# Deformations of circle-valued functions on $2$-torus

Bohdan Feshchenko  Topology laboratory, Department of algebra and topology,
Institute of Mathematics of National Academy of Science of Ukraine,
Tereshchenkivska, 3, Kyiv, 01601, Ukraine<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

In this paper we give an algebraic description of fundamental groups of orbits
of circle-valued smooth functions from some subspace of the space of smooth
functions with isolated singularities on $2$-torus $T^{2}$ with respect to the
action of the group of diffeomorphisms of $T^{2}$.

###### Key words and phrases:

Circle-valued Morse functions, orbits, stabilizers, fundamental groups

##  1\. Introduction

Morse functions on manifolds are one of the main objects in mathematics
nowadays. It is well-known that analytic properties of such functions carry
information about the geometry and topology of the manifold on which they are
defined [32].

Deformational properties of Morse functions were studied by many authors. For
example, homotopy properties of connected components of spaces of Morse
functions on smooth surfaces were studied by V. Sharko [40], H. Zieschang, S.
Matveev, E. Kudryavtseva [17]. Cobordism groups of Morse functions on surfaces
were calculated by К. Ikegami, О. Saeki [14] and B. Kalmar [15].

The paper is devoted to the study of circle-valued smooth functions from some
subspace of smooth functions with isolated singularities on smooth compact
oriented surfaces and homotopy properties of special subspaces of such
functions called orbits. The main example of such functions is circle-valued
Morse functions which are natural generalizations of (ordinary) Morse
functions. Circle-valued Morse functions can be viewed locally as functions
but global properties of such functions are different from real-valued case. A
modern theory of circle-valued Morse functions originates in a series of
papers [33, 34] by S. Novikov in 80’s. It was motivated by the study of multi-
valued Lagrangians in some problems of theoretical physics and leaded him to
develop a generalization of a Morse theory for circle-valued Morse functions
and more generally a theory (now called Morse-Novikov theory) for differential
$1$-forms. This theory has many applications, e.g., in questions of fibrations
of manifolds over $S^{1}$ [37], Lagrangian intersections [8], knot theory
[41], Seiberg-Witten theory [12, 13], etc. The reader can find more on this
theory and its applications in the book by A. Pajitnov [35].

Recall that there is a natural action of the group of diffeomorphisms
$\mathcal{D}(M)$ of a smooth compact surface $M$ on the space of smooth
$P$-valued functions for $P=\mathbb{R}$ or $S^{1}$ given by the rule:

$\gamma:C^{\infty}(M,P)\times\mathcal{D}(M)\to
C^{\infty}(M,P),\quad\gamma(f,h)=f\circ h.$

We consider stabilizers $\mathcal{S}(f)$ and orbits $\mathcal{O}(f)$ of a
smooth function $f\in C^{\infty}(M,P)$ with respect to the action $\gamma$ and
their connected components $\mathcal{S}_{\mathrm{id}}(f)$ and
$\mathcal{O}_{f}(f)$ of the identity map $\mathrm{id}_{M}$ and a connected
component containing $f$ respectively in topologies induced from strong
Whitney topologies on $\mathcal{D}(M)$ and $C^{\infty}(M,P)$ (see definitions
in subsection 2.1).

We restrict out attention on the class of smooth $P$-valued functions
$\mathcal{F}$ on a smooth compact surface which satisfies two conditions:
functions take constant values on each boundary component and near every
critical point can be represented as homogeneous polynomial
$\mathbb{R}^{2}\to\mathbb{R}$ of degree $\geq 2$ without multiple factors
(class of functions $\mathcal{F}$, see subsection 2.2). It is well-known that
the class $\mathcal{F}$ consists of ‘‘generic’’ functions with ‘‘topologically
generic’’ singularities([27, Section 3,4], subsection 2.2). Thus our
restriction is insignificant, since the class $\mathcal{F}$ is ‘‘wide’’
enough.

S. Maksymenko [21, 23, 24, 26, 28] showed that if $f:M\to P$ is a smooth
function from $\mathcal{F}$ and $f$ has at least one saddle point, then
$\pi_{n}\mathcal{O}_{f}(f)=\pi_{n}M$ for $n\geq 3$,
$\pi_{2}\mathcal{O}_{f}(f)=0$ and for $\pi_{1}\mathcal{O}_{f}(f)$ there is a
short exact111Throughout the text $\hookrightarrow$ and $\twoheadrightarrow$
mean mono- and epimorphism respectively. sequence of groups

$\textstyle{\pi_{1}\mathcal{D}_{\mathrm{id}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{1}}$$\textstyle{\pi_{1}\mathcal{O}_{f}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{\pi_{0}\mathcal{S}^{\prime}(f),}$
(2.3)

where $\mathcal{S}^{\prime}(f)$ is the group of $f$-preserving diffeomorphisms
of $M$ which are isotopic to the identity map, see Eq. (1). This exact
sequence is a non-trivial part of long exact sequence of homotopy groups of
some fibration $\zeta_{f}:\mathcal{D}_{\mathrm{id}}(M)\to\mathcal{O}_{f}(f)$
(Theorem 2.5). As the consequence if $M$ is aspherical surface then all
homotopy information is contained in $\pi_{1}\mathcal{O}_{f}(f)$, and the
orbit $\mathcal{O}_{f}(f)$ itself is an Eilenberg–MacLane space
$K(\pi_{1}\mathcal{O}_{f}(f),1).$ So there are two natural questions about
homotopy properties of $\mathcal{O}_{f}(f)$: description of an algebraic
structure of groups $\pi_{1}\mathcal{O}_{f}(f)$ and homotopy type of
$\mathcal{O}_{f}(f).$

An algebraic structure of $\pi_{1}\mathcal{O}_{f}(f)$ ‘‘partially’’ depends on
homotopy properties of $\mathcal{D}_{\mathrm{id}}(M)$ which were studied in
[3, 4, 5, 10]. It is well-known that if $M$ is a closed compact and oriented
surface of genus $\geq 2$, then $\mathcal{D}_{\mathrm{id}}(M)$ is contractible
[3, 4, 5, 10], and so an epimorphism $\partial_{1}$ from (3) is an
isomorphism. Thus in this case the question on an algebraic structure of
$\pi_{1}\mathcal{O}_{f}(f)$ reduces to the study of an algebraic structure of
groups $\pi_{0}\mathcal{S}^{\prime}(f)$, which are easier to compute.

It is also known that the group $\mathcal{D}_{\mathrm{id}}(T^{2})$ for
$2$-torus is not contractible, so the image of
$\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})\cong\mathbb{Z}^{2}$ is non-trivial in
$\pi_{1}\mathcal{O}_{f}(f)$. The sequence (3) for functions on $T^{2}$ in
general does not split [20]. So for this case we need an additional study.

