Deformations of Functions on Surfaces by Isotopic to the Identity Diffeomorphisms

DEFORMATIONS OF FUNCTIONS ON SURFACES BY ISOTOPIC TO THE IDENTITY DIFFEOMORPHISMS

SERGIY MAKSYMENKO

Abstract. Let M be a smooth compact connected surface, P be either the real line R or the circle S1, f : M → P be a smooth map, Γ(f) be the Kronrod-Reeb graph of f, and O(f) be the orbit of f with respect to the right action of the group D(M) of diffeomorphisms of M. Assume that at each of its critical point the map f is equivalent to a homogeneous polynomial R2 → R without multiple factors. In a series of papers the author proved that πnO(f) = πnM for n ≥ 3, π2O(f) = 0, and that π1O(f) contains a free abelian subgroup of finite index, however the information about π1O(f) remains incomplete. The present paper is devoted to study of the group G(f) of automorphisms of Γ(f) induced by the isotopic to the identity diffeomorphisms of M. It is shown that if M is orientable and distinct from sphere and torus, then G(f) can be obtained from some finite cyclic groups by operations of finite direct products and wreath products from the top. As an application we obtain that π1O(f) is solvable.

1. Introduction Study of groups of automorphisms of discrete structures has a long history. One of the first general results was obtained by A. Cayley (1854) and claims that every finite group G of order n is a subgroup of the permutation group of a set consisting of n elements, see also E. Nummela [35] for extension this fact to topological groups and discussions. C. Jordan [14] (1869) described the structure of groups of automorphisms of finite trees and R. Frucht [11] (1939) shown that every finite group can also be realized as a group of symmetries of certain finite graph. Given a closed compact surface M endowed with a cellular decomposition Ξ, (e.g. with a triangulation) one can consider the group of “combinatorial” automorphisms of M. More precisely, say that a homeomorphism h : M → M is cellular or Ξ-homeomorphism, if it maps i-cells to i-cells, and h is Ξ-trivial if it preserves every cell and its orientation. Then the group of combinatorial automorphisms of Ξ is the group of Ξ-homeomorphisms modulo Ξ-trivial ones. Denote this group by Aut(Ξ). It was proved by R. Cori and A. Machi [7] and J. Siráˇnandˇ M. Skovieraˇ [39] that every finite group is isomorphic with Aut(Ξ) for some cellular decomposition of some surface which can be taken equally either orientable or non-orientable. Notice that the 1-skeleton M (1) of Ξ can be regarded as a graph. Suppose each vertex M (1) has even degree. Then in many cases (e.g. when M is orientable) one can construct a smooth function f : M → R such that M (1) is a critical level contaning all saddles

2010 Mathematics Subject Classiﬁcation. 57S05, 20E22, 20F16 . Key words and phrases. wreath product, solvable groups, surface diﬀeomorphisms. 1 2 SERGIY MAKSYMENKO

(i.e. critical points being not local extremes), and the group Aut(Ξ) can be regarded as the group of “combinatorial symmetries” of f. Such a point of view was motivated by work of A. Fomenko on classification of Hamil- tonian systems, see [9, 10]. The group Aut(Ξ) is called the group of symmetries of an “atom” of f. The group of symmetries of Morse functions f was studied by A. Fomenko and A. Bolsinov [3], A. Oshemkov and Yu. Brailov [4], Yu. Brailov and E. Kudryavt- seva [5], A. A. Kadubovsky and A. V. Klimchuk [15], and A. Fomenko, E. Kudryavtseva and I. Nikonov [22]. In [26] the author gave suffucient conditions for a Ξ-homeomorphism to be Ξ-trivial, and in [29] estimated the number of invariant cells of a Ξ-homeomorphism, see Lemma 8.1.1. It was proved by A. Fomenko and E. Kudryavtseva [20, 21] that every finite group is the group of combinatorial symmetries of some Morse function f having critical level containing all saddles. In general, if f : M → R is an arbitrary smooth function having many critical values, then a certain part of its “combinatorial symmetries” is reflected by a so-called Kronrod- Reeb graph Γ(f), see e.g. [18, 2, 23, 19, 38]. The main result of the present paper describes the structure of the group of automorphisms of the Kronrod-Reeb graph Γ(f) induced by isotopic to the identity diffeomorphisms of M. It is based on the observation that the group of combinatorial automorphisms leaving invariant 2-cell is either cyclic or dihedral. The paper continues the series of papers [26, 27, 28, 29, 31] by the author on compu- tation of homotopy types of stabilizers, S(f), and orbits, O(f), of smooth functions on compact surfaces with respect to the right action of the group of diffeomorphisms of M. As an application we obtain that the fundamental group π1O(f) is solvable. 1.1. Preliminaries. Let M be a smooth compact connected surface and P be either the real line R or the circle S1. Consider a natural right action γ : C∞(M,P ) × D(M) −→ C∞(M,P ) of the group of diffeomorphisms D(M) on M on the space C∞(M,P ) defined by γ(f, h) = f ◦ h, (1.1) for f ∈ C∞(M,P ) and h ∈ D(M). Let also S(f) = {h ∈ D(M) | f ◦ h = f}, O(f) = {f ◦ h | h ∈ D(M)} be respectively the stabilizer and the orbit of f. We will endow D(M) and C∞(M,P ) with C∞ Whitney topologies. These topologies induce certain topologies on the spaces S(f) and O(f). Denote by Did(M) and Sid(f) the identity path components of D(M) and S(f) respectively. Let f ∈ C∞(M,P ), z ∈ M, and g :(R2, 0) → (R, 0) be a germ of smooth function. Then f is said to be equivalent at z to g is there exist germs of diffeomorphisms h : (R2, 0) → (M, z) and φ :(R, 0) → (P, f(z)) such that φ ◦ g = f ◦ h. Since both R1 and S1 are orientable, one can say about local minimums and local maximums of f : M → P with respect to that orientation. Also for any a, b ∈ S1 we will denote by [a, b] the arc in between a and b whose orientation from a to b agrees with orientation of S1. DEFORMATIONS OF FUNCTIONS ON SURFACES 3

∞ ∞ Denote by C∂ (M,P ) the subset of C (M,P ) consisting of maps f satisfying the following axiom: Axiom (B). The map f : M → P takes a constant value at each connected component of ∂M, and the set Σf of critical points of f is contained in the interior IntM. ∞ Let Morse(M,P ) ⊂ C∂ (M,P ) be a subset consisting of Morse maps, i.e. maps having only non-degenerate critical points. It is well known that Morse(M,P ) is open and ∞ everywhere dense in C∂ (M,P ). ∞ Let also F(M,P ) be the subset of C∂ (M,P ) consisting of maps f satisfying additional axiom: Axiom (L). For every critical point z of f the germ of f at z is equivalent to some 2 homogeneous polynomial fz : R → R without multiple factors. By Morse lemma a non-degenerate critical point of a function f : M → P is equivalent to a homogeneous polynomial ±x2 ± y2 which, evidently, has no multiple factors, and so satisﬁes (L). Hence Morse(M,P ) ⊂ F(M,P ).

In [26, 27, 28, 29, 31] the author computed the homotopy types of S(f) and O(f) for all f ∈ F(M,P ). It was proven that if f ∈ F(M,P ), then, except for few cases, Sid(f) is contractible, πnO(f) = πnM for n ≥ 3 and π2O(f) = 0. It was also established that the fundamental group π1O(f) contains a normal subgroup k isomorphic to π1Did(M) ⊕ Z for some k ≥ 0, and that the factor group

k G(f) = π1O(f)/(π1Did(M) ⊕ Z ) (1.2) is finite. In a typical situation, when f is generic, i.e. a Morse map taking distinct values at distinct critical points, the group G(f) is trivial and k can be easily expressed via the number of critical points of f, [26, Theorem 1.5]. In particular, we get an isomorphism ∼ k π1O(f) = π1D(M) ⊕ Z . However, if f admits certain “non-trivial symmetries”, the structure of π1O(f) is not well-understood, though some partial results were obtained. In the present paper we characterize the class of all finite groups that arise as the group G(f) in (1.2) for the case when M is orientable and differs from sphere and torus, see Theorem 1.5. For its proof we first show that a finite group G(f) has the form (1.2) for some f ∈ F(M,P ) if and only if G(f) admits a special action on a finite tree. Further, in Theorem 6.6, we prove that groups having that actions are exactly the ones obtained from finite cyclic and dihedral groups by operations of direct product and wreath product from the top. The latter statement can be regarded as an extension of an old result of C. Jordan mentioned above about characterization of groups of automorphisms of finite trees. As a consequence we also obtain that π1O(f) is always solvable, see Theorem 1.6. 4 SERGIY MAKSYMENKO

1.2. Main results. In §2 we will recall from [30] the deﬁnitions of an enhanced Kronrod- Reeb graph Γ(f) of f ∈ F(M,P ) and how S(f) acts on Γ(f). If f is Morse, then Γ(f) coincides with the usual Kronrod-Reeb graph of f. The enhanced graph includes additional edges describing structure of degenerate local extremes of f. The action of S(f) on gives a homomorphism λ : S(f) −→ Aut(Γ(f)). Put G(f) := λ S(f) ∩ Did(M) . (1.3) So G(f) is a group of all automorphisms of the enhanced KR-graph Γ(f) of f induced by isotopic to idM diﬀeomorphisms belonging to S(f). This group in fact coincides with (1.2). Our main result describes the structure of G(f).

Definition 1.3 (Class R(G)). Let G = {Gi | i ∈ Λ} be a family of finite groups containing the unit group {1}. Define inductively the following class of groups R(G): (i) if A, B ∈ R(G), then their direct product A × B belongs to R(G); (ii) if A ∈ R(G) and G ∈ G, then the wreath product A o G belong to R(G), see §4. Since 1 o G = G, it follows from (ii) that G ⊂ R(G). Thus R(G) is the minimal class of groups containing G and closed with respect to finite direct products and wreath products “from the top” by groups belonging to G. Remark 1.4. Emphasize, that if A, B ∈ R(G) and B 6∈ G, then AoB does not necessarily belong to R(G), see Example 4.2 below. Theorem 1.5. Let M be a connected compact orientable surface distinct from the 2- sphere S2 and 2-torus T 2, and P be either R or S1. Then the following classes of groups coincide:

{G(f) | f ∈ Morse(M,P )} = {G(f) | f ∈ F(M,P )} = R({Zn}n≥1). In other words, if M is the same as in Theorem 1.5, then for every f ∈ F(M,P ) the group of automorphisms of Γ(f) induced by isotopic to the identity diffeomorphisms h ∈ S(f) is obtained from finite cyclic groups by the operations of direct product and wreath product from the top. Conversely, every finite group having the above structure can be realized as the group G(f) for some f ∈ F(M,P ) or even f ∈ Morse(M,P ). As a consequence of this theorem we will obtain the following statement about the algebraic structure of π1O(f).

