# Deformational symmetries of smooth functions on non-orientable surfaces

Iryna Kuznietsova  Department of Algebra and Topology, Institute of
Mathematics of NAS of Ukraine, Tereshchenkivska str. 3, Kyiv, 01601, Ukraine
<EMAIL_ADDRESS>and  Sergiy Maksymenko  Department of Algebra and
Topology, Institute of Mathematics of NAS of Ukraine, Tereshchenkivska str. 3,
Kyiv, 01601, Ukraine<EMAIL_ADDRESS>
###### Abstract.

Given a compact surface $M$, consider the natural right action of the group of
diffeomorphisms $\mathcal{D}(M)$ of $M$ on
$\mathcal{C}^{\infty}(M,\mathbb{R})$ given by $(f,h)\mapsto f\circ h$ for
$f\in\mathcal{C}^{\infty}(M,\mathbb{R})$ and $h\in\mathcal{D}(M)$. Denote by
$\mathcal{F}(M)$ the subset of $\mathcal{C}^{\infty}(M,\mathbb{R})$ consisting
of function $f:M\to\mathbb{R}$ taking constant values on connected components
of $\partial{M}$, having no critical points on $\partial{M}$, and such that at
each of its critical points $z$ the function $f$ is $\mathcal{C}^{\infty}$
equivalent to some homogenenous polynomial without multiple factors. In
particular, $\mathcal{F}(M)$ contains all Morse maps. Let also
$\mathcal{O}(f)=\\{f\circ h\mid h\in\mathcal{D}(M)\\}$ be the orbit of $f$.
Previously it was computed the algebraic structure of $\pi_{1}\mathcal{O}(f)$
for all $f\in\mathcal{F}(M)$, where $M$ is any orientable compact surface
distinct from $2$-sphere.

In the present paper we compute the group
$\pi_{0}\mathcal{S}(f,\partial\mathbb{M})$, where $\mathbb{M}$ is a Möbius
band. As a consequence we obtain an explicit algebraic description of
$\pi_{1}\mathcal{O}(f)$ for all non-orientable surfaces distinct from Klein
bottle and projective plane.

###### Key words and phrases:

Diffeomorphism, Morse function, Möbius band, foliation

###### 2000 Mathematics Subject Classification:

57S05, 57R45, 37C05

##  1\. Introduction

The present paper continues a series of works by many authors [23, 24, 27, 31,
32, 33, 29, 6, 14, 15, 16, 17] and others devoted to the study of the natural
right action

$\mathcal{C}^{\infty}(M,P)\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M,P),\qquad(f,h)\mapsto
f\circ h,$

of the diffeomorphism group $\mathcal{D}(M)$ of a compact surface $M$ on the
space $\mathcal{C}^{\infty}(M,P)$ of smooth maps $f:M\to P$, where $P$ is
either the real line $\mathbb{R}$ or the circle $S^{1}$.

For a closed subset $X$ denote by $\mathcal{D}(M,X)$ the subgroup of
$\mathcal{D}(M)$ consisting of diffeomorphisms fixed on $X$, and for every
$f\in\mathcal{C}^{\infty}(M,P)$ let

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

be respectively the stabilizer and the orbit of
$f\in\mathcal{C}^{\infty}(M,P)$ with respect to that action. It will be
convenient to say that the diffeomorphisms from $\mathcal{S}(f)$ preserve $f$.

Endow the spaces $\mathcal{D}(M,X)$ and $\mathcal{C}^{\infty}(M,P)$ with
Whitney $C^{\infty}$-topologies, and their subspaces $\mathcal{S}(f,Y)$,
$\mathcal{O}(f,Y)$ with the induced ones. Let also

  * •

$\mathcal{D}_{\mathrm{id}}(M,X)$ be the identity path component of
$\mathcal{D}(M,X)$ consisting of diffeomorphisms isotopic to $\mathrm{id}_{M}$
rel. $X$,

  * •

$\mathcal{S}_{\mathrm{id}}(f,X)$ be the identity path component of the
stabilizer $\mathcal{S}(f,X)$, and

  * •

$\mathcal{O}_{f}(f,X)$ be the path component of the orbit $\mathcal{O}(f,X)$
containing $f$.

Finally, denote by

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

the subgroup of $\mathcal{S}(f,X)$ consisting of $f$-preserving
diffeomorphisms isotopic to $\mathrm{id}_{M}$ rel. $X$, but such isotopies are
not required to preserve $f$. If $X=\varnothing$, then we will omit it from
notation.

Note that if $(M,X)$ is either a $(D^{2},\partial D^{2})$ or
$(S^{1}\times[0;1],S^{1}\times 0)$ or $(\mathbb{M},\partial\mathbb{M})$, where
$\mathbb{M}$ is Möbius band, then $\mathcal{D}(M,X)$ is well known to be
contractible (and in particular path connected), whence
$\mathcal{D}(M,X)=\mathcal{D}_{\mathrm{id}}(M,X)$ and therefore in those cases
for every $f\in\mathcal{C}^{\infty}(M,P)$ we have that

$\mathcal{S}^{\prime}(f,X):=\mathcal{S}(f)\cap\mathcal{D}_{\mathrm{id}}(M,X)=\mathcal{S}(f)\cap\mathcal{D}(M,X)=\mathcal{S}(f,X).$
(1.1)

We will study the homotopy types of stabilizers and orbits for maps $f:M\to P$
belonging to the following subset
$\mathcal{F}(M,P)\subset\mathcal{C}^{\infty}(M,P)$.

######  Definition 1.1.

Denote by $\mathcal{C}_{\partial}^{\infty}(M,P)$ the subset of
$\mathcal{C}^{\infty}(M,P)$ consisting of maps $f$ such that

  1. (B)

$f$ takes constant values on connected components of $\partial M$ and has no
critical points therein.

Furthermore, let $\mathcal{F}(M,P)$ be the subset of
$\mathcal{C}_{\partial}^{\infty}(M,P)$ consisting of maps $f$ such that

  1. (H)

for every critical point $z$ of $f$ the germ of $f$ at $z$ is
$\mathcal{C}^{\infty}$ equivalent to some non-zero homogeneous polynomial
$g_{z}\colon\mathbb{R}^{2}\to\mathbb{R}$ without multiple factors.

The space $\mathcal{F}(M,P)$ is rather “massive” and (due to Morse lemma and
axiom (H)) contains all maps with non-degenerate critical points. Moreover,
axiom (H) (namely, absence of multiple factors) implies that for each
$f\in\mathcal{F}(M,P)$ its critical points are isolated, and this further
implies that $\mathcal{F}(M,P)$ also contains “up to a homeomorphism” every
$f\in\mathcal{C}_{\partial}^{\infty}(M,P)$ with isolated critical points, see
Remark 4.3.2.

Our main result explicitly computes the group $\pi_{1}\mathcal{O}(f)$ for
every $f\in\mathcal{F}(\mathbb{M},P)$ on the Möbius band $\mathbb{M}$,
(Theorem 2.6). As a consequence this allows to explicitly describe the
homotopy types of path components $\mathcal{O}_{f}(f)$ of orbits for all maps
$f\in\mathcal{F}(M,P)$ on all non-orientable surfaces $M$ distinct from Klein
bottle and projective plane, (Theorem 2.7). For orientable surfaces distinct
from $2$-torus and $2$-sphere such a description was done in S. Maksymenko
[29], while the case of $2$-torus was studied in a series of papers by S.
Maksymenko and B. Feshchenko [31, 32, 33, 6, 8, 9].

###  1.2. Homotopy type of orbits

For $k\geq 0$ let $T^{k}=(S^{1})^{k}$ be the $k$-torus, i.e. the product of
$k$ circles. Say that a map $f\in\mathcal{F}(M,P)$ is generic Morse, if all
critical points of $f$ are non-degenerate, and $f$ takes distinct values at
distinct critical points. By [23, Theorem 1.5] for every generic Morse map
$f:M\to P$ (expect for few types of such maps) we have the following homotopy
equivalences

$\mathcal{O}_{f}(f)\simeq\begin{cases}T^{k}\times SO(3),&\text{if $M=S^{2}$ or
$\mathbb{RP}^{2}$},\\\ T^{k},&\text{otherwise.}\end{cases}$

Moreover, the number $k$ depends on $f$ and is expressed via the numbers of
critical points of $f$ of distinct indices. See also [28, Theorem 5.3.1] for
an explicit construction of the inclusion $T^{k}\subset\mathcal{O}_{f}(f)$
being a homotopy equivalence in the second case.

The above result was extended by E. Kudryavtseva [14, 15, 16, 17] to arbitrary
Morse functions $f:M\to\mathbb{R}$ on orientable surfaces, and in [18] to
functions with singularities of type $A_{\mu}$. She proved that for every such
$f$ (again except for several specific cases) there exists a free action of a
certain finite group $\mathbf{G}(f)$ on a $k$-torus $T^{k}$ for some $k\geq 0$
such that we have the following homotopy equivalences:

$\mathcal{O}_{f}(f)\simeq\begin{cases}(T^{k}/\mathbf{G}(f))\times
SO(3),&\text{if $M=S^{2}$ or $\mathbb{RP}^{2}$},\\\
T^{k}/\mathbf{G}(f),&\text{otherwise.}\end{cases}$

That group $\mathbf{G}(f)$ can be regarded as the group of automorphisms of
the Kronrod-Reeb graph of $f$ induced by diffeomorphisms from
$\mathcal{S}^{\prime}(f)$, and it is also trivial for generic Morse maps.
However, the algebraic structure of $\pi_{1}\mathcal{O}_{f}(f)$ and
$\mathbf{G}(f)$ was not clear. S. Maksymenko [29] explicitly computed
$\pi_{1}\mathcal{O}_{f}(f)$ and $\mathbf{G}(f)$ for all
$f\in\mathcal{F}(M,P)$, where $M$ is an orientable surface distinct from
$2$-torus and $2$-sphere, S. Maksymenko and B. Feshchenko [31, 32, 33, 6, 8,
9] completely described the case when $M$ is a $2$-torus, and also in [30]
explicit models for the actions of $\mathbf{G}(f)$ on $T^{k}$ are given. In
the present paper those computations will be extended to all non-orientable
surfaces distinct from the Klein bottle and projective plane.

Our starting point is the following statement collecting several particular
results from the mentioned above papers.

######  Theorem 1.2.1.

Let $M$ be a connected compact surface, $Y$ be a possibly empty collection of
boundary components of $M$, and $f\in\mathcal{F}(M,P)$.

  1. (1)

Then $\mathcal{O}_{f}(f,Y)=\mathcal{O}_{f}(f)$, i.e. the path components of
$f$ in those orbits coincide as sets, thought it is not true in general that
$\mathcal{O}(f,Y)=\mathcal{O}(f)$.

  2. (2)

Suppose that either $Y\not=\varnothing$, or $Y=\varnothing$ but the Euler
characteristic $\chi(M)<0$. Then

    1. (2a)

$\mathcal{D}_{\mathrm{id}}(M,\partial Y)$ is contractible,
$\mathcal{O}_{f}(f,Y)$ is aspherical, and we have an isomorphism

$\partial:\pi_{1}\mathcal{O}_{f}(f,Y)\to\pi_{0}\mathcal{S}^{\prime}(f,Y);$
(1.2)

    2. (2b)

there exists a compact subsurface $X\subset M$ such that $\overline{M\setminus
X}$ is a disjoint union of compact subsurfaces $L_{1},\ldots,L_{p}$, where
each $L_{i}$ is either a $2$-disk or a cylinder or a Möbius band,
$f|_{L_{i}}\in\mathcal{F}(L_{i},P)$, and the inclusions

$\textstyle{\ \mathcal{S}^{\prime}(f,Y)\ }$$\textstyle{\
\mathcal{S}^{\prime}(f,Y\cup X)\
\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{\
\prod\limits_{i=1}^{p}\mathcal{S}^{\prime}(f|_{L_{i}},\partial L_{i})\ }$

are homotopy equivalences, where $\alpha(h)=(h|_{L_{1}},\ldots,h|_{L_{p}})$.

In particular, we get isomorphisms:

$\pi_{1}\mathcal{O}_{f}(f)\ \cong\ \pi_{1}\mathcal{O}_{f}(f,Y)\ \cong\
\pi_{0}\mathcal{S}^{\prime}(f,Y)\ \cong\
\prod\limits_{i=1}^{p}\pi_{0}\mathcal{S}^{\prime}(f|_{L_{i}},\partial L_{i}).$
(1.3)

###### Remarks to the proof.

Statement (1) proved in [27, Corollary 2.1], and (2)(2b) in [26, Theorem 1.7]

Contractibility of $\mathcal{D}_{\mathrm{id}}(M,Y)$ with $Y\not=\varnothing$
is well-known, e.g. [4, 5, 10]. Further, by statement (c) of Theorem 4.1.1
below, the map $p:\mathcal{D}_{\mathrm{id}}(M,Y)\to\mathcal{O}_{f}(f,Y)$,
$p(h)=f\circ h$, is a locally trivial fibration with fiber
$\mathcal{S}^{\prime}(f,Y)$. Hence, we get the following part of the short
exact sequence of that fibration:

$\pi_{1}\mathcal{D}_{\mathrm{id}}(M,Y)\to\pi_{1}\mathcal{O}_{f}(f,Y)\xrightarrow{\leavevmode\nobreak\
\partial\leavevmode\nobreak\
}\pi_{0}\mathcal{S}^{\prime}(f,Y)\to\pi_{0}\mathcal{D}_{\mathrm{id}}(M,Y).$

Since $\mathcal{D}_{\mathrm{id}}(M,Y)$ is contractible, the first and last
terms are zero, which implies that $\partial$ is an isomorphism. ∎

Thus, by (2)(2a), the group $\pi_{0}\mathcal{S}^{\prime}(f,Y)$ completely
determines the homotopy type of $\mathcal{O}_{f}(f)$. Moreover, due to
(2)(2b), the computation of $\pi_{0}\mathcal{S}^{\prime}(f,Y)$ reduces to the
computation of groups $\pi_{0}\mathcal{S}^{\prime}(g,\partial L)$ for
functions $g\in\mathcal{F}(L,P)$, where $L$ is either a $2$-disk or a cylinder
or a Möbius band.