An algebraic structure of $\pi_{1}\mathcal{O}_{f}(f)$ for real-valued
functions from the class $\mathcal{F}$ on $2$-torus $T^{2}$ was studied in the
series of papers by S. Maksymenko and the author [30, 20, 29, 7]. The
sufficient conditions when sequence (3) splits were proved in [20]. It was
shown that the question on an algebraic structure of
$\pi_{1}\mathcal{O}_{f}(f)$ is reduced to studying algebraic structure of
orbits and stabilizers for restrictions of a given function to subsurfaces of
$T^{2}$ being 2-disks and cylinders, for which this structure is known [28].
These cylinders and $2$-disks are in some sense ‘‘building blocks’’ which
carry ‘‘combinatorial symmetries’’ of the function.

The group $\pi_{0}\mathcal{S}^{\prime}(f)$ contains a lot of information about
an algebraic structure of $\pi_{1}\mathcal{O}_{f}(f)$ and on homotopy type of
$\mathcal{O}_{f}(f)$. Note that each diffeomorphism from
$\mathcal{S}^{\prime}(f)$ induces an automorphism of Kronrod-Reeb graph
$\Gamma_{f}$ of $f$. The group $\pi_{0}\mathcal{S}^{\prime}(f)$ contains a
free abelian subgroup generated by Dehn twists which induce trivial action on
$\Gamma_{f}$ and the corresponding quotient group $G(f)$ contains ‘‘discrete
combinatorial’’ symmetries of the function $f$. So the group
$\pi_{0}\mathcal{S}^{\prime}(f)$ can be viewed as ‘‘homeotopy’’ group for
$f$-preserving and isotopic to the identity diffeomorphisms with $G(f)$ as its
non-trivial counterpart.

It is known that the group $G(f)$ ‘‘controls’’ the homotopy type of
$\mathcal{O}_{f}(f)$. In particular, if $f$ is generic, then the group $G(f)$
is trivial, and $\mathcal{O}_{f}(f)$ is homotopy equivalent to an $m$-torus
$T^{m}$ if $M\neq S^{2}$, $M\neq\mathbb{R}P^{2}$, to $S^{2}$ if $M=S^{2}$ and
$f$ has only two critical points, and to $\mathrm{SO}(3)\times T^{m}$
otherwise, for some $m\geq 0$ depending on $f$. E. Kudryavtseva [18, 19]
calculated the homotopy types of connected components of the space of Morse
functions on compact surfaces and generalized the result on homotopy types of
orbits $\mathcal{O}_{f}(f)$ when the group $G(f)$ is non-trivial.

It is also known that if $f:M\to P$ has exactly $n$ critical points, then
$\mathcal{O}_{f}(f)$ is homotopy equivalent to some covering space of
$n^{\mathrm{th}}$ configuration space of $M$ and $\pi_{1}\mathcal{O}_{f}(f)$
is a subgroup of $n^{\mathrm{th}}$ braid group $B_{n}(M)$ [22, Theorem 2,
Corollary 4]. A good overview the reader can find in [27, 28] where these
results are presented in the form of so-called crystallographic and Bieberbach
sequences.

Our main goal is to generalize our results on algebraic structure of
$\pi_{1}\mathcal{O}_{f}(f)$ to the case of circle-valued functions from the
class $\mathcal{F}$ on $T^{2}.$

Notice that if $f:T^{2}\to S^{1}$ is a smooth function without critical
points, then $f:T^{2}\to S^{1}$ is a locally trivial fibration and the
homotopy types of orbits and stabilizers are known, see [21, Theorem 1.9], so
we will always assume that all functions have at least one critical point.

### General overview of results.

First we show (Theorem 3.2) if $f:M\to S^{1}$ is a null-homotopic function
from $\mathcal{F}$ then $\mathcal{O}_{f}(f)$ is homeomorphic to
$\mathcal{O}_{\tilde{f}}(\tilde{f})$, where $\tilde{f}:M\to\mathbb{R}$ is also
a smooth function from $\mathcal{F}$ which is a lift of $f$ with respect to
the universal cover $p:\mathbb{R}\to S^{1}.$ So for null-homotopic functions
the problem in hand is completely reduced to the real-valued case, which is
known.

Then we show that for circle-valued functions with isolated singularities on
$2$-torus Kronrod-Reeb graphs are trees or contain a unique cycle, and if $f$
is not null-homotopic, then these graphs are not trees (Lemma 3.4).

In our main result (Theorem 4.1) we use an algebraic construction – wreath
product (subsection 3.5) – to describe an algebraic structure of
$\pi_{1}\mathcal{O}_{f}(f)$ for functions from $\mathcal{F}$ on $T^{2}$ whose
graphs contain a cycle via groups of connected components of stabilizers of
the restrictions of a function $f$ onto cylinders.

It is well-known that $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ is free
abelian group of the rank $2$ ([4, 10]); we define these two generators
$\mathbf{L}$ and $\mathbf{M}$ adapted to some coordinate system on $T^{2}$
(see Eq. (6)). The image of $\mathbf{L}$ in $\pi_{1}\mathcal{O}_{f}(f)$ plays
an important role in the proof of Theorem 4.1, but the image of $\mathbf{M}$
is ‘‘invisible’’ in our description of $\pi_{1}\mathcal{O}_{f}(f)$.
Proposition 7.1 contains the information about image of $\mathbf{M}$ in
$\pi_{1}\mathcal{O}_{f}(f).$

The obtained results will be used to further study the orbits of smooth
functions as well as their relationships with braid groups on surfaces
mentioned above or more generally Artin groups.

### Structure of the paper.

In Section 2 we give definitions of stabilizers and orbits of smooth functions
with values in $P$ on surfaces (subsection 2.1) and discuss their homotopy
properties (Theorem 2.5) for functions from the class $\mathcal{F}$.

Section 3 contains auxiliary constructions such as null-homotopic functions to
$S^{1}$ and their orbits (subsection 3.1), topological properties of graphs of
circle-valued functions with isolated singularities on $T^{2}$ (subsection
3.3) and wreath product of special type (subsection 3.5) needed to state our
main result.

Then we state our main result – Theorem 4.1 in Section 4. Section 5 contains
some additional constructions and result needed to the proof of Theorem 4.1 in
Section 6.

Finally, we study the ‘‘place’’ of the generator $\mathbf{M}$ of
$\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$ in $\pi_{1}\mathcal{O}_{f}(f)$
(Section 7).

##  2\. Definitions and useful facts

###  2.1. Orbits and stabilizers of smooth functions

Let $M$ be a smooth compact surface, $X$ be a closed (possible empty) subset
of $M$. By $P$ we also denote $\mathbb{R}$ or $S^{1}$. The group
$\mathcal{D}(M,X)$ of diffeomorphisms of $M$ fixed on $X$ acts from the right
on the space of smooth maps $C^{\infty}(M,P)$ by the rule

$\gamma:C^{\infty}(M,P)\times\mathcal{D}(M,X)\to
C^{\infty}(M,P),\qquad\gamma(f,h)=f\circ h.$

With respect to $\gamma$ we denote by

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

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

$\mathcal{S}^{\prime}(f,X)=\mathcal{S}(f,X)\cap\mathcal{D}_{\mathrm{id}}(M,X).$
(1)

The group $\mathcal{S}^{\prime}(f,X)$ consists of isotopic to the identity
$f$-preserving diffeomorphisms fixed on $X$.

