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# Actions of finite groups and smooth functions on surfaces

## Abstract

Let be a Morse function on a smooth closed surface, be a connected component of some critical level of , and be its atom. Let also be a stabilizer of the function under the right action of the group of diffeomorphisms on the space of smooth functions on and The group acts on the set of connected components of the boundary of Therefore we have a homomorphism . Let also be the image of in

Suppose that the inclusion induces a bijection Let be a subgroup of We present a sufficient condition for existence of a section of the homomorphism so, the action of on lifts to the -action on by -preserving diffeomorphisms of .

This result holds for a larger class of smooth functions having the following property: for each critical point of the germ of at is smoothly equivalent to a homogeneous polynomial without multiple linear factors.

Diffeomorphism, Morse function
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210219

Actions of finite groups and smooth functions on surfaces] Actions of finite groups and smooth functions on surfaces

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[2000]57S05, 57R45, 37C05

## 1 Introduction

Let be a smooth compact surface. The group of diffeomorphisms of acts on the space of smooth functions by the rule

 C∞(M)×D(M)→C∞(M),(f,h)=f∘h. (1.1)

The set is called the stabilizer of the function under action (1.1). Endow and with the corresponding Whitney topologies. The topology on induces a certain topology on the stabilizer

Let be the set of smooth functions satisfying the following two conditions:

• the function takes a constant value at each connected component of , and all critical points of belong to the interior of ;

• for each critical point of the germ of at is smoothly equivalent to some homogeneous polynomial without multiple linear factors.

It is well-known that each homogeneous polynomial splits into a product of linear and irreducible over quadratic factors. Condition (P) means that

 fx=n∏i=1Li⋅m∏j=1Qqjj, (1.2)

where is a linear form, is an irreducible quadratic form such that for , and for . So, if , then is an isolated critical point of

Recall that if is a germ of -function such that is an isolated critical point of , then there is a germ of a homeomorphism such that

 Misplaced &

If is not a local extreme, then the number does not depend of a particular choice of In this case the point will be called a generalized -saddle, or simply an -saddle. The number corresponds to the number of linear factors in (1.2). Examples of level sets foliations near isolated critical points are given in Fig. 1.1.

Let be the space of Morse functions on , which satisfy condition (B), and be a critical point of Then, by Morse Lemma, there exists a coordinate system near such that the function has one of the following forms which are, obviously, homogeneous polynomials without multiple factors. This implies that is a subspace of

Let be a smooth function and be a real number. A connected component of the level set is called critical if it contains at least one critical point, otherwise, is called regular. Let be a foliation of into connected components of level sets of It is well-known that the quotient-space has a structure of -dimensional CW complex. The space is called the Kronrod-Reeb graph, or simply, KR-graph of . We will denote it by . Let be a projection of onto . Then vertices of correspond to connected components of critical level sets of the function

It should be noted that the function can be represented as the composition

 f=ϕ∘pf:M  pf  −−−−→Γf  ϕ′  −−−−→R,

where is the map induced by . Let . Then , and we have for all Hence interchanges connected components of level sets of the function and therefore it induces an automorphism of KR-graph such that the following diagram is commutative:

In other words, we have a homomorphism Let be the image of in It is easy to show that the group is finite.

Let be a vertex of and be the stabilizer of under the action of on An arbitrary connected closed -invariant neighborhood of in containing no other vertices of will be called a star of . We denote it by

The set which consists of restrictions of elements of onto the star is a subgroup of . This group will be called a local stabilizer of Let also be the map defined by for , i.e., is the restriction map.

Let be a vertex of , and be the corresponding connected component of the critical level set

###### Definition 1.1

A vertex of the graph will be called special if there is a bijection between connected components of and . The corresponding connected component will be called special.

It follows from definition of KR-graph that for a special vertex there is a 1–1 correspondence between connected components of complement to in and connected components of

Note that a special component gives a partition of the surface whose -dimensional elements are vertices of -dimensional elements are edges of , and -dimensional elements are connected components of complement of in Since is compact, it follows that has a finite number of elements in each dimension.