Recall that a group G is solvable if there are subgroups {1} = G0 ≤ · · · ≤ Gk = G such that Gi−1 is normal in Gi, and Gi/Gi−1 is an abelian group, for i = 1, 2, . . . , k. It is well known and is easy to see that (1) every abelian group is solvable; (2) if A, B are two solvable groups, then so is A × B; (3) if A is a ﬁnite solvable, and B is solvable, then A o B is also solvable; (4) if A is a normal subgroup of a group B, and two of three groups A, B, A/B are solvable, then so is the third one. DEFORMATIONS OF FUNCTIONS ON SURFACES 5

In particular, it follows from (1)-(3) that each group G ∈ R({Zn}n≥1) is solvable. Theorem 1.6. Suppose M is a connected compact orientable surface distinct from S2 2 and T . Then for every f ∈ F(M,P ) the fundamental group π1O(f) is solvable. 1.7. Structure of the paper. §2 contains the definition of the enhanced Kronrod-Reeb graph of f ∈ F(M,P ) and some ohter groups related with a map f ∈ F(M,P ). In §3 we will show how to deduce Theorem 1.6 from Theorem 1.5. §4 recalls the notion of wreath products of groups. It is well known that if a group G acts on a set A and a group H acts on a set B, then their wreath product G oB H acts in a special way on A × B and preserves partition of the form {A × b}b∈B. In §5 this observation is extended to group actions preserving arbitrary partitions, see Lemma 5.3. On the other hand, it is also well known that wreath products arise as groups of automorphisms of certain trees. Let G = {Gi | i ∈ Λ} be a family of finite groups containing the unit group {1}. In §6 we characterize the class R(G) in terms of special actions on finite trees, see Theorem 6.6. In fact, the family G will be extended to a certain class T (G) of finite groups acting on finite trees (see Definition 6.5.1) and then it will be shown that classes R(G) and T (G) coincide. In §7 we study the structure of level sets of f and then in §8 prove Theorem 1.5 for the case when M is either a 2-disk or a cylinder. Further in §9 we deduce Theorem 1.5 for all other surfaces from the case when M is either a 2-disk or a cylinder. The proof is based on the results of [29].

2. Enhanced KR-graph Let f ∈ F(M,P ). In this section we recall the notion of KR-graph of f, “enhance” it by adding additional edges, and deﬁne the action of S(f) on the obtained “enhanced” graph. 2.1. Notations. Let h : W → W be a homeomorphism of an orientable manifold W . Then we write h(W ) = +W if h preserves orientation and h(W ) = −W otherwise. Similarly, let V be a graph, ω be a path in V (in particular, a cycle or even an edge), and h : V → V be an automorphism. Then we write that h(ω) = ±ω if h leaves ω invariant and preserves or, respectively, reverses its orientation.

2.2. Partition Θf and KR-graph. Let Θf the partition of M whose elements are connected components of level-sets f −1(c) of f, where c runs over all P . An element ω ∈ Θf is called critical if it contains a critical point of f, otherwise, ω is regular, see Figure 2.1.

Figure 2.1. Partition Θf and Kronrod-Reeb graph M/Θf . 6 SERGIY MAKSYMENKO

Let Σ(Θf ) be the union of all critical elements of Θf and all connected components of ∂M. Then the connected components of M \ Σ(Θf ) will be called regular components of Θf . Notice that every local extreme z of f is an element of Θf . It is well known that the factor space M/Θf has a natural structure of a ﬁnite graph, called Kronrod-Reeb graph or simply KR-graph of f: the vertices of M/Θf are critical elements of Θf , and the open edges of M/Θf are regular components of Θf . This graph plays an important role in understanding the structure of f, especially in the case when f is Morse, see e.g. [18, 2, 23, 19, 38]. If fact, if f is simple, i.e. each critical level-set contains only one critical point, then this graph completely determines f up to a topological equivalence. For our purposes of study maps from class F(M,P ) we will add to M/Θf some edges describing structure of degenerate local extremes of f. That new graph will be called the enhanced KR-graph of f. Let us mention that f induces a unique continuous map f : M/Θf → P such that

p f f = f ◦ p : M −−→ M/Θf −→ P.

2.3. Homogeneous polynomials. It is well-known that each homogeneous polynomial g : R2 → R splits into a product of linear and irreducible over R quadratic factors, so

m1 mα n1 nβ g = L1 ··· Lα · Q1 ··· Qβ , (2.1) 2 2 where Li(x, y) = aix + biy is a linear function, Qj(x, y) = cjx + djxy + eiy is an 0 irreducible over R quadratic form, Li/Li0 6= const for i 6= i and Qj/Qj0 6= const for j 6= j0. Let S+(g) denote the group of germs at 0 of orientation preserving C∞ diﬀeomorphisms h :(R2, 0) → R2 such that g ◦ h = g. Then h(0) = 0 for each h ∈ S+(g), and so we have + a homomorphism J : S+(g) → GL (2, R) associating to h ∈ S+(g) its Jacobi matrix J(h) at 0.

Lemma 2.4. Let g : R2 → R be a homogeneous polynomial without multiple factors of degree deg(g) ≥ 2. Then the origin 0 ∈ R2 is a unique critical point of g. This point is non-degenerate if and only if deg(g) = 2. (1) Suppose the origin 0 is a non-degenerate local extreme of g, so g is an irre- + ducible over R quadratic form. Then J(S+(g)) is conjugated in GL (2, R) to SO(2). (2) In all other cases, i.e. when 0 is either not an extreme or a degenerate extreme, J(S+(g)) is conjugated to a subgroup of SO(2) generated by the rotation by the + ∼ angle 2π/n, for some even number n = 2k ≥ 2. In particular, J(S (g)) = Z2k. The number n = 2k will be called the symmetry index of 0. Let f ∈ F(M,P ) and z be a critical point of f, so due to Axiom (L) in some local 2 coordinates at z the map f is equivalent to a homogeneous polynomial fz : R → R without multiple factors. Then by Lemma 2.4 z is isolated. Moreover, if z ∈ Σf is not a non-degenerate local extreme of f, then by (2) of Lemma 2.4 we have a well-deﬁned symmetry index of z which will be denoted by nz. ∗ ∗ ∗ Let Σf be the set of all degenerate local extremes of f. Evidently, h(Σf ) = Σf and ∗ nh(z) = nz for all h ∈ S(f) and z ∈ Σf . DEFORMATIONS OF FUNCTIONS ON SURFACES 7

Deﬁnition 2.5. A framing for f is a family of tangent vectors

0 nz−1 ∗ F = {ξz , . . . , ξz ∈ TzM | z ∈ Σf } satisfying the following conditions.

∗ 0 nz−1 (a) For each z ∈ Σf the vectors ξz , . . . , ξz ∈ TzM are cyclically ordered, so that k 0 there are local coordinates at z in which ξz is obtained from ξz by rotation by angle 2πk . nz z ∗ (b) F is S(f)-invariant, that is Tzh(ξk) ∈ F for each h ∈ S(f), z ∈ Σf , and 0 nz−1 0 nz−1 k = 0, . . . , nz − 1. In this case Tzh maps {ξz , . . . , ξz } onto {ξh(z), . . . , ξh(z) } preserving or reversing the cyclic orders on these vectors. The following lemma is not hard to prove, see [30].

Lemma 2.6. Every f ∈ F(M,P ) admits a framing (not necessarily unique).

0 nz−1 ∗ ∗ Let F = {ξz , . . . , ξz ∈ TzM | z ∈ Σf } be a framing for f. Notice that each z ∈ Σf 0 nz−1 is also a vertex of M/Θf . Regard tangent vectors ξz , . . . , ξz as new nz vertices of ∗ M/Θf and connect them by edges with vertex z. Thus for every z ∈ Σf we add to the KR-graph M/Θf new nz edges

0 1 nz−1 z ξz , z ξz , . . . , z ξz . The obtained graph will be called the enhanced KR-graph of f and denoted by Γ(f), see Figure 2.2.

Figure 2.2. Enhanced KR-graph Γ(f).

2.7. Action of S(f) on Γ(f). Let h ∈ S(f), so f ◦ h = f. Then hf −1(c) = f −1(c) for each c ∈ P . Hence h permutes connected components of f −1(c) being elements of Θf . This implies that h induces a well-defined homeomorphism λ(h): M/Θf → M/Θf . Moreover, λ(h) extends to a homeomorphism of the enhanced KR-graph Γ(f) as fol- ∗ i i lows. Let h ∈ S(f), z ∈ Σf , and ξz ∈ F regarded as a vertex of Γ(f), so zξz is an edge ∗ i of Γ(f). Then h(z) ∈ Σf and by definition of framing we also have that Tzh(ξz) ∈ F . In i particular, Γ(f) contains the edge with vertices h(z) and Tzh(ξz), and so we define i i λ(h)(ξz) = Tzh(ξz), i i where ξz and Tzh(ξz) are now regarded as vertices of Γ(f). 8 SERGIY MAKSYMENKO

∗ Remark. This action can be understood as follows. Let z ∈ Σf . By assumption tangent vectors at z belonging to the framing F are cyclically ordered. Moreover, every h ∈ S(f) interchanges collections of such cyclically ordered vectors also preserving or reversing their cyclic order. Constructing Γ(f) we just attached these cyclically ordered collections to the corresponding vertices of M/Θf , and the action of S(f) on Γ(f) mimics the corresponding action on F . Thus S(f) acts on Γ(f), and so the correspondence λ : S(f) −→ Aut(Γ(f)), h 7→ λ(h), (2.2) is a homomorphism into the group of automorphisms of Γ(f). i Extend f : M/Θf → P to the map f : Γ(f) → P by setting f(z ξz) = f(z) for every i edge z ξz. Also notice that for each h ∈ S(f) the following diagram is commutative: p f M / M/Θf / Γ(f) / P (2.3)

h λ(h) λ(h) p f M / M/Θf / Γ(f) / P

2.8. Partition ∆f . Deﬁne another partition ∆f of M by the rule: a subset ω ⊂ M is an element of ∆f if and only if ω is either a critical point of f or a connected component of the set f −1(c) \ Σ(f), for some c ∈ P , see Figure 2.3. In other words, ∆f is obtained from Θf by dividing critical elements of Θf with critical points of f.

Figure 2.3. Diﬀerence between partitions Θf and ∆f

2.9. Automorphisms of Θf and ∆f . Let D(Θf ) be the group of all diﬀeomorphisms h ∈ D(M) such that

(i) h(ω) = ω for every ω ∈ Θf ; (ii) if ω is a regular element of Θf , then h preserves orientation of ω; ∗ (iii) for every degenerate local extreme z ∈ Σf of f the tangent map Tzh : TzM → TzM is the identity. Let also D(∆f ) = {h ∈ D(Θf ) | h(ω) = ω for each ω ∈ ∆f }.