###  2.2. The space $\mathcal{F}(M,P)$

Let $\mathcal{F}(M,P)$ be a subset of $C^{\infty}(M,P)$ satisfying the
following conditions:

  1. (1)

the map $f$ takes constant values at each boundary component of $M$,

  2. (2)

for every critical point $z$ of $f$ there are local coordinates in which $f$
is a homogeneous polynomial $\mathbb{R}^{2}\to\mathbb{R}$ of degree $\geq 2$
without multiple factors.

Notice that the class $\mathcal{F}$ is ‘‘natural’’ and ‘‘generic’’ class of
smooth functions. It is well-know that every $f\in\mathcal{F}(M,P)$ has only
isolated critical points, and thus the set of critical points of $f$ is
finite. For instance the space of $P$-valued Morse functions, i.e., smooth
maps $f:M\to P$ which has only non-degenerate critical points, satisfying
condition (1) is a subspace of $\mathcal{F}(M,P).$ From the other hand, if a
smooth function $f:M\to P$ has only isolated critical points, then by theorem
proved by P. T. Church and J. G. Timourian [1] and independently by O.
Prishlyak [36], the local topological structure of level sets near any
critical point can be realized by level sets of homogeneous polynomial without
multiple factors. So the space $\mathcal{F}(M,P)$ consists of ‘‘generic’’ maps
with ‘‘topologically generic’’ critical points, see [27, Section 3,4].

###  2.3. $f$-adapted manifolds

To state the results about homotopy properties of orbits and stabilizers for
functions from the class $M$ on surfaces we need the notion called $f$-adapted
manifolds.

Let $f:M\to P$ be a smooth function from $\mathcal{F}$ on compact surface $M.$
A connected component of a level set $f^{-1}(c)$, $c\in P$ is also called a
leaf of $f$. A leaf is called regular if it contains no critical points and
critical otherwise.

Let $K$ be a (regular or critical) leaf of $f$. For $\varepsilon>0$ let
$N_{\varepsilon}$ be a connected component of
$f^{-1}[c-\varepsilon,c+\varepsilon]$ containing $K.$ Then $N_{\varepsilon}$
is called an $f$-regular neighborhood of $K$ if $\varepsilon$ is so small that
$N_{\varepsilon}-K$ contains no critical points and no boundary components.

A submanifold $X\subset M$ is called $f$-adapted if $X=\cup_{i=1}^{a}A_{i}$,
where each $A_{i}$ is either a critical point of $f$, or a regular leaf of
$f$, or an $f$-regular neighborhood of some (regular or critical) leaf of $f$.
Note that if $X$ is a $f$-adapted subsurface, then $f|_{X}:X\to P$ belongs to
$\mathcal{F}(X,P),$ see [27].

For a set $X$ we denote by $|X|$ the number of point in $X$; if $X$ is an
infinite set we put $|X|=\infty$.

###  2.4. Homotopy properties of orbits and stabilizers

The following theorem describes the general homotopy properties of orbits.

######  Theorem 2.5 ([39, 21, 26, 25]).

Let $f\in\mathcal{F}(M,P)$ be a function on a smooth compact surface $M$ and
$X$ be an $f$-adapted submanifold. Then the following statements hold.

  1. (1)

The map

$\zeta_{f}:\mathcal{D}(M,X)\to\mathcal{O}(f,X),\qquad\zeta_{f}(h)=f\circ h$
(2)

is a locally trivial principal fibration with the fiber $\mathcal{S}(f,X)$.
The restriction
$\zeta_{f}|_{\mathcal{D}_{\mathrm{id}}(M,X)}:\mathcal{D}_{\mathrm{id}}(M,X)\to\mathcal{O}_{f}(f,X)$
is also a locally trivial principal fibration with the fiber
$\mathcal{S}^{\prime}(f,X).$ The orbit $\mathcal{O}_{f}(f,X)$ is a Frèchet
manifold, so it has a homotopy type of a CW complex.

  2. (2)

$\mathcal{O}_{f}(f,X)=\mathcal{O}_{f}(f,X\cup\partial M)$, and so

$\pi_{k}\mathcal{O}_{f}(f,X)\cong\pi_{k}\mathcal{O}_{f}(f,X\cup\partial M)$

for $k\geq 1.$

  3. (3)

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

$\textstyle{\pi_{1}\mathcal{D}_{\mathrm{id}}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\zeta_{1}}$$\textstyle{\pi_{1}\mathcal{O}_{f}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\textstyle{\pi_{0}\mathcal{S}^{\prime}(f)}$
(3)

is exact, where $\zeta_{1}$ is a homomorphism induced by $\zeta_{f}$ and
$\partial_{1}$ is a boundary map for long exact sequence of the fibration
$\zeta_{f}$. Moreover, $p(\pi_{1}\mathcal{D}_{\mathrm{id}}(M))$ contains in
the center of $\pi_{1}\mathcal{O}_{f}(f).$

  4. (4)

If $\chi(M)<|X|$ then $\pi_{1}\mathcal{D}_{\mathrm{id}}(M,X)$ is contractible
(in particular, when $X$ is finite or when $\chi(M)<0$),
$\pi_{k}\mathcal{O}_{f}(f,X)=0$ for $k\geq 2$ and the boundary map

$\textstyle{\pi_{1}\mathcal{O}_{f}(f,X)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial_{1}}$$\scriptstyle{\cong}$$\textstyle{\pi_{0}\mathcal{S}^{\prime}(f,X)}$

is an isomorphism.

Notice that a sequence (3) is nonzero part of a long exact sequence of
homotopy groups of the fibration $\zeta_{f}$. We briefly recall the definition
of $\partial_{1}$. Let $\tilde{f}$ be a loop in $\mathcal{O}_{f}(f)$ based in
$f:M\to P$, i.e., $\tilde{f}:[0,1]\to\mathcal{O}_{f}(f)$ with
$\tilde{f}_{0}=\tilde{f}_{1}=f.$ Then by (1) of Theorem 2.5 there exists a
path $h:[0,1]\to\mathcal{D}_{\mathrm{id}}(M)$ such that

$\tilde{f}_{t}=f\circ h_{t},\qquad h_{0}=\mathrm{id}_{M},\qquad
h_{1}=h\in\mathcal{S}^{\prime}(f).$

We denote by $[\tilde{f}]$ and $[h]$ the corresponding homotopy classes
$\tilde{f}$ and $h$ in $\pi_{1}\mathcal{O}_{f}(f)$ and
$\pi_{0}\mathcal{S}^{\prime}(f)$ respectively. Then the boundary homomorphism
is defined by $\partial_{1}[\tilde{f}]=[h].$

Moreover there is an isomorphism

$\varkappa:\pi_{1}(\mathcal{D}_{\mathrm{id}}(M),\mathcal{S}^{\prime}(f))\longrightarrow\pi_{1}\mathcal{O}_{f}(f),\qquad\varkappa:[\\{h_{t}\\}]\longmapsto[\\{f\circ
h_{t}\\}],$ (4)

where $h_{t}:M\to M$, $t\in[0,1]$ is an isotopy of $M$ with
$h_{0}=\mathrm{id}_{M}$ and $h_{1}\in\mathcal{S}^{\prime}(f)$.