The structure of such groups for disks and cylinders is described in [29].
Moreover, it is shown in [19, Theorem 1.4], see Lemma 6.2.1 below, that every
map $f\in\mathcal{F}(\mathbb{M},P)$ allows to additionally decompose the
Möbius band $\mathbb{M}$ into one cylinder $Y_{0}$, several $2$-disks
$Y_{1},\ldots,Y_{k}$, and a certain subsurface $X$. Our main result (Theorem
2.6) shows that $\pi_{1}\mathcal{O}(f)$ can be expressed via the groups
$\pi_{1}\mathcal{O}(f|_{Y_{i}})$ in a certain explicit algebraic way.

##  2\. Main result

In what follows $\mathbf{1}$ denotes the unit group, and the symbols
$\lhook\joinrel\rightarrow$ and $\rightarrow\\!\\!\\!\\!\\!\to$ denote
respectively monomorphism and epimorphism. In particular, the notation
${\color[rgb]{1,0,0}A}\lhook\joinrel\rightarrow{\color[rgb]{1,0,0}B}\rightarrow\\!\\!\\!\\!\\!\to{\color[rgb]{1,0,0}C}$
means a short exact sequence of groups.

First we will need to introduce several types of semidirect products
corresponding to certain non-effective actions of the group $\mathbb{Z}$. All
of them are particular cases of the following construction.

Let ${\color[rgb]{1,0,0}G}$ be a group with unit $e$ and
$\varphi\in\mathrm{Aut}({\color[rgb]{1,0,0}G})$ be any automorphism of
${\color[rgb]{1,0,0}G}$. Then one can define the homomorphism
$\widehat{\varphi}:\mathbb{Z}\to\mathrm{Aut}({\color[rgb]{1,0,0}G})$,
$\widehat{\varphi}(t)=\varphi^{t}$, $t\in\mathbb{Z}$. Hence, we also have the
corresponding semidirect product
${\color[rgb]{1,0,0}G}\rtimes_{\varphi}\mathbb{Z}$ which, by definition, is
the Cartesian product ${\color[rgb]{1,0,0}G}\times\mathbb{Z}$ of sets with the
following multiplication:

$(a,k)(b,l)=(a\,\varphi^{k}(b),k+l)$ (2.1)

for all $a,b\in{\color[rgb]{1,0,0}G}$ and $k,l\in\mathbb{Z}$.

Furthermore, if $\varphi$ is periodic of some order
${\color[rgb]{0.64,0.08,0.98}m}\geq 1$, then $\widehat{\varphi}$ reduces to a
monomorphism
$\overline{\varphi}:\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\to\mathrm{Aut}({\color[rgb]{1,0,0}G})$,
$\overline{\varphi}(t)=\varphi^{t}$,
$t\in\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$, and we also have the
semidirect product
${\color[rgb]{1,0,0}G}\rtimes_{\varphi}\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$,
which again by definition, is a Cartesian product
${\color[rgb]{1,0,0}G}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$ with
the multiplication given by same formula (2.1), but now $k,l$ should be taken
modulo ${\color[rgb]{0.64,0.08,0.98}m}$.

Evidently, there is the following short exact sequence

$\\{e\\}\times{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}\lhook\joinrel\xrightarrow{\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\
}{\color[rgb]{1,0,0}G}\rtimes_{\varphi}\mathbb{Z}\xrightarrow{\leavevmode\nobreak\
(g,\,k)\,\mapsto\,(g,\,k\bmod{\color[rgb]{0.64,0.08,0.98}m})\leavevmode\nobreak\
}\\!\\!\\!\\!\\!\to{\color[rgb]{1,0,0}G}\rtimes_{\varphi}\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}.$

###  2.1. Groups
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ and
${\color[rgb]{1,0,0}G}\wr\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$

Let ${\color[rgb]{0.64,0.08,0.98}m}\in\mathbb{N}$ and
$\varphi:{\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\to{\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}$
be the automorphism cyclically shifting coordinates, i.e.
$\varphi(a_{0},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1})=(a_{1},a_{2},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1},a_{0})$
for all
$(a_{0},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1})\in{\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}$.
Define

$\displaystyle{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
$\displaystyle:={\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\rtimes_{\varphi}\mathbb{Z},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$
$\displaystyle:={\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\rtimes_{\varphi}\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$

to be the corresponding semidirect products. Thus,
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ is a
Cartesian product
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}$ with
the following multiplication:

$(a_{0},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1};k)\cdot(b_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1};l)=(a_{0}b_{k},a_{1}b_{k+1},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1}b_{k-1};k+l),$

where the indices are taken modulo ${\color[rgb]{0.64,0.08,0.98}m}$,
$a_{i},b_{i}\in{\color[rgb]{1,0,0}G}$, $k,l\in\mathbb{Z}$. On the other hand,
${\color[rgb]{1,0,0}G}\wr\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$ is the
set
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$
but with the multiplication given by the same formulas, but now
$k,l\in\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$.

Evidently,
${\color[rgb]{1,0,0}G}\wr\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$ is the
standard wreath product of groups ${\color[rgb]{1,0,0}G}$ and
$\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}$, while
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ is also
the wreath product of ${\color[rgb]{1,0,0}G}$ and $\mathbb{Z}$ with respect to
the noneffective action of $\mathbb{Z}$ on the set
$\\{0,1,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1\\}$ by cyclic shifts to the
left.

###  2.2. Groups
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}}\mathbb{Z}^{2}$
and
${\color[rgb]{1,0,0}G}\wr(\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}})$

More generally, let
${\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}\in\mathbb{N}$
be two numbers. Then the elements of the
${\color[rgb]{0.64,0.08,0.98}m}{\color[rgb]{0.64,0.08,0.98}n}$-power
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}{\color[rgb]{0.64,0.08,0.98}n}}$
of ${\color[rgb]{1,0,0}G}$ can be regarded as
$({\color[rgb]{0.64,0.08,0.98}m}\times{\color[rgb]{0.64,0.08,0.98}n})$-matrices
$\\{g_{i,j}\\}_{i=0,\ldots{\color[rgb]{0.64,0.08,0.98}m}-1,\,j=0,\ldots,{\color[rgb]{0.64,0.08,0.98}n}-1}$
whose entries are elements of ${\color[rgb]{1,0,0}G}$. Then there is a natural
non-effective action of $\mathbb{Z}^{2}$ on
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}{\color[rgb]{0.64,0.08,0.98}n}}$
by cyclic shifts of rows and columns, i.e.

$(k,l)\cdot\\{g_{i,j}\\}=\\{g_{i+k\bmod{\color[rgb]{0.64,0.08,0.98}m},\,j+l\bmod{\color[rgb]{0.64,0.08,0.98}n}}\\}.$

This action reduces to an effective action of
$\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}}$.
Let

$\displaystyle{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}}\mathbb{Z}^{2}:={\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}{\color[rgb]{0.64,0.08,0.98}n}}\rtimes\mathbb{Z}^{2},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr(\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}}):={\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}{\color[rgb]{0.64,0.08,0.98}n}}\rtimes(\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}})$

be the corresponding semidirect products. Then
${\color[rgb]{1,0,0}G}\wr(\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}})$
is the standard wreath product of groups ${\color[rgb]{1,0,0}G}$ and
$\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}_{{\color[rgb]{0.64,0.08,0.98}n}}$,
while
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}}\mathbb{Z}^{2}$
is also the wreath product of ${\color[rgb]{1,0,0}G}$ and $\mathbb{Z}^{2}$
with respect to the action of $\mathbb{Z}^{2}$ on the set
$\\{0,1,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1\\}\times\\{0,1,\ldots,{\color[rgb]{0.64,0.08,0.98}n}-1\\}$
by independent cyclic shifts of coordinates.

###  2.3. Groups
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
and
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}$

Let ${\color[rgb]{1,0,0}H}$ be another group,
${\color[rgb]{0.64,0.08,0.98}m}\in\mathbb{N}$, and
$\gamma\colon{\color[rgb]{1,0,0}H}\to{\color[rgb]{1,0,0}H}$ be an automorphism
such that $\gamma^{2}=\mathrm{id}_{{\color[rgb]{1,0,0}H}}$, so $\gamma$ is
either the identity automorphism or has order $2$. Define the automorphism
$\psi\colon{\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}}\to{\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}}$
by the following formula

$\psi(a_{0},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1};\,b_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1})=(a_{1},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1},a_{0};\,b_{1},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1},\gamma(b_{0})).$
(2.2)

Evidently, $\psi$ is of order $2{\color[rgb]{0.64,0.08,0.98}m}$. Let

$\displaystyle\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}:=({\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}})\rtimes_{\psi}\mathbb{Z},$
$\displaystyle\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}:=({\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}})\rtimes_{\psi}\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}},$

be the corresponding semidirect products associated with $\psi$. Thus,
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
is the Cartesian product
${\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}$
with the following operation. Let
$v=(a_{0},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1};\,b_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1};k)$
and
$w=(c_{0},\ldots,c_{2{\color[rgb]{0.64,0.08,0.98}m}-1};\,d_{0},\ldots,d_{{\color[rgb]{0.64,0.08,0.98}m}-1};l)\in{\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}}\times\mathbb{Z}$.
Denote $k^{\prime}=k\bmod 2{\color[rgb]{0.64,0.08,0.98}m}$. Then

$v\,w=\begin{cases}\bigl{(}a_{0}c_{k},a_{1}c_{k+1},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1}c_{k-1};\\\
\quad
b_{0}d_{k},b_{1}d_{k+1},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-k-1}d_{{\color[rgb]{0.64,0.08,0.98}m}-1},\,b_{{\color[rgb]{0.64,0.08,0.98}m}-k}\gamma(d_{0}),\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1}\gamma(d_{k-1});\,k+l\bigr{)},\\\
\hfill 0\leq k^{\prime}<{\color[rgb]{0.64,0.08,0.98}m},\\\\[2.84526pt]
\bigl{(}a_{0}c_{k},a_{1}c_{k+1},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1}c_{k-1};\\\
\quad
b_{0}\gamma(d_{k}),b_{1}\gamma(d_{k+1}),\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-k-1}\gamma(d_{{\color[rgb]{0.64,0.08,0.98}m}-1}),\,b_{{\color[rgb]{0.64,0.08,0.98}m}-k}d_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1}d_{k-1};\,k+l\bigr{)},\\\
\hfill{\color[rgb]{0.64,0.08,0.98}m}\leq
k^{\prime}<2{\color[rgb]{0.64,0.08,0.98}m},\end{cases}$ (2.3)

where the indices of $a_{*}$ and $c_{*}$ are taken modulo
$2{\color[rgb]{0.64,0.08,0.98}m}$, while the indices of $b_{*}$ and $d_{*}$
are taken modulo ${\color[rgb]{0.64,0.08,0.98}m}$.

Again the multiplication in
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}$
is given by the same formulas (2.3), but one should take
$k,l\in\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}$.

Evidently, for every ${\color[rgb]{0.64,0.08,0.98}m}\in\mathbb{N}$ we have the
following isomorphisms:

$\displaystyle\mathbf{1}\wr_{1}\mathbb{Z}$ $\displaystyle\cong\mathbb{Z},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr_{1}\mathbb{Z}$
$\displaystyle\cong{\color[rgb]{1,0,0}G}\times\mathbb{Z},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr_{1,1}\mathbb{Z}^{2}$
$\displaystyle\cong{\color[rgb]{1,0,0}G}\times\mathbb{Z}^{2},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr_{2{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
$\displaystyle\cong\left({\color[rgb]{1,0,0}G},\mathbf{1}\right)\wr_{\mathrm{id}_{\mathbf{1}},{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z},$
$\displaystyle{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m},1}\mathbb{Z}^{2}$
$\displaystyle\cong({\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z})\times\mathbb{Z}.$

######  Definition 2.4.

Let $\mathcal{G}$ be the minimal class of groups (considered up to
isomorphism) satisfying the following conditions:

  1. 1)

$\mathbf{1}\in\mathcal{G}$;

  2. 2)

if ${\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\in\mathcal{G}$, then
${\color[rgb]{1,0,0}G}\times{\color[rgb]{1,0,0}H}\in\mathcal{G}$;

  3. 3)

if ${\color[rgb]{1,0,0}G}\in\mathcal{G}$ and
${\color[rgb]{0.64,0.08,0.98}m}\geq 1$, then
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}\in\mathcal{G}$.

It is easy to see, [29, Lemma 2.6], that $\mathcal{G}$ consists of groups
${\color[rgb]{1,0,0}G}$ which can be obtained from the unit group $\mathbf{1}$
by finitely many operations of direct products $\times$ and wreath products
$\cdot\wr_{\bullet}\mathbb{Z}$. For example, the following groups belong to
$\mathcal{G}$:

$\displaystyle\mathbb{Z}=\mathbf{1}\wr_{1}\mathbb{Z},$
$\displaystyle\mathbb{Z}^{n},(n\geq 1),$
$\displaystyle\bigl{(}(\mathbb{Z}^{3}\wr_{5}\mathbb{Z})\times(\mathbb{Z}^{17}\wr_{2}\mathbb{Z})\bigr{)}\wr_{11}\mathbb{Z}.$

Notice that such an expression of a group
${\color[rgb]{1,0,0}G}\in\mathcal{G}$ is not unique, e.g.
$\mathbf{1}\wr_{1}\mathbb{Z}\cong(\mathbf{1}\times\mathbf{1})\wr_{1}\mathbb{Z}$.

The following theorem collects some known information about the structure of
$\pi_{1}\mathcal{O}(f)$ for maps $f\in\mathcal{F}(M,P)$ on orientable surfaces
$M$ distinct from $S^{2}$.