Remark. In [30] D(∆f ) denoted the group satisfying only (i) and (ii), while its subgroup N satisfying (iii) was denoted by D (∆f ). We will use here simpliﬁed notation D(∆f ). DEFORMATIONS OF FUNCTIONS ON SURFACES 9

Evidently,

D(∆f ) ⊂ D(Θf ) ⊂ ker λ ⊂ S(f). (2.4) −1 We will now describe the diﬀerence between these four groups. Let c ∈ P , Xc = f (c) be the corresponding level set of f, and

ρ1, . . . , ρa, σ1, . . . , σb, z1, . . . , zr (2.5) be all the connected components of Xc being elements of Θf such that

∗ each ρi is a regular element (i.e. a simple closed curve); ∗ each σj is a critical element being not a degenerate local extreme of f, so σj is a connected graph each of whose vertices has even degree (possibly zero in which case σj is a non-degenerate local extreme of f); ∗ and each zk is a degenerate local extreme.

−1 Figure 2.4. Structure of level set Xc = f (c)

n −1 Moreover, for every z we also have n tangent vectors ξ0 , . . . , ξ zk ∈ T M be- k zk zk z zk longing to the framing F , see Figure 2.4. Then the following statements hold true.

• Every h ∈ S(f) preserves Xc, that is h(Xc) = Xc, and so h can interchange elements of (2.5). • Every h ∈ ker λ preserves each element in (2.5) and the corresponding vectors in the framing, i.e. h(ρ ) = ρ , h(σ ) = σ , h(z ) = z , and T h(ξl ) = ξl . i i j j k k zk zk zk

Recall that by Lemma 2.4 in some local coordinates at zk the tangent map Tzk h

is a rotation. On the other hand Tzk h has nz ≥ 2 ﬁxed vectors, which implies

that Tzk h is the identity map of Tzk M. However, h can change orientations of ρi and also can induce non-trivial automorphisms of σj, that is interchange its vertices and edges. • Every h ∈ D(Θf ) additionally preserves orientation of circles ρi. • Every h ∈ D(∆f ) additionally ﬁxes every vertex of σj and leaves invariant every edge of σj. Since each edge e of σj belongs to the closure of the union of regular elements of ∆f and h preserves orientations of that elements, it follows h also preserves orientation of e. 10 SERGIY MAKSYMENKO

Let D(M, ∂M) be the group of diffeomorphisms of D(M) fixed on ∂M. Let also Did(Θf ) (resp. Did(∆f ), (ker λ)id, Did(M, ∂M)) be the identity path component of D(Θf ) ∞ (resp. D(∆f ), ker λ, D(M, ∂M)) in the C Whitney topology. The following statement is proved in [30]. Lemma 2.10. [30]. Let f ∈ F(M,P ). Then (1) Did(∆f ) = Did(Θf ) = (ker λ)id = Sid(f) and this group is contractible. (2) π0D(∆f ) and π0 D(∆f ) ∩ D(M, ∂M) are finitely generated free abelian groups. (3) If M is orientable, then

D(∆f ) ∩ Did(M) = D(Θf ) ∩ Did(M) = ker λ ∩ Did(M). (2.6) Remark. Statement similar to (2.6) holds for non-orientable surfaces, however one should modify enhanced KR-graph of f by adding more information and thus changing the deﬁnition of homomorphism λ, see [30] for details. In that case we will even have that D(Θf ) = ker λ. However, since our main result concerns only orientable surfaces, Lemma 2.10 is formulated just for them.

3. Proof of Theorem 1.6 Let M be a connected compact orientable surface distinct from S2 and T 2 and f ∈ F(M,P ). Since M is connected, compact, and distinct from S2 and T 2, the group Did(M, ∂M) contractible, see e.g. [12]. In this case by [31, Theorem 2.6] we have an isomorphism ∼ 0 ∂ : π1O(f) = π0S (f, ∂M). 0 Therefore we need to show solvability of π0S (f, ∂M). This will be deduced from Theo- rem 1.5. Denote

0 0 S (f) = S(f) ∩ Did(M), S (f, ∂M) = S(f) ∩ Did(M, ∂M). First we establish one lemma. Lemma 3.1. G(f) ≡ λ(S0(f)) = λ(S0(f, ∂M)). Proof. Since S0(f, ∂M) ⊂ S0(f), it suffices to prove that for each h ∈ S0(f) there exists 0 h1 ∈ S (f, ∂M) such that λ(h) = λ(h1). Let C1,...,Ck be all the connected components of ∂M. Since a regular neighborhood 1 1 of Ci, i = 1, . . . , k, is diffeomorphic to S × I in such a way that the sets S × {t} correspond to level-sets of f, there exists a Dehn twist τi ∈ S(f) fixed on ∂M, see e.g. [25, §6.1] or [26, §6]. In particular, λ(τi) = idΓ(f) for each i = 1, . . . , k. 0 Now let h ∈ S (f) = S(f)∩Did(M). Since h is isotopic to idM , it follows that h(Ci) = +Ci for all i. Hence by [31, Corollary 6.4] h is isotopic in S(f) to a diffeomorphism h0 fixed on ∂M. In particular, we have that λ(h0) = λ(h). However, in general, h0 is not isotopic to idM rel. ∂M, that is h0 ∈ S(f) ∩ D(M, ∂M) 0 but not necessarily to S(f) ∩ Did(M, ∂M) = S (f, ∂M). Nevertheless, it easily follows from [8, Lemma 6.1] that one can find integers m1, . . . , mk such that

m1 mk h1 = h0 ◦ τ1 ◦ · · · ◦ τk DEFORMATIONS OF FUNCTIONS ON SURFACES 11

0 is isotopic to idM relatively ∂M. In other words, h1 ∈ S (f, ∂M). It remains to note that m1 m1 λ(h1) = λ(h0) ◦ λ(τ1 ) ◦ · · · ◦ λ(τk ) = λ(h0) = λ(h), 0 0 whence λ(S (f)) = λ(S (f, ∂M)). 0 0 Consider the natural factor map q : S (f, ∂M) → π0S (f, ∂M). Evidently, the homomorphism λ : S(f) → Aut(Γ(f)) is continuous and its image is discrete (in fact ﬁnite). 0 Hence there exists an epimorphism λb : π0S (f, ∂M) → G(f) such that λ = λb ◦ q, that is the following diagram is commutative:

λ S0(f, ∂M) / λ(S0(f, ∂M)) = λ(S0(f)) ≡ G(f) 4 q ' λb 0 π0S (f, ∂M)

By Theorem 1.5 the group G(f) ∈ R({Zn}n≥1) is solvable. We claim that ˆ ∼ ker(λ) = π0 Did(M, ∂M) ∩ D(∆f ) . By (2) of Lemma 2.10 the latter group is ﬁnitely generated free abelian. This will imply 0 that π0S (f, ∂M) and therefore π1O(f) are solvable as well. Lemma 3.1.1. There exists a natural isomorphism

0 qˆ : π0 S (f, ∂M) ∩ ker λ −→ ker(λb), qˆ([h]) := q(h), 0 0 where [h] ∈ π0 S (f, ∂M) ∩ ker λ) is an isotopy class of some h ∈ S (f, ∂M) ∩ ker λ. The proof is starighforward and we left it for the reader. Now notice that from Did(M, ∂M) ⊂ Did(M) and D(∆f ) ⊂ ker λ ⊂ S(f) it follows that 0 S (f, ∂M) ∩ ker λ = S(f) ∩ Did(M, ∂M) ∩ ker λ

= Did(M, ∂M) ∩ Did(M) ∩ ker λ (2.6) == Did(M, ∂M) ∩ Did(M) ∩ D(∆f )

= Did(M, ∂M) ∩ D(∆f ), whence ∼ 0 ker(λb) = π0 S (f, ∂M) ∩ ker λ ≡ π0(Did(M, ∂M) ∩ D(∆f )). Theorem 1.6 is completed.

4. Wreath products Let B be a set and H be a subgroup of the group of permutations Σ(B) on B, and S be yet another group. Denote by Map(B,S) the group of all maps B → S with respect to the point-wise multiplication. Then H acts on Map(B,S) by following rule:

α · h = α ◦ h : B −−→h B −−→α S, 12 SERGIY MAKSYMENKO for α ∈ Map(B,S) and h ∈ H. Using this action one can deﬁne the semi-direct product Map(B,S) o H which is denoted by S oB H and called the wreath product of S and H over B. Thus S oB H := Map(B,S) o H is the Cartesian product Map(B,S) × H with the multiplication given by the formula: (α1, h1) · (α2, h2) = (α1 ◦ h2) · α2, h1h2 . (4.1) for (α1, h1), (α2, h2) ∈ Map(B,S) × H. The group H is called the top group, and Map(B,S) is the base of S oB H. Let ε : B → S be the constant map into the unit of S. Then the pair (ε, idB) is the unit of S oB H. Moreover, let (α, h) ∈ S oB H andα ¯ ∈ Map(B,S) be the point-wise inverse from α, i.e.α ¯(b) = (α(b))−1 ∈ S for all b ∈ B. Then α¯ ◦ h−1, h−1 is the inverse to (α, h) in S oB H. Further, we have the following exact sequence i π 1 → Map(B,S) −→ S oB H −−→ H → 1, (4.2) where i(α) = (α, e) and π(α, h) = h, in which the map π admits a section:

s : H → S oB H, s(h) = (ε, h). Also notice that the group H naturally acts on itself by right shifts. Hence we can also deﬁne the wreath product S oH H, which in this particular case is denoted by S o H. So

S o H := S oH H = Map(H,S) o H, where Map(H,S) is the set of all maps H → S (not necessarily homomorphisms). It is known, that the operation of wreath product of groups is associative, that is for any groups F, G, H the groups (F o G) o H and F o (G o H) are canonically isomorphic. Therefore we will denote each of them by F o G o H. Wreath products play an outstanding role in group theory. They seem to be indepen- dently appeared in papers by C. Jordan [14], A. Loewy [24], and L. M. Kaluzhnin [16], see also G. P´olya [37, §27] and B. H. Neumann [33] for discussions. The following theorem describes two properties about the structure of wreath products. They can be used to check whether a certain group belongs to a class R(G). Theorem 4.1. [34] Let W = A o B. Then the following statements hold true. (1) W ∼= P × Q for some non-trivial groups P and Q, if and only if B is ﬁnite, of order n, say, and A has a non-trivial abelian direct factor C1 (which may coincide with A) having unique n-th roots for all of its elements, [34, Theorem 7.1]. (2) W ∼= P o Q if and only if P ∼= A and Q ∼= B, [34, Theorem 10.3].

Example 4.2. Let G = {Zn}n≥1 be the class of all ﬁnite cyclic groups. Then

W = Zq o (Zq × Zrq) 6∈ R(G), q ≥ 2, r ≥ 1. Proof. Suppose W ∈ R(G). Then either W = P × Q for some non-trivial groups P,Q ∈ R(G) or W = P o Q for some non-trivial groups P ∈ R(G) and Q ∈ G. ∼ Assume that W = Zq o (Zq × Zr) = P × Q. Then by (1) of the above Theorem 4.1 each 2 2 element of Zq must have a unique root of order |Zq × Zrq| = q r. However, q r · x = 0 DEFORMATIONS OF FUNCTIONS ON SURFACES 13

2 for all x ∈ Zq, and so each non-zero element of Zq has no roots of order q r. Hence W is not decomposable. ∼ ∼ On the other hand, if W = Zq o(Zq ×Zqr) = P oQ, then by (2) of Theorem 4.1, P = Zq ∼ and Q = Zq × Zqr. But we should also have that Q ∈ G, i.e. Zq × Zqr must be a cyclic group which is not true since q divides qr. Thus both cases of W are impossible, and so W 6∈ R(G). As noted in [37, Nr. 54] the following theorem is a consequence of an old result by C. Jordan [14] characterizing groups of automorphisms of ﬁnite trees, see also [1, Propo- sition 1.15] and [17, Lemma 5].