##  3\. Auxiliary constructions

###  3.1. Null-homotopic functions from $\mathcal{F}(M,S^{1})$ and their
orbits

A function $f:M\to S^{1}$ homotopic to a constant map will be called null-
homotopic.

It is easy to see that homotopy properties of null-homotopic functions from
$\mathcal{F}(M,S^{1})$ are the same as for its ‘‘universal’’ lift, an ordinary
function from $\mathcal{F}(M,\mathbb{R})$ arising from a universal cover of
$S^{1}$. To be more precise, consider a universal covering map
$p:\mathbb{R}\to S^{1}$ given by $p(t)=e^{2\pi it}$. Then by a lifting
property for maps [11, Proposition 1.33], there exist a unique smooth function
$\tilde{f}:M\to\mathbb{R}$ (up to a choice of appropriate triple of points)
such that $p\circ\tilde{f}=f$. Note that the function $\tilde{f}$ is also a
function from $\mathcal{F}(M,\mathbb{R})$. This function $\tilde{f}$ will be
called a universal lift of $f$. The fact that homotopy properties of
$\mathcal{O}_{{f}}({f})$ are the same as for
$\mathcal{O}_{\tilde{f}}(\tilde{f})$ follows from the proposition.

######  Proposition 3.2.

Let $f\in\mathcal{F}(M,S^{1})$ be a null-homotopic function and
$\tilde{f}:M\to\mathbb{R}$ be its universal lift. Then $\mathcal{O}(f)$ and
$\mathcal{O}(\tilde{f})$ are homeomorphic, and hence
$\mathcal{O}_{f}(f)\cong\mathcal{O}_{\tilde{f}}(\tilde{f}).$

###### Proof.

Recall that $\zeta_{f}:\mathcal{D}(M)\to\mathcal{O}(f)$ and
$\zeta_{\tilde{f}}:\mathcal{D}(M)\to\mathcal{O}(\tilde{f})$ are locally
trivial principal fibrations with fibers $\mathcal{S}(f)$ and
$\mathcal{S}(\tilde{f})$ respectively. Then $\zeta_{f}$ decomposes as the
composition

$\textstyle{\mathcal{D}(M)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha_{f}}$$\scriptstyle{\zeta_{f}}$$\textstyle{\mathcal{O}(f)}$$\textstyle{\mathcal{D}(M)/{\mathcal{S}(f)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces,}$$\scriptstyle{\beta_{f}}$$\scriptstyle{\cong}$

where $\alpha_{f}(h)=\overline{h}$ is an open quotient map, and
$\beta_{f}(\overline{h})=\zeta_{f}(h)$ is an induced by $\zeta_{f}$
homeomorphism, where $\overline{h}$ is a coset of $h\in\mathcal{D}(M)$ modulo
$\mathcal{S}(f)$. The same decomposition holds for $\zeta_{\tilde{f}}$. So
$\mathcal{O}(f)$ and $\mathcal{O}(\tilde{f})$ are homeomorphic to quotient
groups $\mathcal{D}(M)/\mathcal{S}(f)$ and
$\mathcal{D}(M)/\mathcal{S}(\tilde{f})$ respectively. S. Maksymenko [28, Lemma
5.3] showed that if $f$ is null-homotopic then there is a homeomorphism
$\mathcal{S}(f)\cong\mathcal{S}(\tilde{f})$. Therefore quotient spaces
$\mathcal{D}(M)/\mathcal{S}(f)$ and $\mathcal{D}(M)/\mathcal{S}(\tilde{f})$
are homeomorphic, which implies that $\mathcal{O}(f)$ is homeomorphic to
$\mathcal{O}(\tilde{f}),$ and so
$\mathcal{O}_{f}(f)\cong\mathcal{O}_{\tilde{f}}(\tilde{f}).$ ∎

By Proposition 3.2 homotopy properties of $\mathcal{O}_{f}(f)$ for a null-
homotopic function $f\in\mathcal{F}(M,S^{1})$ are the ‘‘same’’ as for
$\mathcal{O}_{\tilde{f}}(\tilde{f})$ where $\tilde{f}$ is a universal lift of
$f$. Therefore this case is completely reduced to the real-valued case, and so
we will focus on the case when $f$ is not null-homotopic.

###  3.3. Graphs of $P$-valued Morse functions

Kronrod-Reed graphs are important tools to study smooth functions since they
carry a lot of informations about its combinatorial structure. We recall this
definition. Let $f\in\mathcal{F}(M,P)$ be a function on a smooth compact
oriented surface $M$ and $c\in P$. Let $\Xi$ be a partition of $M$ into
connected components of level sets (leaves, see subsection 2.3) of $f$. It is
well-known that the quotient space $M/\Xi$ denoted by $\Gamma_{f}$ has a
structure of an $1$-dimensional CW complex called a Kronrod-Reeb graph of $f$,
or simply a graph of $f$ [38].

Let $q_{f}:M\to\Gamma_{f}$ be a quotient map. Then the map $f:M\to P$ can be
presented as the composition of the projection $q_{f}$ onto $\Gamma_{f}$ and
the map $f_{\Gamma}$ induced by $f$:

$\textstyle{f=f_{\Gamma}\circ
q_{f}:}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{f}}$$\textstyle{\Gamma_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\Gamma}}$$\textstyle{P.}$
(5)

### Graphs of $S^{1}$-valued function on $T^{2}$

Graphs of ordinary Morse functions on $T^{2}$ were studied in [16]. For
functions with isolated singularities on $2$-torus the following lemma holds
true.

######  Lemma 3.4 (cf. Lemma 3.1 [6]).

Let $f:T^{2}\to S^{1}$ be a function with isolated singularities.

(1) The map $q_{f}^{*}:\pi_{1}T^{2}\to\pi_{1}\Gamma_{f}$ induced by $q_{f}$ is
an epimorphism with a non-zero kernel. So $b_{1}(\Gamma_{f})\leq 1$, i.e.,
$\Gamma_{f}$ can be either a tree or contains a unique cycle.

(2) If $f$ is not null-homotopic, then $\Gamma_{f}$ is not a tree.

It is well-known that the inequality $b_{1}(\Gamma_{f})\leq 1$ holds for
generic circle-valued Morse functions on $2$-torus [2, Theorem 3.7] and for
arbitrary circle-valued Morse functions on $2$-torus this inequality follows
from Morse form foliation theory, see [9, Theorem 2.1, Theorem 3.1, Example
3.2].

###### Proof of Lemma 3.4.

(1) This statement for ordinary Morse functions is proved in [6, Lemma 3.1]
and holds true for functions with isolated singularities. The reader can
verify it step-by-step as in [6, Lemma 3.1].

