Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-sphere

TL;DR

This paper characterizes the automorphism groups of Kronrod-Reeb graphs for Morse functions on the 2-sphere, revealing their structure when the fixed subtree contains multiple points, thus advancing understanding of the topological symmetries of such functions.

Contribution

It explicitly computes the groups of graph automorphisms induced by diffeomorphisms for Morse functions on the 2-sphere with non-trivial fixed subtrees.

Findings

The automorphism groups are fully described for Morse functions with fixed subtrees of multiple points.

The structure of these groups depends on the configuration of the fixed subtree.

The results extend previous work on orientable surfaces excluding the 2-sphere and 2-torus.

Abstract

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by $\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$ induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)= \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{O}_{f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this situation $\Gamma_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$ consists of more than one point.

Equations

$$\varphi :{\mathcal{C}}^{\infty }(M,\mathbb{R})\times \mathcal{D}(M)\to {\mathcal{C}}^{\infty }(M,\mathbb{R})$$

$$\mathcal{S}(f)$$

$$M\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\rho :\mathcal{S}(f)\to \mathcal{H}({\Gamma }_f).$$

$${\mathcal{S}}^{\prime }(f)$$

$${\rho }_0:{\pi }_0{\mathcal{S}}^{\prime }(f)\to G_f,$$

$$\varphi (\gamma )=(\gamma {|}_{{\Gamma }_A},\gamma {|}_{{\Gamma }_B})$$

$${\Gamma }_A$$

$${\Gamma }_A\cup {\Gamma }_B$$

$$\rho (h)({\Gamma }_A)$$

$$\rho (h)({\Gamma }_A\cap E)={\Gamma }_A\cap E,$$

$$\rho (h)({\Gamma }_A\cap E)=\rho (h)({\Gamma }_A)\cap \rho (E)={\Gamma }_B\cap E\ne {\Gamma }_A\cap E,$$

$$p(h(A))=\rho (h)(p(A))=\rho (h)({\Gamma }_A)={\Gamma }_A.$$

$$f(x)=f{|}_A(x)=f{|}_A(h{|}_A(x))=f{|}_A(h(x))=f(h(x)),$$

$$\varphi (\delta \circ \omega )$$

$$\displaystyle=\bigl{(}\delta|_{\varGamma_{A}},\ \delta|_{\varGamma_{B}}\bigr{)}\circ\bigl{(}\omega|_{\varGamma_{A}},\ \omega|_{\varGamma_{B}}\bigr{)}=$$

$$\displaystyle=\bigl{(}\delta|_{\varGamma_{A}}\circ\omega|_{\varGamma_{A}},\ \delta|_{\varGamma_{B}}\circ\omega|_{\varGamma_{B}}\bigr{)}=$$

$$\displaystyle=\bigl{(}\delta\circ\omega|_{\varGamma_{A}},\ \delta\circ\omega|_{\varGamma_{B}}\bigr{)},$$

$$h=\{\begin{array}{ll}a(x),& x\in A,\\ b(x),& x\in B.\end{array}$$

Full text

Recently the authors described the structure of groups $G_{f}$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^{2}$ and $2$-sphere $S^{2}$. The present paper is devoted to the case $M=S^{2}$. In this situation $\Gamma_{f}$ is always a tree, and therefore all elements of the group $G_{f}$ have a common fixed subtree $\mathrm{Fix}(G_{f})$, which may even consist of a unique vertex. Our main result calculates the groups $G_{f}$ for all Morse functions $f\colon S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_{f})$ consists of more than one point.

1. Introduction

Let $M$ be a compact two-dimensional manifold and $\mathcal{D}(M)$ the group of diffeomorphisms of $M$. Then there exists a natural right action

[TABLE]

of this group on the space of smooth functions on $M$ defined by the formula $\phi(f,h)=f\circ h$. For $f\in\mathcal{C}^{\infty}(M,\mathbb{R})$ denote by

[TABLE]

its stabilizer with respect to the specified action.

Definition 1.1**.**

Let $\mathcal{F}(M,\mathbb{R})$ be the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisting of maps $f\colon M\to\mathbb{R}$ such

1. (1)

$f$* takes constant values on the connected components of the boundary $\partial M$ and has no critical points on $\partial M$;* 2. (2)

for each critical point $z$ of $f$ there are local coordinates $(x,y)$ in which $z=(0,0)$ and $f(x,y)=f(z)+g_{z}(x,y)$, where $g_{z}\colon\mathbb{R}^{2}\to\mathbb{R}$ is a homogeneous polynomial without multiple factors.

