Deformations of smooth functions on $2$-torus

TL;DR

This paper computes the structure of groups related to Morse functions on the 2-torus, revealing their algebraic properties and how they encode symmetries and deformations of these functions.

Contribution

It explicitly determines the groups of $f$-preserving diffeomorphisms and their automorphisms for Morse functions on the 2-torus, extending understanding of their algebraic and topological structure.

Findings

Computed $oldsymbol{ ext{pi}_0 ext{S'}(f)}$ for Morse functions on $T^2$

Determined the automorphism group $oldsymbol{G(f)}$ for these functions

Analyzed the subgroup $oldsymbol{ ext{pi}_0 ext{Delta'}(f)}$ and its relation to the other groups

Abstract

Let $f $ be a Morse function on a smooth compact surface $M$ and $\mathcal{S}'(f)$ be a group of $f$-preserving diffeomorphisms of $M$ which are isotopic to the identity map. Let also $G(f)$ be a group of automorphisms of the graph of $f$ induced by elements from $\mathcal{S}'(f)$, and $\Delta'$ be a subgroup of $\mathcal{S}'(f)$ of diffeomorphisms which trivially act on the graph of $f$ and are isotopic to the identity map. The group $\pi_0\mathcal{S}'(f)$ can be viewed as an analogue of a mapping class group for $f$-preserved diffeomorphisms of $M$. Groups $\pi_0\Delta'(f)$ and $G(f)$ can be viewed as groups which encode `combinatorially trivial' and `combinatorially nontrivial' counterparts of $\pi_0\mathcal{S}'(f)$ respectively. In the paper we compute groups $\pi_0\mathcal{S}'(f)$, $G(f)$, and $\pi_0\Delta'(f)$ for Morse functions on $2$-torus $T^2$.

Equations

$$\mathcal{S}(f,X)=\{h\in \mathcal{D}(M,X)\mid f\circ h=f\},$$

$$\mathcal{O}(f,X)=\{f\circ h\mid h\in \mathcal{D}(M,X)\}$$

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$${\partial }_1:{\pi }_1{\mathcal{O}}_f(f,X)⟶{\pi }_0{\mathcal{S}}^{\prime }(f,X)$$

$$f=\hat{f}\circ p_f:$$

$$M\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\rho :\mathcal{S}(f,X)\to \mathrm{Aut}({\Gamma }_f).$$

$${\pi }_0{\Delta }^{\prime }(f,X)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\pi }_1{\mathcal{D}}_{\mathrm{id}}(M,X)\times {\pi }_0{\Delta }^{\prime }(f,X)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\iota }_1(\alpha ,\varphi )=[f\circ h^t\circ {\varphi }^t]\in {\pi }_1{\mathcal{O}}_f(f,X).$$

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$$\displaystyle\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 8.65001pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&&\crcr}}}\ignorespaces{\hbox{\kern-8.65001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{A_{1}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 8.65001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 15.44421pt\raise 5.95277pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.6639pt\hbox{$\scriptstyle{i_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 32.65001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 32.65001pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{\ A_{2}\ \ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 61.18909pt\raise 5.1875pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{p_{1}}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 78.95003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern-3.0pt\lower 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 78.95003pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 1.0pt\raise 0.0pt\hbox{$\textstyle{\ A_{3}}$}}}}}}}\ignorespaces}}}}\ignorespaces,$$

$$B_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$A_1\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\displaystyle\bigl{(}(g_{i})_{i=0}^{n-1},a\bigr{)}$$

$$\displaystyle\bigl{(}(g_{ij})_{i,j=0}^{n-1,m-1},(a,b)\bigr{)}$$

$$\displaystyle\bigl{(}(g_{ij})_{i,j=0}^{n-1,m-1},(a^{\prime},b^{\prime})\bigr{)}$$

$$(g,a)\cdot (g^{\prime },a^{\prime })$$

$$(g,(a,b))\cdot (g^{\prime },(a^{\prime },b^{\prime }))$$

$$(g,(c,d))\cdot (g^{\prime },(c^{\prime },d^{\prime }))$$

$${\pi }_1({\Gamma }_f,p_f(x))\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

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0.0pt\hbox{$\textstyle{\mathcal{S}_{\mathbb{D}}\wr(\mathbb{Z}_{n}\times\mathbb{Z}_{nm})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 154.81044pt\raise-61.45609pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\rho}$}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 164.42953pt\raise-71.24052pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\lower-3.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern-3.0pt\raise-81.74052pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 85.7921pt\raise-81.74052pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 133.15076pt\raise-81.74052pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{$\textstyle{G_{\mathbb{D}}\wr(\mathbb{Z}_{n}\times\mathbb{Z}_{nm})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$$

$${\Delta }_{\mathbb{D}}$$

$${\pi }_1{\mathcal{D}}_{\mathrm{id}}(T^2)\times ({\Delta }_{\mathbb{Y}}^n)/H\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Delta }_{\mathbb{Y}}$$

$${\mathcal{S}}_{\mathbb{Y}}$$

$$G_{\mathbb{Y}}$$

$$(\underset{n}{\underbrace{(E_1,m_1,E_2,m_2,\dots ,E_k,m_k),\dots ,(E_1,m_1,E_2,m_2,\dots ,E_k,m_k)}}),$$

$$\gamma (D_{ijk})=D_{i,j+a\mathrm{mod}n,k+b\mathrm{mod}nm}.$$

$$\mathcal{C}=\{h(C_0)\mid h\in {\mathcal{S}}^{\prime }(f)\}$$

$${\pi }_0{\Delta }^{\prime }(f)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$${\Delta }_{\mathbb{Y}}$$

$${\mathcal{S}}_{\mathbb{Y}}$$

$$G_{\mathbb{Y}}$$

Full text

1. Introduction

Homotopy properties of Morse functions on surfaces were studied by many authors. E.g. connected components of the space of Morse functions were computed in the unpublished paper by H. Zieschang, by S. Matveev in the paper by E. Kudryavtseva [6], and V. Sharko [20], cobordism groups of the space of Morse functions on surfaces were described by K. Ikegami and O. Saeki [4], and B. Kalmar [5]. Homotopy groups of stabilizers and orbits of Morse functions on surfaces with respect to the action of diffeomorphism groups were studied by S. Maksymenko [10, 11, 12, 14, 15] and E. Kudryavtseva [7, 8]. We will give an overview of these results.

Let $M$ be a smooth compact surface and $X$ be a closed (possible empty) subset of $M$. The group $\mathcal{D}(M,X)$ of diffeomorphisms fixed on some neighborhood of $X$ acts on the space of smooth functions $C^{\infty}(M)$ by the rule: $C^{\infty}(M)\times\mathcal{D}(M,X)\to C^{\infty}(M)$, $(f,h)\mapsto f\circ h$. With respect to this action we denote by

[TABLE]

the stabilizer and the orbit of $f\in C^{\infty}(M)$ respectively. Endow strong Whitney $C^{\infty}$-topologies on $C^{\infty}(M)$ and $\mathcal{D}(M,X)$. Then for a $f\in C^{\infty}(M)$ these topologies induce some topologies on $\mathcal{S}(f,X)$ and $\mathcal{O}(f,X)$. We denote by $\mathcal{D}_{\mathrm{id}}(M,X)$, $\mathcal{S}_{\mathrm{id}}(f,X)$, and $\mathcal{O}_{f}(f,X)$ connected components of the identity map $\mathrm{id}_{M}$ of $\mathcal{D}(M,X)$, $\mathcal{S}(f,X)$, and the component of $\mathcal{O}(f,X)$ containing $f$ respectively. If $X=\varnothing$ we will omit the symbol ‘‘$\varnothing$ ’’ from our notations, *i.e. *, set $\mathcal{D}(M):=\mathcal{D}(M,\varnothing)$, $\mathcal{S}(f):=\mathcal{S}(f,\varnothing)$, $\mathcal{O}(f):=\mathcal{O}(f,\varnothing)$, and so on.