(2) Assume the converse holds and consider the sequence of fundamental groups
induced by (5):

$\textstyle{\pi_{1}T^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{f}^{*}}$$\textstyle{\pi_{1}\Gamma_{f}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\Gamma}^{*}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces\pi_{1}S^{1}}$$\textstyle{\mathbb{Z}^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{f}^{*}}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{*}_{\Gamma}}$$\textstyle{\mathbb{Z}}$

The homomorphism $f_{\Gamma}^{*}$ is a zero-map, so the homomorphism
$f^{*}=f_{\Gamma}^{*}\circ q_{f}^{*}$ induced by $f$ is also a zero map. This
leads to the contradiction that $f$ is not null-homotopic, so $\Gamma_{f}$
contains a cycle. ∎

The case of functions $f:T^{2}\to P$ from the class $\mathcal{F}$ whose graphs
are trees is a special case – it is only possible when $f$ is null-homotopic.
So in further text we will study only the case of functions whose graphs
contain a cycle. Examples of such functions are given in Figure 1.

Figure 1. Null-homotopic (left) and not null-homotopic (right) functions
$T^{2}\to S^{1}$

Note that functions in Figure 1 allow some ‘‘combinatorial symmetries’’
preserving the given functions which play an essential role in the description
of $\pi_{1}\mathcal{O}_{f}(f)$.

###  3.5. Wreath products

To state our main result we need a notion of wreath product of groups of a
special kind. Let $G$ be a group and $n\geq 1$ be an integer. A semi-direct
product $G^{n}\rtimes\mathbb{Z}$ with respect to a non-effective
$\mathbb{Z}$-action $\alpha$ on $G^{n}$ by cyclic shifts

$\alpha(b_{0},b_{1},\ldots,b_{n-1};k)=(b_{k},b_{1+k},\ldots,b_{n+k-1}),$

where all indexes are taken modulo $n$, will be denoted by
$G\wr_{n}\mathbb{Z}$ and called a wreath product of $G$ with $\mathbb{Z}$
under $n.$ Notice that this definition differs from the standard one, [31].

##  4\. Main result

The following theorem describes an algebraic structure of
$\pi_{1}\mathcal{O}_{f}(f)$ for functions $f:T^{2}\to P$ from the class
$\mathcal{F}$ whose graphs contain a cycle. It will be proved in Section 6.

######  Theorem 4.1.

Let $f$ be a function from $\mathcal{F}(T^{2},P)$ with at least one critical
point and whose graph $\Gamma_{f}$ contains a cycle. Then there exist a
cylinder $Q\subset T^{2}$ and $n\in\mathbb{N}$ such that
$f|_{Q}\in\mathcal{F}(Q,P)$ and there is an isomorphism

$\pi_{1}\mathcal{O}_{f}(f)\cong\pi_{0}\mathcal{S}^{\prime}(f|_{Q},N(\partial
Q))\wr_{n}\mathbb{Z},$

where $N(\partial Q)$ is some $f$-regular neighborhood of $\partial Q$.

######  Remark 4.2.

Theorem 4.1 generalizes the main result of the paper [29]. To prove [29,
Theorem 1.6] we mainly use ‘‘local’’ technique, i.e., properties of
diffeomorphisms of subsurfaces of $T^{2}$ and ‘‘glue’’ them together to obtain
global ones. So Theorem 4.1 can be proved step-by-step by the same arguments
and strategy. In the present paper we give more straightforward proof of this
result separating algebraic methods from topological ones. Proofs of some
known facts will be given only for the sake of completeness.

##  5\. Additional constructions and definitions

###  5.1. Curves on $T^{2}$

Let $f$ be a function from $\mathcal{F}(T^{2},P)$ whose graph $\Gamma_{f}$
contains a unique cycle denoted by $\Lambda$, let also
$q_{f}:T^{2}\to\Gamma_{f}$ be a projection induced by $f.$ Let $z$ be a point
in $\Lambda$, $c=f(q^{-1}_{f}(z))$ be a point in $S^{1}$, and $C$ be a regular
connected component of $f^{-1}(c)$. Note that $f^{-1}(c)$ consists of finitely
many connected components and it is invariant under the action of
$\mathcal{S}^{\prime}(f)$. We set
$\mathcal{C}=\\{h(C)\,|\,h\in\mathcal{S}^{\prime}(f)\\}$. Since $\mathcal{C}$
has finite cardinality we can cyclically enumerate elements of the set
$\mathcal{C}=\\{C_{0}=C,C_{1},C_{2},\ldots,C_{n-1}\\}$ for some
$n\in\mathbb{N}.$ Curves from $\mathcal{C}$ are mutually disjoint and do not
separate $T^{2}$, and each pair $C_{i}$ and $C_{i+1}$ bounds a cylinder
$Q_{i}\subset T^{2}$, where all indexes are taken modulo $n$.

###  5.2. $f$-regular neighborhoods of curves

We regard $S^{1}$ and $T^{2}$ as a quotient spaces $\mathbb{R}/\mathbb{Z}$ and
$\mathbb{R}^{2}/\mathbb{Z}^{2}$ respectively. By a proper choose of
coordinates on $T^{2}$ one can assume that the following conditions hold:

  * •

$C_{i}=\big{\\{}\frac{i}{n}\big{\\}}\times
S^{1}\subset\mathbb{R}^{2}/\mathbb{Z}^{2}=T^{2},$ so we can regard each curve
$C_{i}$ as a meridian of $T^{2}$, and the curve $C^{\prime}=\\{0\\}\times
S^{1}$ as a longitude of $T^{2}$.

  * •

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

This assumption makes possible to define $f$-regular neighborhoods of curves
from $\mathcal{C},$ see Subsection 2.3. So an $f$-regular neighborhood $V$ of
a curve $C$ is saturated neighborhood which has a cylindrical structure.

In this place we fix two families of $f$-regular neighborhoods $V_{i}$ and
$W_{i}$ of $C_{i}$, $i=0,\ldots,n-1$ needed in the further text, so that
$V_{i}\cap V_{j}=\varnothing$, $V_{i}\subset\mathrm{Int}(W_{i})$ for $i\not=j$
and for each $i,j$ there exists $h\in\mathcal{S}^{\prime}(f)$ such that
$h(V_{i})=V_{j}$. In particular, unions $\mathsf{V}=\bigcup_{i=0}^{n-1}V_{i}$
and $\mathsf{W}=\bigcup_{i=0}^{n-1}W_{i}$ are
$\mathcal{S}^{\prime}(f)$-invariant.

