Symplectomorphisms of surfaces preserving a smooth function, I

# Symplectomorphisms of surfaces preserving a smooth function, I

## Abstract.

Let be a compact orientable surface equipped with a volume form , be either or , be a Morse map, and be the Hamiltonian vector field of with respect to . Let also be set of all functions taking constant values along orbits of , and be the identity path component of the group of diffeomorphisms of mutually preserving and .

We construct a canonical map being a homeomorphism whenever has at least one saddle point, and an infinite cyclic covering otherwise. In particular, we obtain that is either contractible or homotopy equivalent to the circle.

Similar results hold in fact for a larger class of maps whose singularities are equivalent to homogeneous polynomials without multiple factors.

###### Key words and phrases:
Morse function, symplectomorphism, surface
###### 2000 Mathematics Subject Classification:
37J05, 57S05, 58B05,
The author is indebted to Bogdan Feshchenko for useful discussions.

## 1. Introduction

Let be a closed oriented surface, be the group of all diffeomorphisms of , and be the identity path component of consisting of all diffeomorphisms isotopic to the identity.

Let also be the space of all volume forms on having volume and . Since , is a closed non-degenerate -form and so it defines a symplectic structure on . Denote by the group of all -preserving diffeomorphisms, and let be its identity path component.

Then Moser’s stability theorem [20] implies that for any family

 {ωt}t∈Dn⊂Vol(M,1)

of volume forms parameterized by points of a closed -dimensional disk , there exists a family of diffeomorphisms

 {ht}t∈Dn⊂Diff0(M)

such that for all . In particular, this implies that the map

 p:Diff0(M)→Vol(M,1), p(h) =h∗ω

is a Serre fibration with fiber , see e.g. [19, §3.2], [2], or [21, §7.2].

Since is convex and therefore contractible, it follows from exact sequence of homotopy groups of the Serre fibration that yields isomorphisms of the corresponding homotopy groups , . Hence the inclusion

 (1.1) Symp0(M,ω) ⊂ Diff0(M)

turns out to be a weak homotopy equivalence. See also [18] for discussions of the inclusion (1.1) for non-compact manifolds.

Moreover, let be the group of orientation preserving diffeomorphisms. Then we have an inclusion . Indeed, if preserves , then it fixes the corresponding cohomology class , and so yields the identity on . In particular, preserves orientation of . Hence (1.1) also implies that yields a monomorphism on the set of isotopy classes.

It is well known that is generated by isotopy classes of Dehn twists, [4], [9], and one easily shows that each Dehn twist can be realized by -preserving diffeomorphism. This implies that is also surjective, and so is a weak homotopy equivalence as well.

On the other hand, let be a Morse function,

 Stab(f)={h∈Diff(M)∣f∘h=f}

be the group of -preserving diffeomorphisms, i.e. the stabilizer of with respect to the right action of on , and be its identity path component. Let also

 O(f)={f∘h∣h∈Diff(M)}

be the corresponding orbit of ,

 Stab(f,ω)=Stab(f)∩Symp(M,ω)

be the group of diffeomorphisms mutually preserving and , and be its identity path component.

In a series of papers the author proved that is either contractible or homotopy equivalent to the circle and computed the higher homotopy groups of , [11], [15]; showed that is homotopy equivalent to a finite-dimensional CW-complex, [12]; and recently described precise algebraic structure of the fundamental group [17]. E. Kudryavtseva, [7], [8], studied the homotopy type of the space of Morse maps on compact surfaces and using similar ideas as in [11], [15] proved that has the homotopy type of a quotient of a torus by a free action of a certain finite group.

The present paper is former in a series subsequent ones devoted to extension of the above results to the case of -preserving diffeomorphisms. We will describe here the homotopy type of . In next papers will study the homotopy type of the subgroup of trivially acting on the Kronrod-Reeb graph of , see §3.2, and describe the precise algebraic structure of .

Notice that if is the Hamiltonian vector field of and is the corresponding Hamiltonian flow, then for all .

More generally, given a function , one can define the map

 Hα:M→M, Hα(x)=H(x,α(x)),

being in general just a map leaving invariant each orbit of , and so preserving . However, is not necessarily a diffeomorphism.

