Orbits of smooth functions on 2-torus and their homotopy types

# Orbits of smooth functions on 2-torus and their homotopy types

## Abstract.

Let be a Morse function on -torus such that its Kronrod-Reeb graph has exactly one cycle, i.e. it is homotopy equivalent to . Under some additional conditions we describe a homotopy type of the orbit of with respect to the action of the group of diffeomorphism of .

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

###### Key words and phrases:
Diffeomorphism, Morse function, homotopy type
###### Key words and phrases:
Diffeomorphism, Morse function, homotopy type
###### 2000 Mathematics Subject Classification:
57S05, 57R45, 37C05
###### 2000 Mathematics Subject Classification:
57S05, 57R45, 37C05

## 1. Introduction

Let be a smooth oriented surface. For a closed (possibly empty) subset denote by the group of diffeomorphisms of fixed on . This group naturally acts from the right on the space of smooth functions by following rule: if and then the result of the action of on is the composition map

 f∘h:M\leavevmode\nobreak \leavevmode\nobreak h\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→M\leavevmode\nobreak \leavevmode\nobreak f\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→R. (1.1)

For let

 S(f,X) ={f∈D(M,X)|f∘h=f}, O(f,X) ={f∘h|h∈D(M,X)}.

be respectively the stabilizer and the orbit of under the action (1.1).

Endow on , and their subspaces and with the corresponding Whitney -topologies. Let also be the path component of the identity map in , be the path component of in , and be the path component of in . If then we omit it from notation and write , , , and so on.

We will assume that all the homotopy groups of will have as a base point, and all homotopy groups of the groups of diffeomorphisms and the corresponding stabilizers of are based at . For instance will always mean . Notice that the latter group is also isomorphic with .

Since and are topological groups, it follows that the homotopy sets , , and have natural groups structures such that

 π0D(M,X) ≅ D(M,X)/Did(M,X),π0S(f,X) ≅ S(f,X)/Sid(f,X),

and in the following part of exact sequence of homotopy groups of the pair

 ⋯→π1D(M,X)\leavevmode\nobreak \leavevmode\nobreak q\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π1(D(M,X),S(f,X))\leavevmode\nobreak \leavevmode\nobreak ∂\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π0S(f,X)\leavevmode\nobreak \leavevmode\nobreak i\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π0D(M,X) (1.2)

all maps are homomorphisms.

Moreover, is contained in the center of .

Recall that two smooth germs are said to be smothly equivalent if there exist germs of diffeomorphisms and such that .

###### Definition 1.1.

Denote by a subset in which consists of functions having the following two properties:

• takes a constant value at each connected components of , and all critical points of are contained in the interior of ;

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

Suppose a smooth germ has a critical point . This point is called non-degenerate if is smoothly equivalent to a homogeneous polynomial of the form .

Denote by the subset of consisting of Morse functions, that is functions having only non-degenerate critical points. It is well known that is open and everywhere dense in . Since has no multiple factors, we get the following inclusion .

###### Remark 1.2.

A homogeneous polynomial has critical points only if , and in this case the origin is always a critical point of . If has no multiple factors, then the origin is a unique critical point. Moreover, is non-degenerate , and degenerate for , see [10, §7].

Now let and . A connected component of the level set is said to be critical if contains at least one critical point of . Otherwise is called regular. Consider a partition of into connected component of level sets of . It is well known that the corresponding factor-space has a structure of a finite one-dimensional -complex and is called Kronrod-Reeb graph or simply KR-graph of the function . In particular, the vertices of are critical components of level sets of .

It is usually said that this graph was introduced by G. Reeb in [16], however in was used before by A. S. Kronrod in [5] for studying functions on surfaces. Applications of to study Morse functions on surfaces are given e.g. in [1, 7, 6, 17, 18, 15].

In a series of papers the first author calculated homotopy types of spaces and for all . These results are summarized in Theorem 1.3 below.