## 2 Main result

Let . Suppose that its Kronrod-Reeb graph contains a special vertex , and be the special component of level set of which corresponds to

Let be a subgroup of leaving invariant. It is easy to see that We denote by the map

 ϕ=r∘ρ:SV(f)  ρ  −−−→Gv  r  −−−→Glocv.

Let be a subgroup of and be a subgroup of We will say that the group has property (C) if the following conditions hold.

• Let and be a -dimensional element of . Suppose that Then for all other , and the map preserves orientation of each element of

###### Lemma 2.1

If has property (C), then acts on the set of all elements of the partition Moreover this action is free on the set of -dimensional elements of

{proof}

Let , and be a diffeomorphism such that Define the map by the following rule

 τ(g,e)=h(e),e∈Ξ.

We claim that this definition does not depend of a particular choice of such Let be diffeomorphisms such that Then where be the unit of By definition of the unit , we have for each -dimensional component of Then, by condition (C), for other Hence So, the map is well-defined. It is easy to see that where is the unit of and for each and Thus is an -action on

Suppose is such that for some -dimensional component of Then, by condition (C), for each -dimensional component of Hence, so the -action on the set of -dimensional components of is free.

Thus condition (C) implies that combinatorially acts of i.e., it ensures invariance of the partition under the action of on Our aim is to prove that in fact this ¡¡combinatorial¿¿ action is induced by a real action of on by diffeomorphisms preserving .

Namely the following theorem holds.

###### Theorem 2.2

Suppose is such that its KR-graph contains a special vertex , and be the local stabilizer of Let also be a subgroup of and be a subgroup of satisfying condition Then there exists a section of the map i.e., the map is a homomorphism satisfying the condition

Group actions which have the property of invariance of some partition of the surfaces are studied by Bolsinov and Fomenko [BolsinovFomenko:1998], Brailov [Brailov:1998], Brailov and Kudryavtseva [BrailovKudryavtseva:VMNU:1999], Kudryavtseva [Kudryavtseva:MatSb:1999], Maksymenko [Maksymenko:def:2009], Kudryavtseva and Fomenko [KudryavtsevaFomenko:DANRAN:2012, KudryavtsevaFomenko:VMU:2013].

### 2.3 Structure of the paper

In Section 3 we recall the definitions and statements that will be used in the text. The topological structure of the atom which corresponds to is described in Section 4. In section 5, we construct an -action on the surface

## 3 Symmetries of homogeneous polynomials

Let be a homogeneous polynomial without multiple linear factors. Suppose the origin is not a local extreme for Let also be a group of orientation preserving linear automorphisms such that . The following lemma holds:

###### Lemma 3.1

([Maksymenko:connected-components:2009], Section 6). After some linear change of coordinates one can assume that

1. if , then the group consists of the linear transformations of the following form

 Misplaced &

see [Maksymenko:connected-components:2009, Section 6, case (B)];

2. if then the group of is a finite cyclic subgroup of [Maksymenko:connected-components:2009, Section 6, case (E)].

We will also need the following lemma:

###### Lemma 3.2

([Maksymenko:connected-components:2009], Corollary 7.4). Let be a germ of a diffeomorphism at and be its tangent map at If , then

{proof}

For the sake of completeness we will recall a short proof from [Maksymenko:connected-components:2009].

Assume that the polynomial is a homogeneous function of degree i.e., for and Then

 fz(x)=fz(tx)tk=fz(h(tx))tk=fz(h(tx)t)−−−−−→  t→0  (fz∘T0h)(x).

Lemma 3.2 is proved.

## 4 Topological structure of the atom EV,a

Let be a smooth function from , and be a connected component of some critical level of .

Let also be a connected component of which contains Assume that the boundary consists of connected components i.e., Since , it follows that belongs to , and so, by (B), takes a constant value at each connected component of the boundary . Assume that , , and Put and . Fix such that

A connected component of which contains will be called an atom of and denoted by .