######  Theorem 2.5 ([29, 20, 8]).

Let $M$ be a compact orientable surface distinct from $2$-sphere and
$f\in\mathcal{F}(M,P)$.

  1. (1)

If $M$ is also distinct from $2$-torus, then there exists
${\color[rgb]{1,0,0}G}\in\mathcal{G}$ such that

$\pi_{1}\mathcal{O}(f)\cong{\color[rgb]{1,0,0}G}.$ (2.4)

  2. (2)

Suppose $M=T^{2}$ is a $2$-torus. If the Kronrod-Reeb graph of $f$ is a tree
then there exist ${\color[rgb]{1,0,0}G}\in\mathcal{G}$ and
${\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}\in\mathbb{N}$
such that

$\pi_{1}\mathcal{O}(f)\cong{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m},{\color[rgb]{0.64,0.08,0.98}n}}\mathbb{Z}^{2}.$
(2.5)

Otherwise, the Kronrod-Reeb graph of $f$ contains a unique cycle and there
exist ${\color[rgb]{1,0,0}G}\in\mathcal{G}$ and
${\color[rgb]{0.64,0.08,0.98}m}\in\mathbb{N}$ such that

$\pi_{1}\mathcal{O}(f)\cong{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}.$
(2.6)

The latter holds e.g. if $P=S^{1}$ and $f:T^{2}\to S^{1}$ is not null
homotopic.

Conversely, for every group as in the r.h.s. of (2.4)-(2.6) there exists
$f\in\mathcal{F}(M,P)$ on the corresponding surface $M$ such that the
corresponding isomorphism holds. Moreover, one can also assume that $f$ is
Morse.

###### Remarks to the proof.

It is shown in [29, Theorem 5.10] that if $M\not=T^{2}$ and $S^{2}$, then for
each $f\in\mathcal{F}(M,P)$ it is possible to explicitly express
$\pi_{1}\mathcal{O}(f)=\pi_{0}\mathcal{S}(f,\partial M)$ in terms of
operations $\times$ and $\cdot\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$.
Conversely, it is proved in [20, Theorem 1.2] that for any finite combination
of such operations giving rise a group ${\color[rgb]{1,0,0}G}$ one can
construct some $f\in\mathcal{F}(M,P)$ with
${\color[rgb]{1,0,0}G}\cong\pi_{1}\mathcal{O}(f)$. In addition, if
$Y_{1},\ldots,Y_{k}$ is some collection of boundary components of $M$, then
(except for few cases) one can even assume that $f$ takes a local minimum of
each $Y_{i}$, and a local maximum on each boundary components from $\partial
M\setminus\cup_{i=1}^{k}Y_{i}$.

Similar considerations were done for the case of torus in [31, 32, 33, 6, 8,
9]. See especially [8, Theorem 3.2] for the general statement on the structure
of $\pi_{1}\mathcal{O}(f)$ and $\pi_{0}\mathcal{S}^{\prime}(f)$ and several
other groups related with $f\in\mathcal{F}(T^{2},P)$. ∎

Our main result is the following theorem which will be proved in Section 7.

######  Theorem 2.6.

Let $\mathbb{M}$ be the Möbius band. Then for every
$f\in\mathcal{F}(\mathbb{M},P)$ either

  * $-$

there exist groups
${\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\in\mathcal{G}$,
an automorphism $\gamma:{\color[rgb]{1,0,0}H}\to{\color[rgb]{1,0,0}H}$ with
$\gamma^{2}=\mathrm{id}_{{\color[rgb]{1,0,0}H}}$, and
${\color[rgb]{0.64,0.08,0.98}m}\geq 1$ such that

$\pi_{1}\mathcal{O}(f)\ \cong\ {\color[rgb]{1,0,0}A}\ \times\
\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z},$
(2.7)

  * $-$

or there exist groups
${\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}G}\in\mathcal{G}$ and an odd number
${\color[rgb]{1,0,0}b}\geq 1$ such that

$\pi_{1}\mathcal{O}(f)\ \cong\ {\color[rgb]{1,0,0}A}\ \times\
{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{1,0,0}b}}\mathbb{Z}.$ (2.8)

In particular, in the second case $\pi_{1}\mathcal{O}(f)\in\mathcal{G}$.

Conversely, for every such tuple
$({\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}b})$ or
$({\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H},\gamma,{\color[rgb]{0.64,0.08,0.98}m})$
there exists $f\in\mathcal{F}(\mathbb{M},P)$ such that we have the
corresponding isomorphism (2.7) or (2.8), and one can assume that $f$ is
Morse.

As a consequence we get the following statement:

######  Theorem 2.7.

Let $M$ be a non-orientable compact surface (possibly with boundary) of non-
orientable genus $g\geq 2$, i.e. distinct from Klein bottle and projective
plane $\mathbb{RP}^{2}$. Then for every $f\in\mathcal{F}(M,P)$ there exist
$k\leq g$, groups
${\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}G}_{1},{\color[rgb]{1,0,0}H}_{1},\ldots,{\color[rgb]{1,0,0}G}_{k},{\color[rgb]{1,0,0}H}_{k}\in\mathcal{G}$,
and for each $j=1,\ldots,k$ an automorphism
$\gamma_{j}:{\color[rgb]{1,0,0}H}_{j}\to{\color[rgb]{1,0,0}H}_{j}$ with
$\gamma_{j}^{2}=\mathrm{id}_{{\color[rgb]{1,0,0}H}_{j}}$ and
${\color[rgb]{0.64,0.08,0.98}m}_{j}\in\mathbb{N}$ such that

$\pi_{1}\mathcal{O}(f)\ \cong\
{\color[rgb]{1,0,0}A}\times\prod_{j=1}^{k}\left({\color[rgb]{1,0,0}G}_{j},{\color[rgb]{1,0,0}H}_{j}\right)\wr_{\gamma_{j},{\color[rgb]{0.64,0.08,0.98}m}_{j}}\mathbb{Z}.$
(2.9)

This formally includes the case $k=0$, when
$\pi_{1}\mathcal{O}(f)\cong{\color[rgb]{1,0,0}A}\in\mathcal{G}$.

###### Proof.

Let $f\in\mathcal{F}(M,P)$. Then by Theorem 1.2.1(2)(2b),
$\pi_{1}\mathcal{O}(f)\cong\prod\limits_{j=1}^{p}\pi_{1}\mathcal{O}(f_{j})$,
where each $L_{j}$ is either a $2$-disk or a cylinder or a Möbius band and
$f_{j}=f|_{L_{j}}\in\mathcal{F}(L_{j},P)$. One can assume that

  * $-$

for some $k<p$ we have that $L_{1},\ldots,L_{k}$ are Möbius bands such that
$\pi_{1}\mathcal{O}(f|_{L_{j}})$ is of type (2.7),

  * $-$

while all others $L_{k+1},\ldots,L_{p}$ are $2$-disks, cylinders, or Möbius
bands, so $\pi_{1}\mathcal{O}(f|_{L_{j}})\in\mathcal{G}$ for all these
subsurfaces.

Recall that the non-orientable genus of $M$ can be defined as the maximal
number of embedded and mutually disjoint Möbius bands in $M$. Therefore, since
$L_{j}$ are mutually disjoint, we should have that $k\leq g$.

Now by Theorem 2.6 and statement (1) of Theorem 2.5 there exist groups
${\color[rgb]{1,0,0}A}_{i},{\color[rgb]{1,0,0}G}_{j},{\color[rgb]{1,0,0}H}_{j}\in\mathcal{G}$,
automorhisms
$\gamma_{j}:{\color[rgb]{1,0,0}H}_{j}\to{\color[rgb]{1,0,0}H}_{j}$ with
$\gamma_{j}^{2}=\mathrm{id}_{{\color[rgb]{1,0,0}H}_{j}}$ and numbers
${\color[rgb]{0.64,0.08,0.98}m}_{j}\in\mathbb{N}$, where $i=1,\ldots,p$ and
$j=1,\ldots,k$, such that

$\pi_{1}\mathcal{O}(f_{j})\cong\begin{cases}{\color[rgb]{1,0,0}A}_{j}\ \times\
\left({\color[rgb]{1,0,0}G}_{j},{\color[rgb]{1,0,0}H}_{j}\right)\wr_{\gamma_{j},{\color[rgb]{0.64,0.08,0.98}m}_{j}}\mathbb{Z},&j=1,\ldots,k,\\\
{\color[rgb]{1,0,0}A}_{j},&j=k+1,\ldots,p.\end{cases}$

Denote
${\color[rgb]{1,0,0}A}=\mathop{\prod}\limits_{i=1}^{p}{\color[rgb]{1,0,0}A}_{i}$.
Since the class $\mathcal{G}$ is closed under direct products, see property 2)
of Definition 2.4, ${\color[rgb]{1,0,0}A}\in\mathcal{G}$ as well, and
therefore
$\pi_{1}\mathcal{O}(f)\cong{\color[rgb]{1,0,0}A}\times\mathop{\prod}\limits_{j=1}^{k}\left({\color[rgb]{1,0,0}G}_{j},{\color[rgb]{1,0,0}H}_{j}\right)\wr_{\gamma_{j},{\color[rgb]{0.64,0.08,0.98}m}_{j}}\mathbb{Z}$,
which agrees with (2.9). ∎

######  Remark 2.7.1.

The question of realization of groups of the form (2.7) as
$\pi_{1}\mathcal{O}_{f}(f)$ for some $f\in\mathcal{F}(M,P)$, where $M$ is a
non-orientable surface as distinct from Klein bottle and projective plane, is
more delicate, and will be studied in another paper.

###  2.8. Structure of the paper

Section 3 contains two results characterizing groups
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ and
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$,
see Lemmas 3.2.1 and 3.2.2.

In Section 4 we present preliminary information about actions of
diffeomorphisms on spaces of functions on surfaces. In particular, we discuss
the codimension (Milnor number) of a singularity used in Lemma 1.2.1, see
Theorem 4.1.1.

Section 5 describes several constructions related with smooth functions on
surfaces.

In Section 6 we recall the result from [19], see Lemma 6.2.1, claiming that
every $f\in\mathcal{F}(\mathbb{M},P)$ yields a certain “almost CW” partition
of $\mathbb{M}$ which is preserved by elements of $\mathcal{S}(f)$. We also
study the action of $\mathcal{S}(f)$ on elements of that partition.

In Section 7 we start to prove Theorem 2.6 reducing it to the case when the
group ${\color[rgb]{1,0,0}A}$ is trivial. We also formulate Theorem 7.2.4 and
in Section 8 deduce from it Theorem 2.6, and, in particular, establish Lemmas
8.1-8.3 giving a more detailed information on the structure of
$\pi_{1}\mathcal{O}(f)$.

Section 9 contains the proof of Theorem 7.2.4 itself.

##  3\. Algebraic preliminaries

###  3.1. Exact $(3\times 3)$-diagrams and epimorphisms onto $\mathbb{Z}$

The following lemmas are easy and well known. We leave them for the reader.

######  Lemma 3.1.1.

Let ${\color[rgb]{1,0,0}B}$ be a group and
${\color[rgb]{1,0,0}A},{\color[rgb]{1,0,0}L}\triangleleft{\color[rgb]{1,0,0}B}$
two normal subgroups, and
${\color[rgb]{1,0,0}K}={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}L}$. Then
we have the following $(3\times 3)$-diagram in which all rows and columns are
short exact sequences:

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0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}B}/{\color[rgb]{1,0,0}A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
89.22037pt\raise-41.50003pt\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<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}A}/{\color[rgb]{1,0,0}K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
16.07635pt\raise-52.00003pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
26.07635pt\raise-52.00003pt\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
26.07635pt\raise-52.00003pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}B}/{\color[rgb]{1,0,0}L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
66.4131pt\raise-52.00003pt\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
66.4131pt\raise-52.00003pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}B}/({\color[rgb]{1,0,0}A}{\color[rgb]{1,0,0}L})}$}}}}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(3.1)

Such diagrams will be called exact $(3\times 3)$-diagram.

######  Lemma 3.1.2.

Let ${\color[rgb]{1,0,0}B}$ be a group,
$\eta:{\color[rgb]{1,0,0}B}\to\mathbb{Z}$ be an epimorphism with kernel
${\color[rgb]{1,0,0}L}$, and $g\in{\color[rgb]{1,0,0}B}$ be any element such
that $\eta(g)=1$. Denote by
$\phi:{\color[rgb]{1,0,0}L}\to{\color[rgb]{1,0,0}L}$, $\phi(v)=g^{-1}vg$, the
automorphism of ${\color[rgb]{1,0,0}L}$ induced by conjugation of
${\color[rgb]{1,0,0}B}$ with $g$. Then the map
$\theta:{\color[rgb]{1,0,0}L}\rtimes_{\phi}\mathbb{Z}\to{\color[rgb]{1,0,0}B}$,
$\theta(v,k)=vg^{k}$, is an isomorphism inducing an isomorphism of the
following short exact sequences:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
6.40279pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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6.40279pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
30.40279pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
30.40279pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}L}\rtimes_{\phi}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
86.65611pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
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37.24335pt\raise-14.67499pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\theta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
46.52945pt\raise-19.51668pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
86.65611pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}}$}}}}}}}{\hbox{\kern-6.40279pt\raise-29.34999pt\hbox{\hbox{\kern
0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
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0.0pt\hbox{\lx@xy@hook{1}}}}}}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
39.48605pt\raise-29.34999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\kern
39.48605pt\raise-29.34999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{{\color[rgb]{1,0,0}B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
65.37663pt\raise-24.16249pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-0.8264pt\hbox{$\scriptstyle{\eta}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
86.65611pt\raise-29.34999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern-3.0pt\lower
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
86.65611pt\raise-29.34999pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\hbox{\kern-1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces}}}}\ignorespaces\end{gathered}$
(3.2)

######  Lemma 3.1.3.

Let ${\color[rgb]{1,0,0}L}$ and ${\color[rgb]{1,0,0}L}^{\prime}$ be two
groups, $\phi\in\mathrm{Aut}({\color[rgb]{1,0,0}L})$ and
$\phi^{\prime}\in\mathrm{Aut}({\color[rgb]{1,0,0}L}^{\prime})$ some of their
automorhisms, and $q:{\color[rgb]{1,0,0}L}\to{\color[rgb]{1,0,0}L}^{\prime}$
be a homomorphism. Then the following conditions are equivalent:

  1. (a)

$q\circ\phi=\phi^{\prime}\circ q$;

  2. (b)

the map
$\psi:{\color[rgb]{1,0,0}L}\rtimes_{\phi}\mathbb{Z}\to{\color[rgb]{1,0,0}L}^{\prime}\rtimes_{\phi^{\prime}}\mathbb{Z}$
defined by $\psi(a,k)=(q(a),k)$ is a homomorphism.