By a Morse function $f$ on a $M$ we will mean a smooth function which satisfies the following conditions:

• •

all critical points of $f$ are non-degenerate and belong to the interior of $M$;

• •

the function $f$ takes constant values on each boundary component of $M$.

Notice that if $N\subset M$ is a subsurface whose boundary components are regular components of some level-sets of a Morse function $f:M\to\mathbb{R}$, then the restriction $f|_{N}$ is a Morse function as well in the sense of the above definition.

Theorem 1.1**.**

[19, 10, 14, 13].* Let $f$ be a Morse function on a smooth compact surface $M$, and $X$ be a closed (possibly empty) subset of $M$ consisting of finitely many connected components of some level-sets of $f$ and some critical points of $f$. Then the following statements hold.*

1. (1)

The map $p:\mathcal{D}_{\mathrm{id}}(M,X)\to\mathcal{O}(f,X)$ defined by $p(h)=f\circ h$ is a Serre fibration with the fiber $\mathcal{S}(f,X)$. Hence $p(\mathcal{D}_{\mathrm{id}}(M))=\mathcal{O}_{f}(f)$, and the restriction $p|_{\mathcal{D}_{\mathrm{id}}(M)}:\mathcal{D}_{\mathrm{id}}(M)\to\mathcal{O}_{f}(f,X)$ is also a Serre fibration with the fiber $\mathcal{S}^{\prime}(f,X)=\mathcal{S}(f)\cap\mathcal{D}_{\mathrm{id}}(M,X)$. 2. (2)

$\mathcal{O}_{f}(f,X)=\mathcal{O}_{f}(f,X\cup\partial M)$, so $\pi_{k}(\mathcal{O}_{f}(f,X))=\pi_{k}(\mathcal{O}_{f}(f,X\cup\partial M))$ for $k\geq 1$. 3. (3)

Suppose that either $f$ has a saddle point or $M$ is a non-orientable surface. Then $\mathcal{S}_{\mathrm{id}}(f)$ is contractible, $\pi_{k}\mathcal{O}_{f}(f)=\pi_{k}M$, $k\geq 3$, $\pi_{2}\mathcal{O}_{f}(f)=0$, and for $\pi_{1}\mathcal{O}_{f}(f)$ we have the following short exact sequence of groups:

[TABLE]

Moreover the group $p_{1}(\pi_{1}\mathcal{D}_{\mathrm{id}}(M))$ is contained in the center of $\pi_{1}\mathcal{O}_{f}(f)$. 4. (4)

If $\chi(M)<|X|$, then $\mathcal{D}_{\mathrm{id}}(M,X)$ is contractible, $\pi_{k}\mathcal{O}_{f}(f,X)=0$ for $k\geq 2$, and the boundary map

[TABLE]

is an isomorphism.

We recall the definition of the map $\partial_{1}$. Let $\omega:[0,1]\to\mathcal{O}_{f}(f)$, $\omega_{0}=\omega_{1}$ be a loop in $\mathcal{O}_{f}(f)$ based in $f$. Since $p$ is a Serre fibration, it follows that there exists an isotopy $h:M\times[0,1]\to M$ such that $\omega_{t}=f\circ h_{t}$, $h_{0}=\mathrm{id}$, and $h_{1}\in\mathcal{S}^{\prime}(f)$, *i.e. *, $h_{1}$ is such that $f\circ h_{1}=f$. Then the map $\partial_{1}$ is defined by the formula $\partial([\omega])=[h_{1}]\in\pi_{0}\mathcal{S}^{\prime}(f)$.

1.2. Automorphisms of graphs of functions on surfaces

Let $f:M\to\mathbb{R}$ be a Morse function on a smooth compact oriented surface $M$ and $c$ be a real number. A connected component $C$ of the level-set $f^{-1}(c)$ is called critical, if $C$ contains at most one critical point of $f$, otherwise $C$ is regular. Let $\Xi$ be a partition of $M$ into connected components of level-sets of $f$. It is well known that the quotient-space $\Gamma_{f}=M/\Xi$ has a structure of a $1$-dimensional CW complex called the graph of $f$, or Kronrod-Reeb graph of $f$. Let also $p_{f}:M\to\Gamma_{f}$ be a projection map. Then $f$ can be represented as the composition:

[TABLE]

Denote by $\mathrm{Aut}(\Gamma_{f})$ the group of homeomorphisms of the graph $\Gamma_{f}$. Note that each $h\in\mathcal{S}(f,X)$ preserves level-sets of $f$. Hence, $h$ induces the homeomorphism $\rho(h)$ of $\Gamma_{f}$ such that the following diagram

[TABLE]

commutes, and the correspondence $h\mapsto\rho(h)$ is a homomorphism

[TABLE]

One can check that the image $\rho(\mathcal{S}(f,X))$ is a finite subgroup in $\mathrm{Aut}(\Gamma_{f})$. The image $\rho(\mathcal{S}^{\prime}(f,X))$ in $\mathrm{Aut}(\Gamma_{f})$ will be denoted by $G(f,X)$.

Let $\Delta(f,X)$ be the normal subgroup of $\mathcal{S}(f,X)$ consisting of diffeomorphisms which leave invariant every connected component of each level set of $f$, and $\Delta^{\prime}(f,X)$ be the following intersection $\Delta(f,X)\cap\mathcal{D}_{\mathrm{id}}(M,X)$. It is known that $\pi_{0}\Delta^{\prime}(f,X)$ is a free abelian group and $\mathrm{Ker}\rho=\pi_{0}\Delta^{\prime}(f,X)$. So the following sequence of groups is exact

[TABLE]

see [15, Section 4]. From [12, Theorem 5.2], we have another short exact sequence

[TABLE]

in which $\iota_{1}$ is defined as follows. Let $\alpha\in\pi_{1}\mathcal{D}_{\mathrm{id}}(M,X)$ be an element represented by some loop $\{h^{t}:M\to M\}_{t\in[0,1]}$ in $\mathcal{D}_{\mathrm{id}}(M,X)$ such that $h^{0}=h^{1}=\mathrm{id}_{M}$, and $\phi\in\Delta^{\prime}(f,X)=\Delta(f)\cap\mathcal{D}_{\mathrm{id}}(M,X)$. Fix any isotopy $\{\phi^{t}:M\to M\}_{t\in[0,1]}$ between $\mathrm{id}_{M}=\phi^{0}$ and $\phi=\phi^{1}$. Then

[TABLE]

1.3. Main diagram

Thus for a given Morse function $f$ on a smooth compact oriented surface $M$ we considered several spaces associated with $f$. If $M\neq S^{2}$, then all non-trivial homotopy information is encoded in the following commutative diagram:

[TABLE]

where $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are projections on the first and the second factor. In diagram (1.3.1) horizontal, vertical and diagonal sequences coincide with sequences (1.1.1), (1.2.1), and (1.2.2) respectively.

Let

[TABLE]

be two exact sequences of groups. Recall that sequences (1.3.4) and (1.3.7) are isomorphic if there exist isomorphisms $\phi=\{\phi_{i}:A_{i}\to B_{i}$, $i=1,2,3\}$ such that the following diagram commutes

[TABLE]

Similarly one can define the notion of an isomorphism for commutative diagrams.

The main aim of the paper is to describe diagram (1.3.1) for Morse functions on $2$-torus up to an isomorphism.

1.4. Acknowledgments

The author is grateful to Sergiy Maksymenko for useful discussions.

1.5. Structure of the paper

Section 2 collects definitions of wreath products, which we need to state our main result, Theorem 3.2. We recall some known results about Morse functions on $2$-torus and their graphs in Section 4, and the fundamental groups of such functions is described in Section 6. Section 5 contains some facts needed for the proofs of our results, and we will prove Theorem 3.2 in Sections 7 and 8.