Let be the algebra of all smooth functions taking constant values along orbits of . Equivalently, is the centralizer of with respect to the Poisson bracket induced by , see §2.3. In Lemma 3.2.1 we also identify with a certain subset of continuous functions on the Kronrod-Reeb graph of . In particular, contains all constant functions.

We will prove in Theorem 3.0.1 that if and only if . Moreover if has at least one saddle critical point, then the correspondence is a homeomorphism with respect to topologies, and so is contractible. Otherwise, that correspondence is an infinite cyclic covering map and is homotopy equivalent to the circle. It will also follow that the inclusion

 Stab0(f,ω) ⊂ Stab0(f)

is a homotopy equivalence. This statement can be regarded as an analogue of (1.1) for -preserving diffeomorphisms.

Again it implies that the inclusion

 j:Stab(f,ω) ⊂ Stab+(f)≡Stab(f)∩Diff+(M)

yields an injection on the sets of isotopy classes. However, now is not necessarily surjective, see §3.3. The reason is that has many invariant subsets, e.g. the sets of the from , , and so if interchanges connected components of , then they must have the same -volume.

In fact, our results hold for a larger class of smooth maps from into and , see §2.4. On the other hand, we also provide in §3.1 an example of a function with isolated critical points for which the above correspondence is not surjective.

The author is indebted to Bogdan Feshchenko for useful discussions.

## 2. Preliminaries

### 2.1. Shift map

Let be a connected -dimensional manifold, be a vector field tangent to and generating a flow . For each define the following map by

 Hα(x)=H(x,α(x)),

for . Evidently, leaves invariant each orbit of and is homotopic to by the homotopy . Also notice that if is a constant function, then is a diffeomorphism belonging to the flow .

For we will denote by the Lie derivative of along .

###### 2.1.1 Lemma.

[10, Theorem 19] Let , , and . Then the tangent map is an isomorphism if and only if .

###### 2.1.2 Remark.

In fact, [10, Lemma 20], if , so is a fixed point of , then the determinant of does not depend on a particular choice of local coordinates at and equals . The general case reduces to by observation that .

To get a global variant of Lemma 2.1.1 notice that the correspondence can also be regarded as the following mapping

 φH:C∞(M,R)→C∞(M,M), φH(α)=Hα.

It will be called the shift map along orbits of , [10], [16]. Consider the following subset of :

 (2.1) ΓH={α∈C∞(M,R)∣1+H(α)>0},

and let be the group of all diffeomorphisms of which leave invariant each orbit of and isotopic to the identity via an orbit preserving isotopy.

###### 2.1.3 Lemma.

[10, Theorem 19] If is compact, then

 (2.2) φ(ΓH)⊂Diff0(H), ΓH=φ−1(Diff0(H)).

In other words, suppose . Then if and only if .

### 2.2. Hamiltonian vector field

Let be a compact orientable surface equipped with a volume form and be either or . Since , is a closed -form, and therefore it defines a symplectic structure on . Then for each map there exists a unique vector field on satisfying

 (2.3) df(z)(u)=ω(u,H(z)),

for each point and a tangent vector . This vector field is called the Hamiltonian vector field of with respect to . For the convenience of the reader we recall its construction as it is usually defined for functions only.

Let . Fix local charts and at and respectively, where is an open subset of the upper half-plane and is an open interval in . Decreasing one can assume that . Then the map is called a local representation of at .

Now if in coordinates on we have that for some non-zero function , then

 (2.4) H(x,y)=1γ(x,y)(−^f′y∂∂x+^f′x∂∂y).

A definition of that does not use local coordinates can be given as follows. Since the restriction of to each tangent space is a non-degenerate skew-symmetric form, it follows that yields a bundle isomorphism

defined by the formula for all and .

Further notice, that the tangent bundle of is trivial, so we have the unit section

 s :P→TP≡P×R, s(q) =(q,1).