In other words, the following diagrams are mutually commutative or non-
commutative:

Under above conditions $q$ is an isomorphism iff so is $\psi$.

Also recall that a group ${\color[rgb]{1,0,0}L}$ is a product of its subgroups
${\color[rgb]{1,0,0}G}_{1},\ldots,{\color[rgb]{1,0,0}G}_{n}<{\color[rgb]{1,0,0}L}$
if the map

$\delta:{\color[rgb]{1,0,0}G}_{1}\times\cdots\times{\color[rgb]{1,0,0}G}_{n}\to{\color[rgb]{1,0,0}L},\qquad\delta(g_{1},\ldots,g_{n})=g_{1}\cdots
g_{n},$

is an isomorphism of groups. Notice that in this definition some groups are
allowed to be unit groups.

###  3.2. Characterization of groups
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ and
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$

Let ${\color[rgb]{1,0,0}G}$ be a group with unit $e$ and
${\color[rgb]{0.64,0.08,0.98}m}\geq 1$. In Section 2.1 we defined the
semidirect product
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
corresponding to the non-effective action of $\mathbb{Z}$ on
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}$ by cyclic left shifts
of coordinates. Notice that
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$ contains
the element $g=(0,\ldots,0,1)$ and the following subgroups

${\color[rgb]{1,0,0}G}_{i}:=e\times\cdots\times
e\times{\color[rgb]{1,0,0}G}\times e\times\cdots\times e\times 0,\quad
i=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1,$

where the multiple ${\color[rgb]{1,0,0}G}$ stands at position $i$. One easily
checks that
$g^{{\color[rgb]{0.64,0.08,0.98}m}}=(0,\ldots,0,{\color[rgb]{0.64,0.08,0.98}m})$
commutes with each ${\color[rgb]{1,0,0}G}_{i}$,
$g{\color[rgb]{1,0,0}G}_{i}g^{-1}={\color[rgb]{1,0,0}G}_{i+1\bmod{\color[rgb]{0.64,0.08,0.98}m}}$
for all $i=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1$, and the kernel
${\color[rgb]{1,0,0}G}^{{\color[rgb]{0.64,0.08,0.98}m}}\times 0$ of the
natural epimorphism
$\eta:{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}\to\mathbb{Z}$,
$\eta(g_{0}\ldots,g_{{\color[rgb]{0.64,0.08,0.98}m}-1},k)=k$, splits into a
direct product of subgroups ${\color[rgb]{1,0,0}G}_{i}$. As the following
lemma shows these properties characterize the group
${\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$.

######  Lemma 3.2.1 ([29, Lemma 2.3]).

Let $\eta:{\color[rgb]{1,0,0}B}\to\mathbb{Z}$ be an epimorphism with kernel
${\color[rgb]{1,0,0}L}:=\ker(\eta)$. Suppose there exist
${\color[rgb]{0.64,0.08,0.98}m}\geq 1$, $g\in{\color[rgb]{1,0,0}B}$, and a
subgroup ${\color[rgb]{1,0,0}G}$ of ${\color[rgb]{1,0,0}L}$ such that

  1. (a)

$\eta(g)=1$ and $g^{{\color[rgb]{0.64,0.08,0.98}m}}$ commutes with
${\color[rgb]{1,0,0}L}$;

  2. (b)

${\color[rgb]{1,0,0}L}$ is a direct product of subgroups
${\color[rgb]{1,0,0}G}_{i}:=g^{i}{\color[rgb]{1,0,0}G}g^{-i}$,
$i=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1$.

Then the following map
$\theta\colon{\color[rgb]{1,0,0}G}\wr_{{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}\to{\color[rgb]{1,0,0}B}$,
$\theta(a_{0},a_{1},\ldots,a_{{\color[rgb]{0.64,0.08,0.98}m}-1};\,k)=\Bigl{(}\mathop{\prod}\limits_{i=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}g^{i}a_{i}g^{-i}\Bigr{)}g^{k}$,
is an isomorphism which also yields an isomorphism of the following short
exact sequences:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
16.88966pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
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3.0pt}}}}}}\ignorespaces{\hbox{\kern
55.83487pt\raise-17.49161pt\hbox{\hbox{\kern 0.0pt\raise
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1.0pt\raise 0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}\ignorespaces{}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx@xy@droprule}}}}{\hbox{\hbox{\kern 1.0pt\raise
0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\hbox{\lx<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
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0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
48.79147pt\raise-27.32492pt\hbox{\hbox{\kern 0.0pt\raise
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94.78008pt\raise-27.32492pt\hbox{\hbox{\kern 0.0pt\raise
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(3.3)

Let also ${\color[rgb]{1,0,0}A}$ be a normal subgroup of
${\color[rgb]{1,0,0}B}$. Denote
${\color[rgb]{1,0,0}K}:={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}L}$,
${\color[rgb]{1,0,0}P}:={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}G}$, and
suppose that

  1. (c)

$\eta({\color[rgb]{1,0,0}A})={\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}$,
$g^{{\color[rgb]{0.64,0.08,0.98}m}}\in{\color[rgb]{1,0,0}A}$,

  2. (d)

${\color[rgb]{1,0,0}K}$ is generated by subgroups
${\color[rgb]{1,0,0}P}_{i}:=g^{i}{\color[rgb]{1,0,0}P}g^{-i}$,
$i=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1$.

Then
$\theta({\color[rgb]{1,0,0}P}^{{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z})={\color[rgb]{1,0,0}A}$,
whence $\theta$ induces an isomorphism of the following exact $(3\times
3)$-diagrams:

$\begin{gathered}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
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(3.4)

and that isomorphism is the identity on the lower row.

The following lemma gives a similar characterization of groups
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$.

######  Lemma 3.2.2.

Let $\eta:{\color[rgb]{1,0,0}B}\to\mathbb{Z}$ be an epimorphism with kernel
${\color[rgb]{1,0,0}L}:=\ker(\eta)$. Let also $g\in{\color[rgb]{1,0,0}B}$ and
$\xi:{\color[rgb]{1,0,0}L}\to{\color[rgb]{1,0,0}L}$, $\xi(l)=g^{-1}lg$, be the
conjugation by $g$ automorphism of ${\color[rgb]{1,0,0}L}$. Suppose further
that there exist ${\color[rgb]{0.64,0.08,0.98}m}\geq 1$, and two subgroups
${\color[rgb]{1,0,0}G}$ and ${\color[rgb]{1,0,0}H}$ of ${\color[rgb]{1,0,0}L}$
such that

  1. (a)

$\eta(g)=1$, and $g^{2{\color[rgb]{0.64,0.08,0.98}m}}$ commutes with
${\color[rgb]{1,0,0}L}$, i.e.
$\xi^{2{\color[rgb]{0.64,0.08,0.98}m}}=\mathrm{id}_{L}$;

  2. (b)

$\xi^{{\color[rgb]{0.64,0.08,0.98}m}}({\color[rgb]{1,0,0}H})={\color[rgb]{1,0,0}H}$,
so we have an automorphism
$\gamma=\xi^{{\color[rgb]{0.64,0.08,0.98}m}}|_{{\color[rgb]{1,0,0}H}}:{\color[rgb]{1,0,0}H}\to{\color[rgb]{1,0,0}H}$
with $\gamma^{2}=\mathrm{id}_{{\color[rgb]{1,0,0}H}}$;

  3. (c)

${\color[rgb]{1,0,0}L}$ is a direct product of subgroups111The signs of powers
of $g$ are chosen so that $\xi$ will act on the tuple of subgroups
$({\color[rgb]{1,0,0}G}_{0},{\color[rgb]{1,0,0}G}_{1},\ldots,{\color[rgb]{1,0,0}G}_{2{\color[rgb]{0.64,0.08,0.98}m}-1})$
by cyclically shifting them to the left. Similar observations hold for
${\color[rgb]{1,0,0}H}_{0},\ldots,{\color[rgb]{1,0,0}H}_{{\color[rgb]{0.64,0.08,0.98}m}-1}$.
Note that this agrees with the behavior of $\psi$ from Section 2.3.

$\displaystyle{\color[rgb]{1,0,0}G}_{i}$
$\displaystyle:=\xi^{-i}({\color[rgb]{1,0,0}G})\equiv
g^{i}{\color[rgb]{1,0,0}G}g^{-i},\
(i=0,\ldots,2{\color[rgb]{0.64,0.08,0.98}m}-1),$
$\displaystyle{\color[rgb]{1,0,0}H}_{j}$
$\displaystyle:=\xi^{-j}({\color[rgb]{1,0,0}H})\equiv
g^{j}{\color[rgb]{1,0,0}H}g^{-j},\
(j=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1).$

Then the following map
$\theta\colon\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}\to{\color[rgb]{1,0,0}B}$,

$\displaystyle\theta(a_{0},a_{1},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1};\,b_{0},b_{1},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1};\,k)=\Bigl{(}\prod_{i=0}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-i}(a_{i})\cdot\prod_{j=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-j}(b_{j})\Bigr{)}g^{k},$
(3.5)

is an isomorphism which also yields an isomorphism of the following short
exact sequences:

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(3.6)

Let also ${\color[rgb]{1,0,0}A}$ be a normal subgroup of
${\color[rgb]{1,0,0}B}$. Denote
${\color[rgb]{1,0,0}K}:={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}L}$,
${\color[rgb]{1,0,0}P}:={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}G}$,
${\color[rgb]{1,0,0}Q}:={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}H}$, and
suppose that

  1. (d)

$\eta({\color[rgb]{1,0,0}A})=2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}$,
$g^{2{\color[rgb]{0.64,0.08,0.98}m}}\in{\color[rgb]{1,0,0}A}$;

  2. (e)

${\color[rgb]{1,0,0}K}$ is generated by subgroups

$\begin{gathered}{\color[rgb]{1,0,0}P}_{i}:=\xi^{i}({\color[rgb]{1,0,0}P})\equiv
g^{-i}{\color[rgb]{1,0,0}P}g^{i}={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}G}_{i},\
i=0,\ldots,2{\color[rgb]{0.64,0.08,0.98}m}-1,\\\
{\color[rgb]{1,0,0}Q}_{j}:=\xi^{j}({\color[rgb]{1,0,0}Q})\equiv
g^{-j}{\color[rgb]{1,0,0}Q}g^{j}={\color[rgb]{1,0,0}A}\cap{\color[rgb]{1,0,0}H}_{j},\
j=0,\ldots,{\color[rgb]{0.64,0.08,0.98}m}-1.\end{gathered}$ (3.7)

Then
$\theta({\color[rgb]{1,0,0}P}^{{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}Q}^{{\color[rgb]{0.64,0.08,0.98}m}}\times
2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z})={\color[rgb]{1,0,0}A}$, whence
$\theta$ also induces an isomorphism of the following exact $(3\times
3)$-diagrams:

$\textstyle{{\color[rgb]{1,0,0}P}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}Q}^{{\color[rgb]{0.64,0.08,0.98}m}}\times
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}}\times
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{({\color[rgb]{1,0,0}G}/{\color[rgb]{1,0,0}P})^{2{\color[rgb]{0.64,0.08,0.98}m}}\times({\color[rgb]{1,0,0}H}/{\color[rgb]{1,0,0}Q})^{{\color[rgb]{0.64,0.08,0.98}m}}\times
0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}P}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}Q}^{{\color[rgb]{0.64,0.08,0.98}m}}\times
2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\left({\color[rgb]{1,0,0}G}/{\color[rgb]{1,0,0}P},{\color[rgb]{1,0,0}H}/{\color[rgb]{1,0,0}Q}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}}$
$\textstyle{\cong}$
$\textstyle{{\color[rgb]{1,0,0}K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}L}/{\color[rgb]{1,0,0}K}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}B}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{\color[rgb]{1,0,0}B}/{\color[rgb]{1,0,0}A}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}_{2{\color[rgb]{0.64,0.08,0.98}m}}}$

and that isomorphism is the identity on the lower row.

###### Proof.

First note that due to (c) the product in round brackets in (3.5) does not
depend on their order, and therefore the map $\theta$ is well defined.