2. Wreath products

To state our results we need special kinds of wreath products of groups with cyclic groups which we describe below. Let $G$ be a group and $n,m\geq 1$ be integers. We will consider the following wreath products:

• •

$G\wr_{n}\mathbb{Z}:=G^{n}\rtimes_{\alpha}\mathbb{Z}$,

• •

$G\wr\mathbb{Z}_{n}:=G^{n}\rtimes_{\beta}\mathbb{Z}_{n}$,

• •

$G\wr_{n,m}\mathbb{Z}^{2}:=G^{{nm}}\rtimes_{\gamma}\mathbb{Z}^{2}$,

• •

$G\wr(\mathbb{Z}_{n}\times\mathbb{Z}_{m}):=G^{{nm}}\rtimes_{\delta}(\mathbb{Z}_{n}\times\mathbb{Z}_{m})$,

where $\alpha:G^{n}\times\mathbb{Z}\to G^{n}$ and $\beta:G\times\mathbb{Z}_{n}\to G^{n}$ correspond to a non-effective $\mathbb{Z}$-action and an effective $\mathbb{Z}_{n}$-action on $G^{n}$ by cyclic shifts of coordinates defined by formulas:

[TABLE]

where all indexes are taken modulo $n$, $g_{i}\in G$, $a\in\mathbb{Z}$, $b\in\mathbb{Z}_{n}$. Similarly $\gamma:G^{{nm}}\times\mathbb{Z}^{2}\to G^{{nm}}$ and $\delta:G^{{nm}}\times(\mathbb{Z}_{n}\times\mathbb{Z}_{m})\to G^{{nm}}$ correspond to a non-effective $\mathbb{Z}^{2}$-action and an effective $\mathbb{Z}_{n}\times\mathbb{Z}_{m}$-action on $G^{{nm}}$ by cyclic shifts of the corresponding coordinates defined by formulas

[TABLE]

where the indexes $i$ and $j$ takes modulo $n$ and $m$ respectively, $(a,b)\in\mathbb{Z}^{2}$, $(a^{\prime},b^{\prime})\in\mathbb{Z}_{n}\times\mathbb{Z}_{m}$.

So $G\wr_{n}\mathbb{Z}$ and $G\wr\mathbb{Z}_{n}$ are direct products of sets $G^{n}\times\mathbb{Z}$ and $G^{n}\times\mathbb{Z}_{n}$ with the following multiplications

[TABLE]

for $g,g^{\prime}\in G^{n}$, $a,a^{\prime}\in\mathbb{Z}$, and $b,b^{\prime}\in\mathbb{Z}_{n}$. Similarly $G\wr_{n,m}\mathbb{Z}^{2}$ and $G\wr(\mathbb{Z}_{n}\times\mathbb{Z}_{m})$ are direct products of sets $G^{nm}\times\mathbb{Z}^{2}$ and $G^{nm}\times\mathbb{Z}_{n}\times\mathbb{Z}_{m}$ respectively with multiplications

[TABLE]

for $g,g^{\prime}\in G^{{nm}}$, $a,a^{\prime}b,b^{\prime}\in\mathbb{Z}$, $c,c^{\prime}\in\mathbb{Z}_{n}$, and $d,d^{\prime}\in\mathbb{Z}_{m}$.

The general definition of wreath product and its properties the reader can find in [18].

3. Main result

Let $f$ be a Morse function on $T^{2}$, $\Gamma_{f}$ be its graph, and $p_{f}:T^{2}\to\Gamma_{f}$ be the projection map induced by $f$.

Lemma 3.1**.**

The map $p_{f}^{*}:\pi_{1}T^{2}\to\pi_{1}\Gamma_{f}$ induced by $p_{f}$ is an epimorphism with a nonzero kernel. Hence the graph of $f$ is either a tree, or has a unique circuit.

Proof.

Let $x\in T^{2}$ be an arbitrary point. Note that $\pi_{1}(\Gamma_{f},p_{f}(x))=F_{k}$, where $F_{k}$ is a free groups of some rank $k$ equals the number of independent cycles in $\Gamma_{f}$, and $\pi_{1}(T^{2},x)$ is isomorphic to $\mathbb{Z}^{2}$. It is easy to see that there exists a map $s:(\Gamma_{f},p_{f}(x))\to(T^{2},x)$ such that the composition $p_{f}\circ s$ is homotopic to the identity map $\mathrm{id}_{\Gamma_{f}}$ relatively base point. Then the composition

[TABLE]

is the identity isomorphism.

Moreover $p_{f}^{*}$ is a surjective map, and $\pi_{1}(\Gamma_{f},p_{f}(x))=F_{k}$ is a subgroup of the group $\pi_{1}(T^{2},x)=\mathbb{Z}^{2}$. Since $\pi_{1}(T^{2},x)$ is commutative, it follows that $F_{k}$ as a subgroup of $\pi_{1}(T^{2},x)$ is also a commutative group. This is possible only when $k=0$ or $k=1$. So $\Gamma_{f}$ is either a tree if $k=0$, or $\Gamma_{f}$ has a unique circuit if $k=1$. In both of these cases $p_{f}^{*}$ has a nonzero kernel. ∎

Denote by $\mathscr{F}_{0}$ the class of Morse functions on $T^{2}$ whose graphs are trees, and by $\mathscr{F}_{1}$ the class of Morse functions on $T^{2}$ whose graphs contain circuits111Thus the index $i$ in $\mathscr{F}_{i}$ refers to the rank of homology group $H_{1}(\Gamma_{f},\mathbb{Z})$..

Our main result is the following theorem.

Theorem 3.2**.**

Let $f$ be a Morse function on $T^{2}$ and $\Gamma_{f}$ be its graph.

1. (1)

Assume that $f$ belongs to $\mathscr{F}_{0}$. Then there exists a set of mutually disjoint $2$-disks $\mathbb{D}=\{D_{i}\}_{i=1}^{r}\subset T^{2}$ for some $r\geq 1$ such that each restriction $f|_{D_{i}}:D_{i}\to\mathbb{R}$, $i=1,\ldots,r$ is also a Morse function, and the diagram (1.3.1) is isomorphic to

[TABLE]

for some $n,m\in\mathbb{N}$, where

[TABLE] 2. (2)

Assume that $f$ belongs to $\mathscr{F}_{1}$. Then there exists a set of mutually disjoint $2$-disks $\mathbb{Y}=\{Y_{ij}\}_{i=0,\ldots,k}^{j=0,\ldots,c_{i}}\subset T^{2}$ for certain $k,c_{i}\in\mathbb{N}$, and $m_{i}\in\mathbb{N}$, $i=0,\ldots,k-1$, such that diagram (1.3.1) is isomorphic to

[TABLE]

for some $n\in\mathbb{N}$, where

[TABLE]

and the group $H$ is a normal subgroup of $\Delta_{\mathbb{Y}}^{n}$ isomorphic to $\mathbb{Z}$ and generated by the element

[TABLE]

where $E_{i}$ is the unit of the group $(\prod_{j=1}^{c_{i}}\Delta_{Y_{ij}})^{m_{i}}$, $i=0,1\ldots,k$. The group $H$ is contained in the center of $\mathcal{S}_{\mathbb{Y}}\wr_{n}\mathbb{Z}$ via ${j_{0}}(h)=(h,0)$ for $h\in H$.