Denote also

 S′(f) =S(f)∩Did(M), S′(f,X) =S(f)∩Did(M,X).

Thus consists of diffeomorphisms preserving , fixed on and isotopic to relatively , though the isotopy between and is not required to be -preserving.

###### Theorem 1.3.

[9, 11, 12]. Let and be a finite (possibly empty) union of regular components of certain level sets of function . Then the following statements hold true.

(1) , and so

 πkO(f,X) ≅ πkO(f,X∪∂M),k≥1.

(2) The following map

 p:D(M,X)⟶O(f,X),p(h)=f∘h.

is a Serre fibration with fiber , i.e. it has homotopy lifting property for CW-complexes. This implies that

• ;

• the restriction map

 p|Did(M,X):Did(M,X)⟶Of(f,X) (1.3)

is also a Serre fibration with fiber ;

• for each we have an isomorphism defined by for a continuous map , and making commutative the following diagram

see for example [4, § 4.1, Theorem 4.1].

(3) Suppose either has a critical point which is not a nondegenerate local extremum or is a non-oriented surface. Then is contractible, for , , and for we have the following short exact sequence of fibration :

 1⟶π1D(M)\leavevmode\nobreak \leavevmode\nobreak p\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π1O(f)\leavevmode\nobreak \leavevmode\nobreak ∂∘j−11\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π0S′(f)⟶1. (1.4)

Moreover, is contained in the center of .

(4) Suppose either or . Then and are contractible, whence from the exact sequence of homotopy groups of the fibration (1.3) we get for , and that the boundary map

 ∂∘j−11:π1O(f,X) ⟶ π0S′(f,X)

is an isomorphism.

Suppose is orientable and differs from the sphere and the torus , and let . Then and satisfy assumptions of (4) of Theorem 1.3. Therefore from (1) of that theorem we get the following isomorphism

 π1O(f) \lx@stackrel(1)≅ π1O(f,∂M) \lx@stackrel(4)≅ π0S′(f,∂M).

A possible structure of for this case is completely described in [13].

However when is a sphere or a torus the situation is more complicated, as and from the short exact sequence (1.4) we get only that is an extension of with .

## 2. Main result

Suppose . Then it can easily be shown that for each its KR-graph is either a tree or has exactly one simple cycle. Moreover, , see [2, 3], and therefore the sequence (1.4) can be rewritten as follows:

 1⟶Z2\leavevmode\nobreak \leavevmode\nobreak p\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π1Of(f)\leavevmode\nobreak \leavevmode\nobreak ∂\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π0S′(f)⟶1. (2.1)

In [14] the authors studied the case when is a tree and proved that under certain ‘’triviality of -action’’ assumptions on the sequence (2.1) splits and we get an isomorphism .

In the present paper we consider the situation when is has exactly one simple cycle and under another ‘’triviality of -action’’ assumption describe the homotopy type of in terms of for some regular component of some level-set of , see Definition 2.3 and Theorem 2.6.

First we mention the following two simple lemmas which are left for the reader.

###### Lemma 2.1.

Let . Then the following conditions are equivalent:

• is a tree;

• every point separates ;

• every connected component of every level set of separates .

###### Lemma 2.2.

Assume that has exactly one simple cycle . Let also be any point belonging to some open edge of and be the corresponding regular component of certain level set of . Then the following conditions are equivalent:

• ;

• does not separate ;

• does not separate .

Thus if is not a tree, then there exists a connected component of some level set of that does not separate , and this curve corresponds to some point on an open edge of cycle .

For simplicity we will fix once and for all such and , and use the following notation:

 Did :=Did(T2), O :=Of(f), S :=S′(f), Sid :=Sid(T2), DidC :=Did(T2,C), OC :=Of(f,C), SC :=S′(f,C), SidC :=Sid(f,C)

Let also , so and is isotopic to . Then , and therefore interchanges connected components of . In particular, is also a connected component of . However, in general, does not coincide with , see Figure 2.1.