Let be a subgroup of and We will need the following lemma.

###### Lemma 4.1

Let be an atom of a special critical component , be a connected component of , and Assume that the group has property (C). If then preserves the orientation of .

{proof}

Fix a Riemannian metric on Let be a gradient vector field of the function in this Riemannian metric. Let also be a set of points such that there exists an integral curve of which joins the point with some point Then is a union of open intervals in and the map , is an embedding. The image of is a cycle in So, the connected component of defines the cycle in Moreover the orientation of induces the orientation of and vice versa, see [Oshemkov].

Assume that has property (C). Let and be a -dimensional element of such that Then by (C), for all other In particular and preserves orientation of . Then and preserves orientation of .

## 5 Proof of Theorem 2.2

Suppose is such that its KR-graph contains a special vertex be the corresponding special component of some level set, which corresponds to , and be the local stabilizer of

Let be a subgroup of such that has property (C). We will construct a lifting of the -action on to the action of the group on the surface

By Lemma 2.1 there is an action of on the set of vertices of defined by the rule:

 σ0(g,z)=h(z),

where is any diffeomorphism such that

### Step 1

Now we will extend the action to the -action on the set of neighborhoods of vertices of Assume that the action has orbits for some and let be the union of vertices of

Then, by definition of the class for each there exists a chart which contains such that the map is a homogeneous polynomial without multiple linear factors. We can also assume that is a -disk with the center at and radius , and the group has the properties described in Lemma 3.1. Fix any diffeomorphisms

 hri∈Hsuchthathri(zr0)=zri,i=1,2,…,k(r), (5.1)

and define charts for the points , in the following way:

• ;

• the map is defined from the diagram:

 Misplaced &

i.e., .

Reducing , we can assume that for

Thus the chart is chosen so that the map is a homogeneous polynomial without multiple linear factors which coincides with given polynomial for the chart We also put , and .

###### Lemma 5.1

There exist a homomorphism and a monomorphism such that the following diagram is commutative:

 Misplaced &
{proof}

(1) First we construct a map Let be such that for some and Let also be a diffeomorphism of It is easy to see that the map preserves the polynomial By Lemma 3.2 the tangent map also preserves the polynomial , so Define a linear map as follows: if , then, by Lemma 3.1,

 Misplaced &

and we set

 Misplaced &

If then by assumption and Lemma 3.1, is a cyclic subgroup of In this case we put

 Ah=T0γh.

We define the diffeomorphism by the rule:

 λ1(h)|Uri=q−1rj∘Ah∘qri. (5.2)

(2) Now we prove that the map is a homomorphism. Suppose are such that and By (5.2), we have

 λ1(h1)|Uri=q−1rj∘Ah1∘qri,λ1(h2)|Urj=q−1rk∘Ah2∘qrj,

and

 λ1(h2)|Urj∘λ1(h1)|Uri=q−1rk∘Ah2∘Ah1∘qri.

On the other hand, we have

 λ1(h2∘h1)=q−1rk∘Ah2∘h1∘qri.

It follows from the definition of the linear map , that Hence

 λ1(h2∘h1)=λ1(h2)∘λ1(h1).

So, the map is a homomorphism.

(3) Let and be such that Then we define the map by the rule

 χ1(g)=λ1(h).

Obviously that is a homomorphism. It remains to prove that the map is a monomorphism. It is sufficient to check that i.e., iff trivially acts on the set of -dimensional elements of

Suppose that trivially acts on the set of -dimensional elements of By condition (C), trivially acts on set of vertices and edges of . Since for all and , it follows from (5.2) that

Suppose is such that Then for each edge of , and preserves the orientation of Hence by Lemma 4.1, leaves invariant each connected component of with its orientation. Therefore trivially acts on the set of -dimensional elements of

Let be a map defined by the formula

 σ1(g,x)=χ1(g)(x),x∈U.