Remark 3.3*.*

1. (1)

It is known that the inclusion $T^{2}\hookrightarrow\mathcal{D}_{\mathrm{id}}(T^{2})$ is a homotopy equivalence, see [1, 3]. So, the group $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ is isomorphic to $\mathbb{Z}^{2}$. 2. (2)

S. Maksymenko showed that the group $\pi_{1}\mathcal{O}_{f}(f)$ is a subgroup of the braid group of $T^{2}$, see for example [15, Theorem 1.1]. The element (3.2.2) can be regarded as a certain analogue of the Garside element in this braid group. 3. (3)

In a series of papers [17, 9, 16, 2] S. Maksymenko and the author described groups $\pi_{1}\mathcal{O}_{f}(f)$ for functions on $2$-torus. We shortly review these results in Section 6. 4. (4)

Since the structure of groups $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{1}\mathcal{O}_{f}(f)$ are known, it follows that for description of diagram (1.3.1) we need to find a structure of groups $\pi_{0}\mathcal{S}^{\prime}(f)$, $G(f)$, and $\pi_{0}\Delta^{\prime}(f)$.

4. Functions on $2$-torus, their graphs, and homotopy properties

In this section we give a short overview of known results about graphs of smooth functions on $2$-torus, see [9, 17, 16, 2]. Let $f$ be a Morse function on $T^{2}$, $\Gamma_{f}$ be its graph, and $p_{f}:T^{2}\to\Gamma_{f}$ be the projection map induced by $f$.

Case 1: Functions from $\mathscr{F}_{0}$

The following lemma holds.

Lemma 4.1**.**

[9, Proposition 1].* Let $f$ be a function from $\mathscr{F}_{0}$, and $\Gamma_{f}$ be its graph. Then there exits a unique vertex $v$ in $\Gamma_{f}$ such that each component of the complement $T^{2}\setminus p_{f}^{-1}(v)$ is an open $2$-disk. Such a vertex $v$ of $\Gamma_{f}$ is called special. ∎*

Let $v$ be a special vertex of $\Gamma_{f}$, $G_{v}$ be the stabilizer of $v$ with respect to the action of the group $G(f)$ acting on $\Gamma_{f}$. Since $f$ has a unique special vertex, it follows that $G_{v}$ coincides with $G(f)$. Let $\mathrm{st}(v)$ be a $G_{v}$-invariant connected neighborhood of $v$ containing no other vertices of $\Gamma_{f}$. The set $G_{v}^{loc}=\{g|_{\mathrm{st}(v)}\mid g\in G_{v}\}$ consisting of restrictions of elements of $G_{v}$ onto $\mathrm{st}(v)$ is a subgroup of $\mathrm{Aut}(\mathrm{st}(v))$. We will call $G_{v}^{loc}$ the local stabilizer of $v$. Let also $r:G_{v}\to G_{v}^{loc}$ be the restriction map.

Lemma 4.2**.**

[2, Theorem 2.5].* Let $f\in\mathscr{F}_{0}$, and $v$ be the special vertex of $\Gamma_{f}$.*

1. (1)

Then $G_{v}^{loc}$ is isomorphic to $\mathbb{Z}_{n}\times\mathbb{Z}_{nm}$ for some $n,m\in\mathbb{N}$. 2. (2)

There exists a section $s:G_{v}^{loc}\to\mathcal{S}^{\prime}(f)$ of the map $r\circ\rho$ such that $s(G_{v}^{loc})$ freely acts on $T^{2}$, so the map $q:T^{2}\to T^{2}/s(G_{v}^{loc})$ is a covering projection. Moreover the space of orbits $T^{2}/s(G_{v}^{loc})$ is also diffeomorphic to $T^{2}$.

Remark 4.3*.*

1. (i)

Throughout the paper by $G_{v}^{loc}$-action on $T^{2}$ we will mean its action via the section $s$ from (2). So we will write $T^{2}/G_{v}^{loc}$ instead of $T^{2}/s(G_{v}^{loc})$. 2. (ii)

Let $V=p_{f}^{-1}(v)$ be a critical component of a level set of $f$ which corresponds to the special vertex $v$. Since $G_{v}^{loc}$ freely acts on $T^{2}$, it follows that connected components of $T^{2}\setminus p_{f}^{-1}(v)$ can be enumerated by three indexes $D_{ijk}$, $i=1,2,\ldots,r,$ $j=0,\ldots,n-1$, and $k=0,\ldots,{nm}-1$. So, if $\gamma=(a,b)\in G_{v}^{loc}=\mathbb{Z}_{n}\times\mathbb{Z}_{{nm}}$ then

[TABLE] 3. (iii)

The set $\mathbb{D}=\{D_{i00}\}_{i=1}^{r}$ from (1) of Theorem 3.2 will be called a fundamental set of this $G_{v}^{loc}$-action. Numbers $n,m$ and $r$ in (1) of Theorem 3.2 are also determined from (i).

Case 2: Functions from $\mathscr{F}_{1}$

Let $f\in\mathscr{F}_{1}$ and $\Theta$ be a unique circuit in $\Gamma_{f}$. Let also $C_{0}\subset T^{2}$ be a regular connected component of some level set $f^{-1}(c)$, $c\in\mathbb{R}$, such that $z=p_{f}(C_{0})$ is the point belonging to $\Theta$.

Notice that $f^{-1}(c)$ consists of finitely many connected components and is invariant under the action of any $h\in\mathcal{S}(f)$. Let

[TABLE]

be the set of all images of $C_{0}$ under the action of elements from $\mathcal{S}^{\prime}(f)$. Evidently, the curves from $\mathcal{C}$ are pairwise disjoint. Since $C_{0}$ does not separate $T^{2}$, it follows that each $C_{i}$ does not separate $T^{2}$ as well. Two such curves will be called parallel, and the set $\mathcal{C}$ will be called parallel family on $T^{2}$.

We can also assume that they are cyclically enumerated along $T^{2}$, so that $C_{i}$ and $C_{i+1}$ bound a cylinder $Q_{i}$ such that the interior of $Q_{i}$ does not intersect $\mathcal{C}$, where all indexes are taken modulo $n$. The number $n$ from (2) of Theorem 3.2 is the number of curves in $\mathcal{C}$; we will call it the cyclic index of $f$. For other details see [16].

5. Preliminaries

5.1. Homotopy properties of Morse functions on cylinder

The following lemma holds.

Lemma 5.2**.**

[15, Theorem 5.8, Lemma 5.1 III (a)].* Let $f$ be a Morse function on cylinder $Q=S^{1}\times[0,1]$. Then there exists a family of mutually disjoint $2$-disks $\mathbb{Y}=\{Y_{ij}\}_{i=1,\ldots,k}^{j=1,\ldots,c_{i}}\subset Q$ for some $k,c_{i}\in\mathbb{N}$, and $m_{i}\in\mathbb{N}$, $i=1,\ldots,k$ such that*

1. (1)

$f|_{Y_{ij}}$* is also a Morse function for each $j=1,\ldots,c_{i}$, $i=1,\ldots,k$;* 2. (2)

there exists an isomorphism $\zeta=(\zeta_{1},\zeta_{2},\zeta_{3})$ of the following short exact sequences:

[TABLE]

where the first row is a sequence (1.2.1) with $X=\varnothing$, and

[TABLE] 3. (3)

The kernel of the homomorphism $\pi_{0}\mathcal{S}^{\prime}(f,\partial Q)\to\pi_{0}\mathcal{S}^{\prime}(f)$ induced by the inclusion is isomorphic to $\mathbb{Z}$ and is generated by the isotopy class of $\tau_{0}\circ\tau_{1}^{-1}$, where $\tau_{i}\in S^{\prime}(f)$ is a Dehn twist along $S^{1}\times\{i\}$, $i=0,1$. Moreover, in the notation of (2) $[\tau_{0}\circ\tau_{1}^{-1}]$ corresponds to the element

[TABLE]

where $E_{i}$ is the unit of $(\prod_{j=1}^{c_{i}}\Delta_{Y_{ij}})^{m_{i}}$, $i=1,\ldots,k$, **[15, Lemma 5.1 III (a)]**.