Notice that every critical point of $f\in\mathcal{F}(M,\mathbb{R})$ is isolated.

A function $f\in\mathcal{F}(M,\mathbb{R})$ is called Morse, if $\deg g_{z}=2$ for each critical point $z$ of $f$. In that case, due to Morse Lemma, one can assume that $g_{z}(x,y)=\pm x^{2}\pm y^{2}$.

We will denote by $\mathcal{M}(M,\mathbb{R})$ the space of all Morse maps $M\to\mathbb{R}$.

Homotopy types of stabilizers and orbits of Morse functions and functions from $\mathcal{F}(M,\mathbb{R})$ were studied in [8], [9], [10], [1], [2], [3], [4], [5], [6].

Let $f\in\mathcal{C}^{\infty}(M,\mathbb{R})$, $\varGamma_{f}$ be a partition of the surface $M$ into the connected components of level sets of this function, and $p\colon M\to\varGamma_{f}$ be the canonical factor-mapping, associating to each $x\in M$ the connected component of the level set $f^{-1}(f(x))$ containing that point.

Endow $\varGamma_{f}$ with the factor topology with respect to the mapping $p$: so a subset $A\subset\varGamma_{f}$ will be regarded as open if and only if its inverse image $p^{-1}(A)$ is open in $M$. Then $f$ induces the function $\hat{f}\colon\varGamma_{f}\to\mathbb{R}$, such that $f=\hat{f}\circ p$.

It is well known, that if $f\in\mathcal{F}(M,\mathbb{R})$, then $\varGamma_{f}$ has a structure of a one-dimensional CW-complex called the Kronrod-Reeb graph, or simply the graph of $f$. The vertices of this graph correspond to critical connected components of level sets of $f$ and connected components of the boundary of the surface. By the edge of $\varGamma_{f}$ we will mean an open edge, that is, a one-dimensional cell.

Denote by $\mathcal{H}(\varGamma_{f})$ the group of homeomorphisms of $\varGamma_{f}$. Notice that each element of the stabilizer $h\in\mathcal{S}(f)$ leaves invariant each level set of $f$, and therefore induces a homeomorphism $\rho(h)$ of the graph of $f$, so that the following diagram is commutative:

[TABLE]

Moreover, the correspondence $h\mapsto g(h)$ is a homomorphism of groups

[TABLE]

Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the path component of the identity map $\mathrm{id}_{M}$ in $\mathcal{D}(M)$. Put

[TABLE]

Thus, $G_{f}$ is the group of automorphisms of the Kronrod-Reeb graph of $f$ induced by diffeomorphisms of the surface preserving the function and isotopic identity.

Remark 1.2**.**

Since $\hat{f}$ is monotone on edges of $\varGamma_{f}$, it is easy to show that $G_{f}$ is a finite group. Moreover, if $g(E)=E$, for some $g\in G$ and an edge $E$ of the graph $\varGamma_{f}$, then $g(x)=x$ for all $x\in E$.

Since $G_{f}$ is finite and $\rho$ is continuous, it follows that $\rho$ reduces to an epimorphism

[TABLE]

of the group $\pi_{0}\mathcal{S}^{\prime}(f)$ path components of $\mathcal{S}^{\prime}(f)$ being an analogue of the mapping class group for $f$-preserving diffeomorphisms.

Algebraic structure of the group $\pi_{0}\mathcal{S}^{\prime}(f)$ of connected components of $\mathcal{S}^{\prime}(f)$ for all $f\in\mathcal{F}(M,\mathbb{R})$ on orientable surfaces $M$ distinct from $2$-torus and $2$-sphere is described in [11], and the structure of its factor group $G_{f}$ is investigated in [7]. These groups play an important role in computing the homotopy type of the path component $\mathcal{O}_{f}(f)$ of the orbit of $f$, see also [8], [9], [1], [2], [3].

The purpose of this note is to describe the groups $G_{f}$ for a certain class of smooth functions on $2$-sphere $S^{2}$.