Now for a map its differential and the Hamiltonian vector field are unique maps for which the following diagram is commutative:

Thus , and . It follows that

 (2.5) H(z)(f)=ω(H(z),H(z))=0,

as is skew-symmetric, and so is tangent to level curves of .

Suppose, in addition, that takes constant values at boundary components of . Then, due to (2.5), is tangent to , and therefore it yields a flow . It also follows from (2.5) that each diffeomorphism preserves , in the sense that . Moreover, the well known Liouville’s theorem claims that each diffeomorphism also preserves . In fact, that theorem is a simple consequence of Cartan’s identity:

 (2.6) LHω =d(ιHω)+ιHdω=d(df)+ιH0=0,

since by (2.3), and as .

### 2.3. Poisson multiplication

Let be another one-dimensional manifold without boundary, so is either or as well as . Then yields a Poisson multiplication

 (2.7) {⋅,⋅}:C∞(M,P)×C∞(M,Q)⟶C∞(M,R)

defined by one of the following equivalent formulas:

 (2.8) {f,g}:=ω(Hf,Hg)=ψ(Hf)(Hg)=Hf(g)=−Hg(f),

where and are Hamiltonian vector fields of and respectively.

In particular, for each one can define its annulator with respect to (2.8) by

 (2.9) ZQω(f)={g∈C∞(M,Q)∣Hf(g)={f,g}=0}.

Thus consists of all maps taking constant values along orbits of the Hamiltonian vector field . It follows from (2.8) that iff .

When , this multiplication is the usual Poisson bracket, and is the centralizer of , see [19, §3].

### 2.4. Class F(M,P)

Let be the subspace of consisting of maps satisfying the following two axioms:

Axiom (B) The map takes a constant value at each connected component of and has no critical points on .

Axiom (L) For every critical point of there is a local presentation of near in which is a homogeneous polynomial without multiple factors.

In particular, since the polynomial (a non-degenerate singularity) is homogeneous and has no multiple factors, we see that contains an open and everywhere dense subset consisting of maps satisfying Axiom (B) and having non-degenerate critical points only.

Figure 2.1 describes possible singularities satisfying Axiom (L).

###### 2.4.1 Definition.

We will say that a vector field on is Hamiltonian like for if

1. , and, in particular, is tangent to and generates a flow on ;

2. if and only if is a critical point of ;

3. for each critical point of there exists a local representation of as a homogeneous polynomial without multiple factors such that in these coordinates .

One can easily prove that for each there exists a Hamiltonian like vector field, [11, Lemma 5.1].

Notice also that every Hamiltonian vector field of has properties 1 and 2 of Definition 2.4.1. Moreover, if is also a Hamiltonian like, then due to (2.4) in the corresponding coordinates satisfying property 3 of Definition 2.4.1 we have that .

###### 2.4.2 Lemma.

Let be any Hamiltonian like vector field for , and be the Hamiltonian vector field for with respect to . Then there exists an everywhere non-zero function such that .

###### Proof.

Denote by the set of critical point of , being also the set of zeros of as well as of . Since and are parallel and non-zero on , it follows that there exists a non-zero function such that . It remains to show that can be defined by non-zero values on to give a function on all of .

Let be a critical point of . Then by definition of Hamiltonian like vector field there exists a local representation of such that , is a homogeneous polynomial without multiple factors, and .

Then for some non-zero function , and by formula (2.4), we have on . Hence , and so smoothly extends to all of by . ∎

The following statement is a particular case of results of [13] on parameter rigidity.

###### 2.4.3 Corollary.

c.f. [13, §4 & Theorem 11.1] For any two Hamiltonian like vector fields and there exists an everywhere non-zero function such that .

###### Proof.

It follows from Lemma 2.4.2 that for some everywhere non-zero functions . Hence . ∎

### 2.5. Topological type of Stab0(f)

Let , be a Hamiltonian like vector field for , and be the corresponding Hamiltonian flow.

###### 2.5.1 Theorem.

[14], [15], [16]. Let be the shift map along orbits of and

 Γ={α∈C∞(M,R)∣1+H(α)>0},

see (2.1).

1.  and  .