5.3. Dehn twists and slides

Let $\alpha,\beta:[-1,1]\to[0,1]$ be two smooth functions such that $\alpha=0$ on $[-1,-1/2]$ and $\alpha=1$ on $[1/2,1]$, while $\beta=0$ on $[-1,-2/3]\cup[2/3,1]$, and $\beta=1$ on $[-1/3,1/3]$.

Let $Q=S^{1}\times[-1,1]$ be a cylinder and $C=S^{1}\times 0$. Define the following two diffeomorphisms of $Q$ by the formulas

[TABLE]

for $(z,t)\in S^{1}\times[-1,1]$. The diffeomorphisms $\tau$ and $\theta$ are called a Dehn twist and a slide along the curve $C$ respectively. Note that $\tau$ is fixed on some neighborhood of $\partial Q$ and $\theta$ is fixed on some neighborhood of $C\cup\partial Q$.

Let $M$ be a smooth surface and $C\subset M$ be a simple closed curve. Suppose that $C$ is a two-sided curve, i.e., $C$ has a regular neighborhood $W$ diffeomorphic to a cylinder $Q$. Fix any diffeomorphism $\phi:Q\to W$ such that $\phi(S^{1}\times 0)=C$. Since $\tau$ is fixed on some neighborhood of $\partial Q$, it is easy to see that $\phi\circ\tau\circ\phi^{-1}:W\to W$ extends by the identity to a unique diffeomorphism $\overline{\tau}$ of $M$. Any diffeomorphism $h:M\to M$ isotopic to $\overline{\tau}$ or $\overline{\tau}^{-1}$ will be called a Dehn twist along $C$.

Similarly $\phi\circ\theta\circ\phi^{-1}:W\to W$ extends by the identity map to the diffeomorphism $\overline{\theta}$ of $M$. Any diffeomorphism $h:M\to M$ fixed on some neighborhood of $C$ supported on some cylindrical neighborhood $W$ of $C$ and isotopic to $\overline{\theta}$ or $\overline{\theta}^{-1}$ relatively to some neighborhood of $C\cup\overline{M\setminus Q}$ will be called a slide along $C$. For more details see [16].

5.4. Groups $\pi_{0}\Delta^{\prime}(f)$ and their special subgroups

Let $f$ be a Morse function from $\mathscr{F}_{1}$ with cyclic index $n$, so there exists an $\mathcal{S}^{\prime}(f)$-invariant family

[TABLE]

of connected components of the same level set of $f$ and neither of those curves separate $T^{2}$, $i=0,\ldots,n-1$. We let denote by $Q_{i}$ the cylinder bounded by curves $C_{i}$ and $C_{i+1\ \mathrm{mod}\ n}$.

A neighborhood $V$ of $C\in\mathcal{C}$ will be called $f$-adapted if

• •

$V$ is diffeomorphic to $S^{1}\times[0,1]$ via a diffeomorphism, say $\phi$,

• •

$\phi^{-1}(S^{1}\times\{t\})$ is a connected component of some level set of $f$, $t\in[0,1]$,

• •

$V$ does not contain critical points of $f$.

Fix an $f$-adapted neighborhood ${V}_{i}$ of $C_{i}$, $i=0,\ldots,n-1$, so that

• – 

${V}_{i}\cap{V}_{j}=\varnothing$ for $i\not=j$;

• – 

for each $i,j$ there exists $h\in\mathcal{S}^{\prime}(f)$ such that $h({V}_{i})={V}_{j}$.

In particular, the union

[TABLE]

is $\mathcal{S}^{\prime}(f)$-invariant.

Notice that by definition the group $\Delta^{\prime}(f,\mathsf{{V}})$ consists of diffeomorphisms of $T^{2}$ which are isotopic to $\mathrm{id}_{T^{2}}$, fixed on an $f$-adapted neighborhood $\mathsf{{V}}$ of $\mathcal{C}$ and induce trivial homeomorphisms of $\Gamma_{f}$. Let $j:\Delta^{\prime}(f,\mathsf{{V}})\to\Delta^{\prime}(f)$ be the natural inclusion. It is known and is easy to see that the homomorphism

[TABLE]

induced by $j$ is an epimorphism, see [15, Lemma 5.1]. Let also ${W}_{i}$ be an $f$-adapted neighborhood of $C_{i}$ satisfying ${V}_{i}\subset\mathrm{Int}{W}_{i}$ for $i=0,1\ldots,n-1$. Put $\mathsf{{W}}=\bigcup_{i=0}^{n-1}{W}_{i},$ Then, [15, Corollary 7.2], the natural inclusion

[TABLE]

is the homotopy equivalence and groups $\pi_{0}\Delta^{\prime}(f)$ and $\pi_{0}\Delta^{\prime}(f,\mathsf{{V}})$ are abelian groups.

Recall that a vector field $F$ on a smooth oriented surface $M$ is called Hamiltonian-like for a Morse function $f$ if the following conditions hold:

• •

singular points of $F$ correspond to critical points of $f$,

• •

$f$ is constant along $F$,

• •

Let $z$ be a critical point of $f$. Then there exists a local coordinate system $(x,y)$ such that $f(z)=0$, $f(x,y)=\pm x^{2}\pm y^{2}$ near $z$, and in these coordinates $F$ has the form $F(x,y)=-f^{\prime}_{y}\frac{\partial}{\partial x}+f^{\prime}_{x}\frac{\partial}{\partial x}$.

By [10, Lemma 5.1] for every Morse function $f:M\to\mathbb{R}$ there exists a Hamiltonian-like vector field of $f$ if $M$ is smooth and orientable surface.

Fix a Hamiltonian-like vector field $F$ for the given function Morse function $f$ on $T^{2}$, and let $\mathbf{F}$ be the flow of $F$. Then $\mathsf{{W}}$ is $\mathbf{F}$-invariant and consists of periodic orbits. Therefore one can assume that periods of all trajectories of $\mathbf{F}$ is equal to $1$ on $\mathsf{{W}}$.

Let $\theta_{i}:T^{2}\to T^{2}$ be a slide along $C_{i}$ supported in ${W}_{i}\setminus{V}_{i}$, $i=0,\ldots,n-1$, and

[TABLE]

Then, [16, Lemma 5.2], there exists a smooth function $\sigma:T^{2}\to\mathbb{R}$ such that

• •

$\sigma$ is constant along trajectories of $\mathbf{F}$,

• •

$\sigma=1$ on $\mathsf{{V}}$, $\sigma=0$ on $T^{2}\setminus\mathsf{{W}}$, and

• •

$\theta=\mathbf{F}_{\sigma}$,

Then for $k\in\mathbb{Z}$ we have $\theta^{k}=\mathbf{F}_{k\sigma}$. From this definition it immediately follows that $\theta$ belongs to $\Delta^{\prime}(f,\mathsf{{V}})$. A free abelian subgroup of $\pi_{0}\Delta^{\prime}(f,\mathsf{{V}})$ generated by $\theta$ will be denoted by $\langle\theta\rangle$. The following theorem is our main result of this section.

Theorem 5.5**.**

For $f\in\mathscr{F}_{1}$ the following statements hold true.

$\mathrm{Ker}j_{0}\cong\langle\theta\rangle\cong\mathbb{Z}$. In other words, each $h\in\Delta^{\prime}(f,\mathsf{{V}})$ is isotopic to $\theta^{k(h)}$ for some $k(h)\in\mathbb{Z}$ relatively $\mathsf{{V}}$ whenever $[h]\in\mathrm{Ker}j_{0}.$ 2. 2)

The following exact sequence of abelian groups

[TABLE]

splits. In particular, we there is an isomorphism

[TABLE]

Proof.