Theorem 4.3. (C. Jordan’ 1869). Let Σn, n ≥ 1, be the permutation group of n elements. Then a group G is isomorphic to a group of automorphisms of a ﬁnite tree if and only if G belongs to the class R({Σn}n≥1).

Recall that our main result (Theorem 1.5) characterizes class R({Zn}n≥1) generated by cyclic groups. In §6 this class will be described as a class of groups having special actions on ﬁnite trees. See also Remark 6.7.

5. Partition preserving group actions

Let A be a set, τ = {Tb}b∈B be a partition of A and q : A → B be the natural projection, so q(a) = b for some a ∈ A and b ∈ B if and only if a ∈ Tb. Let G be a group eﬀectively acting on A and preserving partition τ, that is g(T ) ∈ τ for all g ∈ G and T ∈ τ. Then this action reduces to the action of G on B: we have a unique homomorphism p : G → Σ(B) into the group of permutation of B such that

g(Tb) = Tp(g)(b) for every g ∈ G and b ∈ B. In particular, the following diagram is commutative: g A −−−→ A   q q y y p(g) B −−−→ B For each b ∈ B deﬁne the following subgroup of Σ(Tb):

Sb = { g|Tb | g ∈ G, g(Tb) = Tb }, so Sb consists of all permutations of Tb induced by elements of G preserving Tb. Let also H = p(G) 0 be the image of G in Σ(B) under p. Evidently, if g(Tb) = Tb0 for some b, b ∈ B, i.e. b 0 and b belongs to the same H-orbit, then the groups Sb and Sb0 are isomorphic. Let {Bλ}λ∈B/H be the partition of B by orbits of the action of H. For each λ ∈ B/H

ﬁx some element bλ ∈ Bλ and consider the group Map(Bλ,Sbλ ) of all maps Bλ → Sbλ with respect to a point-wise multiplication. Then we can deﬁne the standard right action of H on the product × Map(Bλ,Sbλ ) as follows. Let λ∈B/H

α = {αλ}λ∈B/H ∈ × Map(Bλ,Sbλ ) λ∈B/H 14 SERGIY MAKSYMENKO

where each αλ : Bλ → Sbλ is a map, and h ∈ H is regarded as a bijection h : Bλ → Bλ. Then the result of the action of h on α is

α ◦ h := {αλ ◦ h|Bλ }λ∈B/H , where h| Bλ α αλ ◦ h|Bλ : Bλ −−−−→ Bλ −−→ Sbλ is a usual composition map. Therefore we have a well-deﬁned semi-direct product

C := × Map(Bλ,Sbλ ) o H λ∈B/H with respect to this action, and so C is a product of sets

× Map(Bλ,Sbλ ) × H λ∈B/H with the following operation: if (α, h1), (β, h2) ∈ C, then (α, h1) · (β, h2) = (α ◦ h2) · β, h1h2 .

Also notice that × Map(Bλ,Sbλ ) can be regarded as the set of maps λ∈B/H a α : B −→ Sbλ λ∈B/H into a disjoint union of Sb , such that α(Bλ) ⊂ Sb for all λ ∈ B/H, and the action of H λ λ ` on this product is induced from the standard right action of H on Map(B, λ∈B/H Sbλ ). Lemma 5.1. 1) If H acts transitively on B, and b ∈ B, then ∼ C = Sb oB H. 2) Suppose the action of H is free, then ∼ C = × Sbλ o H. (5.1) λ∈B/H 3) Let F be the set of all orbits in B/H corresponding to ﬁxed points, and N be its complement in B/H. Then ∼ C = × Sbλ × × Map(Bλ,Sbλ ) o H . λ∈F λ∈N Proof. Statement 1) is trivial. 2) Suppose the action of H is free. Then for each λ ∈ B/H the correspondence H 3 h 7→ h(bλ) ∈ Bλ is a bijection which leads to the following identiﬁcations: ∼ ∼ × Map(Bλ,Sbλ ) = × Map(H,Sbλ ) = Map H, × Sbλ . λ∈B/H λ∈B/H λ∈B/H Therefore C is a certain semi-direct product Map H, × Sbλ o H. λ∈B/H It is easy to check that the multiplication in C coincides with the wreath product multiplication which gives an isomorphism (5.1). DEFORMATIONS OF FUNCTIONS ON SURFACES 15

3) By assumption every orbit λ ∈ F consists of a unique point. Therefore Bλ = {bλ}, ∼ Map(Bλ,Sbλ ) = Sbλ , and the right action of H on Map(Bλ,Sbλ ) is trivial. This easily implies that the following correspondence ({αλ}λ∈B/H , h) 7→ {αλ}λ∈F , ({αλ}λ∈N , h) , is an isomorphism of C onto × Sbλ × × Map(Bλ,Sbλ ) o H . We leave the λ∈F λ∈N details for the reader.

Deﬁnition 5.2. Let X ⊂ B be a subset, τ = {Tb}b∈B be a partition on A. We will say that a τ-preserving action of G on A is τ-decomposable over X if for any family {gb}b∈X of (not necessarily mutually distinct) elements of G enumerated by points of X such that gb(Tb) = Tb for all b ∈ X the following permutation g : A → A of A deﬁned by ( gb(a), a ∈ Tb, b ∈ X, g(a) = −1 a, a ∈ A \ ∪b∈X Tb = A \ q (X). belongs to G. An action which is τ-decomposable over any subset X ⊂ B will be called τ-decomposable. The following lemma characterizes τ-decomposable actions. Lemma 5.3. (1) There exists a monomorphism ν : G −→ C making commutative the following diagram:

ν G / C = × Map(Bλ,Sbλ ) o H λ∈B/H

p π ( H v where π is the natural projection. (2) The following conditions are equivalent: (a) ν is an isomorphism; (b) the action of G on A is τ-decomposable. −1 (c) the subgroup π (idB) = × Map(Bλ,Sbλ ) × idB is contained in ν(G); λ∈B/H

Proof. (1) Since H transitively acts on Bλ, for each u ∈ Bλ we can ﬁx some φu ∈ G such that φu(Tbλ ) = Tu.

Let g ∈ G. For each λ ∈ B/H deﬁne the map αλ : Bλ → Sbλ by the formula: −1 αλ(u) = (φ ◦ g ◦ φu)|T , u ∈ Bλ. (5.2) p(g)(u) bλ Notice that we have the following commutative diagram: g Tu −−−→ Tp(g)(u) x x φ  φ u  p(g)(u)

αλ(u) Tbλ −−−−→ Tbλ 16 SERGIY MAKSYMENKO

Now deﬁne ν : G → C by the formula: ν(g) = {αλ}λ∈B/H , p(g) .

First we prove that ν is a homomorphism. Let g1, g2 ∈ G, and

ν(g1) = ({αλ}λ∈B/H , p(g1)), ν(g2) = ({βλ}λ∈B/H , p(g2)),

ν(g1g2) = ({γλ}λ∈B/H , p(g1g2)). Due to (4.1) we need to show that

γλ(u) = αλ(h2(u)) · βλ(u), h = h1 ◦ h2, for all u ∈ Bλ. Since p is a homomorphism, we get

h = p(g1g2) = p(g1) ◦ p(g2) = h1 ◦ h2. Moreover, −1 −1 αλ(h2(u)) = (φ g1 φh (u))|T = (φ g1 φh (u))|T , h1◦h2(u) 2 bλ h(u) 2 bλ −1 βλ(u) = φ g2 φu |T , h2(u) bλ −1 γλ(u) = φ g1g2 φu |T . h(u) bλ

Hence γλ(u) = αλ(h2(u)) · βλ(u), and so ν is a homomorphism.

Let us prove that the kernel of ν is trivial. Let ελ : Bλ → Sbλ be the constant map into the unit ελ of Sbλ , ε = {ελ}λ∈B/H , and idB : B → B be the identity map, so the pair (ε, idB) is the unit in C. Suppose ν(g) = (ε, idB). Then p(g) = idB, i.e. g(Tu) = Tu for all u ∈ B. Moreover, −1 ελ(u) = (φ ◦ g ◦ φu)|T = ελ ∈ Sb , u ∈ Bλ, u bλ λ whence g|Tu : Tu → Tu is the identity map, and so g trivially acts on all of A. Since G is eﬀective, we see that g is a unit of G. Thus ν is a monomorphism. (2) We should establish equivalence of (a)-(c).

(a)⇒(b) Suppose ν is an isomorphism. Let X ⊂ B be any subset, {gb}b∈X be any family of elements of G such that gb(Tb) = Tb, and g : A → A be the permutation deﬁned −1 by the rule: g|Tb = gb|Tb for b ∈ X, and g = id on A \ q (X). We should prove that g ∈ G. This will imply that the action of G on A is τ-decomposable.

For each λ ∈ B/H deﬁne the map αλ : Bλ → Sbλ by the formula ( −1 φ ◦ gb ◦ φb|T , b ∈ Bλ ∩ X, b bλ αλ(b) = idT , b ∈ Bλ \ X, bλ and consider the following element γ = {αλ}λ∈B/H , idB ∈ C. Then it is easy to see that ν−1(γ) coincides with g. Thus g ∈ G, and so the action of G on A is τ-decomposable. (b)⇒(c). Suppose the action of G on A is τ-decomposable. Let γ = {αλ}λ∈B/H , idB −1 be any element of π (idB). We should prove that γ = ν(g) for some g ∈ G. DEFORMATIONS OF FUNCTIONS ON SURFACES 17

By definition Sbλ consists of restrictions of elements from G to Tbλ , whence for each b ∈ Bλ there exists gb ∈ G such that gb|T = αλ(b). In particular, bλ −1 φb ◦ gb ◦ φb (Tb) = Tb. (5.3) Consider now the family of elements of G: −1 {φb ◦ gb ◦ φb }b∈B. As G is τ-decomposable, it follows from (5.3) that there existsg ˆ ∈ G such that −1 gˆ|Tb = φb ◦ gb ◦ φb |Tb . Now we get from (5.2) that ν(ˆg) = γ. (c)⇒(a). Let γ = {αλ}λ∈B/H , h be any element of C. Since p : G → H is surjective, there exists g such that h = p(g). Then it follows from the multiplication rule in C that the element δ = ν(g−1) · γ ∈ C has the following form: δ = {αλ}λ∈B/H , idB , −1 −1 i.e. belongs to π (idB). But by (c), π (idB) ⊂ ν(G), so δ = ν(ˆg) for someg ˆ ∈ G, whence γ = ν(g ◦ gˆ). Thus ν is surjective, and therefore it is an isomorphism. 6. Groups acting on finite trees Let T be a finite graph and G be a subgroup of Aut(T ). If g ∈ G, then an edge e of T is fixed under g if g(e) = e and g preserves orientation of e. We will denote by Fix(G) the subgraph consisting of common fixed vertices and edges G. For each vertex u ∈ T let

Gu := {g ∈ G | g(u) = u} be the stabilizer of u with respect to G. Similarly, for an edge uv of T we deﬁne its stabilizer by Guv := Gu ∩ Gv = {g ∈ G | g(u) = u, g(v) = v}.