The main result Theorem 1.4 reduces computation of $G_{f}$ to computations of similar groups for restrictions of $f$ to some disks in $S^{2}$. As noted above the latter calculations were described in [7].

First we recall a variant of the well known fact about automorphisms of finite trees from graphs theory.

Lemma 1.3**.**

Let $\Gamma$ be a finite contractible one-dimensional CW-complex (<<a topological tree>>), $G$ be a finite group of its cellular homeomorphisms, and $\mathrm{Fix}(G)$ be the set of common fixed points of all elements of the group $G$. Then $\mathrm{Fix}(G)$ is either a contractible subcomplex or consists of a single point belonging to some edge $E$ an open 1-cell), and in the latter case there exists $g\in G$ such that $g(E)=E$ and $g$ changes the orientation of $E$.

Suppose $f\colon\mathcal{S}^{2}\to{\mathbb{R}}$ belongs to $\mathcal{F}(M,\mathbb{R})$. Then it is easy to show that $\varGamma_{f}$ is a tree, i.e., a finite contractible one-dimensional CW-complex, and by Remark 1.2 $G_{f}$ is a finite group of cellular homeomorphisms of $\varGamma_{f}$. Therefore, for $G_{f}$, the conditions of Lemma 1.3 are satisfied. Note that according to Remark 1.2 the second case of Lemma 1.3 is impossible, and hence $G_{f}$ has a fixed subtree.

In this paper we consider the case when the fixed subtree of the group $G_{f}$ contains more than one vertex, i.e. has at least one edge.

Let us also mention that $\mathcal{D}_{\mathrm{id}}(S^{2})$ coincides with the group $\mathcal{D}^{+}(S^{2})$ of diffeomorphisms of the sphere preserving orientation, [12]. Therefore $\mathcal{S}^{\prime}(f)$ consists of diffeomorphisms of the sphere preserving the function $f$ and the orientation of $\mathcal{S}^{2}$.

Theorem 1.4**.**

Let $f\in\mathcal{F}(M,\mathbb{R})$. Suppose that all elements of the group $G_{f}$ have a common fixed edge $E$. Let $x\in E$ be an arbitrary point and $A$ and $B$ be the closures of the connected components of $S^{2}\setminus p^{-1}(x)$. Then

1. (1)

$A$* and $B$ are 2-disks being invariant with respect to $\mathcal{S}^{\prime}(f)$;* 2. (2)

the restrictions $f|_{A}\in\mathcal{F}(M,\mathbb{R})$ and $f|_{B}\in\mathcal{F}(M,\mathbb{R})$; 3. (3)

the map $\phi\colon G_{f}\to G_{f|_{A}}\times G_{f|_{B}}$ defined by the formula

[TABLE]

is an isomorphism of groups.

Proof.

(1) By assumption $x$ belongs to the open edge $E$. Therefore $p^{-1}(x)$ is a regular connected component of some level set of the function $f$, that is, a simple closed curve. Then, by Jordan Theorem, $p^{-1}(x)$ divides the sphere into two connected components whose closures are homeomorphic to two-dimensional disks. Consequently, $A$ and $B$ are two-dimensional disks.

Let as show that $A$ and $B$ are invariant with respect to $\mathcal{S}^{\prime}(f)$, i.e., $h(A)=A$ and $h(B)=B$ for each $h\in\mathcal{S}^{\prime}(f)$. Denote

[TABLE]

Then

[TABLE]

By definition, $\rho(h)(x)=x$, whence $\rho(h)$ either preserves both $\varGamma_{A}$ and $\varGamma_{B}$ or interchange them. We claim that

[TABLE]

Indeed suppose $\rho(h)(\varGamma_{A})=\varGamma_{B}$. Since $\rho(h)$ is fixed on $E$, it follows that

[TABLE]

whence

[TABLE]

which contradicts to our assumption. Thus $\varGamma_{A}$ and $\varGamma_{B}$ are invariant with respect to the group $G_{f}$.

Now we can show that $A$ and $B$ are also invariant with respect to $h$. By virtue of the commutativity of the diagram (1.1) $\rho(h)(p(y))=p(h(y))$ for all $y\in\varGamma$. In particular:

[TABLE]

Therefore, $h(A)=p^{-1}(\varGamma_{A})=A$. The proof for $B$ is similar. Thus, $A$ and $B$ are invariant with respect to $\mathcal{S}^{\prime}(f)$.

(2) Notice that the function $f$ takes a constant value on the simple closed curve $p^{-1}(x)$ being a common boundary of disks $A$ and $B$, and does not contain critical points of $f$. Therefore, the restrictions $f|_{A},f|_{B}$ satisfy the conditions 1) and 2) the Definition 1.1, and so they belong to $\mathcal{F}(M,\mathbb{R})$ and $\mathcal{F}(M,\mathbb{R})$ respectively.

(3) We should prove that the map $\phi\colon G_{f}\to G_{f|_{A}}\times G_{f|_{B}}$ defined by formula $\phi(\gamma)=(\gamma|_{\varGamma_{A}},\gamma|_{\varGamma_{B}})$ is an isomorphism.

First we will show that $\phi$ is correctly defined. Let $\gamma\in G_{f}=\rho(\mathcal{S}^{\prime}({f}))$, that is, $\gamma=\rho(h)$, where $h$ is a diffeomorphism of the sphere preserving the function $f$ and isotopic to the identity.

We claim that $h|_{A}\in\mathcal{S}^{\prime}({f|_{A}})=\mathcal{S}(f|_{A})\cap\mathcal{D}_{\mathrm{id}}(A)$. Indeed, for each point $x\in A$ we have that:

[TABLE]

which means that $h|_{A}\in\mathcal{S}(f|_{A})$.

Moreover, since $h$ preserves the orientation of the sphere, it follows that $h|_{A}$ preserves the orientation of the disk $A$, and therefore by [12], $h|_{A}\in\mathcal{D}_{\mathrm{id}}(A)$. Thus $\gamma|_{\varGamma_{A}}\in G_{f|_{A}}$. Similarly $\gamma|_{\varGamma_{B}}\in G_{f|_{B}}$, and so $\phi$ is well defined.

Let us now verify that $\phi$ is an isomorphism of groups, that is, a bijective homomorphism.

Let $\delta,\omega\in G_{f}$. Then

[TABLE]

sp $\phi$ is a homomorphism.

Let us show that $\ker\phi=\{\mathrm{id}_{\varGamma}\}$. Indeed, suppose $\gamma\in\ker\phi$, that is $\gamma|_{\varGamma_{A}}=\mathrm{id}_{\varGamma_{A}}$ and $\gamma|_{\varGamma_{B}}=\mathrm{id}_{\varGamma_{B}}$. Then $\gamma$ is fixed on $\varGamma_{A}\cup\varGamma_{B}=\varGamma$, and hence it is the identity map.

Surjectivity of $\phi\colon G_{f}\to G_{f|_{A}}\times G_{f|_{B}}$ is implied by the following simple lemma whose proof we leave to the reader.

Lemma 1.5**.**

Suppose $f\colon D^{2}\to\mathbb{R}$ belgns to the space $\mathcal{F}(M,\mathbb{R})$. Then for arbitrary $\alpha\in G_{f}$, there exists $a\in\mathcal{S}^{\prime}({f})$ fixed near the boundary $\partial D^{2}$ and such that $\alpha=\rho(a)$.∎

Let $(\alpha,\beta)\in G_{f|_{A}}\times G_{f|_{B}}$, then by Lemma 1.5 there exist $a\in\mathcal{S}^{\prime}({f|_{A}})$ and $b\in\mathcal{S}^{\prime}({f|_{B}})$ fixed near $\partial A=\partial B=p^{-1}(x)$ and such that $\alpha=\rho_{A}(a)$ and $\beta=\rho_{B}(b)$. Define $h$ by the following formula:

[TABLE]

Then, $h$ is a diffeomorphism of the sphere, preserving the function and orientation, whence $h\in\mathcal{S}^{\prime}(f)$.

Moreover if we put $\gamma=\rho(h)\in G_{f}$, then $\gamma|_{\varGamma_{A}}=\rho(h|_{A})=\alpha$ and $\gamma|_{\varGamma_{B}}=\rho(h|_{B})=\beta$. In other words, $\phi(\gamma)=(\gamma|_{\varGamma_{A}},\gamma|_{\varGamma_{B}})=(\alpha,\beta)$, i.e., $\phi$ is surjective and therefore an isomorphism. ∎