###### Definition 2.3.

We will say that trivially acts on if for all .

###### Remark 2.4.

Emphasize that the above definition only require that for all diffeomorphisms that preserve and are isotopic to . We do not put any assumptions on diffeomorphisms that are not isotopic to .

###### Remark 2.5.

It is easy to see that if is another non-separating regular component of some level-set , then trivially acts on if and only if trivially acts on .

###### Theorem 2.6.

Let be such that has exactly one cycle, and be a regular connected component of certain level set of which does not separate . Suppose trivially acts on . Then there is a homotopy equivalence

 O≃OC×S1.

In particular, we have the following isomorphisms:

 π1O ≅ π1OC×Z \lx@stackrelj×idZ≅ π0SC×Z.

The proof of this theorem will be given in § 4.

## 3. Preliminaries

### 3.1. Algebraic lemma

###### Lemma 3.2.

Let be four groups. Suppose there exists a short exact sequence

 1→L×M\leavevmode\nobreak \leavevmode\nobreak q\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→T\leavevmode\nobreak \leavevmode\nobreak ∂\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→S→1 (3.1)

and a homomorphism such that

• is the identity map and

• .

Then we have the following exact sequence:

 1⟶1×M\leavevmode\nobreak \leavevmode\nobreak q\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ker(φ)\leavevmode\nobreak \leavevmode\nobreak ∂\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→S⟶1.
###### Proof.

It suffices to prove that

• ,

• .

1) Let . We have for find such that . Since , there exists such that . Put and . Then

 φ(^t) =φ∘q∘φ(t)=φ(t), ∂(^t) =∂∘q∘φ(t)=1.

Hence and

 ∂(b)=∂(t)∂(^t)−1=∂(t)=s.

2) Let . We should find such that .

As , so there exist such that . But , whence

 (1,1)=φ(a)=φ(q(l,m))=φ(q(l,1))⋅φ(q(1,m))=φ(q(l,1))=(l,1).

Hence , and so . ∎

### 3.3. Isotopies of T2 fixed on a curve

We will need the following general lemma claiming that if a diffeomorphism of is fixed on a non-separating simple closed curve and is isotopic to , then an isotopy between and can be made fixed on .

###### Lemma 3.4.

Let be a not null-homotopic smooth simple closed curve. Then

 Did(T2,C)=Did(T2)∩D(T2,C). (3.2)
###### Proof.

The inclusion is evident. Therefore we have to establish the inverse one.

Let , so is fixed on and is isotopic to . We have to prove that , i.e. it is isotopic to via an isotopy fixed on .

Let be a simple closed curve isotopic to and disjoint from , and be a Dehn twist along fixed on . Cut the torus along and denote the resulting cylinder by .

Notice that the restrictions are fixed on . It is well-known that the isotopy class generates the group . Hence there exists such that is isotopic to relatively to . This isotopy induces an isotopy between and fixed on .

By assumption is isotopic to , while is isotopic to only for . Hence is isotopic to via an isotopy fixed of . ∎

### 3.5. Smooth shifts along trajectories of a flow

Let be a smooth flow on a manifold . Then for every smooth function one can define the following map by the formula:

 Fα(z)=F(z,α(z)),z∈M. (3.3)
###### Lemma 3.6.

If is a diffeomorphism then for each the map

 Ftα:M→M,Ftα(z)=F(z,tα(z))

is a diffeomorphism as well. In particular, is an isotopy between and .

### 3.7. Some constructions associated with f

In the sequel we will regard the circle and the torus as the corresponding factor-groups and . Let be the unit of . We will always assume that is a base point for all homotopy groups related with and its subsets. For let

 Jε=(−ε,ε)⊂S1

be an open -neighbourhood of .