###  5.3. Generators of $\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})$

Let $\mathbf{L},\mathbf{M}:T^{2}\times[0,1]\to T^{2}$ be two isotopies defined
by

$\mathbf{L}(x,y,t)=(x+t\,\mathrm{mod}\,1,y),\qquad\mathbf{M}(x,y,t)=(x,y+t\,\mathrm{mod}\,1)$
(6)

for $x\in C^{\prime}$, $y\in C_{k}$, $k=0,1,\ldots,n-1$. Geometrically
$\mathbf{L}$ is a rotation of $T^{2}$ along its longitude, and $\mathbf{M}$ is
a rotation along meridians. Isotopies $\mathbf{L}$ and $\mathbf{M}$ can be
regarded as loops in $\mathcal{D}_{\mathrm{id}}(T^{2})$ It is well known that
$\mathbf{L}$ and $\mathbf{M}$ commute and
$\pi_{1}\mathcal{D}_{\mathrm{id}}(T^{2})=\langle\mathbf{L}\rangle\times\langle\mathbf{M}\rangle$,
see [4, 10].

###  5.4. Dehn twists and slides along curves from $\mathcal{C}$

Let $Q=S^{1}\times[0,1]$ be a cylinder and $C$ be the curve
$S^{1}\times\\{0\\}$, and $\alpha,\beta:[-1,1]\to[0,1]$ be two smooth
functions such that

$\alpha(x)=\begin{cases}0,&x\in[-1,-1/2],\\\
1,&x\in[1/2,1],\end{cases}\qquad\beta(x)=\begin{cases}0,&x\in[-1,-2/3]\cup[2/3,1],\\\
1,&x\in[-1/3,1/3].\end{cases}$

Define two diffeomorphisms of $Q$ by formulas:

$\tau(z,t)=(ze^{\alpha(t)},t)\qquad\theta(z,t)=(ze^{\beta(t)},t),\qquad(z,t)\in
Q;$

the diffeomorphisms $\tau$ and $\theta$ are called a Dehn twist and a slide
along $C=S^{1}\times\\{0\\}.$ Note that $\tau$ is fixed on some neighborhood
of $\partial Q$, and $\theta$ is fixed on some neighborhood of $C\cup\partial
Q$. A diffeomorphism of a smooth surface $M$ supported in some cylindrical
neighborhood of a simple closed and two-sided curve $C\subset M$ isotopic to a
Dehn twist with respect to the boundary of this neighborhood will be called a
Dehn twist along $C$ on $M$. Similarly the notion of slide along $C$ can be
extended to the case of surfaces.

Recall that a vector field $F$ on a smooth oriented surface $M$ is called
Hamiltonian-like for a function $f$ from the class $\mathcal{F}$ if the
following conditions satisfied:

  * •

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 this coordinates $F$ has the form $F(x,y)=-f^{\prime}_{y}\partial/\partial
x+f^{\prime}_{x}\partial/\partial y.$

Fix a Hamiltonian-like vector field $F$ of the given function $f:T^{2}\to
S^{1}$, and let $\mathbf{F}_{t}:T^{2}\to T^{2}$, $t\in\mathbb{R}$ be its flow.
The set $\mathsf{W}$ does not consist singular points of $\mathbf{F}$ and it
is $\mathbf{F}$-invariant and consists of periodic orbits. So one can assume
that periods of all trajectories of $\mathbf{F}$ are equal to $1$ on
$\mathsf{W}$.

Let $\theta_{i}:T^{2}\to T^{2}$ be a slide along $C_{i}$ supported on
$W_{i}-V_{i}$, $i=0,1,\ldots,n-1$, and set
$\theta=\theta_{0}\circ\theta_{1}\circ\ldots\circ\theta_{n-1}.$ We proved that
there exists a smooth function $\sigma:T^{2}\to\mathbb{R}$ which satisfies

  * •

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

  * •

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

  * •

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

and therefore $\theta^{k}=\mathbf{F}_{k\sigma}$, see [29, Lemma 5.2]. A free
abelian group generated by $\theta$ will be denoted by $\langle\theta\rangle$.

###  5.5. Characterization of direct and wreath products

In this paragraph we recall conditions when the group $G$ splits into a direct
product of its subgroups and discuss when $G$ splits into a wreath product as
in subsection 3.5. Let $G$ be a group and $G_{1},\ldots,G_{n}$ be their
subgroups. It is well-known that the group $G$ splits into a direct product
$G_{1}\times G_{2}\times\ldots\times G_{n}$ if the following three conditions
satisfied:

  1. (D1)

$G_{i}\cap G_{j}=\\{e\\}$, for $i\neq j=1,2,\ldots,n$, where $e$ is the unit
of $G$,

  2. (D2)

$G_{i}G_{j}=G_{j}G_{i}$ for all $i,j=1,2,\ldots,n,$

  3. (D3)

groups $G_{1}$, $G_{2}$ and $G_{n}$ generate $G$.

The following lemma gives conditions when the group $G$ splits into a wreath
product $L_{0}\wr_{n}\mathbb{Z}$ for some subgroup $L_{0}\subset G.$

######  Lemma 5.6 (Lemma 2.3 [28]).

Let $\phi:G\to\mathbb{Z}$ be an epimorphism and $L_{0}$ be a subgroup of
$\ker\phi.$ Let also $g\in G$ be with $\phi(g)=1$. Assume that for some
$n\in\mathbb{Z}$ the following conditions hold:

  1. (1)

$g^{n}$ commutes with $\ker\phi$,

  2. (2)

$\ker\phi$ splits into a direct product of

$L_{0},\quad L_{1}=g^{-1}L_{0}g^{1},\quad\ldots,\quad
L_{m-1}=g^{-(n-1)}L_{0}g^{n-1}.$

Then the map $\xi:L_{0}\wr_{n}\mathbb{Z}\to G$ given by the formula

$\displaystyle\xi(b_{0},b_{1},\ldots,b_{n-1},k)$
$\displaystyle=b_{0}(g^{-1}b_{1}g^{1})(g^{-2}b_{2}g^{2})\ldots(g^{-n+1}b_{n-1}g^{n-1})g^{k}$
$\displaystyle=b_{0}g^{-1}b_{1}\ldots g^{-1}b_{n-1}b^{-1+n+k}$ (7)

is an isomorphism.

##  6\. Proof of Theorem 4.1

###  6.1. Structure of the proof

Our main proof-tool is Lemma 5.6. So we need to define a data of a ‘‘natural’’
epimorphism $\phi:\pi_{1}\mathcal{O}_{f}(f)\to\mathbb{Z}$, an element $g$ from
$\ker\phi$, and groups $L_{i}$, $i=0,1,\ldots,n-1$ such in Lemma 5.6.

###  6.2. Epimorphism $\phi$ and its kernel

Let $V_{i}$ and $W_{i}$ be fixed $f$-regular neighborhoods of
$C_{i}\in\mathcal{C}$, $i=0,1,\ldots,n-1$ such in Subsection 5.2.

######  Proposition 6.3 (Theorem 6.1 [29]).

There exists an epimorphism $\phi:\pi_{1}\mathcal{O}_{f}(f)\to\mathbb{Z}$ with
the kernel isomorphic to $\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W})$, i.e.,
the following sequence of groups

$\textstyle{\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{1}\mathcal{O}_{f}(f)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \phi}$$\textstyle{\mathbb{Z}}$

is exact.