Since is a homomorphism, it follows that is an -action on

### Step 2

In this step we extend the action to the -action on the atom We start with some preliminaries. Let be the chart on which contains defined above. The projection map induces the map between tangent bundles of and Fix a Riemannian metric on such that the following diagram is commutative

 Misplaced &

where and are gradient fields of and in Riemannian metrics on and on respectively. Let also be the flow of on

### Another description of the diffeomorphism λ1(h)

Let be a point, , and be its image under Let also and be the trajectories of the gradient flow such that and Since preserves trajectories of the flow in , it follows that By definition of we have that In particular, if the trajectory intersects some edge of at some point and then where is the trajectory of , which passes through the point . Namely the image of is uniquely defined by the image of the point

By Lemma 2.1, the group acts on the set of all edges of . Assume that this action has orbits for some and We also put For each edge fix

• a -diffeomorphism such that restrictions and are isometries,

where is the radius of the disk defined in Step 1.

###### Lemma 5.2

There exist a homomorphism and a monomorphism such that the following diagram is commutative:

 Misplaced &

and

{proof}

Let We will extend the diffeomorphism to a diffeomorphism of the atom . Let be any point. If for some , then we put

Suppose that . Let be a trajectory of the flow passing through the point . Then we have one of the following two cases: the trajectory either

1. intersects some edge of at a point, say , or

2. converges to some vertex of .

In the case (1) let , and be maps, defined by (I) for and respectively, and

 h′=ℓ′∘ℓ−1:R  ℓ−1  −−−−→(−1,1)  ℓ′  −−−→R′.

Let also , be the trajectory of which passes through and be a unique point in such that . Then we put

Consider the case (2). Let be the neighborhood of , defined in Step 1, be the corresponding point in be the trajectory of such that and be a unique point in such that . In this case we define by the rule: .

By definition Let be the map defined as follows: for and such that , we put It is easy to check that the map is a homomorphism. Moreover iff Therefore is a monomorphism. Define the map by the rule

 σ(g,x)=χ2(g)(x).

Since is the homomorphism, it follows that the map is an -action on the atom

### Step 3

In this step we extend the -action on the atom to the -action on the surface We start with some preliminaries. Let be a set of -dimensional elements of By Lemma 2.1 the group acts on the set . Assume that this action has orbits , , and We also put Fix diffeomorphisms such that

Let Since is a special vertex, it follows that the set is a cylinder. We put , and

We choose such that the set is also an atom of . Let

 Zr=Er0∩f−1([−a1,a/2]∪[a/2,a1])∩EV,a1.

By definition, we have that and does not contain critical points of . We also put and

Fix a vector field on such that its orbits coincide with connected components of level sets of the restriction and let be the flow of Then for each smooth function we can define the following map

 Fα:M→M,Fα(x)=F(x,α(x)).

Such maps have been studied in [Maktymenko:smooth-shifts:2003].

Since all orbits of are closed, it follows from [Maktymenko:smooth-shifts:2003, Theorem 19] that the map is a diffeomorphism, iff the Lie derivative of along satisfies the condition: Moreover we have that , where

 ξ=−α∘F−1α. (5.3)
###### Lemma 5.3

For each the map extends to a diffeomorphism , so that the correspondence is a homomorphism .

{proof}

We will need the following two lemmas.

###### Lemma 5.4

Let and be such that , and Then there exists a unique -function such that

 χ2(g)|Yr=hri|Yr∘Fξri:Yr→Yr.

In particular, the function depends only on

###### Lemma 5.5

The diffeomorphism extends to a diffeomorphism such that on

We prove Lemma 5.4 and Lemma 5.5 bellow, and now we will complete Theorem 2.2.

Define a diffeomorphism by the formula:

 ~hri=hri∘wri.

Let . Define the diffeomorphism by the rule: if then

 λ3(h)|Eri=~hrj∘~h−1ri:Eri  ~h−1ri  −−−−−→Er0  ~hrj  −−−−→Erj.

It follows from Lemma 5.5 that coincides with on

Now we will check that the correspondence is a homomorphism. Let and be homeomorphisms from such that and By definition , and Then