1) Let $h\in\Delta^{\prime}(f,\mathsf{{V}})$. We have to show that if $[h]\in\mathrm{Ker}j_{0}$, then $h$ is isotopic to $\theta^{k(h)}$ relatively $\mathsf{{V}}$ for some $k(h)\in\mathbb{Z}$.

Recall that the identity component of $\Delta^{\prime}(f)$ is $\mathcal{S}_{\mathrm{id}}(f)$, see [15, Lemma 4.1]. Hence if $h\in\Delta^{\prime}(f,\mathsf{{V}})$ is such that $[h]\in\mathrm{Ker}j_{0}$, then $h\in\mathcal{S}_{\mathrm{id}}(f)$, and by [15, Lemma 6.1] there exists a unique smooth function $\alpha:T^{2}\to\mathbb{R}$ such that $h=\mathbf{F}_{\alpha}$.

We claim that such $h$ admits some ‘‘simplification’’ on $T^{2}\setminus\mathsf{{W}}$, i.e., $h$ can be deformed relatively $\mathsf{{V}}$ to a diffeomorphism $h^{\prime}$ such that $[h^{\prime}]\in\mathrm{Ker}(j_{0})$ and $h^{\prime}$ is also fixed on $T^{2}\setminus\mathsf{{W}}$. Indeed, fix any smooth function $\delta:T^{2}\to\mathbb{R}$ satisfying

• •

$\delta|_{\mathsf{{V}}}=0$,

• •

$\delta|_{T^{2}\setminus\mathsf{{W}}}=1$,

• •

$\delta$ is constant along trajectories of $\mathbf{F}$,

and define the following homotopy $H^{t}:T^{2}\to T^{2}$, $t\in[0,1]$, by the formula

[TABLE]

By [15, Lemma 6.1] a family $\{H^{t}\}$ is in fact an isotopy between $H^{0}=h$, $h^{\prime}:=H^{1}(h)=\mathbf{F}^{-1}_{\delta\alpha}\circ h$ and each diffeomorphism $H^{t}(h)$ belongs to $\Delta^{\prime}(f,\mathsf{{V}})$. In particular, the isotopy classes of $h$ and $h^{\prime}$ in $\pi_{0}\Delta^{\prime}(f,\mathsf{{V}})$ coincide, and $h^{\prime}$ is fixed on $T^{2}\setminus\mathsf{{W}}$.

So we can redenote $h^{\prime}$ with $h$ and additionally assume further that $h$ is fixed on $T^{2}\setminus\mathsf{{W}}$. The restriction of $\alpha$ and $h$ onto ${V}_{i}$ will be also denoted by $\alpha_{i}$ and $h_{i}$ respectively $i=0,\ldots,n-1$. By assumptions $h$ is fixed on $\mathsf{{V}}$, that is

[TABLE]

Since the period of trajectories of $\mathbf{F}$ is equal to $1$ on $\mathsf{{W}}$, it follows that $\alpha_{i}$ takes an integer value, say $k_{i}(h)\in\mathbb{Z}$, depending on $h$ for each $x\in{V}_{i}$.

We claim that in fact $\alpha_{i}$ takes the same value $k\in\mathbb{Z}$ on each ${V}_{i}$, so all of those numbers $k_{i}(h)$ must coincide. Indeed, recall $h|_{Q_{i}}$ is isotopic relatively $\mathsf{{V}}\cap Q_{i}$ to the Dehn twist $\tau^{a_{i}}$ supported on $\mathsf{{V}}\cap Q_{i}$, where

[TABLE]

However, as $h\in\Delta^{\prime}(f,\mathsf{{V}})$, it follows that $h|_{Q_{i}}$ is isotopic relatively $\mathsf{{V}}\cap Q_{i}$ to $\mathrm{id}_{Q_{i}}=\tau^{0}$ for each $i=0,\ldots,n-1$, that is

[TABLE]

and so $k_{i+1}(h)=k_{i}(h)$ for each $i=0,\ldots,n-1$. Thus we can denote the common values of all $k_{i}(h)$ simply by $k(h)$.

Define now an isotopy $H^{t}(h):T^{2}\to T^{2}$ between $h=\mathbf{F}_{\alpha}$ and $\theta^{k}=\mathbf{F}_{k(h)\sigma}$ by the formula

[TABLE]

Then $H^{t}(h)$ is fixed on $\mathsf{{V}}$ for all $t\in[0,1]$ and so each $h$ with $[h]\in\mathrm{Ker}j_{0}$ is isotopic to $\theta^{k(h)}$ relatively to $\mathsf{{V}}$. In other words, $\mathrm{Ker}j_{0}\subset\langle\theta\rangle$.

The inverse inclusion is easy $\langle\theta\rangle\subset\mathrm{Ker}j_{0}$, and so $\mathrm{Ker}j_{0}=\langle\theta\rangle$ is an free abelian group generated by the element $\theta$.

2) Now it follows from 1) and surjectivity of $j_{0}$ that we have a short exact sequence (5.5.1):

[TABLE]

Moreover, [10], it consists of abelian groups and the group $\pi_{0}\Delta^{\prime}(f)$ is a free abelian. Therefore the above sequence splits, i.e., there is an isomorphism $\pi_{0}\Delta^{\prime}(f,\mathsf{{V}})\cong\pi_{0}\Delta^{\prime}(f)\times\langle\theta\rangle$. ∎

6. Orbits of Morse functions on 2-torus

The following theorem describes the fundamental groups of orbits of Morse functions on $2$-torus. We showed that they can be computed using zero homotopy groups of stabilizers of restrictions of the given function onto subsurfaces of $T^{2}$ being $2$-disks and cylinders.

Theorem 6.1**.**

[17, 9, 16, 2].* Let $f$ be a Morse function on $T^{2}$, and $\Gamma_{f}$ be its graph.*

1. (1)

Assume that $f$ belongs to $\mathscr{F}_{0}$ and $G_{v}^{loc}\cong\mathbb{Z}_{n}\times\mathbb{Z}_{nm}$ for some $n,m\in\mathbb{N}$. Then there exists a set of mutually disjoint $2$-disks $\mathbb{D}=\{D_{i}\}_{i=1}^{r}\subset T^{2}$ for some $r\in\mathbb{N}$ which is a fundamental set of the free $G_{v}^{loc}$-action on $T^{2}$ such that the restriction $f|_{D_{i}}$, $i=1,\ldots,r$, is a Morse function, and there is an isomorphism

[TABLE]

where $\mathcal{S}_{\mathbb{D}}:=\prod\limits_{i=1}^{r}\pi_{0}\mathcal{S}^{\prime}(f|_{D_{i}},\partial D_{i})$. Moreover $r$ is the number of orbits of free $G_{v}^{loc}$-action on $T^{2}$. 2. (2)

Assume that $f$ belongs to $\mathscr{F}_{1}$ and has a cyclic index $n\in\mathbb{N}$. Then there exists a cylinder $Q\subset T^{2}$ such that $f|_{Q}$ is also a Morse function and we have an isomorphism

[TABLE]

where $\mathcal{S}_{Q}:=\pi_{0}\mathcal{S}^{\prime}(f|_{Q},\partial Q)$.

6.2. Strategy of the proof of Theorem 3.2

First we recall explicit definitions of isomorphisms

[TABLE]

from Theorem 6.1 in Subsections 7.1 and 8.1 respectively. Since groups $\pi_{0}\mathcal{S}^{\prime}(f)$ and $G(f)$ are quotient-groups

[TABLE]

see diagram (1.3.1), it follows that in order to describe them and the diagram (1.3.1) we need to characterize images of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ with respect to the maps $\xi_{1}^{-1}$ and $\xi_{2}^{-1}$ and take the corresponding quotient-groups. This will be done in Subsection 7.2 and Subsection 8.2.