Regarding T as a CW-complex, let Star(u) be a small regular Gu-invariant neighborhood of u containing no other vertices of T except for u, see Figure 6.1.

Figure 6.1

Then the group of restrictions of Gu to Star(u) will be called the local stabilizer of u loc with respect to G and denoted by Gu , that is loc Gu := {g|Star(u) | g ∈ Gu} = {g|Star(u) | g ∈ G, g(u) = u}. loc Let also e be an edge of Star(u). Then the stabilizer of e with respect to Gu will be loc denoted by Gu,e, so loc loc Gu,e := {γ ∈ Gu | γ(e) = e} = {g|Star(u) | g ∈ G, g(u) = u, g(e) = e}. 18 SERGIY MAKSYMENKO

The following lemma is evident. Lemma 6.1. Let X ⊂ T be a G-invariant subgraph, so G also acts on X. Suppose w ∈ X is a vertex such that Star(w) ⊂ X. Then the local stabilizers of w with respect to the action of G on T and on X coincide. From now on we will assume that T is a ﬁnite tree.

Deﬁnition 6.2. For an edge uv of T let Tu v be the closure of the connected component of T \{u} containing v. We will call Tu v the subtree growing out of the vertex u and containing uv, see Figure 6.2. Let also Tbu v be the subtree tree obtained from Tu v by removing the vertex u and the open edge uv.

Figure 6.2. Subtrees Tu v, Tbu v, Tv u, and Tbv u.

6.3. t-decomposable actions. Let uv be an edge of T and g ∈ Gu v, so g(u) = u and g(v) = v. Then g(Tu v) = Tu v and g(Tv u) = Tv u. Therefore we can deﬁne the following automorphism ru v(g) of T by the rule: ( g on Tu v ru v(g) = (6.1) id on Tv u.

Notice that, in general, ru v(g) does not belong to G. It is also evident, that the correspondence g 7→ ru v(g) is a homomorphism:

ru v : Gu v −→ Aut(T ). (6.2) Deﬁnition 6.3.1. The action of G on T will be called t-decomposable if

ru v(Gu v) ⊂ G for every edge uv of T .

Lemma 6.3.2. Suppose ru v(Gu v) ⊂ G for some edge uv. Then rv u(Gv u) ⊂ G as well.

Proof. It is evident that g = ru v(g) ◦ rv u(g) for any g ∈ Gv u = Gu v = Gu ∩ Gv. By −1 assumption g, ru v(g) ∈ G. Therefore rv u(g) = g ◦ ru v(g) ∈ G as well. For each edge uv deﬁne the following subgroup of G:

Su v = {g ∈ G | supp (g) ⊂ Tbu v},

Evidently, Su v ⊂ Gu v. DEFORMATIONS OF FUNCTIONS ON SURFACES 19

Lemma 6.3.3. Suppose the action of G on T is t-decomposable. Let also uv be an edge of T . Then

(1) ru v(Gu v) = Su v, and the induced map ru v : Gu v → Su v is a retraction, that is ru v ◦ ru v = ru v and the restriction of ru v to Su v is the identity. (2) Let g ∈ Gu v and A ⊂ T be a subset such that g(A) = A. Then ru v(g)(A) = A. In particular, ru v(Gw ∩ Gu v) ⊂ Gw for any vertex w ∈ T . (3) ru v does not increase supports, that is supp (ru v(g)) ⊂ supp (g) for each g ∈ Gu v. (4) The induced action of Su v on T is t-decomposable. (5) If G = Su v, so Tv u ⊂ Fix(G), then the induced action of G on Tbu v is also t-decomposable. Proof. Statements (1)-(3) are easy and we left them for the reader. (4) Let wz be an edge in T . We should prove that rwz((Su v)wz) ⊂ Su v, that is for every g ∈ G satisfying supp (g) ⊂ Tbu v, g(w) = w, and g(z) = z we have that supp (rwz(g)) ⊂ Tbu v as well. But this follows from (3). (5) Let wz be an edge of Tbu v, g ∈ Gwz, andr ˆwz(g) be an automorphism of Tbu v deﬁned similarly to (6.2), so ( g on Tw z ∩ Tbu v, rˆwz(g) = id on Tz w ∩ Tbu v. We have to show thatr ˆ (g) = h| for some h ∈ G. In fact, one can put h = r (g). wz Tbu v wz 6.4. Relation between τ- and t-decomposability. Let u be a vertex of T and y1, . . . , yc be all the vertices adjacent to u. Then the set T \{u} admits a partition

τu = {Tu yi \{u}}i=1,...,c.

Notice that Gu(T \{u}) = T \{u} and Gu preserves partition τu. Since T is a tree, there is a bijection between the edges in Star(u) and vertices adjacent loc to u. In particular, one can assume that Gu acts on vertices adjacent to u. The following lemma is a direct consequence of deﬁnitions. Lemma 6.4.1. The action of G on T is t-decomposable if and only if for every vertex u ∈ T the action of the stabilizer Gu on T \{u} is τu-decomposable in the sense of Deﬁnition 5.2.

Corollary 6.4.2. Let u be a vertex of T , v1, . . . , va, w1, . . . , wb be the set of all vertices adjacent to u such that v1, . . . , va are all vertices ﬁxed with respect to the local stabilizer loc loc Gu . Assume that the restriction of Gu to the set W = {w1, . . . , wb} has exactly p ≤ b orbits W1,...,Wp so that wj ∈ Wj for j = 1, . . . , p. If the action of G on T is t- decomposable, then a p ∼ loc Gu = × Su vi × × Map(Wj, Su wj ) o Gu , (6.3) i=1 j=1 loc where the multiplication in the semi-direct product is induced by the right action of Gu loc on each set Wj. In particular, if the action of Gu on W is free, then a p ∼ loc Gu = × Su vi × × Su wj o Gu . (6.4) i=1 j=1 20 SERGIY MAKSYMENKO

Proof. By Lemma 6.4.1 the action of the stabilizer Gu on T \{u} is τu-decomposable in the sense of Definition 5.2. Therefore by Lemma 5.3 we have an isomorphism (6.3). Relation (6.4) follows from statements 2) and 3) of Lemma 5.1. 6.5. Class T (G). Recall that an effective action of a group G on a set X is semi- free if the restriction of this action to the set of non-fixed points, X \ Fix(G), is free. Equivalently, the action of G is semi-free if Fix(g) = Fix(h) for all g, h ∈ G \{1}. Evidently, that in this case for any subset F ⊂ Fix(G), the set X \ F is G-invariant, and the induced action of G on X \ F is semi-free as well.

Definition 6.5.1. Let G = {Gi | i ∈ Λ} be a family of finite groups containing the unit group {1}. We will say that a group G belongs to the class T (G) if there exists an effective action of G on a finite tree T satisfying the following conditions: (a) Fix(G) 6= ∅; (b) the action of G is t-decomposable; loc (c) for every vertex u of T the local stabilizer Gu of u belongs to G and its action on the set of edges of Star(u) is semi-free. Lemma 6.5.2. Suppose G acts of T so that the conditions (a)-(c) of Definition 6.5.1 hold, and uv is an edge of T such that Gw ⊂ Gu for every vertex w of Tbu v. Then (i) the action of Su v on T and (ii) the induced action of Su v to Tbu v also satisfy conditions (a)-(c) of Definition 6.5.1 for the same family of groups G.

Proof. (a) Notice that v ∈ Fix(Su v) ∩ Tbu v 6= ∅. This proves that both actions of Su v have non-empty set of ﬁxed points.

Condition (b) for the actions of Su v on T and on Tbu v follow from (4) and (5) of Lemma 6.3.3 respectively.

(c) First we check this condition for the action of Su v on T . loc Let w be any vertex of T . We should prove that the local stabilizer (Su v)w of w with respect to Su v belongs to G and its action on Star(w) is semi-free.

If w ∈ Tbv u, then Star(w) ⊂ Tv u. Hence any g ∈ Su v is fixed on Star(w). This means loc loc that (Su v)w is a trivial group, and so it belongs to G. In particular, the action of (Su v)w on the set of edges in Star(w) is free. loc Suppose w ∈ Tbu v, so Star(w) ⊂ Tu v. It suffices to show that in this case (Su v)w = loc loc loc Gw . Then condition (c) for (Su v)w will follow from the condition (c) for Gw . As loc loc Su v ⊂ G, it suffices to show that (Su v)w ⊃ Gw , that is for each g ∈ Gw there exists gˆ ∈ (Su v)w such that g|Star(w) =g ˆ|Star(w). Let g ∈ Gw. By assumption Gw ⊂ Gu, so g(u) = u. We claim that g(v) = v as well, i.e. g ∈ Gv. Indeed, let u = z0, z1, . . . , zk = w be a unique simple path in Tu v between u and z. Since uv is a unique edge in Tu v adjacent to u, it follows that z1 = v. Moreover, as g fixes u and w, it follows from uniqueness of this path that g(zi) = zi for all i. In particular, g(v) = v. Thus g ∈ Gu ∩ Gv. Putg ˆ = ru v(g). Theng ˆ ∈ Su v, andg ˆ = g on Tu v. In particular, gˆ = g on Star(w). DEFORMATIONS OF FUNCTIONS ON SURFACES 21

Now let us check condition (c) for the restriction of Su v to Tbu v. Let w ∈ Tbu v. Consider two cases. 1) Suppose w 6= v. Then Star(w) ⊂ Tbu v, whence by Lemma 6.1 the local stabilizers of w with respect to the actions of Su v on T and Tbu v coincide. 2) Suppose w = v. Then Star(w) \ uv is a regular neighborhood of v in Tbu v. As the edge uv is ﬁxed under Su v, it follows that the local stabilizers of v with respect to the actions of Su v on T and Tbu v are isomorphic, and the action of the local stabilizer of v with respect to the action of Su v on Tbu v is also semi-free. Our main result claims that classes R(G) and T (G) coincide, see Deﬁnition 1.3.

Theorem 6.6. R(G) = T (G) for any family G = {Gi | i ∈ Λ} of finite groups containing the unit group {1}. Proof. First we show that T (G) ⊂ R(G). Let G ∈ T (G), so there is an effective action of G on a finite tree T satisfying conditions (a)-(c) of Definition 6.5.1. We should prove that G ∈ R(G). Let us use the induction on the number of vertices in a tree T on which G can act satisfying that conditions (a)-(c). If T is a vertex, the group G is trivial, and so it belongs to R(G). In general case, due to (a) there exists a fixed vertex u ∈ Fix(G) 6= ∅, whence G = Gu. Then using notations of Corollary 6.4.2 we have an isomorphism (6.4):

a p ∼ loc G = Gu = × Su vi × × Su wj o Gu , i=1 j=1

loc where Gu ∈ G. We claim that Su z ∈ R(G) for each z ∈ {v1, . . . , va, wi, . . . , wp}. This will imply that G ∈ R(G) as well. Notice that Gx ⊂ Gu = G for every vertex x of Tu z. Therefore by Lemma 6.5.2 the action of Su z on Tbu z satisfies conditions (a)-(c) of Definition 6.5.1. But u 6∈ Tbu z, so Tbu z contains less vertices than T , whence by inductive assumption Su z ∈ R(G). Now we will establish the inverse inclusion R(G) ⊂ T (G). (i) Suppose A1,A2 ∈ T (G). Then Ai, i = 1, 2, acts on some tree Ti so that the conditions (a)-(c) of Definition 6.5.1 hold true. We should prove that A1 × A2 ∈ T (G). For i = 1, 2 choose a fixed point ui ∈ Fix(Ai). Not loosing generality, one can assume that ui has degree 1 in Ai. Let T = T1 ∨u1=u2 T2 be the tree obtained from T1 and T2 by identifying u1 with u2. Then each (g1, g2) ∈ A1 × A2 defines a unique automorphism fg1,g2 of T equal to gi on Ai, i = 1, 2. It is easy to check that the obtained action of A1 × A2 satisfies conditions (a)-(c) of Definition 6.5.1. We left the details for the reader. (ii) Suppose A ∈ T (G) and G ∈ G. We have to show that A o G ∈ T (G). By assumption A acts on some tree T so that the conditions (a)-(c) of Definition 6.5.1 hold true. Let u be any fixed point of this action. Again one can assume that deg u = 1. For each g ∈ G take a copy Tg of the tree T and fix an isomorphism φg : T → Tg. Let also ug = φg(u) be a vertex of Tg corresponding to u. 22 SERGIY MAKSYMENKO

Let S = W T be a tree obtained from disjoint union of all T by identifying {ug | g∈G} g g all points ug. Now for every (α, g) ∈ A o G = Map(G, A) o G deﬁne an automorphism fα,g of S by the following rule:

fα,g(Th) = Tgh and −1 fα,g|Th = φgh ◦ α(h) ◦ φh . Again it is not hard to verify that the latter formula gives an action of A o G on T satisfying conditions (a)-(c) of Deﬁnition 6.5.1, i.e. A o G ∈ T (G). Thus T (G) = R(G). Theorem 6.6 completed. Remark 6.7. Let T be a ﬁnite tree and G = Aut(T ) be the group of its automorphisms. Then by Jordan’s Theorem 4.3 and by Theorem 6.6

G ∈ R({Sn}n≥1) = T ({Sn}n≥1). In particular, G admits an action on a finite tree T 0 satisfying conditions (a)-(c) of Definition 6.5.1. On the other hand we have a natural action of G on T . It is easy to see that this action is t-decomposable, however other conditions (a) and (c) of Definition 6.5.1 can fail. Therefore, in general, T 0 6= T .

7. Critical level-sets Let f ∈ F(M,P ). For every subset Y ⊂ M deﬁne the following groups 0 S(f, Y ) := {h ∈ S(f) | h(Y ) = Y }, S (f, Y ) := S(f, Y ) ∩ Did(M). If Y is an orientable submanifold of M, then we also set S(f, +Y ) := {h ∈ S(f) | h(Y ) = +Y }.

Let U be a critical element of partition Θf and c = f(U) ∈ P be the value of f on U. Fix ε > 0 such that the closed interval [c − ε, c + ε] ⊂ P contains no critical values of f except for c, and denote by N the connected component of f −1[c − ε, c + ε] containing U, see Figure 7.1. If N ∩ ∂M = ∅ (7.1) then we will call N a regular ε-neighborhood of U. Such a neighbourhood is often called an atom of U, see e.g. [9, 10, 2]. In fact, if ε is suﬃciently small, then (7.1) always hold. So assume that N is a regular ε-neighborhood of U. Denote −1 −1 ∂−N := N ∩ f (c − ε), ∂+N := N ∩ f (c + ε). Then

∂N = ∂−N ∪ ∂+N.

Lemma 7.1. Let B be a connected component of ∂N. Then S(f, B) ⊂ S(f, U) = S(f, N). DEFORMATIONS OF FUNCTIONS ON SURFACES 23

Figure 7.1

Proof. First we prove that S(f, B), S(f, U) ⊂ S(f, N). (7.2) Denote Q = f −1[c − ε, c + ε]. Then h(Q) = Q for every h ∈ S(f), as f ◦ h = f. Notice that N is a connected component of Q containing U and B. Hence if h ∈ S(f) is such that either h(V ) = V or h(B) = B, then h(N) = N. This completes (7.2). It remains to show that S(f, N) ⊂ S(f, U). Let h ∈ S(f, N). From U = N ∩ f −1(c), it follows that h(U) = h(N ∩ f −1(c)) = h(N) ∩ h(f −1(c)) = N ∩ f −1(c) = U, i.e. h ∈ S(f, U). Thus S(f, N) ⊂ S(f, U). 7.2. Action of S(f, N). Let λ : S(f) → Aut(Γ(f)) be the action homomorphism of S(f) on the enhanced KR-graph Γ(f) of f, and u = p(U) be the vertex of Γ(f) corresponding to U. Denote G := λ(S(f)), G := λ(S0(f)).

Evidently, N is the union of some elements of the partition Θf , and the restriction f|N of f to N belongs to F(N,P ). Therefore image of N under p coincides with the enhanced KR-graph of f|N , that is

Γ(f|N ) = p(N).

It is easy to see that Γ(f|N ) is a closed neighborhood of u in Γ(f), and so it can be regarded as a regular neighborhood Star(u) of u in Γ(f), see Figure 7.1, where Γ(f|N ) is drawn with bold lines.

In particular, one can deﬁne the partition ∆f|N which clearly consists of all elements of ∆f contained in N. Let

rN : S(f, N) → S(f|N ), rN (h) = h|N , be “the restriction to N map” and

λN : S(f|N ) −→ Aut(Γ(f|N )) ≡ Aut(Star(u)) be the corresponding action homomorphism. Notice that p induces a bijection between the connected components of ∂N and edges of Γ(f|N ). Let B be a boundary component of ∂N and e be the edge of Γ(f|N ) corresponding to B. The following lemma is easy and we left it for the reader: 24 SERGIY MAKSYMENKO

Lemma 7.2.1. The following identities hold true:

λ(S(f, U)) = Gu ≡ {g ∈ G | g(u) = u}, loc λN ◦ rN (S(f, U)) = Gu ≡ {g|Γ(f|N ) | g ∈ Gu}, loc loc λN ◦ rN (S(f, B)) = Gu,e ≡ {gˆ ∈ Gu | gˆ(e) = e}, and similarly for the group G: 0 0 loc 0 loc λ(S (f, U)) = Gu, λN ◦ rN (S (f, U)) = Gu , λN ◦ rN (S (f, B)) = Gu,e.

7.3. Cycle ω(B). Let NB be the connected component of N \ U containing B, and let

C(B) := U ∩ NB, see Figure 7.2.

Figure 7.2. Cycle ω(B).

Lemma 7.3.1. 1) C(B) is a connected subcomplex of U and for every open edge δ of U there exists a unique connected component B− of ∂−N such that δ ⊂ C(B−) and a unique connected component B+ of ∂+N such that δ ⊂ C(B+). Suppose U contains more than one point, so it is not a local extreme. Then there exists an oriented Eulerian cycle ω(B) in C(B) (i.e. a closed path visiting every edge exactly once), see Figure 7.2, such that the following statements hold.

2) Let h : N → N be a homeomorphism such that h(U) = U, B0,B1 be two oriented connected components of ∂N, and ε = + or −. Then h(B0) = εB1 if and only if h(ω(B0)) = εω(B1).

3) Let h : N → N be a homeomorphism such that h(U) = U and h(∂−N) = ∂−N. Suppose there exists an open edge δ of U such that h(δ) = +δ. Then h is a trivial automorphism of U, whence, by 2), h(B) = +B for every connected component of ∂N. Moreover, if f ◦ h = f, then h preserves each element of ∆f with its orientation. Proof. Statements 1) and 2) are well-known. They are widely used e.g. in [13, 6, 36, 19, 38, 15, 32] and others. Statement 3) is contained in [26, Claim 7.1.1]. 7.4. Critical levels with more than one point. Assume that U contains more than one point. Let Aut(ω(B)) be the group of combinatorial automorphisms of the graph C(B) which preserve or reverse the cyclic order of edges in the cycle ω(B), and Aut+(ω(B)) be the subgroups of Aut(ω(B)) consisting of automorphism preserving orientation of ω(B). Then Aut(ω(B)) is a subgroup of a dihedral group Dn, where n is the number of edges in C(B). In particular, Aut(ω(B)) is either a cyclic or a dihedral group, while Aut+(ω(B)) is a cyclic group, and the orders of both groups divide n. DEFORMATIONS OF FUNCTIONS ON SURFACES 25

As noted above, f|N ∈ F(M,P ), so we can deﬁne the partition ∆f|N . Clearly, ∆f|N consists of all elements of ∆f contained in N. Let D(∆f|N ) be the group of its automorphisms, see §2.9.

Lemma 7.4.1. 1) There exists a homomorphism α : S(f|N ,B) −→ Aut(ω(B)) such that

rN λN S(f, B) / S(f|N ,B) / λN S(f|N ,B) 6 α β

' α S(f|N ,B) Hence (i) the group λN ◦ rN S(f, B)) is either ﬁnite cyclic or dihedral, while (ii) the group λN ◦ rN S(f, +B)) is always cyclic.

Proof. 1) Let h ∈ S(f|N ,B). Then due to 2) of Lemma 7.3.1, h(ω(B)) = ω(B), so the restriction h|ω(B) induces a combinatorial automorphism of ω(B) which we will denote by α(h). Evidently, the correspondence h 7→ α(h) is the desired homomorphism α.

(a) Since the vertices and edges of ω(B) are elements of ∆f , we see that D(∆f|N ) ⊂ ker(α). Conversely, let h ∈ ker(α), so h preserves every vertex and every edge of ω(B) with their orientation. Then, by 3) of Lemma 7.3.1, h ∈ D(∆f|N ).

(b) If h ∈ S(f|N , +B), so h preserves orientation of B, then by 2) of Lemma 7.3.1 α(h) ∈ Aut+(ω(B)).

(a) (2.4) 2) Existence of β follows from the relation ker(α) == D(∆f| ) ⊂ ker(λN ). N Statements (i) and (ii) follow from the description of α S(f|N ,B)) and α S(f|N , +B)) given in 1). loc Corollary 7.4.2. 1) The group Gu,e = λN ◦ rN S(f, B) is either cyclic or dihedral. loc 0 2) If M is orientable, then the group Gu,e = λN ◦ rN (S (f, B)) is cyclic. Proof. Statement 1) coincides with 2(i) of Lemma 7.4.1. 2) Suppose M is orientable. We claim that then S0(f, B) ⊂ S(f, +B) (7.3)

loc which will imply by 2(ii) of Lemma 7.4.1 that Gu,e is a subgroup of the cyclic group λN ◦ rN (S(f, +B)). 26 SERGIY MAKSYMENKO

Let q = f(B), W be a small cylindrical neighborhood of B, such that W \ B consists of two connected components W1 and W2 with f(W1) ∩ f(W2) = ∅. We can assume that W is a connected component of a set f −1[q − µ, q + µ] for some small µ. 0 Let h ∈ S (f, B) = S(f, B) ∩ Did(M). Since f ◦ h = f and h(B) = B, it follows that h(W ) = W , and h(Wi) = Wi, i = 1, 2. Moreover, as h is isotopic to idM , it preserves orientation of M, and so we must have that h(B) = +B. This proves (7.3). 7.5. Local extremes. Suppose U = {z} is a local extreme of f. Then the vertex u = p(z) has degree 1 in the KR-graph M/Θf . Let also v be a unique vertex of M/Θf adjacent to u.

loc Lemma 7.6. 1) If z is a non-degenerate local extreme, then the local stabilizers Gu and Gu are trivial. 2) Suppose z is a degenerate local extreme having symmetry index n. Then Gu is a either cyclic group Zn or a dihedral group Dn. loc ∼ If M is orientable, then Gu = Zn and its action on Star(u) is semi-free. Proof. 1) If z is a non-degenerate local extreme, then u has degree 1 in the enhanced KR-graph Γ(f). Therefore its Star(u) consists of a unique edge uv, and so both groups loc Gu ⊂ Gu are trivial. 2) Suppose z is a degenerate local extreme having symmetry index n. Thus besides v there are also n vertices adjacent to u in the enhanced KR-graph Γ(f): 0 n−1 ξz , . . . , ξz , (7.4) i where each ξz is regarded as a tangent vector in TzM belonging to the framing F . By Definition 2.5 for each h ∈ S(f, U) its tangent map Tzh preserves or reverses cyclic order of vectors (7.4), whence g = λ(h) ∈ Gu preserves cyclic order of vertices (7.4), and so g(v) = v. Moreover, by (2) of Lemma 2.4 for each i = 1, . . . , n − 1 there exists 0 i h ∈ S(f, U) such that Tzh(ξz ) = ξz, so Gu acts on vertices (7.4) transitively. loc ∼ This implies that Gu = Dn if there exists h ∈ S(f, U) reversing cyclic order of (7.4) loc ∼ and Gu = Zn otherwise. 0 Suppose M is orientable. Then each h ∈ S (f, U) being isotopic to idM preserves ˆ 0 ˆ 0 1 cyclic order of vectors (7.4). We will find h ∈ S (f, U) such that Tzh(ξz ) = ξz . This will loc ∼ imply that Gu = Zn. Let h ∈ S(f, U) be any diffeomorphism such that Tzh preserves cyclic order of vec- 0 1 tors (7.4), and Tzh(ξz ) = ξz . Let N be a regular ε-neighborhood of z. Then N is a 2-disk, h(∂N) = ∂N, and h preserve orientation of ∂N. Therefore h is isotopic in S(f) to a diffeomorphism hˆ fixed on some neighborhood of ∂N. Change hˆ on M \ N by the identity. Then hˆ ∈ S0(f, U) and hˆ = h on N. loc Since each non-trivial element g ∈ Gu cyclically permutes vertices (7.4) and fixes v, loc we see that the action of Gu on Star(u) is semi-free.

8. Proof of Theorem 1.5. Case M = D2 and S1 × I Let M be either a 2-disk D2 or a cylinder S1 × I. We will prove that

R({Zn}n≥1) = {G(f) | f ∈ F(M,P )}. DEFORMATIONS OF FUNCTIONS ON SURFACES 27

The proof consists of two Lemmas 8.1 and 8.2 showing that each of the classes contains the other.

Lemma 8.1. Let M be either a 2-disk D2 or a cylinder S1 × I, and let f ∈ F(M,P ). Then the action of the group G(f) on Γ(f) satisfies conditions (a)-(c) of Definition 6.5.1 for G = {Zn}n≥1 being the family of all finite cyclic groups. In particular, by Theo- rem 6.6,

G(f) ∈ T ({Zn}n≥1) = R({Zn}n≥1), and so {G(f) | f ∈ F(M,P )} ⊂ R({Zn}n≥1). Proof. (a) Since M is either D2 or S1 × I, the enhanced KR-graph Γ(f) is a finite tree. Then only one of the following two possibilities hold, see e.g. [14] or [37, §46]: (i) either Fix(G(f)) 6= ∅, (ii) or there exists a common invariant edge uv for all elements of g ∈ G, so g(uv) = uv or g(uv) = vu and both cases are realized. Notice that the edges of Γ(f) are oriented via orientation of P , and G(f) preserves those orientations. This implies, we see that the case (ii) is impossible. Hence Fix(G(f)) 6= ∅. (b) We should prove that the action of G(f) on Γ(f) is t-decomposable. Let uv be an edge of enhanced KR-graph Γ(f), Tu v be the tree growing out of u, g ∈ G(f) be such that g(u) = u and g(v) = v, h ∈ S0(f) be such that g = λ(h), and ˆ 0 gˆ = ru v(g). We have to show thatg ˆ ∈ G(f) as well, that is there exists h ∈ S (f) such thatg ˆ = λ(hˆ). Due to Lemma 6.3.2 we can exchange roles of vertices u and v. Therefore we can assume that u is a vertex of the (usual) KR-graph M/Θf of f, i.e. it corresponds to a critical element of Θf but not to a tangent vector belonging the framing F . Let also N be a regular ε-neighborhood of U. Consider two cases. (b1) Suppose v corresponds to a tangent vector of the framing F . In other words, u i is a vertex corresponding to a degenerate local extreme z of f and v = ξz for some i. Hence Tu v is the edge uv, and thereforeg ˆ = ru v(g) = idΓ(f) ∈ G(f). (b2) Suppose v is a vertex of KR-graph M/Θf , and B be the boundary component of N corresponding to the edge uv. Notice that B separates M. Denote by Q the closure of the connected component of M \ B that does not contain N. Then h(Q) = Q. Not loosing generality one can assume that h is fixed on some neighborhood of B. Define a diffeomorphism hˆ : M → M by the rule: hˆ = h on Q and hˆ = id on M \ Q. Then hˆ ∈ S(f), and it is evident that λ(hˆ) =g ˆ. ˆ ˆ Therefore it remains to show that h is isotopic to idM , which will imply that h ∈ 0 S (f) = S(f) ∩ Did(M). Notice that hˆ is fixed on the non-empty open set M \Q, whence it preserves orientation of M. Moreover, hˆ preserves connected components of ∂M. As M is either D2 or S1 ×I, ˆ it follows that h is isotopic to idM . 28 SERGIY MAKSYMENKO

(c) Let u be a vertex of the enhanced KR-graph Γ(f). We should prove that its loc local stabilizer Gu is a cyclic group and its action on the set of adjacent edges to u is semi-free. Consider three cases. i (c1) Suppose u = ξz corresponds to a tangent vector at some degenerate local extreme loc z of f. Then Star(u) consists of a unique edge uz, whence Gu is a unit group. In particular, this group is cyclic and its action on Star(u) is semi-free. (c2) Suppose u corresponds to a degenerate local extreme z of f. Then our statement is contained in 2) of Lemma 7.6. (c3) Now assume that u is a vertex of Γ(f) corresponding to a critical element U of Θf being not a degenerate local extreme. First we recall one result from [29]. Let Ξ be a cellular decomposition of a closed surface N and h : N → N be a homeomorphism. If h(α) = α for some cell α of Ξ, then we will say that α is h-invariant. Moreover, α will be called h+-invariant if either dim α = 0, or dim α ≥ 1 and h preserves orientation of α. We will say that h is a Ξ-homeomorphism if for every cell α of Ξ its image h(α) is also a cell of Ξ. We also say that a homeomorphism h is Ξ-trivial if every cell of Ξ is h+-invariant. Lemma 8.1.1. [29, Proposition 5.4.] Let N be a closed, orientable surface endowed with some cellular decomposition Ξ and h : N → N a Ξ-homeomorphism preserving orientation of N. Then either (1) h is Ξ-trivial, or (2) the number of h-invariant cells of Ξ is equal to the Lefschetz number L(h) of h, so in this case L(h) ≥ 0.

Let Mc be a closed surface obtained by gluing each boundary component of M with a 2-disk, so Mc is a 2-sphere, Let also U be the critical element of Θf corresponding to u and N be a regular ε- neighborhood of U, and Ξ be the CW-subdivision of Mc by vertices and edges of U and connected components of Mc \ U. Evidently, there is a natural bijection between 2-cells of Ξ and edges of Star(u). loc 0 0 Let g ∈ Gu = λN ◦ rN S (f, U) be a non-unit element, and h ∈ S (f, U) be such that

g = λN ◦ rN (h).

Since h is isotopic to idM , it leaves invariant every connected component of ∂M and preserves its orientation. Therefore we can extend h onto the union Mc \ M of glued 2-disks to some homeomorphism bh of Mc. As h(U) = bh(U) = U, bh induces a certain permutation of cells of Ξ, whence bh is a Ξ-homeomorphism. Evidently, ﬁxed edges of g correspond to bh-invariant 2-cells of Ξ and vice versa. In particular, as g is a non-unit element, we see that bh is not Ξ-trivial. Moreover, h preserves orientation of M = S2, and therefore it is isotopic to id . Hence b c Mc we get from (ii) of Lemma 8.1.1 that the number of bh-invariant cells is equal to

2 L(bh) = L(idS2 ) = χ(S ) = 2. DEFORMATIONS OF FUNCTIONS ON SURFACES 29

Let V be a connected component of ∂M, α be a 2-cell of Ξ containing V , and e be the edge of Star(u) corresponding to α. Then bh(α) = α, and so g(e) = e. Since α loc does not depend on g, we see that e is a fixed edge of Gu , whence by Corollary 7.4.2 loc loc ∼ Gu = Gu,e = Zn for some n ≥ 1. loc It remains to show that the action of Gu on the set of edges of Star(u) is semi-free. loc For g ∈ Gu let Fix(g) be the set of fixed edges of g on Star(u). Then it suffices to loc establish that Fix(g) = Fix(g1) for all g, g1 ∈ Gu \{1}. loc ∼ Using the notation above assume that g is a generator of Gu = Zn. Let α1 be another bh-invariant cell of Ξ. Then for k = 1, . . . , n − 1 the homeomorphism bhk is not Ξ-trivial, and by the same arguments as above has exactly two invariant cells. Therefore k k these cells must be α and α1. Thus Fix(g ) = {e, e1} if dim α1 = 2, and Fix(g ) = {e} otherwise. Lemma 8.2. Let M = D2 or S1 × I. Denote M = {G(f) | f ∈ Morse(M,P )}. Then 1) Zn ∈ M for each n ≥ 1; 2) if G0, G1 ∈ M, then G0 × G1 ∈ M; 3) if G ∈ M and n ≥ 1, then G o Zn ∈ M as well. Hence R({Zn}n≥1) ⊂ M ⊂ {G(f) | f ∈ F(M,P )}. ∼ Proof. 1) Structure of critical level-sets of the function f satisfying G(f) = Zn is shown in Figure 8.1 (a) for M = D2 and in (b) for M = S1 × I.

(a) M = D2 (b) M = S1 × I ∼ Figure 8.1. Maps f : M → P with G(f) = Zn, for n = 3.

2) Suppose G0, G1 ∈ M. We have to construct a mapping f ∈ Morse(M,P ) such ∼ that G(f) = G0 × G1. Such a map schematically shown in Figure 8.2. Let us describe its structure in details.

(a) M = D2 (b) M = S1 × I ∼ Figure 8.2. Maps f : M → P with G(f) = G(f1) × G(f2)

Suppose M = D2. Let f ∈ Morse(M,P ) be a Morse function reaching its minimum on ∂M, having 1 saddle and two local maximums z0 and z1 and such that f(z0) 6= f(z1). Let also Yi, i = 0, 1, be a small neighborhood of z1 diﬀeomorphic to 2-disk and such that 30 SERGIY MAKSYMENKO

∂Yi is a connected component of some level-set of f, and Y0 ∩ Y1 = ∅. Replace f on Yi so that G(f|Yi ) = Gi, and f(Y0) ∩ f(Y1) = ∅, see Figure 8.2(a). If M = S1 × I, then let f ∈ Morse(M,P ) without critical points, so we can assume 1 1 that f takes its minimum of S × 0 and its maximum on S × 1. Let Yi, i = 0, 1, be a 1 regular neighborhood of S × i diﬀeomorphic to a cylinder and such that ∂Yi consists of connected components of some level-set of f, and Y0 ∩ Y1 = ∅. Replace f on Yi so that

G(f|Yi ) = Gi and f(Y0) ∩ f(Y1) = ∅, see Figure 8.2(b). ∼ We claim that in both cases G(f) = G(f0) × G(f1). Indeed, ﬁrst notice that we can regard Γ(fi) as a subgraph of Γ(f). Let λi : S(fi) → Aut(Γ(fi)) be the corresponding action homomorphism, and λ : S(f) → Aut(Γ(f)). 0 Let h ∈ S (f) and g = λ(h). Since f(Y0) ∩ f(Y1) = ∅, it follows that g is supported in Γ(f0) ∪ Γ(f1), and

g|Γ(f1) = λi(h|Yi ), i = 0, 1. Therefore we have a homomorphism η : G(f) −→ G(f0) × G(f1), η(g) = g|Γ(f0), g|Γ(f1) . Let us show that η is an isomorphism. As g is supported in Γ(f0) ∪ Γ(f1), we see that g is a monomorphism. 0 To show that η is surjective let gi ∈ G(fi) = λi S (fi, ∂Yi) , i = 0, 1. Thus gi = λi(hi), 0 where hi ∈ S (fi, ∂Yi), so we can assume that hi is fixed on some neighborhood of ∂Yi. Now define h ∈ S(f) by h = hi on Yi, and h = id on M \ (Y0 ∪ Y1). Then h preserves orientation of M and leaves invariant every connected component of ∂M. Therefore h 0 is also isotopic to idM , that is h ∈ S(f) ∩ Did(M) = S (f). Denote g = λ(h). Then it follows that η(g) = (g0, g1). Hence η is an isomorphism. 3) Let G ∈ M and n ≥ 1. We have find a mapping f ∈ Morse(M,P ) such that ∼ G(f) = G o Zn. Construction of such a map f is shown in Figure 8.3.

(a) M = D2 (b) M = S1 × I ∼ Figure 8.3. Maps f : M → P with G(f) = G(g) o Zn ∼ Let g ∈ Morse(M,P ) be a map constructed in 1) such that G(g) = Zn. This map has n local maximums z1, . . . , zn such that zi is contained in the domain denoted by i, see Figure 8.1. One can also assume that there exists h ∈ S(f) such that h preserves i n orientation of M, h (z1) = zi, and h = idM . Now let Y1 be a small disk neighborhood of z1 such that ∂Y1 is a connected component of a level set of f, and Y1 does not contain i other critical values except for f(z1). Then Yi = h (Y1) is a neighborhood of zi with similar properties. ∼ i Deﬁne a new function f ∈ Morse(M,P ) such that G(f|Y1 ) = G, f|Yi = g ◦ h |Y1 , n ∼ i = 2, . . . , n, and f = g on M \ ∪i=1Yi . Then it is easy to check that G(f|Yi ) = G for ∼ all i, and G(f) = G o Zn. We leave the details for the reader. DEFORMATIONS OF FUNCTIONS ON SURFACES 31

9. Proof of Theorem 1.5. All other surfaces It remains to consider the case when M is a connected compact orientable surface distinct from S2, T 2, D2 and S1 × I, and so χ(M) < 0. We have to prove that

R({Zn}n≥1) = {G(f) | f ∈ F(M,P )}. The proof reduces the situation to the case M = D2 and S1 × I considered in §8.

Lemma 9.1. Let f ∈ F(M,P ). Then there exist finitely many surfaces Y1,...,Yk each 2 1 diffeomorphic either to a 2-disk D or to a cylinder S × I, and maps fi ∈ F(Yi,P ) such that ∼ G(f) = G(f1) × · · · × G(fk), whence G(f) ∈ R({Zn}n≥1), and so {G(f) | f ∈ F(M,P )} ⊂ R({Zn}n≥1). Proof. Since χ(M) < 0, it follows from [29, Theorem 1.7] that there exists finitely many k disjoint subsurfaces Y1,...,Yk having the following properties. Denote Y = ∪i=1Yi. Then 2 1 (a) each Yi is diffeomorphic either to D or to S × I;

(b) the restriction f|Yi satisﬁes axioms (B) and (L) for each i = 1, . . . , k; 0 0 (c) every h ∈ S (f) is isotopic in S (f) to a diﬀeomorphism h1 is supported in IntY , and, in particular, λ(h) = λ(h1); 0 (d) if h ∈ S (f) is supported in IntY , then for each i = 1, . . . , k the restriction h|Yi

is isotopic in D(Yi) to idYi with respect to some neighborhood of ∂Yi. In other 0 words, h|Yi ∈ S (f|Yi , ∂Yi).

Let fi = f|Yi : Yi → P be the restriction of f to Yi,Γi = Γ(fi) be the enhanced KR-graph of fi, λi : S(fi) −→ Aut(Γi) be the corresponding action homomorphism, and

0 Lemma 3.1 0 Gi := G(fi) ≡ λi S (fi) ======λi S (fi, ∂Yi) .

It follows from (a) that Γi is a subtree of Γ(f) with respect to some CW-partition. To complete Lemma 9.1 it suffices to prove that there is an isomorphism ∼ η : G(f) = G1 × · · · × Gk. Let g ∈ G(f), so g = λ(h) for some h ∈ S0(f). Then it follows from (c) that g is k supported in ∪i=1Γi, and by (d) the restriction of g to Γi belongs to Gi. Hence we have a well-defined homomorphism: η : G(f) −→ G1 × · · · × Gk, η(g) = g|Γ1 , . . . , g|Γk . We claim that η is an isomorphism. k Since each g ∈ G(f) is supported in ∪ Γi, it follows that η is a monomorphism. i=1 0 Let (g1, . . . , gk) ∈ G1 × · · · × Gk and for each i = 1, . . . , k let hi ∈ S (fi, ∂Yi) be such that λi(hi) = gi. Sine hi is fixed near ∂Yi, it extends by the identity on M \ Yi to a diffeomorphism supported in Yi and isotopic to idM . 32 SERGIY MAKSYMENKO

0 Denote h = h1 ◦ · · · ◦ hk ∈ S (f). Then η(λ(h)) = λ1(h1), . . . , λk(hk) = (g1, . . . , gk), and so η is surjective, and therefore it is an isomorphism.

Now by Lemma 8.1 each Gi ∈ R({Zn}n≥1), and so G(f) ∈ R({Zn}n≥1) as well. Lemma 9.2. Let M be a connected compact orientable surface with χ(M) < 0. Then

R({Zn}n≥1) ⊂ {G(f) | f ∈ Morse(M,P )}.

Proof. Let G ∈ R({Zn}n≥1). Take a generic map f ∈ Morse(M,P ) having a unique local maximum z and such that the group G(f) is trivial, see e.g. Figure 9.1.

∼ ∼ Figure 9.1. Maps f : M → P with G(f) = G(f|Y ) = G.

Let Y be a disk-neighborhood of z such that ∂Y is a connected component of some level set of f and Y contains no critical points except for z. ∼ By Lemma 8.2 we can change f on the disk Y so that G(f|Y ) = G. We claim that then G(f) ∼= G. The proof is similar to 3) of Lemma 8.2. Let h ∈ S0(f) and g = λ(h). Since h is isotopic to idM and the factor map p : M → Γ(f) induces a surjection on the first homology groups p∗ : H1(M, Z) → H1(Γ(f), Z), it follows that g induces the identity automorphism of H1(Γ(f), Z). This implies that g is supported in Γ(f|Y ). Now let ∼ η : G(f) −→ G(f|Y ) = G, g 7→ g|Γ(f|Y ) be the restriction map. We claim that η is an isomorphism. Indeed, since g is supported in Γ(f|Y ), we obtain that η is a monomorphism. ˆ To prove that η is surjective, let λ : S(f|Y ) → Aut(Γ(f|Y )) be the corresponding ˆ 0 ˆ ˆ ˆ action homomorphism. Letg ˆ ∈ G(f|Y ) = λ S (f|Y , ∂Y ) , sog ˆ = λ(h) for some h ∈ 0 ˆ S (f|Y , ∂Y ). We can even assume that h is fixed near ∂Y . Then it extends by the identity on M \ Y to a diffeomorphism h ∈ S(f). Moreover, as h is supported in a disk 0 Y , it is isotopic to idM , and so h ∈ S (f). It remains to note that η(λ(h)) =g ˆ. Thus η is an isomorphism. References [1] LászlóBabai, Automorphism groups, isomorphism, reconstruction, Handbook of combinatorics, Vol. 1, 2, Elsevier, Amsterdam, 1995, pp. 1447–1540. MR 1373683 (97j:05029) [2] A. V. Bolsinov and A. T. Fomenko, Vvedenie v topologiyu integriruemykh gamiltonovykh sistem (Introduction to the topology of integrable hamiltonian systems), “Nauka”, Moscow, 1997 (Russian). MR MR1664068 (2000g:37079) DEFORMATIONS OF FUNCTIONS ON SURFACES 33

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Topology dept., Institute of Mathematics of NAS of Ukraine, Tereshchenkivska st. 3, Kyiv, 01601 Ukraine E-mail address: [email protected] URL: http://www.imath.kiev.ua/~maks