Let be a function such that its KR-graph has only one cycle, and let be a regular connected component of certain level set of not separating . For this situation we will now define several constructions ‘‘adopted’’ with .

Special coordinates. Since is non-separating and if a regular component of , one can assume (by a proper choice of coordinates on ) that the following two conditions hold:

1.  ;

2. there exists such that for all the curve is a regular connected component of some level set of .

It is convenient to regard as a meridian of . Let be the corresponding parallel. Then , see Figure 3.1. Consider also the following loops defined by

 λ(t) =(t,0), μ(t) =(0,t). (3.4)

They represent the homotopy classes of and in respectively.

Let us also mention that is a subgroup of the group . Therefore has a natural groups structure.

Let be the inclusion map. Then the corresponding homomorphism is injective. Since is also connected, i.e. , we get the following short exact sequence of homotopy groups of the pair :

 1⟶π1C\leavevmode\nobreak \leavevmode\nobreak k\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π1T2\leavevmode\nobreak \leavevmode\nobreak r\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→π1(T2,C)⟶1. (3.5)

As and , it follows that and this group is generated by the image of the homotopy class of the parallel . In particular, there exists a section

 s:π1(T2,C)⟶π1T2 (3.6)

such that , so is the identity map of .

An inclusion . Notice that is a connected Lie group. Therefore it acts on itself by smooth left translations. This yields the following embedding : if , then is a diffeomorphism given by the formula:

 ξ(a,b)(x,y)=(x+a mod 1,  y+b mod 1). (3.7)

It is well known that is a homotopy equivalence, see e.g. [3].

Notice also that yields the following map

 ξ:C((I,∂I),(T2,e))⟶C((I,∂I),(Did,idT2)) (3.8)

between the spaces of loops defined as follows: if is a continuous map, then

 ξ(ω)=ξ∘ω: I⟶Did.

It is well known that this map is continuous with respect to compact open topologies. Moreover the corresponding map between the path components is just the homomorphism of fundamental groups:

 ξ:π1T2⟶π1Did. (3.9)

Since is a homotopy equivalence, the homomorphism (3.9) is in fact an isomorphism.

To simplify notation we denoted all these maps with the same letter . However this will never lead to confusion.

Isotopies and . Let

 L =ξ(λ), M =ξ(μ) (3.10)

be the images of the loops and in under the map Eq. (3.8) Evidently, they can be regarded as isotopies defined by

 L(x,y,t) =(x+t mod 1, y), M(x,y,t) =(x, y+t mod 1), (3.11)

for , , and . Geometrically, is a ‘’rotation’’ of the torus along its parallels and is a rotation along its meridians.

Denote by and the subgroups of generated by loops and respectively. Since is a homotopy equivalence, and the loops and freely generate , it follows that and are commuting free cyclic groups, and so we get an isomorphism:

 π1Did≅L×M.

Also notice that and can be also regarded as flows defined by the same formulas Eq. (3.11) for . All orbits of the flows and are periodic of period . We will denoted these flows by the same letters as the corresponding loops (3.10), however this will never lead to confusion.

A flow . As is an orientable surface, there exists a flow having the following properties, see e.g. [9, Lemma 5.1]:

• a point is fixed for if and only if is a critical point of ;

• is constant along orbits of , that is for all and .

It follows that every critical point of and every regular components of every level set of is an orbit of .

In particular, each curve for is an orbit of . On the other hand, this curve is also an orbit of the flow . Therefore, we can always choose so that

 M(x,y,t)=F(x,y,t),(x,y,t)∈Jε×S1×R. (3.12)
###### Lemma 3.8.

[9, Lemma 5.1]. Suppose a flow satisfies the above conditions 1) and 2) and let . Then if and only if there exists a function such that , see Eq. (3.3). Such a function is unique and the family of maps constitute an isotopy between and . ∎

## 4. Proof of Theorem 2.6

Let be such that its KR-graph has only one cycle, and let be a non-separating regular connected component of certain level set of . Assume also that trivially acts of . We have to prove that there exists a homotopy equivalence .