7. Proof of (1) of Theorem 3.2

Due to our strategy, see Subsection 6.2, first we give the explicit description of the isomorphism $\xi_{1}:\mathcal{S}_{\mathbb{D}}\wr_{n,nm}\mathbb{Z}^{2}\to\pi_{1}\mathcal{O}_{f}(f)$.

7.1. Isomorphism from (1) of Theorem 6.1

Let $f$ be a Morse function on $T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$ such that its graph $\Gamma_{f}$ is a tree, $v$ be the special vertex of $\Gamma_{f}$, and $G_{v}^{loc}=\mathbb{Z}_{n}\times\mathbb{Z}_{nm}$ be the local stabilizer of $v$. Then the free action of $\mathbb{Z}_{n}\times\mathbb{Z}_{nm}$ on $T^{2}$ can be given by:

[TABLE]

for $(a,b)\in\mathbb{Z}_{n}\times\mathbb{Z}_{nm}$ and $(x,y)\in T^{2}$, and the quotient space $T^{2}/G_{v}^{loc}$ is diffeomorphic to $T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}$, so that the quotient map $q:T^{2}\to T^{2}/G_{v}^{loc}$ is given by the formula

[TABLE]

Let $y$ be a point in $T^{2}$, and $z=q(y)\in T^{2}/G_{v}^{loc}$. Then we obtain the following commutative diagram with exact rows

[TABLE]

where maps $q_{1}$ and $\partial$ are given as follows

[TABLE]

Let $\mathsf{L},\mathsf{M}:T^{2}\times\mathbb{R}\to T^{2}$ be smooth flows on $T^{2}$ defined by

[TABLE]

Then they commute each with other, and

[TABLE]

for $a,b\in\mathbb{R}$. Moreover, $\mathsf{L}_{n}=\mathsf{M}_{nm}=\mathrm{id}_{T^{2}}$, and the restrictions

[TABLE]

can be regarded as loops in $\mathcal{D}_{\mathrm{id}}(T^{2})$ constituting also a basis $(1,0)$, $(0,1)$ of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})\cong\mathbb{Z}^{2}$.

Then, [2], the isomorphism $\xi_{1}:\mathcal{S}_{\mathbb{D}}\wr_{n,{nm}}\mathbb{Z}^{2}\to\pi_{1}\mathcal{O}_{f}(f)$, see (6.1.1), can be defined as follows. Let

[TABLE]

where $h_{ijk}\in\mathcal{S}^{\prime}(f|_{D_{i00}},\partial D_{i00})$, $(a,b)\in\mathbb{Z}^{2}$, $i=1,\ldots,r$, $j=0,\ldots,n-1$, $k=0,\ldots,nm-1$. For each triple $(i,j,k)$ fix any isotopy $h^{t}_{ijk}:D_{i00}\to D_{i00}$, $t\in[0,1]$ between $h^{0}_{ijk}=\mathrm{id}_{D_{i00}}$ and $h^{1}_{ijk}=h_{ijk}$ relatively some neighborhood of $\partial D_{i00}$. Then

[TABLE]

where $h^{t}:T^{2}\to T^{2}$, $t\in[0,1]$, is given by the formula

[TABLE]

$D_{ijk}$ is a $2$-disk defined in (ii) of Remark 4.3, and $N$ is a regular neighborhood of the critical level-set $V=p_{f}^{-1}(v)$ containing no other critical points.

7.2. Images of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ in $\mathcal{S}_{\mathbb{D}}\wr_{n,nm}\mathbb{Z}^{2}$

The following lemma describes the images of the groups $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ with respect to the map $\xi_{1}^{-1}$.

Lemma 7.3**.**

1. (1)

Let $Z_{n,0}=(\underbrace{e,\ldots,e}_{{nm}n},n,0)$ and $Z_{0,nm}=(\underbrace{e,\ldots,e}_{{nm}n},0,nm)$ be elements from $\mathcal{S}_{\mathbb{D}}\wr_{n,nm}\mathbb{Z}^{2}$, where $e$ is the unit of $\mathcal{S}_{\mathbb{D}}$. Then

[TABLE] 2. (2)

The isomorphism $\xi_{1}$ induces an isomorphism

[TABLE]

Proof.

(1) Indeed, $\xi_{1}(Z_{n,0})$ is a loop $f\circ h^{t},$ where $h^{t}$ is given by (7.1.3) with $a=1$ and $b=0$. Then by (7.1.3) the isotopy $h$ has the form $h^{t}=\mathsf{L}_{t}$.

The case of $p_{1}(\mathsf{M})$ can be checked similarly and we leave this verification to the reader. So we have $p_{1}(\langle\mathsf{L}\rangle)=\xi_{1}(\langle Z_{n,0}\rangle)$ and $p_{1}(\langle\mathsf{M}\rangle)=\xi_{1}(\langle Z_{0,nm}\rangle)$.

(2) Let $\iota:\pi_{0}\Delta^{\prime}(f)\to\pi_{1}\mathcal{O}_{f}(f)$ be a restriction of $\iota_{1}$ onto ${\pi_{0}\Delta^{\prime}(f)}$. We need to show that the subgroup $(\Delta^{nmn}_{\mathbb{D}},0,0)\subset\mathcal{S}_{\mathbb{D}}\wr_{n,nm}\mathbb{Z}^{2}$ is mapped via $\xi_{1}$ isomorphically to $\iota(\pi_{0}\Delta^{\prime}(f))$. Obviously, an isomorphism $\xi_{1}$ induces a monomorphism $\xi_{1}|_{\Delta^{nmn}_{\mathbb{D}}}:\Delta^{nmn}_{\mathbb{D}}\to\iota_{1}(\pi_{0}\Delta^{\prime}(f))$. It remains to show that $\xi_{1}|_{\Delta^{nmn}_{\mathbb{D}}}$ is an epimorphism.

Let $[\tau]$ be a generator of $\pi_{0}\Delta^{\prime}(f)$. Then $\tau$ be a Dehn twist supported on some $2$-disk $D\in\{D_{i,j,k}\}_{i=1,\ldots,r,\,j=0,\ldots,n-1,\,k=0,\ldots,nm-1}$. Fix an isotopy $\tau^{t}:T^{2}\to T^{2}$ between $\tau^{1}=\tau$ and $\tau^{0}=\mathrm{id}_{T^{2}}$. The image $\iota([\tau])$ is given by the class of the loop $[f\circ\tau^{t}]$. It follows from (7.1.3) that there exist a diffeomorphism $\tilde{h}$ of $D$ and an isotopy $\tilde{h}^{t}:D\to D$ between $\tilde{h}^{1}=\tilde{h}$ and $\tilde{h}^{0}=\mathrm{id}_{D}$ such that $\xi_{1}(\{[\tilde{h}]\},0,0)=[f\circ\tau^{t}]$. Also we get from (7.1.3) that $\tilde{h}$ is conjugate to $\tau|_{D}$, and so $\tilde{h}$ is a Dehn twist. Hence $\tilde{h}$ belongs to $\Delta_{\mathbb{D}}^{nmn}$. Thus $\xi_{1}|_{\Delta^{nmn}_{\mathbb{D}}}$ is an isomorphism. ∎

7.4. Final remarks for the proof of (1) of Theorem 3.2

Recall that each third term of every short exact sequence of groups is uniquely defined by other two. So groups $\pi_{0}\mathcal{S}^{\prime}(f)$ and $G(f)$ are the corresponding quotient-groups $\pi_{1}\mathcal{O}_{f}(f)/\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $G(f)=\pi_{0}\mathcal{S}^{\prime}(f)/\pi_{0}\Delta^{\prime}(f)$, see diagram (1.3.1). The images of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ in $\pi_{1}\mathcal{O}_{f}(f)$ are known, see Subsection 7.2. So it is easy to prove that an isomorphism $\xi_{1}$ induces the following isomorphisms

[TABLE]

as well as isomorphism of diagrams from (1) of Theorem 3.2.