###### Proof.

This result is proved in [29], but for completeness of our exposition we will
recall the construction of an epimorphism $\phi$. Let $q:\mathbb{R}\times
S^{1}\to S^{1}\times S^{1}=T^{2}$ be a covering map given by
$q(x,y)=(\frac{x}{n}\mathrm{mod}\,1,y)$. Then $q(\\{i\\}\times
S^{1})=C_{i\,\mathrm{mod}\,n}$, $i\in\mathbb{Z}$ and
$q^{-1}(\mathcal{C})=\mathbb{Z}\times S^{1}.$

Let $\omega:[0,1]\to\mathcal{O}_{f}(f)$ be a loop and $h:T^{2}\times[0,1]\to
T^{2}$ be an isotopy such that $\omega_{t}=f\circ h_{t}$ and
$h_{0}=\mathrm{id}_{T^{2}}$, $h_{1}\in\mathcal{S}^{\prime}(f)$. There exist an
isotopy $\tilde{h}:(\mathbb{R}\times S^{1})\times[0,1]=\mathbb{R}\times S^{1}$
such that $\tilde{h}_{0}=\mathrm{id}_{\mathbb{R}\times S^{1}}$ and
$q\circ\tilde{h}_{t}=h_{t}\circ q$ for all $t\in[0,1].$ Since
$h_{1}(\mathcal{C})=\mathcal{C}$ then from the definition of $q$ we have
$\tilde{h}_{1}(\mathbb{Z}\times S^{1})=\mathbb{Z}\times S^{1}$. Then there
exists an integer $\phi_{h}$ such that

$\tilde{h}_{1}(\\{i\\}\times S^{1})=\\{i+\phi_{h}\\}\times S^{1}.$ (8)

The number $\phi_{h}$ depends only on the homotopy class of $h$, so on the
isotopy class of the loop $\omega$, and the correspondence

$\textstyle{[\omega]\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\phi}$$\textstyle{\phi_{h}}$

defined by (8) is an epimorphism
$\phi:\pi_{1}\mathcal{O}_{f}(f)\to\mathbb{Z}$.

The kernel of $\phi$ consists of homotopy classes of isotopies
$h:T^{2}\times[0,1]\to T^{2}$ such that $h_{1}$ leaves invariant each curve
$C_{i}$ from $\mathcal{C}$, i.e., $h_{1}(C_{i})=C_{i}.$ It was shown [21,
Lemma 4.14] that each such $h_{1}$ can be isotoped in
$\mathcal{S}^{\prime}(f)$ to the diffeomorphism $h_{1}^{\prime}$ which is
fixed on an $f$-cylindrical neighborhood $\mathsf{W}$ of $\mathcal{C}$, so
$h^{\prime}_{1}\in\mathcal{S}^{\prime}(f,\mathsf{W}).$ By (4) of Theorem 2.5
and equation (4) the composition

$\textstyle{\pi_{1}(\mathcal{D}_{\mathrm{id}}(T^{2},\mathsf{W}),\mathcal{S}^{\prime}(f,\mathsf{W}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \varkappa^{-1}}$$\scriptstyle{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
\cong}$$\textstyle{\pi_{1}\mathcal{O}_{f}(f,\mathsf{W})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\scriptstyle{\cong}$$\textstyle{\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W})}$

is an isomorphism. So the kernel of $\phi$ is isomorphic to
$\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W}).$ ∎

###  6.4. Special isotopy and subgroups of $\ker\phi$

The following proposition holds true.

######  Proposition 6.5 (Theorem 6.1 (c), [29]).

There exist an isotopy $g:T^{2}\times[0,1]\to T^{2}$ satisfying the following
conditions

  1. (1)

$g_{1}\in\mathcal{S}^{\prime}(f,\mathsf{W})$,

  2. (2)

$g_{1}^{n}=\mathrm{id}_{T^{2}}$,

  3. (3)

$g_{1}(Q_{i})=Q_{i+1}$, $i=0,1,\ldots,n-1$,

  4. (4)

$\phi([f\circ g_{t}])=1$.

###### Proof.

By definition of the set $\mathcal{C}$, there exists
$g_{1}\in\mathcal{S}^{\prime}(f)$ such that $g(Q_{i})=Q_{i+1}.$ So (3) is
obvious. To prove (1) and (2) we need to replace $g_{1}$ to
$\sigma=\mathbf{L}_{1/n}$ on $\mathsf{W}$ in $\mathcal{S}^{\prime}(f)$, where
$\mathbf{L}$ is defined in (6); this was done in (c) of [29, Theorem 6.1]. The
resulting diffeomorphism we also denote by $g_{1}$.

(4) Let $g:T^{2}\times[0,1]\to T^{2}$ be an isotopy between $g_{1}$ and
$\mathrm{id}_{T^{2}}$. So $\phi([g])=an+1$ for some $n\in\mathbb{Z}.$ If
$a\neq 0$ then we replace $[g_{t}]$ by $[g_{t}\circ\mathbf{L}_{t}^{-a}]$ in
order to have $\phi([f\circ g_{t}])=1.$ ∎

Denote by $X_{i}^{-}$ and $X_{i}^{+}$ the following intersections $Q_{i}\cap
W_{i}$, $Q_{i}\cap W_{i+1}$, and set $X_{i}=X_{i}^{-}\cup X_{i}^{+}$,
$U_{i}=Q_{i}-X_{i}$ for $i=0,1,\ldots,n-1$. The set $X_{i}$ is an $f$-adapted
neighborhood of the boundary $\partial Q_{i}$ of the cylinder $Q_{i}\subset
T^{2}$, see Figure 2.

Figure 2.

We denote by $L^{\prime}_{i}$ the following subgroup of the kernel of $\phi$:

$L^{\prime}_{i}=\pi_{0}\mathcal{S}^{\prime}(f,T^{2}-U_{i}),\qquad
i=0,1,\ldots,n-1.$ (9)

Elements of $L^{\prime}_{i}$ are isotopy classes of diffeomorphisms supported
on $U_{i}$. Let $g:T^{2}\times[0,1]\to T^{2}$ be an isotopy from Proposition
6.5. Since $g_{1}$ satisfies (1)-(4) of Proposition 6.5, it follows that

$L^{\prime}_{i}=[g_{1}^{-i}]L^{\prime}_{0}[g_{1}^{i}],\qquad
i=0,1,\ldots,n-1.$

Each diffeomorphism $h\in\mathcal{S}^{\prime}(f,T^{2}-U_{i})$ is fixed on
$T^{2}-U_{i}$, so the restriction $h\to h|_{Q_{i}}$ induces an isomorphism

$\beta_{i}:\pi_{0}\mathcal{S}^{\prime}(f,T^{2}-U_{i})\to\pi_{0}\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i})$
(10)

given by the map $\beta_{i}([h])=[h|_{Q_{i}}]$. Let $h_{i}$ be a
diffeomorphism from $\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i})$, and
$h^{\prime}_{i}$ be its extension to $T^{2}$ by the identity map. Then the
inverse of $\beta_{i}$ is given by the rule:
$\beta_{i}^{-1}([h_{i}])=[h^{\prime}_{i}].$ So we will not distinguish groups
$\pi_{0}\mathcal{S}^{\prime}(f,T^{2}-U_{i})$ and
$\pi_{0}\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i})$, and we set
$L_{i}=\pi_{0}\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i}).$

###  6.6. The end of the proof

The following lemma completes the proof.