Recall further that
$\left({\color[rgb]{1,0,0}G},{\color[rgb]{1,0,0}H}\right)\wr_{\gamma,{\color[rgb]{0.64,0.08,0.98}m}}\mathbb{Z}$
is the semidirect product
$({\color[rgb]{1,0,0}G}^{2{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}H}^{{\color[rgb]{0.64,0.08,0.98}m}})\rtimes_{\psi}\mathbb{Z}$
corresponding to the isomorphism $\psi$ given by (2.2). Also, by Lemma 3.1.2,
${\color[rgb]{1,0,0}B}$ is isomorphic to
${\color[rgb]{1,0,0}L}\rtimes_{\xi}\mathbb{Z}$. Hence, due to Lemma 3.1.3, for
the proof that $\theta$ is an isomorphism, it suffices to check that
$\theta\circ\psi=\xi\circ\theta$ on ${\color[rgb]{1,0,0}L}$. Indeed, since
$\gamma=\xi^{{\color[rgb]{0.64,0.08,0.98}m}}$ we have that

$\displaystyle\theta\circ\psi(a_{0},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1},b_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1})=\theta(a_{1},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1},a_{0};\,b_{1},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1},\gamma(b_{0}))=$
$\displaystyle\quad=\prod_{i=1}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-(i-1)}(a_{i})\
\cdot\ \xi^{-(2{\color[rgb]{0.64,0.08,0.98}m}-1)}(a_{0})\ \cdot\
\prod_{j=1}^{{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-(j-1)}(b_{j})\ \cdot\
\xi^{-({\color[rgb]{0.64,0.08,0.98}m}-1)+{\color[rgb]{0.64,0.08,0.98}m}}(b_{0})=$
$\displaystyle\quad=\prod_{i=0}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-i+1}(a_{i})\
\cdot\
\prod_{j=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-j+1}(b_{j})=\xi\Bigl{(}\prod_{i=0}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-i}(a_{i})\
\cdot\ \prod_{j=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}\xi^{-j}(b_{j})\Bigr{)}=$
$\displaystyle\quad=\xi\circ\theta(a_{0},\ldots,a_{2{\color[rgb]{0.64,0.08,0.98}m}-1},b_{0},\ldots,b_{{\color[rgb]{0.64,0.08,0.98}m}-1}).$

Suppose (d) and (e) hold. Then ${\color[rgb]{1,0,0}A}$ splits into the direct
product of subgroups ${\color[rgb]{1,0,0}K}\times\langle
g^{2{\color[rgb]{0.64,0.08,0.98}m}}\rangle$. Moreover, by (c),
${\color[rgb]{1,0,0}K}$ is a direct product
$\mathop{\prod}\limits_{i=0}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}{\color[rgb]{1,0,0}P}_{i}\times\mathop{\prod}\limits_{j=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}{\color[rgb]{1,0,0}Q}_{j}$
of its subgroups (3.7). It now follows from (3.5) that
$\theta({\color[rgb]{1,0,0}P}^{{\color[rgb]{0.64,0.08,0.98}m}}\times{\color[rgb]{1,0,0}Q}^{{\color[rgb]{0.64,0.08,0.98}m}}\times
2{\color[rgb]{0.64,0.08,0.98}m}\mathbb{Z})$ is exactly
$\mathop{\prod}\limits_{i=0}^{2{\color[rgb]{0.64,0.08,0.98}m}-1}{\color[rgb]{1,0,0}P}_{i}\times\mathop{\prod}\limits_{j=0}^{{\color[rgb]{0.64,0.08,0.98}m}-1}{\color[rgb]{1,0,0}Q}_{j}\times\langle
g^{2{\color[rgb]{0.64,0.08,0.98}m}}\rangle={\color[rgb]{1,0,0}A}$. ∎

##  4\. Codimension of singularities

In this Section we give a short and elementary proof of finiteness of Milnor
numbers of homogeneous polynomials $g\in\mathbb{R}[x,y]$ without multiple
factors, (Lemma 4.3.1), which is a principal property allowing to compute the
homotopy types of orbits of maps from $\mathcal{F}(M,P)$. In fact, that
statement follows from general results of algebraic geometry, see Lemma 4.2.1,
and used in mentioned above papers, e.g. [29], however the authors did not
find its explicit proof in available literature.

We will also discuss structure of level-sets of isolated critical points on
surfaces. These results will note be directly used in the proofs of main
results and may be skipped on first reading.

###  4.1. Maps of finite codimension

Let $\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$ be the algebra of
germs at the origin $\mathsf{0}\in\mathbb{R}^{n}$ of $\mathcal{C}^{\infty}$
functions $f:\mathbb{R}^{n}\to\mathbb{R}$. For
$f\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$ denote by
$J_{\mathbb{R}}(f)$ the ideal in
$\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$ generated by germs of
partial derivatives of $f$. Then the real codimension of $J_{\mathbb{R}}(f)$
in $\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$:

$\mu_{\mathbb{R}}(f):=\dim_{\mathbb{R}}\bigl{(}\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})/J_{\mathbb{R}}(f)\bigr{)}$
(4.1)

is called the codimension or Milnor number or multiplicity of $f$ at the
critical point $\mathsf{0}$ with respect to the algebra
$\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$.

The well-known Tougeron theorem [38] claims that if $\mathsf{0}$ is a critical
point of a finite codimension $k$ of a $\mathcal{C}^{\infty}$ germ
$f:(\mathbb{R}^{n},\mathsf{0})\to(\mathbb{R},\mathsf{0})$, then $f$ is
$\mathcal{C}^{\infty}$ equivalent to its Taylor polynomial of order $k+1$.

It is easy to see that $\mu_{\mathbb{R}}(f)$ does not depend on local
coordinates at $\mathsf{0}\in\mathbb{R}^{n}$, and therefore it can be defined
for $\mathcal{C}^{\infty}$ functions on manifolds and even for
$\mathcal{C}^{\infty}$ circle-values maps.

Now let $M$ be a smooth compact manifold and $P$ be either $\mathbb{R}$ or
$S^{1}$. Say that a map $f\in\mathcal{C}_{\partial}^{\infty}(M,P)$ is of
finite codimension if it has only finitely many critical points and at each of
them $f$ has finite codimension (4.1).

######  Theorem 4.1.1 ([37, 23]).

Let $f\in\mathcal{C}_{\partial}^{\infty}(M,P)$ be a map of finite codimension
and $Y$ be any collection of boundary components of $M$. Then

  1. (a)

the corresponding orbit $\mathcal{O}(f,Y)$ is a Fréchet submanifold of finite
codimension of the Fréchet manifold $\mathcal{C}_{\partial}^{\infty}(M,P)$,

  2. (b)

the natural map $\nu:\mathcal{D}(M,Y)\to\mathcal{O}(f,Y)$, $h\mapsto f\circ
h$, is a principal locally trivial fibration with fiber $\mathcal{S}(f,Y)$;

  3. (c)

$\nu(\mathcal{D}_{\mathrm{id}}(M,Y))=\mathcal{O}_{f}(f,Y)$ and the restriction
map $\nu:\mathcal{D}_{\mathrm{id}}(M,Y)\to\mathcal{O}_{f}(f,Y)$, $h\mapsto
f\circ h$, is also principal locally trivial fibration with fiber
$\mathcal{S}^{\prime}(f,Y):=\mathcal{S}(f,Y)\cap\mathcal{D}_{\mathrm{id}}(M,Y)$.

###### Remarks to the proof.

Note that (c) directly follows from (b). For the statements (a) and (b)
consider first the so-called left-right action

$\mathcal{D}(\mathbb{R})\times\mathcal{C}^{\infty}(M,\mathbb{R})\times\mathcal{D}(M)\to\mathcal{C}^{\infty}(M,\mathbb{R}),\qquad(\phi,f,h)\mapsto\phi\circ
f\circ h,$

of the group $\mathcal{D}(\mathbb{R})\times\mathcal{D}(M)$ on
$\mathcal{C}^{\infty}(M,\mathbb{R})$. F. Sergeraert [37] proved that if
$f\in\mathcal{C}^{\infty}(M,\mathbb{R})$ is a smooth function of finite
codimension, then the corresponding orbit

$\widetilde{\mathcal{O}}(f)=\\{\phi\circ f\circ
h\mid(\phi,h)\in\mathcal{D}(\mathbb{R})\times\mathcal{D}(M)\\}$

is a Fréchet submanifold of finite codimension of the Fréchet space
$\mathcal{C}^{\infty}(M,\mathbb{R})$, while the natural map
$\nu:\mathcal{D}(\mathbb{R})\times\mathcal{D}(M)\to\widetilde{\mathcal{O}}(f)$,
$(\phi,h)\mapsto\phi\circ f\circ h$, is a principal locally trivial fibration.

In [23, Theorem 11.7] such a result was extended to “tame actions of tame Lie
groups of tame Fréchet manifolds”, where tameness is understood in the sense
of R. Hamilton [12]. In particular, this implied that (a) and (b) hold for $f$
being a Morse map and $Y=\varnothing$, see [23, Section 11.30].

However, almost literally the same arguments show that the same result holds
for the above map $\nu:\mathcal{D}(M,Y)\to\mathcal{O}(f,Y)$, where $Y$ is any
(possibly empty) collection of boundary components of $M$ and $f:M\to P$ is a
map of finite codimension. One should just

  * $-$

write in Eq. (2) of [23, Section 11.30] $g(x)$ as
$g(x)=\sum_{i=1}^{k}c_{i}\lambda_{i}(x)+\sum\beta_{i}(x)g^{\prime}_{x_{i}}(x)$,
where $c_{i}\in\mathbb{R}$ are some constants, and $\lambda_{i}$ some
$\mathcal{C}^{\infty}$ functions which span the (finite-dimensional)
complement to the Jacobi ideal of $f$ at $z_{i}$;

  * $-$

note that in Eq. (3) of [23, Section 11.30] the vector field $H_{j}$ vanishes
on the boundary component $B_{i}$, and thus belong to the tangent space at
$\mathrm{id}_{M}$ of the group $\mathcal{D}(M,Y)$ not only of
$\mathcal{D}(M)$. ∎

###  4.2. Finiteness of Milnor number

Denote by $\mathcal{O}(\mathbb{C}^{n})$ the $\mathbb{C}$-algebra of germs of
analytic maps $\mathbb{C}^{n}\to\mathbb{C}$ at $\mathsf{0}\in\mathbb{C}^{2}$,
i.e. maps represented by series covering on some neighborhood of
$\mathsf{0}\in\mathbb{C}^{n}$. In particular, we can regard the ring of
polynomials $\mathbb{C}[z_{1},\ldots,z_{n}]$ as a subalgebra of
$\mathcal{O}(\mathbb{C}^{n})$. Let also

$\mathfrak{m}=\\{f\in\mathcal{O}(\mathbb{C}^{n})\mid f(\mathsf{0})=0\\}$

be the unique maximal ideal in $\mathcal{O}(\mathbb{C}^{n})$. Note that for
each $f\in\mathcal{O}(\mathbb{C}^{n})$ one can define its Jacobi ideal
$J_{\mathbb{C}}(f)=(f^{\prime}_{z_{1}},\ldots,f^{\prime}_{z_{n}})$ in
$\mathcal{O}(\mathbb{C}^{n})$ generated by partial derivatives of $f$. Its
codimension,
$\mu_{\mathbb{C}}(f):=\dim_{\mathbb{C}}\bigl{(}\mathcal{O}(\mathbb{C}^{n})/J_{\mathbb{C}}(f)\bigr{)}$,
is called the Milnor number of $f$ with respect to
$\mathcal{O}(\mathbb{C}^{n})$.

######  Lemma 4.2.1.

For $f\in\mathcal{O}(\mathbb{C}^{n})$ the following conditions are equivalent:

  1. $(\mu 1)$

there exists $m\geq 1$ such that $z_{i}^{m}\in J_{\mathbb{C}}(g)$ for each
$i=1,\ldots,n$;

  2. $(\mu 2)$

there exists $p\geq 1$ such that $z_{1}^{a_{1}}\cdots z_{n}^{a_{n}}\in
J_{\mathbb{C}}(g)$ if $a_{1}+\cdots+a_{n}\geq p$;

  3. $(\mu 3)$

there exists $p\geq 1$ such that $\mathfrak{m}^{p}\subset J_{\mathbb{C}}(g)$;

  4. $(\mu 4)$

$0\in\mathbb{C}^{n}$ is an isolated critical point of $f$;

  5. $(\mu 5)$

$\mu_{\mathbb{C}}(f)<\infty$.

###### Proof.

The equivalence $(\mu 1)$$\Leftrightarrow$$(\mu 2)$$\Leftrightarrow$$(\mu 3)$
is evident.

$(\mu 3)$$\Rightarrow$$(\mu 5)$. Note that for each $p\geq 1$ the vector space
$\mathcal{O}(\mathbb{C}^{n})/\mathfrak{m}^{p}$ over $\mathbb{C}$ is generated
by finitely many monomials $\\{z_{1}^{a_{1}}\cdots z_{n}^{a_{n}}\mid 0\leq
a_{1}+\cdots a_{n}<p\\}$. Hence, if $\mathfrak{m}^{p}\subset
J_{\mathbb{C}}(g)$, then

$\mu_{\mathbb{C}}(f):=\dim_{\mathbb{C}}\bigl{(}\mathcal{O}(\mathbb{C}^{n})/J_{\mathbb{C}}(f)\bigr{)}\leq\dim_{\mathbb{C}}\bigl{(}\mathcal{O}(\mathbb{C}^{n})/\mathfrak{m}^{p}\bigr{)}<\infty.$

The equivalence $(\mu 4)$$\Leftrightarrow$$(\mu 5)$ is a principal non-trivial
statement and can be found in [11, Lemma 2.3], which in turn is based [11,
Proposition 1.70].

$(\mu 4)$$\Rightarrow$$(\mu 3)$ This implication follows in fact from the
proof of the implication (d)$\Rightarrow$(b) of that [11, Proposition 1.70].
Namely, in that proof it is actually shown that isolateness of $\mathsf{0}$
implies that $\mathfrak{m}^{p}\subset J_{\mathbb{C}}(g)$ for some $p\geq 1$,
i.e. that $(\mu 4)$ implies $(\mu 3)$. ∎

######  Corollary 4.2.2.

Let $f\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$ be an analytic
germ, i.e. a series with real coefficients converging on some neighborhood of
$\mathsf{0}\in\mathbb{R}^{n}$; for instance $f$ can be a polynomial. Regard
$f$ as complex series with real coefficients, i.e. as an element of
$\mathcal{O}(\mathbb{C}^{n})$. Then either of the conditions $(\mu 1)$-$(\mu
5)$ implies that $\mu_{\mathbb{R}}(f)<\infty$.

###### Proof.

Let $\phi\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$. Then by
Hadamard lemma there are $\mathcal{C}^{\infty}$ germs
$\alpha_{1},\ldots,\alpha_{n}\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$
and $b_{0}=\phi(\mathsf{0})$ such that
$\phi(\mathbf{x})=b_{0}+\sum_{j=1}^{n}\beta_{j}(\mathbf{x})x_{j}$, where
$\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}$. Applying the same
statement to each $\beta_{j}$ and so on we get that for each $p$ we have the
identity

$\phi(x,y)=\sum_{0\leq a_{1}+\cdots+a_{n}<p}b_{ij}x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}+\sum_{a_{1}+\cdots+a_{n}=p}\beta_{a_{1},\ldots,a_{n}}(\mathbf{x})x_{1}^{a_{1}}\cdots
x_{n}^{a_{n}}$ (4.2)

for some $b_{ij}\in\mathbb{R}$ and
$\beta_{a_{1},\ldots,a_{n}}\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$.
Hence, as in the proof of $(\mu 3)$$\Rightarrow$$(\mu 5)$ in Lemma 4.2.1, in
order to prove that $\mu_{\mathbb{R}}(f)<\infty$, it suffices to show that
there exists $m\geq 1$ such that $x_{i}^{m}\in J_{\mathbb{R}}(g)$ for all
$i=1,\ldots,n$. Then for some large $p$, the second term in (4.2) will belong
to $J_{\mathbb{R}}(g)$, whence the vector space
$\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})/J_{\mathbb{R}}(g)$ over
$\mathbb{R}$ will be generated by a finite set of monomials of degree $<p$.

Regard $\mathbb{R}^{n}$ as a subspace of $\mathbb{C}^{n}$ of real coordinates.
By assumption $f$, as an element of $\mathcal{O}(\mathbb{C}^{n})$, satisfies
condition $(\mu 1)$, so there exist $m\geq 1$ and
$\tau_{1},\ldots,\tau_{n}\in\mathcal{O}(\mathbb{C}^{n})$ such that

$z_{i}^{m}=\tau_{1}(\mathbf{z})f^{\prime}_{z_{1}}(\mathbf{z})+\cdots+\tau_{n}(\mathbf{z})f^{\prime}_{z_{n}}(\mathbf{z}),$
(4.3)

for all $i=1,\ldots,n$, where
$\mathbf{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}$. Let
$\tau_{j}=\alpha_{j}+i\beta_{j}$ be the decomposition of $\tau_{j}$ into the
real and imaginary parts. Then $\alpha_{j}$ and $\beta_{j}$ are converging
series near $\mathsf{0}$ in $\mathbb{C}^{n}$ with real coefficients, and
$\alpha_{j}|_{\mathbb{R}^{n}},\beta_{j}|_{\mathbb{R}^{n}}\in\mathcal{C}^{\infty}_{\mathsf{0}}(\mathbb{R}^{n})$.
Since $f$ has real coefficients, we also have that
$f^{\prime}_{z_{j}}(\mathbf{x})=f^{\prime}_{x_{i}}(\mathbf{x})$ for all $j$.
Now taking the real parts of both sides of (4.3) and restricting them to the
real subspace $\mathbb{R}^{n}\subset\mathbb{C}^{n}$, i.e. substituting
$\mathbf{x}$ instead of $\mathbf{z}$, we get that

$x_{i}^{m}=\alpha_{j}(\mathbf{x})f^{\prime}_{x_{1}}(\mathbf{x})+\cdots+\alpha_{n}(\mathbf{x})f^{\prime}_{x_{n}}(\mathbf{x})\in
J_{\mathbb{R}}(f).\qed$

###  4.3. Homogeneous polynomials in two variables

Let $g:\mathbb{R}^{2}\to\mathbb{R}$ be a real homogeneous polynomial with
$\deg g\geq 2$. Then it directly follows from fundamental theorem of algebra
that $g$ splits into a product of finitely many linear and irreducible over
$\mathbb{R}$ quadratic factors:

$g(x,y)=\prod_{i=1}^{p}(a_{i}x+b_{i}y)\cdot\prod_{j=1}^{q}(c_{j}x^{2}+2d_{j}xy+e_{j}y^{2}).$
(4.4)

Evidently,

  1. (1)

$g^{-1}(0)$ is a union of straight lines $\\{a_{i}x+b_{i}y=0\\}$ corresponding
to the linear factors;

  2. (2)

if $g$ has two proportional linear factors, then all points of the
corresponding line are critical for $g$;

  3. (3)

hence, $g$ has no multiple (non-proportional) linear factors if and only if
the origin $\mathsf{0}\in\mathbb{R}^{2}$ is an isolated critical point; in
this case $g^{-1}(0)$ consists of $2p$ rays starting from the origin;

  4. (4)

the following conditions are equivalent due to Lemma 4.2.1:

    1. (4a)

$g$ has no multiple factors at all;

    2. (4b)

the partial derivatives $g^{\prime}_{x}$ and $g^{\prime}_{x}$ of $g$ have no
common factors;

    3. (4c)

the origin $\mathsf{0}\in\mathbb{C}^{2}$ is an isolated critical point of
$g:\mathbb{C}^{2}\to\mathbb{C}$ as a complex polynomial with real
coefficients;

    4. (4d)

$\mu_{\mathbb{C}}(g)<\infty$.

Moreover, by Corollary 4.2.2, they also imply that
$\mu_{\mathbb{R}}(g)<\infty$.

For the completeness and for the convenience of the reader we present a short
explicit and not based on Lemma 4.2.1 proof that (4)(4a) implies
$\mu_{\mathbb{R}}(g)<\infty$.

######  Lemma 4.3.1.

Suppose $g\in\mathbb{R}[x,y]$ is a homogeneous polynomial without multiple
factors and $\deg g\geq 2$. Then $x^{m},y^{m}\in J_{\mathbb{R}}(f)$ for some
large $m\geq 1$, and therefore $\mu_{\mathbb{R}}(g)<\infty$.

In particular, if $M$ is a compact surface, then each $f\in\mathcal{F}(M,P)$
is of finite codimension, and Theorem 4.1.1 holds for $f$.

###### Proof.

Put $k=\deg g-1$. To simplify notation also redenote $A:=g^{\prime}_{x}$ and
$B:=g^{\prime}_{y}$. Then $A$ and $B$ are homogeneous polynomials of degree
$k$. As mentioned above, the assumption that $g$ has no multiple factors
implies that $A$ and $B$ are relatively prime in $\mathbb{R}[x,y]$. We will
find homogeneous polynomials $P,Q\in\mathbb{R}[x,y]$ such that $AP+BQ=x^{m}$
for some $m\geq 1$. This will mean that $x^{m}\in J_{\mathbb{R}}(f)$. The
proof that $y^{m}\in J_{\mathbb{R}}(f)$ for some $m$ is similar.

Consider the following polynomials $\alpha(t):=A(1,t)$,
$\beta(t):=B(1,t)\in\mathbb{R}[t]$. Since $A$ and $B$ are homogeneous of
degree $k$, we have that

$\displaystyle A(x,y)$ $\displaystyle=\alpha(y/x)x^{k},$ $\displaystyle
B(x,y)$ $\displaystyle=\beta(y/x)x^{k}.$

Moreover, $\mathrm{gcd}(\alpha(t),\beta(t))=1$ in $\mathbb{R}[t]$, since
$\mathrm{gcd}(A,B)=1$ in $\mathbb{R}[x,y]$. In particular, using Euclid
division algorithm, one can find polynomials $p,q\in\mathbb{R}[t]$ such that

$\displaystyle\alpha(t)p(t)+\beta(t)q(t)=1.$ (4.5)

This implies that $\deg(\alpha p)=\deg(\beta q)$, and we will denote this
common degree by $m$. It follows that $P:=p(y/x)x^{\deg p}$ and
$Q:=q(y/x)x^{\deg q}$ are homogeneous polynomials in $\mathbb{R}[x,y]$.
Multiplying (4.5) by $x^{m}$ and substituting $t=y/x$ we get:

$\bigl{(}\alpha(y/x)x^{\deg\alpha}\bigr{)}\,\bigl{(}p(y/x)x^{\deg p}\bigr{)}\
+\ \bigl{(}\beta(y/x)x^{\deg\beta}\bigr{)}\,\bigl{(}q(y/x)x^{\deg q}\bigr{)}\
=\ x^{m},$

i.e. $AP+BQ=x^{m}\in J_{\mathbb{R}}(f)$.

Now, let $M$ be a compact surface and $f\in\mathcal{F}(M,P)$. Then $f$ has
finitely many critical points and at each of them its germ is
$\mathcal{C}^{\infty}$ equivalent to a homogeneous polynomial without multiple
factors. As just proved the Milnor number of $f$ at each critical point is
finite. Therefore, $f$ is also of finite codimension. ∎

######  Remark 4.3.2.

Let $M$ be a $2$-dimensional manifold and $f\in\mathcal{C}^{\infty}(M,P)$.
Suppose $w\in\mathrm{Int}{M}$ is an isolated critical point of $f$. Then there
are germs of homeomorphisms $h:(\mathbb{C},0)\to(M,w)$ and
$\phi:(\mathbb{R},0)\to(\mathbb{R},0)$ such that

$\phi\circ f\circ h(z)=\begin{cases}|z|^{2},&\text{if $0\in\mathbb{C}$ is a
\sf\small local extreme of $f$,
\cite[cite]{[\@@bibref{}{Dancer:2:JRAM:1987}{}{}]}},\\\ Re(z^{p}),&\text{for
some $p\geq 1$ otherwise, \cite[cite]{[\@@bibref{}{ChurchTimourian:PJM:1973,
Prishlyak:TA:2002}{}{}]}}.\end{cases}$ (4.6)

Moreover, one can make $h$ to be $\mathcal{C}^{\infty}$ out of an arbitrary
small neighborhood of $w$. Figure 4.1 shows examples of level sets of smooth
functions near isolated critical points.

|  |  |  |  |  |
---|---|---|---|---|---|---
Figure 4.1. Topological structure of level-sets of isolated singularities

Note that every function in (4.6) is a homogeneous polynomial without multiple
factors. Hence, for any compact surface $M$ the class $\mathcal{F}(M,P)$ is
not only “massive” but also contains “up to homeomorphism” each map
$f\in\mathcal{C}_{\partial}^{\infty}(M,P)$ with isolated critical points.

##  5\. Functions on surfaces

Let $M$ be a compact surface, $f\in\mathcal{F}(M,P)$, and $\Sigma_{f}$ be the
set of all critical points of $f$. Then every connected component of each
level set of $f$ will be called a contour of $f$. A countour $X$ of $f$ is
critical if it contains critical points of $f$, and regular otherwise.
Evidently, each regular contour is a submanifold of $M$ diffeomorphic to the
circle. On the other hand, due to Axiom (H), see also Figure 4.1, a critical
contour has a structure of a $1$-dimensional CW-complex whose $0$-cells are
critical points of $f$ belonging to $X$.

Denote by $\Gamma_{f}$ the partition of $M$ into contours of $f$, and let
$p:M\to\Gamma_{f}$ be the quotient map associating to each $x\in M$ the
contour of $f$ containing $x$. Endow $\Gamma_{f}$ with the corresponding
quotient topology with respect to $p$. Then it is well known that $\Gamma_{f}$
has a structure of a finite $1$-dimensional CW-complex and is called Kronrod-
Reeb graph of $f$, [36, 1, 13].

A subset $X\subset M$ will be called $f$-saturated if it is a union of
(possibly uncountably many) contours of $f$. In other words, $X=p^{-1}(p(X))$.

For instance, a compact $1$-dimensional submanifold $X$ of $M$ is
$f$-saturated whenever it is a union of finitely many regular contours. Also,
a compact $2$-dimensional submanifold $X$ of $M$ is $f$-saturated iff
$\partial X$ is $f$-saturated, i.e. every boundary component of $X$ is a
contour. For example, if $f\in\mathcal{F}(M,\mathbb{R})$ and $a<b$ are two
regular values of $f$, then $f^{-1}\bigl{(}[a;b]\bigr{)}$ is $f$-saturated.

Let $X\subset M$ be a connected $f$-saturated submanifold. Then by an
$f$-regular neighborhood of $X$ we will mean an arbitrary connected
$f$-saturated subsurface $U_{X}$ such that $U_{X}$ is a neighborhood of $X$
and $U_{X}\setminus X$ contains no critical points of $f$.

More generally, suppose $X=\mathop{\cup}\limits_{i=1}^{k}X_{i}$ is a disjoint
union of connected $f$-saturated submanifolds $X_{i}$. For every
$i=1,\ldots,k$ choose any $f$-regular neighborhood $U_{X_{i}}$ of $X_{i}$ such
that $U_{X_{i}}\cap U_{X_{j}}=\varnothing$ for $i\not=j$. Then their union
$U_{X}=\mathop{\cup}\limits_{i=1}^{k}U_{X_{i}}$, will be called an $f$-regular
neighborhood of $X$.

Denote by $\Delta(f)$ the subgroup of $\mathcal{S}(f)$ consisting of
diffeomorphisms $h$ having the following properties:

  1. (1)

$h$ leaves invariant every connected component of every level set of $f$, i.e.
it preserves the elements of $\Gamma_{f}$;

  2. (2)

if $z$ is a degenerate local extreme, and thus $h(z)=z$, then the tangent map
$T_{z}h:T_{z}M\to T_{z}M$ is the identity $\mathrm{id}_{T_{z}M}$.

Evidently, $\Delta(f)$ is a normal subgroup of $\mathcal{S}(f)$. Moreover, it
follows from [25, Lemma 6.2], describing local linear symmetries of
homogeneous polynomials, that the quotient $\mathcal{S}(f)/\Delta(f)$ is
finite. In fact, it can be regarded as a group of automorphisms of an
“enhanced” Kronrod-Reeb graph of $f$ induced by elements of $\mathcal{S}(f)$,
see [29, Section 4]. That graph is obtained from $\Gamma_{f}$ by adding a
certain finite number edges to each vertex corresponding to a degenerate local
extreme of $f$, so that every $h\in\mathcal{S}(f)$ “cyclically rotates” those
new edges. In particular, if $f$ is Morse, so it has no degenerated critical
points, then that “enhanced” graph coincides with $\Gamma_{f}$, and in that
case $\mathcal{S}(f)/\Delta(f)$ is the group of automorphisms of $\Gamma_{f}$
induced by $\mathcal{S}(f)$. However, we will not use that interpretation in
the present paper.