By (3) and (4) of Theorem 1.3 the orbits and are aspherical, as well as , i.e. their homotopy groups vanish for . Therefore, by Whitehead Theorem [4, § 4.1, Theorem 4.5], it suffices to show that there exists an isomorphism . Such an isomorphism will induce a required homotopy equivalence.

Moreover, due to (2) of Theorem 1.3 we have isomorphisms:

 π1(DidC,SC) ≅ π1OC, π1(Did,S) ≅ π1O.

Therefore it remains to find the following isomorphism:

 π1(DidC,SC)×π1S1 ≅ π1(Did,S). (4.1)

Notice that every smooth function always have critical points being not local extremes, since otherwise would be diffeomorphic with a -sphere . Therefore by (3) and (4) of Theorem 2.6 the spaces , , and are contractible. Moreover, as noted above, is homotopy equivalent to .

Let be the inclusion map. It yields a morphism between the exact sequences of homotopy groups of these pairs. The non-trivial part of this morphism is contained in the following commutative diagram:

 (4.2)

The proof of Theorem 2.6 is based on the following two Propositions 4.1 and 4.3 below.

###### Proposition 4.1.

Under assumptions of Theorem 2.6 there exists an epimorphism

 φ:π1(Did,S)⟶L

such that

1. is a left inverse for , that is

2. .

###### Corollary 4.2.
1. The map defined by for , is a groups isomorphism.

2. The following sequence is exact:  .

###### Proof.

Statement a) follows from 1) of Proposition 4.1 and the fact that is contained in the center of , see (3) of Theorem 2.6. Statement b) is a direct consequence of statements 1) and 2) of Proposition 4.1 and Lemma 3.2 applied to the lower exact sequence of Eq. (4.2). We leave the details for the reader. ∎

Due to 2) and 3) of Proposition 4.1 and 3) of Corollary 4.2 we see that diagram Eq. (4.2) reduces to the following one:

 1−−−−→1−−−−→π1(DidC,SC)∂C−−−−→≅π0SC−−−−→1⏐⏐↓i1⏐⏐↓i0⏐⏐↓1−−−−→Mq−−−−→kerφ∂−−−−→π0S−−−−→1 (4.3)

Thus to complete Theorem 2.6 it suffices to prove that the middle vertical arrow, , in Eq. (4.3) is an isomorphism between and . As we will then get the required isomorphism

 π1(DidC,SC)×Z ≅ kerφ×L \leavevmode\nobreak \leavevmode\nobreak θ\leavevmode\nobreak \leavevmode\nobreak −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ π1(Did,S).

To show that is an isomorphism notice that is also an isomorphism by (4) of Theorem 1.3. Therefore from the latter diagram Eq. (4.3) we get the following one:

 1−−−−→keri0−−−−→π0SCi0−−−−→π0S⏐⏐↓i1∘∂−1C⏐⏐↓∥∥1−−−−→q(M)−−−−→kerφ∂−−−−→π0S@>>>1
###### Proposition 4.3.

Homomorphism is surjective, and the induced map

 i1∘∂−1C:keri0⟶q(M)

is an isomorphism.

In other words, Proposition 4.3 claims that we have the following morphism between short exact sequences:

 1−−−−→keri0−−−−→π0SCi0−−−−→π0S−−−−→1≅⏐⏐↓i1∘∂−1C⏐⏐↓∥∥1−−−−→q(M)−−−−→kerφ∂−−−−→π0S@>>>1

Since left and right vertical arrows are isomorphisms, it will follow from five lemma, [4, § 2.1], that is an isomorphism as well. This completes Theorem 2.6 modulo Propositions 4.1 and 4.3. The next two sections are devoted to the proof of those propositions.

###### Remark 4.4.

It easily follows from statement a) of Corollary 4.2 that the map defined by: