Automorphisms of Kronrod-Reeb graphs of Morse functions on 2-torus

This paper is devoted to the study of special subgroups of the automorphism groups of Kronrod-Reeb graphs of a Morse functions on $2$-torus $T^2$ which arise from the action of diffeomorphisms preserving a given Morse function on $T^2$. In this paper we give a full description of such classes of groups.


Introduction
Topological graphs naturally arise from the study of smooth functions on smooth manifolds as the powerful tools which contain "combinatorial" information about a given smooth function and hence the information about a topology of a smooth manifolds. Kronrod-Reeb graphs of Morse functions on compact manifolds, named after G. Reeb by R. Thom and A. Kronrod by V. Sharko are the famous example of such graphs. They were studied by many authors. E.g. the problem of realization of a graph as a Kronrod-Reeb graph of a Morse (in general smooth) function on a given smooth compact manifold was proposed by V. Sharko and were studied in papers of V. Sharko [31], Y. Masumoto and O. Saeki [24], L. Michalak [26], K. Cole-McLaughlin et al [1], M. Kaluba, W. Marzantowicz, N. Silva [7] and others. This problem is closely related to other important problem of topologically conjugacy for Morse functions on smooth manifolds, see e.g. E. Kulinich [14], V. Sharko [30], D. Lychak and O. Prishlyak [15]. E. Polulyakh [27,28] generalized the notion of Kronrod-Reeb "graph" for functions on non-compact surfaces.
Homotopy properties of Morse functions on smooth surfaces were studied by V. Sharko [29], H. Zieschang, S. Matveev, E. Kudryavtseva [11], K. Ikegami and O. Saeki [5], B. Kalmar [6], S. Maksymenko. We give a short overview of the results of S. Maksymenko [17,18,19,20,21] and E. Kudryavtseva devoted to the study of homotopy properties of orbits and stabilizers of smooth functions on compact surfaces under the action of their diffeomorphism groups. And we will see that subgroups of the automorphism groups of Kronrod-Reeb graphs plays the essential role in the description of homotopy types of these spaces.
Let M be a smooth compact surface and X be a closed ( In this paper we will consider Kronrod-Reeb graphs of Morse functions on smooth compact surfaces. Since these surfaces may have the boundary we need to specify what will be meant by a Morse function. By a Morse function f on a surface M we will mean a smooth function f ∈ C ∞ (M ) on a surface M such that f takes constant values on the connected components of the boundary ∂M and each critical point of f is non-degenerate and is contained in Int(M ). A Morse function f on M is called simple if every critical connected component of every critical level-set contains a unique critical point, and f is called generic if each level-set of f contains no more than one critical point.
S. Maksymenko, [17,18,19,20,21], showed that if f has at least one saddle point, then for some k ≥ 0 and a finite group G(f ) which is a group of automorphisms of a Kronrod-Reeb graph of f induced by isotopic to id M and f -preserving diffeomorphisms, i.e., diffeomorphisms from where G(f ) freely acts on (S 1 ) m , and the number m is the rank of the abelian group π 1 D id (M ) ⊕ Z k , see (1). Note that G(f ) is the holonomy group of the compact manifold (S 1 ) m /G(f ). An algebraic structure of π 1 O f (f ) for the Morse functions on 2-torus was described in the series of papers by S. Maksymenko and the second author [23,16,22,4]. This is one of non-trivial cases since D id (T 2 ) is not contractible, so the image Since groups G(f ) play the essential role in the description of homotopy types of O f (f ) S. Maksymenko and A. Kravchenko [10] studied classes of such groups and their subsets. Let G (M ) be the isomorphisms class of groups G(f ) for all Morse functions f on M . By G smp (M ) and G gen (M ) will be denoted subclasses of G (M ) which corresponds to isomorphisms classes of groups G(f ) for all simple and generic Morse functions on M respectively. The first author and S. Maksymenko gave a full algebraic description of classes G (M ) and G smp (M ) for all compact oriented surface M which are distinct from 2-sphere S 2 and 2-torus T 2 , and proved that the class G gen (M ) is trivial, i.e., contains only trivial group {1} for all compact oriented surface M , see Theorem 2.2, or [10]. We also mention that an algebraic structure of G(f ) for Morse function on S 2 is partially understood [8], and in general this case is more complicated than the case of Morse functions on compact surfaces of genus ≥ 1. Recently S. Maksymenko and A. Kravchenko [9] described special subgroups of G(f ) for Morse functions on S 2 .
The aim of this paper is to extend the result of [10] by given a full description of the classes G (T 2 ) and G smp (T 2 ) for a Morse functions on 2-torus.
1.1. Acknowledgments. Authors would like to express their gratitude to Sergiy Maksymenko for advices and discussions.

Conventions and notations.
To state our main result we need the notion of wreath products of groups with cyclic groups. So we recall these definitions. Let G be a group, m, n ≥ 1 be integers. Consider two effective actions α : G n × Z n → G n , and β : G nm × (Z n × Z m ) → G nm of Z n and Z n × Z mn on G n and G nm given by the formulas: where all indexes are taken modulo n and n, m respectively. With respect to these actions we define semi-direct products G ≀ Z n := G n ⋊ α Z n and G ≀ (Z n × Z m ) := G nm ⋊ β (Z n ×Z m ) and call them wreath products of G with Z n and G with (Z n ×Z m ) respectively. More general definition the reader can find in the book [25].
1.3. Structure of the paper. Section 2 is devoted to out main result -Theorem 2.5. Sections 3 and 4 include some preliminary facts about automorphisms of graphs of Morse functions on surfaces, and "combinatorial" structure of such functions on 2-torus. The proof of Theorem 2.5 is contained in Section 5.

Main result
2.1. The class P. First we recall the main result of [10]. For n ∈ N, let P n be a minimal set of isomorphism classes of groups which satisfies the following two conditions: (1) the unit group {1} belongs to P, (2) groups A × B and A ≀ Z n belong to P whenever A, B ∈ P, and n ∈ N. Let also P be a minimal set of isomorphism classes of groups which contains P n as a set for each n ∈ N. The following theorem gives the description of the classes G (M ), G smp (M ) and G gen (M ) for all compact surfaces M = S 2 , T 2 .

2.3.
Classes of groups E 1 , E 2 and E 2 . To describe classes G (T 2 ) and G smp (T 2 ) we introduce three classes E 0 , E 1 and E 2 in the following way. Let E i be a minimal set of isomorphism classes of groups such that So classes P and E 1 in general coincide, but we will consider the class E 1 due to the convenience of a unique presentation of a group G ∈ E 1 in the form G = H ≀ Z n for some n ≥ 1 and H ∈ P.
It is well known that the Kronrod-Reeb graph of Morse functions on 2-torus is either a tree or contains a unique circuit, see Lemma 3.1 in [3]. The class of groups G(f ) for Morse functions f on T 2 whose graph are trees we will denote by G 0 (T 2 ), otherwise, in the case of circuits, by G 1 (T 2 ) 1 . The following theorem is our main result.
Theorem 2.5. The following classes coincide: We prove Theorem 2.5 in Section 5. The proof is divided into three separate cases: for G 0 (T 2 ), G 1 (T 2 ) and G gen (T 2 ).

Automorphisms of graphs of functions on surfaces
In this section we want to show a precise way how the group G(f ) arises from the action of Denote by Aut(Γ f ) the group of homeomorphisms of the graph Γ f . Note that each h ∈ S ′ (f ) preserves level-sets of f . Hence, h ∈ S ′ (f ) induces the homeomorphism ρ(h) of Γ f such that the following diagram is a finite group in Aut(Γ f ); we will denote it by G(f ).

Combinatorial generalities on Morse functions on 2-torus and their graphs
We are interested in the "combinatorial" structure of Morse functions on T 2 , so we will recollect some useful for us results on the structure of such functions. The following lemma holds.  1 [3]). Let f be a Morse function on T 2 and Γ f be its graph. Then Γ f is either a tree or contains a unique circuit.
We describe these two cases separately. 1 Here indexes 0 and 1 correspond to the rank of H 1 (Γ f , Z) 4.2. Γ f contains a circuit. Let Θ be a circuit in Γ f . Let C 0 ⊂ T 2 be a regular connected component of some level set f −1 (c), c ∈ R, and z be a point in Γ f corresponding to C 0 . Obviously, z belongs to the cycle Θ in Γ f , iff C 0 does not separate T 2 . Note that the level-set f −1 (c) consists of a finite number of connected components, and is invariant under the action of any h ∈ S ′ (f ). Let C be the set {h(C 0 ) | h ∈ S ′ (f )} of all images of C 0 under the action of elements from S ′ (f ).
Then the set C consists of a finite number of components {C 0 , C 1 , . . . , C n−1 } of the set f −1 (c) for some n ≥ 1. Curves from C are pairwise disjoint, and since C 0 does not separate T 2 , it follows that each C i also does not separate T 2 . Then C i and C i+1 bounds a cylinder Q i such that the interior of Q i does not intersects with C. Note that the group Z n freely acts on the set of cylinders {Q i } by cyclic permutations. More about combinatorial description of Morse functions on 2-torus whose graphs contain circuits the reader can find in [23,22] [3]). Let f be a Morse function on T 2 , and Γ f be its graph.
(1) If Γ f contains a unique circle, then the group Z n freely acts on the set of cylinders {Q i } n−1 i=0 and there is an isomorphism G(f ) = G(f | Q0 ) ≀ Z n , where n is a cyclic index of f ,and Q 0 is a cylinder bounded by parallel curves C 0 and C 1 .
It is well known that χ(T 2 ) = 0, so Morse equality for Morse functions on T 2 has the form: c 0 (f ) + c 2 (f ) = c 1 (f ).

Proof of Theorem 2.5
Obviously that the inclusion G i (T 2 ) ֒→ E i for i = 0, 1 directly follows from Lemma 4.5 because of the structure of the class E i . So to prove Theorem 2.5 we have to establish the reverse inclusion E i ֒→ G i (T 2 ) for i = 0, 1. In other words we need to prove that for each A ∈ E i there exists a Morse function f on T 2 such that A ∼ = G(f ). These will be done bellow in Subsections 5.1 and 5.2. The case of simple Morse functions will be considered in Subsection 5.3. 5.1. Case 1: Functions, whose graphs have circuits. Let A be a group from E 1 . So the group A has the form B ≀ Z n for some B ∈ P and some n ≥ 1. We divide the proof of the inclusion E 1 ֒→ G 1 (T 2 ) into two steps. First we have to define a Morse function function f 0 : T 2 → R such that Γ f0 contains a circuit and G(f 0 ) ∼ = Z n for n as above. Then for the given group B we change f 0 on the neighborhood of maxima of f 0 to obtain another Morse function f : Such Morse function f 0 one can define by the following procedure. Let Q i ∼ = S 1 × [0, 1], i = 0, . . . , n − 1 be a cylinder and g i : Q i → R be a Morse function such that g i has one maximum l i , one minimum p i , two saddles s i1 , s i2 and g(∂Q i ) = 0, g(l i ) = 1, g(p i ) = −1, g(s ij ) = ±1/2, j = 1, 2.
Consider the same orientation on each Q i ; it canonically induces the orientations of the connected components of the boundary If n = 1 attach the boundaries of Q 0 by identity diffeomorphism. The resulting surface is a 2-torus. Since the values of g on ∂Q 0 0 and ∂Q 1 0 coincides, it follows that g 0 induces a unique Morse function f 0 on T 2 . If n ≥ 1 attach all Q i together in cyclic order by identity diffeomorphism of ∂Q 1 i → ∂Q 0 i+1 where the index i takes modulo n. The resulting surface is obviously a 2-torus, and since the values g i on connected components of the boundary of Q i coincides, it follows that g 0 induces a unique smooth function f 0 on T 2 such that Γ f0 contains a circuit and G(f 0 ) ∼ = Z n . Points l i , p i and s ij are the corresponding maxima, minima and saddle points of f 0 , i = 0, . . . n − 1, j = 1, 2.
Let D i be a regular neighborhood of maximum l i which does not contain other critical points. For the given group B by Theorem 2.2 there exists a smooth function f i on D i such that G(f i ) ∼ = B, where B ∈ P. Next we change the function f 0 on D i by replacing f 0 | Di to f i on D i ; the resulting function we denote by f . By Lemma 4.5, f is such that G(f ) ∼ = B ≀ Z n . So we proved that the inclusion E 1 ֒→ G 1 (T 2 ) holds.

5.2.
Case 2: Functions, whose graphs are trees. Let A be a group from E 0 . We need to show that there exists a Morse functions f on T 2 such that Γ f is a tree and G(f ) ∼ = A. From the definition of the class E 0 there exist n, m ≥ 1 and B ∈ P such that A = B ≀ (Z n × Z mn ). As in the Case 1 we divide our proof into two steps. First for a given n, m ≥ 1 we define a Morse function f 0 : T 2 → R such that Γ f0 is a tree and G(f 0 ) = Z n × Z mn , and finally for the given group B we change the function f 0 to obtain the Morse function f : To realize the first step we need some preliminaries. Let γ : R 2 × Z 2 → R 2 be a free action of Z 2 given by the formula γ((x, y), (1, 1)) = (x + 2, y + 2). For n, m ≥ 1 as above this action induce a free action δ of the subgroup nZ × mnZ of Z 2 by the formula δ((x, y), (1, 1)) = (x + 2n, y + 2mn). Note that the rectangle P = [0, 2n] × [0, 2mn] is a fundamental domain for the action δ, and the quotient space of R 2 /δ is a 2-torus. Let D kl be a square [k, k + 1] × [l, l + 1] ⊂ R 2 , k, l ∈ Z, and q : R 2 → R 2 /δ be the projection map, and B be the set of images of D kl with respect to q. The action δ induces the free action σ on Z n × Z mn on the set B. The number of orbits of this action is equal to Then there exists a double-periodic Morse function g 0 : R 2 → R such that g 0 • δ = g 0 , and on P it satisfies: • g 0 ({k, l}) = 0 is a saddle point, k = 0, 1, . . . , 2n − 1, l = 0, 1, . . . , 2mn − 1, • g 0 has a unique maximum at m r00 for r = 1, 2, and a unique minimum for r = 3, 4 and such that g 0 (m 100 ) = 1, g 0 (m 200 ) = 2, g 0 (m 300 ) = −1, g 0 (m 400 ) = −2.
It is easy to see that g 0 induces a unique Morse function on T 2 = R 2 /δ, which we denote by f 0 . By construction, Γ f0 is a tree, and G(f 0 ) = Z n × Z mn . It remains to modify f 0 using the group B as above. Since the action σ on B has 4 orbits, choose one of them, say when r = 1. By Theorem 2.2 there exists a Morse function f 1ij on D 1ij such that G(f 1ij ) ∼ = B for all i = 0, . . . , n − 1 and j = 0, . . . , mn − 1. We change the function f 0 on D 1ij such that f 0 | D1ij = f 1ij , and the resulting functions we will denote by f . Note that G(f | Drij ) = 1 for r = 2, 3, 4 and i = 0, 1 . . . , n − 1, j = 0, 1, . . . , nm − 1 by definition of f 0 . Then by Lemma 4.5 So we proved that the inclusion E 0 ֒→ G 0 (T 2 ) holds.

Case 3: Simple Morse functions.
It is easy to see that there exist simple Morse functions on T 2 in the case when its graph contains a unique circuit. Indeed, a Morse function f on T 2 such that Γ f contains a unique circuit is simple if the restriction f | Qi is simple for each i = 0, 1 . . . , n. The following lemma shows that this is the only case.
Lemma 5.4. Let f be a simple Morse function on T 2 . Then Γ f is not a tree.
Proof. Assume that f is simple and Γ f is a tree. Then there exists a unique special vertex v of the tree Γ f . Let V = p −1 f (v) be a special component of critical level-set of f which corresponds to v. Note that the special component V contains "many" saddles of f . Let also N be a regular neighborhood of V which consists of level-sets of f and does not contain other critical points. Since v is a special vertex, it follows that T 2 − N is a disjoint union of 2-disks, say {D i } n i=1 for some n ∈ N. Note that the restriction f | Di for all i = 0, 1, . . . n is also a simple Morse function since f is simple.
Next we change the function f in the following way: replace f | Di by the function on 2-disk D i which has the only one critical point, i = 1, . . . , n; the resulting function we denote by g. After these changes the function g satisfies the following conditions: (1) Γ g is also a tree, since all changes of f were on the connected components of the complement of N being 2-disks, (2) V is also a critical level-set of g, but all saddles of g belong to V , (3) g is simple since f was simple. Since g is simple (by assumption f and by construction), it follows that the number of saddles belonging to V must be equal to 1, and so c 1 (g) = 1. From Morse equality (Theorem 4.7) we have c 1 (g) = c 0 (g) + c 2 (g), and so 1 = c 0 (g) + c 2 (g). This is an inconsistency. Hence if f is simple, then Γ f is not a tree. Now we need to show that the classes G smp (T 2 ) and E 2 coincide. First we show the inclusion G smp (T 2 ) ֒→ E 2 holds. Let f be a simple Morse function on T 2 . By Lemma 5.4 the graph Γ f has a unique circle. The restriction f | Q0 is simple Morse function on cylinder Q 0 so G(f | Q0 ) belongs to the class P 2 . By (1) Lemma 4.5 the group G(f ) is isomorphic to G(f | Q0 ) ≀ Z n for some n which depends on the function f , and so G(f ) belongs to E 2 by definition of the class E 2 .
The reverse inclusion E 2 ֒→ G smp (T 2 ) follows from the procedure defined in Case 1 with B ∈ P 2 as sub-case. Theorem is proved.