For a closed subset $X\subset M$ denote

$\displaystyle\Delta(f,X):=\Delta(f)\cap\mathcal{D}(M,X),$
$\displaystyle\Delta^{\prime}(f,X):=\Delta(f)\cap\mathcal{D}_{\mathrm{id}}(M,X),$
$\displaystyle\mathbf{G}(f,X):=\mathcal{S}(f,X)/\Delta(f,X),$
$\displaystyle\mathbf{G}^{\prime}(f,X):=\mathcal{S}^{\prime}(f,X)/\Delta^{\prime}(f,X).$

Then we have the following short exact sequences

$\displaystyle\Delta(f,X)\lhook\joinrel\rightarrow\mathcal{S}(f,X)\rightarrow\\!\\!\\!\\!\\!\to\mathbf{G}(f,X),$
$\displaystyle\Delta^{\prime}(f,X)\lhook\joinrel\rightarrow\mathcal{S}^{\prime}(f,X)\rightarrow\\!\\!\\!\\!\\!\to\mathbf{G}^{\prime}(f,X).$

The second sequence will play the most important role, and we will call it the
Bieberbach (short exact) sequence of $(f,X)$, see [29]. Our results will be
given in terms of such sequences, see proofs of Lemmas 8.1-8.3.

Finally, denote by $\mathcal{D}_{{\mathrm{nb}}}(M,X)$ the group of
diffeomorphisms of $M$ fixed on some neighborhood of $X$ (so each
$h\in\mathcal{D}_{{\mathrm{nb}}}(M,X)$ is fixed on its own neighborhood of
$X$), and put

$\displaystyle\Delta_{\mathrm{nb}}(f,X):=\Delta(f)\cap\mathcal{D}_{{\mathrm{nb}}}(M,X),$
$\displaystyle\mathcal{S}_{{\mathrm{nb}}}(f,X):=\mathcal{S}(f)\cap\mathcal{D}_{{\mathrm{nb}}}(M,X).$

######  Lemma 5.1 ([29, Corollary 7.2]).

Let $X$ be an $f$-saturated submanifold, and $U_{X}$ be an $f$-saturated
neighborhood of $X$. Then the following inclusions of pairs are homotopy
equivalences:

$\displaystyle\bigl{(}\mathcal{S}(f,U_{X}),\Delta(f,U_{X})\bigr{)}\subset\bigl{(}\mathcal{S}_{{\mathrm{nb}}}(f,X),\Delta_{\mathrm{nb}}(f,X)\bigr{)}\subset\bigl{(}\mathcal{S}(f,X),\Delta(f,X)\bigr{)}.$

###### Remarks to the proof.

Though this statement is established in [29, Corollary 7.2] for orientable
surfaces, the proof actually uses the fact that $f$-regular neighborhoods of
regular contours of $f$ (being either connected components of $X$ or of
$\partial X$) are always cylinders, and therefore they are orientable, see
[19, Corollary 6.3]. ∎

We will also need the following two simple statements which we leave for the
reader. The first lemma is a straightforward consequence of definitions, while
the second one easily follows from axiom (B).

######  Lemma 5.2.

Let $f\in\mathcal{F}(M,P)$, $h\in\mathcal{S}(f)$, and
$\\{Y_{i}\\}_{i\in\Lambda}$ be a family of $f$-saturated subsurfaces invariant
under $h$ and such that $M=\cup_{i\in\Lambda}Y_{i}$ (these subsurfaces may
intersect each other). Then $h\in\Delta(f)$ if and only if
$h|_{Y_{i}}\in\Delta(f|_{Y_{i}})$ for each $i\in\Lambda$.

######  Lemma 5.3.

Let $f\in\mathcal{F}(M,P)$, $X$ be any collection of boundary components of
$\partial\mathbb{M}$, and $U_{X}$ be any $f$-regular neighborhood of $X$. Let
also $h\in\mathcal{S}_{\mathrm{id}}(f)$ be a diffeomorphism fixed near $X$.
Then $h$ is isotopic in $\mathcal{S}_{{\mathrm{nb}}}(f,X)$ to a diffeomorphism
supported in $U_{X}$.

##  6\. Functions on Möbius band

###  6.1. Reduction to the case $P=\mathbb{R}$

Let $\mathbb{M}$ be a Möbius band and $f\in\mathcal{F}(\mathbb{M},S^{1})$ be a
map into $S^{1}$. Since $f$ takes constant value on $\partial\mathbb{M}$, it
follows that $f$ is null-homotopic. Let $p:\mathbb{R}\to S^{1}$, $p(t)=e^{2\pi
it}$, be the universal covering map and $\widehat{f}:\mathbb{M}\to\mathbb{R}$
be any lifting of $f$, so $f=\widehat{f}\circ p$. Then, see [29, Lemma
5.3(3)], $\widehat{f}\in\mathcal{F}(\mathbb{M},\mathbb{R})$, $f$ and
$\widehat{f}$ have the same critical points and the same partitions into level
sets, which also implies that $\mathcal{S}(f)=\mathcal{S}(\widehat{f})$.
Hence, for studying the stabilizer of $f\in\mathcal{F}(\mathbb{M},S^{1})$ we
can replace $f$ with $\widehat{f}$.

Therefore in what follows we will assume that
$f\in\mathcal{F}(\mathbb{M},\mathbb{R})$.

###  6.2. Special $f$-decomposition of $\mathbb{M}$

The following lemma provides an additional (to statement (2)(2b) of Theorem
1.2.1) decomposition of a Möbius band $\mathbb{M}$ with respect to $f$.

######  Lemma 6.2.1 ([19, Theorem 1.4]).

Every $f\in\mathcal{F}(\mathbb{M},\mathbb{R})$ has a unique critical contour
$K$ with the following property: there exists an $f$-regular neighborhood
$U_{K}\subset\mathrm{Int}{\mathbb{M}}$ of $K$ invariant under
$\mathcal{S}(f,\partial\mathbb{M})$ and such that

  1. (a)

the unique connected component $Y_{0}$ of $\overline{\mathbb{M}\setminus
U_{K}}$ containing $\partial\mathbb{M}$ is a cylinder;

  2. (b)

all other components $Y_{1},\ldots,Y_{n}$ of $\overline{\mathbb{M}\setminus
U_{K}}$ are $2$-disks, see Figure 6.1.

Properties (a) and (b) then hold for arbitrary
$\mathcal{S}(f,\partial\mathbb{M})$-invariant $f$-regular neighborhood of $K$.

Figure 6.1. Schematic decomposition of a Möbius band associated with
$f\in\mathcal{F}(\mathbb{M},\mathbb{R})$

We will call $K$ a special contour of $f$, and
$\xi(f):=\\{U_{K},Y_{0},Y_{1},\ldots,Y_{n}\\}$ a $f$-special decomposition of
$\mathbb{M}$ (associated with the
$\mathcal{S}(f,\partial\mathbb{M})$-invariant $f$-regular neighborhood $U_{K}$
of $K$).

For $i=0,1\ldots,n$, let $C_{i}$ be the connected component of
$\mathbb{M}\setminus K$ containing $Y_{i}$, and
$Q_{i}:=\overline{C_{i}}\setminus\Sigma_{f}$. One easily checks the following
statements.

  1. (i)

$Q_{0}$ is diffeomorphic to $(S^{1}\times[0;1])\setminus(F\times 1)$, where
$F$ is a finite set. So it is a closed cylinder out of finitely many points
removed from one of its boundary components. Another boundary component
corresponds thus to $\partial\mathbb{M}$, see Figure 6.2.

  2. (ii)

On the other hand, each $Q_{i}$, $i=1,\ldots,n$, is a closed $2$-disk out of
finitely many points removed from its boundary.

  3. (iii)

Moreover, for each $i=0,\ldots,n$, the intersection $Q_{i}\cap K$ is a finite
collection of arcs (edges of $K$) which also constitute a cycle. We will call
them boundary edges of $Q_{i}$.

  4. (iv)

Each edge of $K$ belongs to exactly two distinct components $Q_{i}$ and
$Q_{j}$ one of which is “upper” and another one “lower” in the sense that the
images $f(C_{i})$ and $f(C_{j})$ are contained in distinct components of
$\mathbb{R}\setminus f(K)$.

###  6.3. Quasi-cones

Let $CS^{1}=\bigl{(}S^{1}\times[0;1]\bigr{)}/\bigl{(}S^{1}\times 0\bigr{)}$ be
a cone over a circle, i.e. just a $2$-disk. The equivalence class
$[S^{1}\times 0]$ is called the vertex of the cone $CS^{1}$. Let also
$F_{1}\sqcup\cdots\sqcup F_{k}\subset S^{1}\times 1$ be a finite collection of
mutually disjoint finite subsets. Shrink each $F_{i}$ into a single point.
Then the corresponding quotient space $L:=CS^{1}/(F_{1}\sqcup\cdots\sqcup
F_{k})$ will be called a quasi-cone over $S^{1}$. Let $p:CS^{1}\to L$ be the
respective quotient map. Then $p(S^{1}\times 1)$ will be called the base of
the quasi-cone $L$ and denoted by $bL$.

One easily checks that if $h:L_{1}\to L_{2}$ is a homeomorphism between quasi-
cones over $S^{1}$, then it lifts to a unique homeomorphism $\hat{h}:CS^{1}\to
CS^{1}$ such that $h\circ p_{1}=p_{2}\circ\hat{h}$, where $p_{i}:CS^{1}\to
L_{i}$, $i=1,2$, is the corresponding quotient map. Moreover, as
$\hat{h}(S^{1}\times 1)=S^{1}\times 1$, we have a homeomorphism
$\hat{h}_{1}:S^{1}\to S^{1}$ such that $\hat{h}(x,1)=(\hat{h}_{1}(x),1)$ for
all $x\in S^{1}$. Hence, one can also define another homeomorphism
$\hat{h}^{\prime}:CS^{1}\to CS^{1}$,
$\hat{h}^{\prime}(x,t)=(\hat{h}_{1}(x),t)$, called a cone over $\hat{h}_{1}$.
It coincides with $h$ on the bases and sends the vertex of $L_{1}$ to the
vertex of $L_{2}$. Moreover, $\hat{h}^{\prime}$ induces a homeomorphism
$h^{\prime}:L_{1}\to L_{2}$ which coincides with $h$ on the base $bL_{1}$, and
is evidently determined by $h:bL_{1}\to bL_{2}$. We will call $h^{\prime}$ the
“cone change” of $h$.

###  6.4. CW-decompositions of $S^{2}$ and $\mathbb{RP}^{2}$ associated with
$f$

Let $\mathbb{RP}^{2}=\mathbb{M}/\partial\mathbb{M}$ be a surface obtained by
shrinking the boundary $\partial\mathbb{M}$ into one point which we will
denote by $x^{*}$. Then $\mathbb{RP}^{2}$ is a projective plane, and we get a
CW-partition $\Xi$ of $\mathbb{RP}^{2}$ whose

  * •

$0$-cells are critical points of $f$ belonging to $K$;

  * •

$1$-cells are connected components of $K\setminus\Sigma_{f}$;

  * •

$2$-cells are connected components
$C_{0}/\partial\mathbb{M},\,C_{1},\,\ldots,\,C_{n}$ of
$\mathbb{RP}^{2}\setminus K$.

Thus, we can regard $\mathbb{RP}^{2}$ as a one point compactification of
$\mathrm{Int}{\mathbb{M}}$, so
$\mathbb{RP}^{2}=\mathrm{Int}{\mathbb{M}}\sqcup\\{x^{*}\\}$. Denote by
$c_{i}$, $i=0,1,2$, the total number of $i$-cells of $\Xi$. In particular,
$c_{2}=n+1$.

Moreover, let $p:S^{2}\to\mathbb{RP}^{2}$ be the universal cover and
$\xi:S^{2}\to S^{2}$ be the reversing orientation involution without fixed
points generating the group $\mathbb{Z}_{2}$ of covering transformations. Then
$S^{2}$ has a CW-structure $\widetilde{\Xi}$ whose $i$-cells are connected
components of the inverses of $i$-cells of $\Xi$. In fact, for each $i$-cell
$e\in\Xi$ its inverse image $p^{-1}(e)$ is a disjoint union of two $i$-cells
$\widetilde{e}_{0}$ and $\widetilde{e}_{1}$ such that $\xi$ exchanges them. We
will call them liftings of $e$.

Notice that it follows from (i) and (ii) above that the closure of each
$2$-cell of $\Xi$ is a quasi-cone over $S^{1}$. We can also assume that
$x^{*}$ is the vertex of $C_{0}/\partial\mathbb{M}$.

###  6.5. Lefschetz number

It follows from Lemma 6.2.1 that each $h\in\mathcal{S}(f,\partial\mathbb{M})$
yields a cellular (i.e. sending cells to cells) homeomorphism
$\mathbf{h}:\mathbb{RP}^{2}\to\mathbb{RP}^{2}$ which fixes $x^{*}$ and
coincides with $h$ on $\mathrm{Int}{\mathbb{M}}$. Thus, if $e$ is an $i$-cell
of $\Xi$, then $\mathbf{h}(e)$ is also an $i$-cell of $\Xi$. Then for $i=1,2$
one can take to account whether the restriction map
$\mathbf{h}|_{e}:e\to\mathbf{h}(e)$ preserves or reverses chosen orientations.
For the case $i=0$ we will always assume that
$\mathbf{h}|_{e}:e\to\mathbf{h}(e)$ preserves the orientation.

We will say that $e$ is $\mathbf{h}^{+}$-invariant (resp.
$\mathbf{h}^{-}$-invariant) if $\mathbf{h}(e)=e$ and the restriction map
$\mathbf{h}:e\to e$ preserves (resp. reverses) orientation. Denote by
$c^{+}_{i}(\mathbf{h})$ and $c^{-}_{i}(\mathbf{h})$ the number of
$\mathbf{h}^{+}$-invariant (resp. $\mathbf{h}^{-}$-invariant) cells of $\Xi$.
Then $c^{-}_{0}(\mathbf{h})=0$.