2. Suppose all critical points of are non-degenerate local extremes, so, in particular, . Then the restriction map is an infinite cyclic covering, and so is homotopy equivalent to the circle. More precisely, in this case there exists such that

1. on all of ;

2. each non-constant orbit of is periodic, and takes a constant value on being an positive integral multiple of the period of ;

3. there exists a free action of on defined by , for and , such that the map is a composite

 (2.10) φ:Γ      p      −−−−−−−−−→Γ/Z       r       −−−−−−−−−−→≅Stab0(f),

where is a projection onto the factor space endowed with the corresponding final topology, and is a homeomorphism.

3. Suppose has a critical point being not a non-degenerate local extreme. Then is a homeomorphism, and so is contractible.

###### Proof.

In fact, Theorem 2.5.1 is stated and proved in [15] for any Hamiltonian like vector field of . The advantage of using Hamiltonian like vector fields is that we have precise formulas for near critical points of .

Let be every where non-zero function and . We will deduce from results of [14] that Theorem 2.5.1 also holds for . Due to Lemma 2.4.2 this includes the case when is Hamiltonian.

Let be the flows of and respectively,

 φF,φH:C∞(M,R)→C∞(M,M)

be the corresponding shift maps, , be their images in , and , be corresponding the subsets of defined by (2.1). Define the following function

 σ:M×R→R, σ(x,s)= ∫s0λ(H(x,t))dt.

Then it is well known and easy to see, e.g. [14], that for each we have that

 (2.11) H(x,α(x))=F(x,σ(x,α(x))).

Consider the map

 γ:C∞(M,R)→C∞(M,R), γ(α)(x) =σ(x,α(x)).

Evidently, is continuous with respect to topologies. Moreover, (2.11) means that for all . Hence , , and we get the following commutative diagram:

Since everywhere, one can interchange and . Hence by the same arguments as above we get that and is a homeomorphism. Also notice that the orbit structures of and coincide. Hence , and so

 ΓH \lx@stackrel(???)===φ−1H(image(φH)∩Diff0(H))=φ−1H(image(φF)∩Diff0(F)) =γ−1∘φ−1F(image(φF)∩Diff0(F))\lx@stackrel(???)===γ−1(ΓF).

Thus yields a homeomorphism of onto . Since Theorem 2.5.1 holds for , we get the following commutative diagram

implying that has the same topological properties 1-3 as , and so Theorem 2.5.1 holds for as well. ∎

###### 2.5.2 Remark.

Let us discus the case 2 of Theorem 2.5.1 which is realized precisely for the following four types of Morse maps, see [11, Theorem 1.9]:

1. is a -sphere and has exactly two critical points: non-degenerate local minimum and maximum;

2. is a -disk and has exactly one critical point being a non-degenerate local extreme;

3. is a cylinder and has no critical points;

4. is a -torus, is a circle, and has no critical points.

Due to 2a and 2b each regular point of is periodic of some period , and there exists depending on such that . Hence

 Hθ(x)=H(x,θ(x))=H(x,kxPer(γ))=x,

and so . Moreover, if , then

 Hα+nθ(x) =H(x,α(x)+nθ(x))=H(H(x,nθ(x)),α(x))= =H(x,α(x))=Hα(x).

This implies correctness of the -action from 2c of Theorem 2.5.1 and existence of decomposition (2.10) with continuous and . The principal difficulty was to prove that is a homeomorphism.

The aim of the present paper is to deduce from Theorem 2.5.1 a description of the homotopy type of , see Theorem 3.0.1 below.

## 3. Main result

Let be a compact orientable surface equipped with a volume form , , be the Hamiltonian vector field of with respect to , be the corresponding Hamiltonian flow, and

 φ:C∞(M,R)→C∞(M,M), φ(α)(x)=H(x,α(x))

be the shift map along orbits of . Let also

 Z=ZRω(f)={α∈C∞(M,R)∣H(α)=0}

be the space of functions taking constant values along orbits of , see (2.9). Then is a linear subspace of and is contained in , see (2.1). In particular, is contractible as well as .

###### 3.0.1 Theorem.

The following statements hold true.

1. and .

2. If all critical points of are non-degenerate local extremes, then the restriction is an infinite cyclic covering, and is homotopy equivalent to the circle.

3. Otherwise, is a homeomorphism, and so is contractible.

4. The inclusion is a homotopy equivalence.

5. The inclusion map induces an injection .

###### Proof.

First we need the following lemma.

###### 3.0.2 Lemma.

Let . Then the action of on is given by

 (3.1) H∗αω =(1+H(α))⋅ω.
###### Proof.

Since the set of critical points is finite and so nowhere dense, it suffices to check this relation at regular points of only.

So let be a regular point of . Then , whence there are local coordinates at in which , , and for sufficiently small . In particular, . We also have that for some function .

Notice that one may also assume that . Indeed, let . Then . Since preserves , see (2.6), it follows that

 H∗αω=H∗α−b∘H∗bω=H∗α−bω.

Thus suppose , whence . Then

 H∗αω(x,y) =γ∘Hα(x,y)d(x+α)∧dy =γ(x+α,y)(1+α′x)dx∧dy =γ(x+α,y)(1+H(α))dx∧dy.

In particular, at we have that

 H∗αω(p)=(1+H(α)(p))⋅ω(p),

which proves (3.1). ∎

Now we can complete Theorem 3.0.1.

1 Let us check that

 (3.2) φ(Z)⊂Stab0(f,ω).

Let . As leaves invariant each orbit of , and therefore it preserves , we have that .

Moreover, by formula (3.1), , so .

Now notice that for all , and so as well. Thus the homotopy , , is in fact an isotopy in between and . Hence .

Further we claim that

 (3.3) Z ⊃ φ−1(Stab0(f)∩Stab(f,ω)).

Indeed, let . Then by 1 of Theorem 2.5.1 for some . As is everywhere non-zero on , it follows from formula (3.1) that on all of , that is .

Hence

 φ(Z) \lx@stackrel(???)⊂ Stab0(f,ω)  ⊂  Stab0(f)∩Stab(f,ω)  \lx@stackrel(???)⊂ φ(Z),
 Z \lx@stackrel(???)⊂ φ−1(Stab0(f,ω)) ⊂ φ−1(Stab0(f)∩Stab(f,ω)) \lx@stackrel(???)⊂ Z.

This proves 1.

2, 4 Suppose is an infinite cyclic covering map, and let be the function from 2 of Theorem 2.5.1.

Then due to property 2b in Theorem 2.5.1 takes constant values along orbits of , and therefore . Since, in addition, is a group, it follows that is invariant with respect to the -action on , i.e. for all . Therefore . Hence is a -covering as well as . As is contractible, we obtain that the quotient is homotopy equivalent to the circle.

Consider the following path , . Then is a loop in , since

 φ∘τ(1)(x)=F(x,θ(x))=x=F(x,0)=φ∘τ(0)(x).

This loop is a generator of as well as a generator of . Hence the inclusion yields an isomorphism of fundamental groups. Since these spaces homotopy equivalent to the circle, we obtain that is a homotopy equivalence.

3, 4 If is a homeomorphism, then due to 1 it yields a homeomorphism of onto . In particular, both and are contractible, and so the inclusion is a homotopy equivalence.

5 Injectivity of follows from the relation . Theorem 3.0.1 is completed. ∎

###### 3.0.3 Remark.

Though the inclusion is a homotopy equivalence, it seems to be a highly non-trivial task to find precise formulas for a strong deformation retraction of onto . For the case 3 of Theorem 3.0.1 this is equivalent to a construction of a strong deformation retraction of onto . In fact, it suffices to find a retraction , so to associate to each a function taking constant values along orbits of so that each remains unchanged. Then a strong deformation , , of onto can be given by .

### 3.1. Counterexample for maps g∉F(M,P)

Let be the unit disk in the complex plane and be the standard symplectic form. Consider the following two functions defined by

 f(x,y) =x2+y2=|z|2, g(x,y) =(x2+y2)2=|z|4.

Then the foliations by level sets of and coincide, whence

 ZRω(f) =ZRω(g), Stab0(f,ω) =Stab0(g,ω),