Symplectomorphisms of surfaces preserving a smooth function, I

# Symplectomorphisms of surfaces preserving a smooth function, I

Sergiy Maksymenko Institute of Mathematics of NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01004 Ukraine
###### 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 [Moser:TrAMS:1965] 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. [McDuffSalamon:SympTop:1995, §3.2], [Banyaga:CMH:1978], or [Polterovich:SympDiff:2001, §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 [McDuff:ProcAMS:1985] 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, [Dehn:AM:1938], [Lickorish:PCPS:1964], 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 , [Maksymenko:AGAG:2006], [Maksymenko:ProcIM:ENG:2010]; showed that is homotopy equivalent to a finite-dimensional CW-complex, [Maksymenko:TrMath:2008]; and recently described precise algebraic structure of the fundamental group [Maksymenko:DefFuncI:2014]. E. Kudryavtseva, [Kudryavtseva:MathNotes:2012], [Kudryavtseva:MatSb:2013], studied the homotopy type of the space of Morse maps on compact surfaces and using similar ideas as in [Maksymenko:AGAG:2006], [Maksymenko:ProcIM:ENG:2010] 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 §LABEL:sect:Kronrod_Reeb_graph, 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 §LABEL:sect:Poisson_mult. In Lemma LABEL:lm:Z_as_func_on_KR 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 LABEL:th:charact_Sid_f_omega 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 §LABEL:sect:j0_non_surj. 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 §LABEL:sect:class_F. On the other hand, we also provide in §LABEL:counterexample:Zg_Sg 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.

[Maksymenko:TA:2003, Theorem 19] Let , , and . Then the tangent map is an isomorphism if and only if .

###### 2.1.2 Remark.

In fact, [Maksymenko:TA:2003, 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 , [Maksymenko:TA:2003], [Maksymenko:OsakaJM:2011]. 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.

[Maksymenko:TA:2003, 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

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