######  Lemma 6.7.

The data of an epimorphism $\phi$ and the element $g$ from Proposition 6.5,
and groups $L_{i}=\pi_{0}\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i})$ from (9)
satisfy conditions of Lemma 5.6. So $\pi_{1}\mathcal{O}_{f}(f)$ is isomorphic
to $\pi_{0}\mathcal{S}^{\prime}(f|_{Q_{0}},X_{0})\wr_{n}\mathbb{Z}$ and this
isomorphism is given by the formula (5.6).

###### Proof.

Let $g:T^{2}\times[0,1]\to T^{2}$ is an isotopy defined from Proposition 6.5.
By (2) of Proposition 6.5 the diffeomorphism $g^{n}_{1}=\mathrm{id}_{T^{2}}$,
and so it commutes with $\ker\phi.$ Thus (1) of Lemma 5.6 holds true.

To verify (2) of Lemma 5.6 we need to check that the three conditions
(D1)–(D3) from subsection 5.5 satisfied:

  1. (1)

$L_{i}\cap L_{j}=[\mathrm{id}_{T^{2}}]$, $i\neq j$

  2. (2)

$L_{i}L_{j}=L_{j}L_{i}$,

  3. (3)

groups $L_{0}$, $L_{1},\ldots,L_{n-1}$ generate $\ker\phi$,

for all $i,j=0,1,\ldots,n-1$. Conditions (1) and (2) follows from the fact
that $\mathrm{supp}(h_{i})\cap\mathrm{supp}(h_{j})=\varnothing$ for
$[h_{i}]\in L_{i}$, $[h_{j}]\in L_{j}$, $i\neq j=0,1,\ldots,n-1$.

Let $[h]$ belongs to $\ker\phi.$ Then
$h|_{Q_{i}}\in\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i})$ and there is a unique
decomposition

$[h]=[h|_{Q_{0}}][h|_{Q_{1}}]\ldots[h|_{Q_{n-1}}].$

From the other hand let $h_{i}$ be a diffeomorphism from
$\mathcal{S}^{\prime}(f|_{Q_{i}},X_{i}),$ $i=0,1,\ldots,n-1$. Then

$\beta_{0}^{-1}([h_{0}])\circ\beta_{1}^{-1}([h_{1}])\circ\ldots\circ\beta_{n-1}^{-1}([h_{n-1}])$

represents an element of $\ker\phi.$ So (3) is true i.e., groups
$L_{0},\,L_{1},\ldots,L_{n-1}$ generate $\ker\phi.$ ∎

##  7\. The kernel of $\phi$ and an isotopy $\mathbf{M}$

Note that the group $\langle\mathbf{M}\rangle$ contains in the kernel of
$\phi$, but to prove Theorem 4.1 we do not need the explicit form of the
element of $\ker\phi$ representing $\mathbf{M}.$ Let $\mathsf{V}$ and
$\mathsf{W}$ be fixed in subsection 5.2 $f$-regular neighborhoods of
$\mathcal{C}.$ Next proposition describes the relationship between $\ker\phi$
and the group $\langle\mathbf{M}\rangle$.

######  Proposition 7.1.

There is an isomorphism
$\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W})\cong\langle\mathbf{M}\rangle\times\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{V})$.

###### Proof.

In [29] we showed that the isotopy class of $\mathbf{M}$ can be represented as
the isotopy class of
$\theta=\theta_{0}\circ\theta_{1}\circ\ldots\circ\theta_{n-1}$, where
$\theta_{i}$ is a slide along $C_{i}$ supported in $W_{i}-V_{i}$,
$i=0,1,2\ldots,n-1.$

To prove this proposition we need to check that conditions (D1)–(D3) from
subsection 5.5 hold for subgroups $\langle\theta\rangle$ and
$\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{V})$ of
$\pi_{0}\mathcal{S}^{\prime}(f,\mathsf{W}).$ Conditions (D1) and (D2)
obviously hold since $\mathrm{supp}(\theta)\cap\mathrm{supp}(h)=\varnothing$,
where $h\in\mathcal{S}^{\prime}(f,\mathsf{V}).$ So, it remains to show that
for each $h$ in $\mathcal{S}^{\prime}(f,\mathsf{W})$ there is a unique
decomposition

$[h]=[\theta^{k(h)}]\circ[h^{\prime}],$ (11)

where $h^{\prime}\in\mathcal{S}^{\prime}(f,\mathsf{V})$ and
$k(h)\in\mathbb{Z}.$ To define this decomposition we use the same arguments
such in [6, Theorem 5.5].

Let $h$ be a diffeomorphism from $\mathcal{S}^{\prime}(f,\mathsf{W}).$ Since
$f$ is fixed on $\mathsf{W}$, it follows from [28, Lemma 6.1], there exists a
smooth function $\alpha:\mathsf{W}\to\mathbb{R}$ such that
$h=\mathbf{F}_{\alpha}.$ The restriction of $h$ and $\alpha$ onto $W_{i}$ are
denoted by $h_{i}$ and $\alpha_{i}$ respectively. Since periods of all
trajectories of $\mathbf{F}|_{\mathsf{W}}$ is equal to $1$, it follows that
$\alpha_{i}$ takes an integer number $k_{i}(h)\in\mathbb{Z}$, $x\in W_{i}.$
The diffeomorphism $h|_{Q_{i}}$ is isotopic relative $\mathsf{W}\cap Q_{i}$ to
a Dehn twist $\tau^{a_{i}}$ supported on $\mathsf{W}\cap Q_{i}$, where
$a_{i}=\alpha(C_{i+1})-\alpha(C_{i})$, $i=0,1,\ldots,n-1$. Since
$h\in\mathcal{S}^{\prime}(f,\mathsf{W})$, it follows that $h|_{Q_{i}}$ is
isotopic to the identity map of $Q_{i}.$ Hence
$a_{i}=\alpha(C_{i+1})-\alpha(C_{i})=k_{i+1}(h)-k_{i}(h)=0.$ Then numbers
$k_{i}(h)$ pairwise equal for $i=0,1,\ldots,n-1$, so they all coincide, i.e.
$k_{i}(h)=k(h).$

Define an isotopy $H^{t}:T^{2}\to T^{2}$ between $h$ and $\theta^{-k(h)}\circ
h$ by the formula

$H^{t}(h)=\mathbf{F}^{-1}_{tk(h)\sigma}\circ h.$

A diffeomorphism $H^{t}(h)$ is fixed on $\mathsf{V}$ for all $t\in[0,1].$ Then
we have the following decomposition

$[h]=[\theta^{k(h)}]\circ[\theta^{-k(h)}\circ
h]=[\theta^{k(h)}]\circ[H^{1}(h)]$

which coincides with (11) for $h^{\prime}=H^{1}(h).$ ∎

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