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.
210219
Actions of finite groups and smooth functions on surfaces] Actions of finite groups and smooth functions on surfaces
[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
(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 wellknown that each homogeneous polynomial splits into a product of linear and irreducible over quadratic factors. Condition (P) means that
(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
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 wellknown that the quotientspace has a structure of dimensional CW complex. The space is called the KronrodReeb graph, or simply, KRgraph 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
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 KRgraph 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 KRgraph 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 KronrodReeb 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
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
Let , and be a diffeomorphism such that Define the map by the following rule
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 welldefined. 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 KRgraph 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
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:connectedcomponents:2009], Section 6). After some linear change of coordinates one can assume that

if , then the group consists of the linear transformations of the following form
see [Maksymenko:connectedcomponents:2009, Section 6, case (B)];

if then the group of is a finite cyclic subgroup of [Maksymenko:connectedcomponents:2009, Section 6, case (E)].
We will also need the following lemma:
Lemma 3.2
([Maksymenko:connectedcomponents:2009], Corollary 7.4). Let be a germ of a diffeomorphism at and be its tangent map at If , then
For the sake of completeness we will recall a short proof from [Maksymenko:connectedcomponents:2009].
Assume that the polynomial is a homogeneous function of degree i.e., for and Then
Lemma 3.2 is proved.
4 Topological structure of the atom
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 .
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 KRgraph 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:
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
(5.1) 
and define charts for the points , in the following way:

;

the map is defined from the diagram:
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:
(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,
and we set
If then by assumption and Lemma 3.1, is a cyclic subgroup of In this case we put
We define the diffeomorphism by the rule:
(5.2) 
(2) Now we prove that the map is a homomorphism. Suppose are such that and By (5.2), we have
and
On the other hand, we have
It follows from the definition of the linear map , that Hence
So, the map is a homomorphism.
(3) Let and be such that Then we define the map by the rule
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
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
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
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:
and
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

intersects some edge of at a point, say , or

converges to some vertex of .
In the case (1) let , and be maps, defined by (I) for and respectively, and
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
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
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
Such maps have been studied in [Maktymenko:smoothshifts:2003].
Since all orbits of are closed, it follows from [Maktymenko:smoothshifts:2003, Theorem 19] that the map is a diffeomorphism, iff the Lie derivative of along satisfies the condition: Moreover we have that , where
(5.3) 
Lemma 5.3
For each the map extends to a diffeomorphism , so that the correspondence is a homomorphism .
We will need the following two lemmas.
Lemma 5.4
Let and be such that , and Then there exists a unique function such that
In particular, the function depends only on
Lemma 5.5
The diffeomorphism extends to a diffeomorphism such that on
Define a diffeomorphism by the formula:
Let . Define the diffeomorphism by the rule: if then
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