))$ and compare this to the set of all points in the preimages of some vertices in ${g_{k_{D}}}^{-1}((t_{1},t_{2}))$. The size of this set is $i+1$ times the size of the number of all vertices in ${g_{k_{D}}}^{-1}((t_{1},t_{2}))$.

We can easily see these properties. We explain about the third and the fourth properties here mainly. By the symmetry, it is sufficient to consider the case C5.1. The preimage of $\{p_{t_{1},1},p_{t_{1},2}\}$ considered for ${\phi}_{f_{i}}$ is a two-point set with the discrete topology and consists of two vertices. The preimage of $\{p_{t_{2}}\}$ considered for ${\phi}_{f_{i}}$ is a one-point set and consists of some vertex. Consider the preimage of the union of the two given embedded arcs $e_{t_{1},t_{2},1}$ and $e_{t_{1},t_{2},2}$. The restriction of the map ${\phi}_{f_{i}}$ to the interior of this 1-dimensional set is regarded as an ($i+1$)-fold covering. The interior of this $1$-dimensional set is obtained by removing the preimage of $\{p_{t_{1},1},p_{t_{1},2}\}$ considered for ${\phi}_{f_{i}}$ and the preimage of $\{p_{t_{2}}\}$ considered for ${\phi}_{f_{i}}$.

We consider the case C5.1 further to complete the proof. Consider a (presented) small connected embedded arc in $e_{t_{1},t_{2},1}\bigcap e_{t_{1},t_{2},2}$ containing $p_{t_{2}}$ and remove the point $p_{t_{2}}$. We consider the preimage of the resulting arc for ${\phi}_{f_{i}}$. There ${\phi}_{f_{i}}$ is an ($i+1$)-fold covering. Assume that $\bar{f_{i}}:W_{f_{i}}\rightarrow\mathbb{R}$ is represented as the composition of some embedding $g_{K_{D,t_{1},t_{2},i},{\mathbb{R}}^{2}}:W_{f_{i}}\rightarrow{\mathbb{R}}^{2}$ with the projection to the first component and we explain about the contradiction. We have $i+1$ distinct points in the image $g_{K_{D,t_{1},t_{2},i},{\mathbb{R}}^{2}}({\rm Int}\ (e_{t_{1},t_{2},1}\bigcap e_{t_{1},t_{2},2}))\subset{\mathbb{R}}^{2}$ of the interior of the intersection $e_{t_{1},t_{2},1}\bigcap e_{t_{1},t_{2},2}$ where the values of the projection to the first component are a same real number $t_{1,2}\in(t_{1},t_{2})$. Let $p_{e,j}$ denote the $j$-th point of these $i+1$ points. We also have $i+1$ pairs of embedded arcs in $W_{f_{i}}$ with the following.

• •

  For the $j$-th pair, one arc connects $p_{t_{1},1}$ and $p_{e,j}$ and the other arc connects $p_{t_{1},2}$ and $p_{e,j}$.

• •

  As presented, the values of the projection to the first component at $p_{e,j}\in{\mathbb{R}}^{2}$ are all $t_{1,2}$.

We can see the contradiction, implying that $g_{K_{D,t_{1},t_{2},i_{1}},{\mathbb{R}}^{2}}$ is not an embedding.

This completes the proof. ∎

We abuse the notation as in our proof of Main Theorem 1. As presented in our proof of Main Theorem 1, we can similarly obtain a smooth real algebraic map $f_{D}:M_{D}\rightarrow{\mathbb{R}}^{k+1}$ by using Proposition 1. We consider the composition of $f_{D}$ with the projection to the first component and its Reeb graph. We have a smooth real algebraic function, denoted by $f_{i}:{M_{i}}^{\prime}\rightarrow{\mathbb{R}}^{k+1}$ with $\dim{M_{i}}^{\prime}={m_{i}}^{\prime}$. We should note that instead of using an $(i,t_{1},t_{2})$-type domain with circles, we use an $(i+7,t_{1},t_{2})$-type domain for obtaining our new open set $D$ and map $f_{D}$.

We respect the condition C4^′ to have the following.

• •

  Remember that $\bar{{f_{i}}^{\prime}}:W_{{f_{i}}^{\prime}}\rightarrow\mathbb{R}$ enjoys ${f_{i}}^{\prime}=\bar{{f_{i}}^{\prime}}\circ q_{{f_{i}}^{\prime}}$. $\bar{{f_{i}}^{\prime}}$ is represented as the composition of a suitable piecewise smooth map ${\phi}_{{f_{i}}^{\prime}}$ onto $K_{D}$ with the function $g_{K_{D}}$.

• •

  The restriction of ${\phi}_{{f_{i}}^{\prime}}$ to the preimage ${{\phi}_{{f_{i}}^{\prime}}}^{-1}({g_{K_{D}}}^{-1}(\mathbb{R}-(t_{1},t_{2})))$ of the complementary set of the open interval $(t_{1},t_{2})$ in $\mathbb{R}$ is regarded to give an isomorphism of the two graphs.

• •

  The restriction of ${\phi}_{{f_{i}}^{\prime}}$ to the preimage ${{\phi}_{{f_{i}}^{\prime}}}^{-1}({g_{K_{D}}}^{-1}((t_{1},t_{2})))$ of the open interval $(t_{1},t_{2})$ is regarded as an ($i+8$)-fold covering.

• •

  Consider the set of all vertices of the graph $W_{{f_{i}}^{\prime}}$ in the preimage ${{\phi}_{{f_{i}}^{\prime}}}^{-1}({g_{K_{D}}}^{-1}((t_{1},t_{2})))$ and compare this to the set of all points in the preimages of some vertices in ${g_{k_{D}}}^{-1}((t_{1},t_{2}))$. The size of this set is $i+8$ times the size of the number of all vertices in ${g_{k_{D}}}^{-1}((t_{1},t_{2}))$.

We can easily see these properties. We explain about the third and the fourth properties here mainly. We respect the condition C5^′. The preimage of $\{{p_{t_{1},1}}^{\prime},{p_{t_{1},2}^{\prime}},{p_{t_{1},3}}^{\prime}\}$ considered for ${\phi}_{{f_{i}}^{\prime}}$ is a three-point set with the discrete topology and consists of three vertices. The preimage of $\{{p_{t_{2},1}}^{\prime},{p_{t_{2},2}^{\prime}},{p_{t_{2},3}}^{\prime}\}$ considered for ${\phi}_{{f_{i}}^{\prime}}$ is a three-point set with the discrete topology and consists of three vertices. Consider the preimage of the union of the given embedded arcs $e_{t_{1},t_{2},j_{1},j_{2}}$. The restriction of the map ${\phi}_{{f_{i}}^{\prime}}$ to the interior of this 1-dimensional set is regarded as an ($i+8$)-fold covering. The interior of this $1$-dimensional set is obtained by removing the preimage of $\{{p_{t_{1},1}}^{\prime},{p_{t_{1},2}}^{\prime},{p_{t_{1},3}}^{\prime}\}$ considered for ${\phi}_{{f_{i}}^{\prime}}$ and the preimage of $\{{p_{t_{2},1}}^{\prime},{p_{t_{2},2}}^{\prime},{p_{t_{2},3}^{\prime}}\}$ considered for ${\phi}_{{f_{i}}^{\prime}}$. From this argument, we can find a so-called $K_{3,3}$-graph as a subgraph in the graph ${K_{D,t_{1},t_{2},i}}^{\prime}$. It is well-known that we cannot embed such a graph into ${\mathbb{R}}^{2}$. This completes the proof. ∎