8. Proof of (2) of Theorem 3.2

First we give the explicit description of an isomorphism $\xi_{2}:\mathcal{S}_{Q}\wr_{n}\mathbb{Z}\to\pi_{1}\mathcal{O}_{f}(f)$.

8.1. Isomorphism from (2) of Theorem 6.1

Let $f$ be a Morse function from $\mathscr{F}_{1}$ with a cyclic index $n$ and circuit $\Theta$ in the graph $\Gamma_{f}$. As curves from $\mathcal{C}$ are ‘‘parallel’’, one can assume that the following conditions hold:

(a) $C_{i}=\frac{i}{n}\times S^{1}\subset\mathbb{R}^{2}/\mathbb{Z}^{2}=T^{2}$;

(b) there exists $\varepsilon\geq 0$ such that for all $t\in(\frac{i}{n}-\varepsilon,\frac{i}{n}+\varepsilon)$ the curve $t\times S^{1}$ is a regular connected component of some level set of $f$.

Let $\mathbb{L},\mathbb{M}:T^{2}\times\mathbb{R}\to T^{2}$ be two flows defined by formulas:

[TABLE]

$x\in C^{\prime}$, $y\in C_{0}$, and $t\in\mathbb{R}$.

Denote by $Q$ the cylinder bounded by $C_{0}$ and $C_{1}$. Then the isomorphism $\xi_{2}:\mathcal{S}_{Q}\wr_{n}\mathbb{Z}\to\pi_{1}\mathcal{O}_{f}(f)$ can be defined as follows, [16, Section 8]. Let

[TABLE]

where $h_{i}\in\mathcal{S}_{Q}$, $i=0,\ldots,n$, and $a\in\mathbb{Z}$. Fix any isotopy $h_{i}^{t}$ between $h^{0}_{i}=\mathrm{id}_{Q}$ and $h^{1}_{i}=h_{i}$. Then

[TABLE]

where $h^{t}$ is defined by the formula

[TABLE]

8.2. Images of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ in $\mathcal{S}_{Q}\wr_{n}\mathbb{Z}$

The following lemma easily follows from the definition of the isomorphism $\xi_{2}$ similarly to (1) of Lemma 7.3.

Lemma 8.3**.**

Let $Z=(\underbrace{e^{\prime},\ldots,e^{\prime}}_{n},n)$ be the element from $S_{Q}\wr_{n}\mathbb{Z}$, where $e^{\prime}$ is the unit of $\mathcal{S}_{Q}$. Then $p_{1}(\mathbb{L})=\xi_{2}(Z)$. ∎

The image of another generator $\mathbb{M}$ in $\pi_{1}\mathcal{O}_{f}(f)$ is ‘‘invisible’’ in our description of the group $\pi_{1}\mathcal{O}_{f}(f)$ via $\mathcal{S}_{Q}\wr_{n}\mathbb{Z}$, but in [16, Theorem 6] we showed that $p_{1}(\mathbb{M})$ in $\pi_{1}\mathcal{O}_{f}(f)$ is given by $[f\circ\theta]$, where $\theta=\theta_{0}\circ\ldots\circ\theta_{n-1}$ is the generator of $H=\langle\theta\rangle$, and $\theta_{i}$ is a slide along $C_{i}$, see Section 5.3. Then by Theorem 5.5 groups $\pi_{0}\Delta^{\prime}(f)\times\langle\mathbb{M}\rangle$ and $\pi_{0}\Delta^{\prime}(f,\mathcal{C})$ are isomorphic as subgroups of $\pi_{1}\mathcal{O}_{f}(f)$. So $\pi_{0}\Delta^{\prime}(f)\cong\pi_{0}\Delta^{\prime}(f,\mathcal{C})/H$.

Lemma 8.4**.**

An isomorphism $\xi_{2}$ induces an isomorphism

[TABLE]

where $\Delta_{Q}=\pi_{0}\Delta^{\prime}(f|_{Q},\partial Q)$. So $\pi_{0}\Delta^{\prime}(f)$ is isomorphic to $\Delta_{Q}^{n}/H$.

Proof.

Obviously, the isomorphism $\xi_{2}$ induces a monomorphism $\xi_{2}|_{\Delta_{Q}^{n}}$. Let $[\tau]$ be a generator of $\pi_{0}\Delta^{\prime}(f,\mathcal{C})$. Then $\tau$ is a Dehn twist supported in $Q\in\{Q_{i}\}_{i=0,\ldots,n-1}$. Fix an isotopy $\tau^{t}:T^{2}\to T^{2}$ such that $\tau^{1}=\tau$ and $\tau^{0}=\mathrm{id}$. Then $\iota([\tau])=[f\circ\tau^{t}]$. From (8.1.1) there exists a diffeomorphism $\tilde{h}:Q\to Q$ and an isotopy $\tilde{h}^{t}:Q\to Q$ between $\tilde{h}^{1}=\tilde{h}$ and $\tilde{h}^{0}=\mathrm{id}_{Q}$ such that $\xi_{2}([\tilde{h}],0)=[f\circ\tau^{t}]$. By (8.1.2) $\tilde{h}$ is conjugate to $\tau_{Q}$, then $\tilde{h}$ is also a Dehn twist. Hence $\tilde{h}$ belongs to $\Delta_{Q}^{n}$. So $\xi_{2}|_{\Delta_{Q}^{n}}$ is an isomorphism, and hence $\pi_{0}\Delta^{\prime}(f)$ is isomorphic to $\Delta_{Q}^{n}/H$. ∎

The following corollary directly follows from Lemma 5.2 and describes $\pi_{1}\mathcal{O}_{f}(f)$ for functions from $\mathscr{F}_{1}$ via embedded $2$-disks and the image of the generator $\mathbb{M}$ in $\pi_{1}\mathcal{O}_{f}(f)$.

Corollary 8.5**.**

1. (1)

There exist a cylinder $Q$ and a set of mutually disjoint $2$-disks $\mathbb{Y}=\{Y_{ij}\}_{i=0,\ldots,k}^{j=0,\ldots,c_{i}}\subset Q$ for some $k,c_{i}\in\mathbb{N}$, and $m_{i}\in\mathbb{N}$, $i=1,\ldots,k$ such that the following groups are isomorphic

[TABLE]

where $\zeta^{\prime}=\underbrace{\zeta\times\ldots\times\zeta\times\mathrm{id}_{\mathbb{Z}}}.$ 2. (2)

The group $H$ is normal in $\Delta_{\mathbb{Y}}^{n}$ is generated by

[TABLE]

where $E_{i}$ is the unit of the group $(\prod_{j=1}^{c_{i}}\Delta_{Y_{ij}})^{m_{i}}$, and the image of $H$ in $\mathcal{S}_{\mathbb{Y}}\wr_{n}\mathbb{Z}$ is generated by

[TABLE]

8.6. Final remarks for the proof of (2) of Theorem 3.2

Similarly to the proof of (1) of Theorem 3.2, groups $\pi_{0}\mathcal{S}^{\prime}(f)$ and $G(f)$ are the corresponding quotient-groups $\pi_{1}\mathcal{O}_{f}(f)/\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $G(f)=\pi_{0}\mathcal{S}^{\prime}(f)/\pi_{0}\Delta^{\prime}(f)$, see diagram (1.3.1). The images of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ and $\pi_{0}\Delta^{\prime}(f)$ in $\pi_{1}\mathcal{O}_{f}(f)$ are known, see Subsection 8.2. So it is easy to prove that an isomorphism $\zeta^{\prime}\circ\xi_{2}$ induces the following isomorphisms

[TABLE]

and so it induces an isomorphism of diagrams from (2) of Theorem 3.2.