Orbits of smooth functions on torus and their homotopy types
Abstract.
Let be a Morse function on torus such that its KronrodReeb 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 typeKey words and phrases:
Diffeomorphism, Morse function, homotopy type2000 Mathematics Subject Classification:
57S05, 57R45, 37C052000 Mathematics Subject Classification:
57S05, 57R45, 37C051. 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
(1.1) 
For let
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
and in the following part of exact sequence of homotopy groups of the pair
(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 nondegenerate 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 nondegenerate 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 nondegenerate , 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 factorspace has a structure of a finite onedimensional complex and is called KronrodReeb graph or simply KRgraph 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
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
(2) The following map
is a Serre fibration with fiber , i.e. it has homotopy lifting property for CWcomplexes. This implies that

;

the restriction map
(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 nonoriented surface. Then is contractible, for , , and for we have the following short exact sequence of fibration :
(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
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
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 KRgraph 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:
(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 levelset 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:
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 nonseparating regular component of some levelset , 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
In particular, we have the following isomorphisms:
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
(3.1) 
and a homomorphism such that

is the identity map and

.
Then we have the following exact sequence:
Proof.
It suffices to prove that

,

.
1) Let . We have for find such that . Since , there exists such that . Put and . Then
Hence and
2) Let . We should find such that .
As , so there exist such that . But , whence
Hence , and so . ∎
3.3. Isotopies of fixed on a curve
We will need the following general lemma claiming that if a diffeomorphism of is fixed on a nonseparating simple closed curve and is isotopic to , then an isotopy between and can be made fixed on .
Lemma 3.4.
Let be a not nullhomotopic smooth simple closed curve. Then
(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 wellknown 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:
(3.3) 
Lemma 3.6.
If is a diffeomorphism then for each the map
is a diffeomorphism as well. In particular, is an isotopy between and .
3.7. Some constructions associated with
In the sequel we will regard the circle and the torus as the corresponding factorgroups 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
be an open neighbourhood of .
Let be a function such that its KRgraph 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 nonseparating and if a regular component of , one can assume (by a proper choice of coordinates on ) that the following two conditions hold:

;

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
(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 :
(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
(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:
(3.7) 
It is well known that is a homotopy equivalence, see e.g. [3].
Notice also that yields the following map
(3.8) 
between the spaces of loops defined as follows: if is a continuous map, then
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:
(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
(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
(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:
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
(3.12) 
4. Proof of Theorem 2.6
Let be such that its KRgraph has only one cycle, and let be a nonseparating 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:
Therefore it remains to find the following isomorphism:
(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 nontrivial 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
such that

is a left inverse for , that is


.
Corollary 4.2.

The map defined by for , is a groups isomorphism.

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:
(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
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:
Proposition 4.3.
Homomorphism is surjective, and the induced map
is an isomorphism.
In other words, Proposition 4.3 claims that we have the following morphism between short exact sequences:
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: