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

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.


Introduction
f (x, y) = f (z) + g z (x, y), where g z : R 2 → R is a homogeneous polynomial without multiple factors.
for each critical point z of f . In that case, due to Morse Lemma, one can assume that g z (x, y) = ±x 2 ± y 2 .
Let f ∈ C ∞ (M, R), Γ f be a partition of the surface M into the connected components of level sets of this function, and p : M → Γ f be the canonical factor-mapping, associating to each x ∈ M the connected component of the level set f −1 (f (x)) containing that point.
Endow Γ f with the factor topology with respect to the mapping p: so a subset A ⊂ Γ f will be regarded as open if and only if its inverse image It is well known, that if f ∈ F (M, R), then Γ 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 Γ f we will mean an open edge, that is, a one-dimensional cell.
Denote by H(Γ f ) the group of homeomorphisms of Γ f . Notice that each element of the stabilizer h ∈ S(f ) leaves invariant each level set of f , and therefore induces a homeomorphism ρ(h) of the graph of f , so that the following diagram is commutative: Let also D id (M) be the path component of the identity map id M in D(M). Put 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.
Since G f is finite and ρ is continuous, it follows that ρ reduces to an epimorphism of the group π 0 S ′ (f ) path components of S ′ (f ) being an analogue of the mapping class group for f -preserving diffeomorphisms.
Algebraic structure of the group π 0 S ′ (f ) of connected components of S ′ (f ) for all f ∈ F (M, 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 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 Γ be a finite contractible one-dimensional CW-complex ( a topological tree ), G be a finite group of its cellular homeomorphisms, and Fix(G) be the set of common fixed points of all elements of the group G. Then 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 ∈ G such that g(E) = E and g changes the orientation of E.
Suppose f : S 2 → R belongs to F (M, R). Then it is easy to show that Γ 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 Γ 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 D id (S 2 ) coincides with the group D + (S 2 ) of diffeomorphisms of the sphere preserving orientation, [12]. Therefore S ′ (f ) consists of diffeomorphisms of the sphere preserving the function f and the orientation of S 2 .
is an isomorphism of groups.
Proof. Then By definition, ρ(h)(x) = x, whence ρ(h) either preserves both Γ A and Γ B or interchange them. We claim that , which contradicts to our assumption. Thus Γ A and Γ 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) ρ(h)(p(y)) = p(h(y)) for all y ∈ Γ . In particular: Therefore, h(A) = p −1 (Γ A ) = A. The proof for B is similar. Thus, A and B are invariant with respect to S ′ (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 F (M, R) and F (M, R) respectively.
(3) We should prove that the map φ : First we will show that φ is correctly defined. Let γ ∈ G f = ρ(S ′ (f )), that is, γ = ρ(h), where h is a diffeomorphism of the sphere preserving the function f and isotopic to the identity.
We claim that h| A ∈ S ′ (f | A ) = S(f | A ) ∩ D id (A). Indeed, for each point x ∈ A we have that: 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], Similarly γ| Γ B ∈ G f | B , and so φ is well defined.
Let us now verify that φ is an isomorphism of groups, that is, a bijective homomorphism. Let δ, ω ∈ G f . Then sp φ is a homomorphism. Let us show that ker φ = {id Γ }. Indeed, suppose γ ∈ ker φ, that is γ| Γ A = id Γ A and γ| Γ B = id Γ B . Then γ is fixed on Γ A ∪ Γ B = Γ , and hence it is the identity map.
is implied by the following simple lemma whose proof we leave to the reader. Lemma 1.5. Suppose f : D 2 → R belgns to the space F (M, R). Then for arbitrary α ∈ G f , there exists a ∈ S ′ (f ) fixed near the boundary ∂D 2 and such that α = ρ(a).
Let (α, β) ∈ G f | A × G f | B , then by Lemma 1.5 there exist a ∈ S ′ (f | A ) and b ∈ S ′ (f | B ) fixed near ∂A = ∂B = p −1 (x) and such that α = ρ A (a) and β = ρ B (b). Define h by the following formula: h = a(x), x ∈ A, b(x), x ∈ B. Then, h is a diffeomorphism of the sphere, preserving the function and orientation, whence h ∈ S ′ (f ).