Now fix some orientation of each cell. This which allows to define the group
$C_{i}(\mathbb{RP}^{2},\mathbb{Z})$, $i=0,1,2$, of integral chains of the CW-
partition $\Xi$, being just a free abelian group whose basis consists of
$i$-cells. Let also $\mathbf{h}_{i}:C_{i}(\mathbb{RP}^{2},\mathbb{Z})\to
C_{i}(\mathbb{RP}^{2},\mathbb{Z})$ be the induced automorphism of
$C_{i}(\mathbb{RP}^{2},\mathbb{Z})$. Since $\mathbf{h}$ is cellular,
$\mathbf{h}_{i}$ is given by some non-degenerate $(c_{i}\times c_{i})$-matrix
$A_{i}$ and each column and each row of $A_{i}$ contains only one non-zero
being either $+1$ or $-1$. Notice also that non-zero diagonal elements of
$A_{i}$ correspond to invariant $i$-cells under $\mathbf{h}$. Moreover, such
an element equals $+1$ (resp. $-1$) is $\mathbf{h}$ preserves (resp. reverses)
orientation of the invariant cell. In particular, due to the above convention,
$A_{0}$ may consist of $0$ and $1$. This implies that
$\mathrm{tr}(A_{i})=c^{+}_{i}(\mathbf{h})-c^{-}_{i}(\mathbf{h})$, $i=0,1,2$.

It is also well known that the following number
$L(\mathbf{h})=\mathrm{tr}(A_{0})-\mathrm{tr}(A_{1})+\mathrm{tr}(A_{2})$,
called the Lefschetz number of $\mathbf{h}$, depends in fact only on the
homotopy class of $\mathbf{h}$. In particular, since $h$ is isotopic to
$\mathrm{id}_{\mathbb{M}}$ relatively $\partial\mathbb{M}$, it follows that
$\mathbf{h}$ is isotopic to $\mathrm{id}_{\mathbb{RP}^{2}}$, whence

$L(\mathbf{h})=L(\mathrm{id}_{\mathbb{RP}^{2}})=c_{0}-c_{1}+c_{2}=\chi(\mathbb{RP}^{2})=1.$

Thus, we finally get the following identity:

$1=c_{0}-c_{1}+c_{2}=c^{+}_{0}(\mathbf{h})-\bigl{(}c^{+}_{1}(\mathbf{h})-c^{-}_{1}(\mathbf{h})\bigr{)}+\bigl{(}c^{+}_{2}(\mathbf{h})-c^{-}_{2}(\mathbf{h})\bigr{)}.$
(6.1)

Also note that since $\mathbf{h}$ is homotopic to
$\mathrm{id}_{\mathbb{RP}^{2}}$ there exists a unique lifting
$\widetilde{\mathbf{h}}:S^{2}\to S^{2}$ such that

  * •

$\widetilde{\mathbf{h}}$ is isotopic to $\mathrm{id}_{S^{2}}$ and thus
preserves orientation of $S^{2}$;

  * •

$\widetilde{\mathbf{h}}$ is also $\widetilde{\Xi}$-cellular.

######  Lemma 6.5.1.

Let $e\in\Xi$ be an $i$-cell with $i=1,2$, and
$\widetilde{e}_{0},\widetilde{e}_{1}\in\widetilde{\Xi}$ be its liftings. If
$e$ is $\mathbf{h}^{+}$-invariant, then both $\widetilde{e}_{0}$ and
$\widetilde{e}_{1}$ are $\widetilde{\mathbf{h}}^{+}$-invaraint. If $e$ is
$\mathbf{h}^{-}$-invariant, then $\widetilde{\mathbf{h}}$ exchanges
$\widetilde{e}_{0}$ and $\widetilde{e}_{1}$, so they are not invariant under
$\widetilde{\mathbf{h}}$. In particular,

$\displaystyle c^{+}_{i}(\widetilde{\mathbf{h}})$
$\displaystyle=2c^{+}_{i}(\mathbf{h}),$ $\displaystyle
c^{-}_{i}(\widetilde{\mathbf{h}})$ $\displaystyle=0$

and therefore we have the following identity:

$L(\widetilde{\mathbf{h}})=\chi(S^{2})=2=c^{+}_{0}(\widetilde{\mathbf{h}})-2c^{+}_{1}(\mathbf{h})+2c^{+}_{2}(\mathbf{h}).$
(6.2)

###### Proof.

The statements about $\widetilde{\mathbf{h}}^{\pm}$-invariantness follow from
the fact that $\xi$ exchanges $\widetilde{e}_{0}$ and $\widetilde{e}_{1}$, and
$\xi:\widetilde{e}_{k}\to\widetilde{e}_{1-k}$, $k=0,1$, reverses orientation.
The last formula is a variant of the Lefschetz formula (6.1) for
$\widetilde{\mathbf{h}}$. We leave the details for the reader. ∎

###  6.6. Epimorphism
$\eta\colon\pi_{0}\mathcal{S}(f,\partial\mathbb{M})\to\mathbb{Z}$

Let $\tilde{Q}_{0}=(\mathbb{R}\times[0;1])\setminus(\mathbb{Z}\times 1)$, and
$J_{k}=(k;k+1)\times 1$ for $k\in\mathbb{Z}$. Then we have the universal
covering map $p:\tilde{Q}_{0}\to Q_{0}$, and hence for each
$h\in\mathcal{S}(f,\partial\mathbb{M})$ there exists a unique lifting
$\mathbf{h}:\tilde{Q}_{0}\to\tilde{Q}_{0}$ of the restriction $h|_{Q_{0}}$
fixed on $\mathbb{R}\times 0$. Evidently, $\mathbf{h}$ should shift the
intervals $J_{k}$, so there exists unique integer
$\widehat{\eta}(h)\in\mathbb{Z}$ such that
$\mathbf{h}(J_{k})=J_{k+\widehat{\eta}(h)}$ for all $k\in\mathbb{Z}$. One
easily checks that the correspondence $h\mapsto\widehat{\eta}(h)$ is a well-
defined homomorphism
$\widehat{\eta}:\mathcal{S}(f,\partial\mathbb{M})\to\mathbb{Z}$.

Figure 6.2.

Denote also by $c$ the total number of intervals in $Q_{0}\cap K$ which is the
same as the number of points in $F$. Moreover, let
$\tau\in\mathcal{S}(f,\partial\mathbb{M})$ be a Dehn twist along
$\partial\mathbb{M}$ supported in the cylinder $Y_{0}\subset C_{0}$. Thus, by
definition, $\tau$ is a “one rotation along $\partial\mathbb{M}$”, which means
that $\widehat{\eta}(\tau)=\pm c$. Replacing $\tau$ with $\tau^{-1}$ we can
assume that $\widehat{\eta}(\tau)=c$. In particular, $\widehat{\eta}$ is a
non-trivial homomorphism, and thus
$\widehat{\eta}(\mathcal{S}(f,\partial\mathbb{M}))=a\mathbb{Z}$ for some
$a\geq 1$.

Hence, we get an epimorphism

$\eta:\mathcal{S}(f,\partial\mathbb{M})\to\mathbb{Z},\qquad\eta(h)=\widehat{\eta}(h)/a.$

Notice also that the value $\eta(h)$ depends only on the isotopy class of $h$
in $\pi_{0}\mathcal{S}(f,\partial\mathbb{M})$, and therefore $\eta$ factors to
the desired epimorphism
$\eta:\pi_{0}\mathcal{S}(f,\partial\mathbb{M})\to\mathbb{Z}$, which (for
simplicity) we will denote by the same letter $\eta$. Put

${\color[rgb]{1,0,0}b}:=\eta(\tau)=c/a.$ (6.3)

Thus, roughly speaking, every $h\in\mathcal{S}(f,\partial\mathbb{M})$
cyclically shifts the boundary edges of $Q_{0}$, so that all shifts are
integer multiples of $a$, and the period of that shift is $c$.

###  6.7. Action of $\mathcal{S}(f,\partial\mathbb{M})$ on disks $Y_{i}$

Let $\mathbf{Y}=\\{Y_{1},\ldots,Y_{n}\\}$ be the family of $2$-disks from the
$f$-special decomposition of $\mathbb{M}$. Since
$\mathop{\cup}\limits_{k=1}^{n}Y_{k}$ is invariant with respect to
$\mathcal{S}(f,\partial\mathbb{M})$, we have a natural action of
$\mathcal{S}(f,\partial\mathbb{M})$ on $\mathbf{Y}$ by permutations.

Let us fix some orientation of each $Y_{k}$, $k=1,\ldots,n$, and put
$\hat{\mathbf{Y}}=\mathbf{Y}\times\\{\pm 1\\}$. Then the action of
$\mathcal{S}(f,\partial\mathbb{M})$ on $\mathbf{Y}$ extends further to an
action on $\hat{\mathbf{Y}}$ defined by the following rule:

  1. $(enumi)$

if $h\in\mathcal{S}(f,\partial\mathbb{M})$ and $Y_{k}\in\mathbf{Y}$, then
$h(Y_{k},+1)=(h(Y_{k}),\delta)$ and $h(Y_{k},-1)=(h(Y_{k}),-\delta)$, where

$\delta=\begin{cases}+1,&\text{if the restriction $h|_{Y_{k}}:Y_{k}\to
h(Y_{k})$ preserves orientation},\\\ -1,&\text{otherwise}.\end{cases}$

Let also $\mathcal{S}(f;\hat{\mathbf{Y}})$ be the kernel of non-effectiveness
of that action of $\mathcal{S}(f,\partial\mathbb{M})$ on $\hat{\mathbf{Y}}$,
i.e. $\mathcal{S}(f;\hat{\mathbf{Y}})$ consists of diffeomorphisms $h$ such
that $h(Y_{i})=Y_{i}$, for all $i=1,\ldots,n$, and $h$ also preserves
orientation of $Y_{i}$.

######  Lemma 6.7.1.

Let $h\in\mathcal{S}(f,\partial\mathbb{M})$. Then the following statements are
equivalent:

  1. (a)

$h\in\mathcal{S}(f;\hat{\mathbf{Y}})$, in other words $h$ preserves each
$Y_{k}$ with its orientation;

  2. (b)

$h$ preserves each connected component of $\mathbb{M}\setminus K$ with its
orientation, i.e. $c^{+}_{2}(\mathbf{h})=n+1$ and $c^{-}_{2}(\mathbf{h})=0$;

  3. (c)

$h$ preserves at least one connected component of $\mathbb{M}\setminus K$
distinct from $C_{0}$ with its orientation, i.e. $c^{+}_{2}(\mathbf{h})\geq
2$;

  4. (d)

$h$ each preserves each cell of $\Xi$ with its orientation;

  5. (e)

$h$ preserves each edge of $K$ with its orientation, i.e.
$c^{+}_{1}(\mathbf{h})=c_{1}$ and $c^{-}_{1}(\mathbf{h})=0$;

  6. (f)

$h\in\eta^{-1}(b\mathbb{Z})$, in other words $h$ preserves each boundary edge
of $Q_{0}$ with its orientation, see (6.3);

  7. (g)

$h$ preserves at least one edge of $K$ with its orientation, i.e.
$c^{+}_{1}(\mathbf{h})>0$.

In particular,

  * $-$

(f) implies that
$\frac{\mathcal{S}(f,\partial\mathbb{M})}{\mathcal{S}(f;\hat{\mathbf{Y}})}\cong\frac{\eta(\mathcal{S}(f,\partial\mathbb{M}))}{\eta(\mathcal{S}(f;\hat{\mathbf{Y}}))}=\mathbb{Z}/{\color[rgb]{1,0,0}b}\mathbb{Z}\cong\mathbb{Z}_{{\color[rgb]{1,0,0}b}}$;

  * $-$

the equivalence (a)$\Leftrightarrow$(b) implies that the action of
$\mathcal{S}(f,\partial\mathbb{M})/\mathcal{S}(f;\hat{\mathbf{Y}})\cong\mathbb{Z}_{{\color[rgb]{1,0,0}b}}$
on $\hat{\mathbf{Y}}$ is free;

  * $-$

(d) implies that two diffeomorphisms
$h_{1},h_{2}\in\mathcal{S}(f,\partial\mathbb{M})$ induce the same permutation
of cells of $\Xi$ if and only if $h_{1}^{-1}\circ
h_{2}\in\mathcal{S}(f;\hat{\mathbf{Y}})$, i.e. they belong to the same
adjacent class
$\mathcal{S}(f,\partial\mathbb{M})/\mathcal{S}(f;\hat{\mathbf{Y}})$.

###### Proof.

The equivalence (a)$\Leftrightarrow$(b) follows from the equality $h(K)=K$,
while the implications (b)$\Rightarrow$(c) and
(d)$\Rightarrow$(e)$\Rightarrow$(f)$\Rightarrow$(g) are evident.

The implication (g)$\Rightarrow$(e) can be found in [23, Claim 7.1.1]. Let us
recall the arguments, Since $h$ preserves some edge of $K$ with its
orientation, $h$ also preserves the closure of this edge with its orientation,
i.e. $h$ fixes the vertices on the boundary of this edge. But $h$ also
preserves the cyclic order of edges incident to both of its ends, and
therefore $h$ should preserve all these edges with their orientations. It now
follows from the connectedness of $K$ that $h$ preserves all edges of $K$ with
their orientations.

(e)$\Rightarrow$(b) Let $C_{i}$ be a connected component of
$\mathbb{M}\setminus K$, and $e\subset K$ be a boundary edge of
$Q_{i}=\overline{C_{i}}\setminus\Sigma_{f}$. Let also $C_{j}$ be another
component of $\mathbb{M}\setminus K$ such that $e\subset Q_{j}$. By (e),
$h(e)=e$ whence $h$ should leave invariant $Q_{i}\cup Q_{j}$, and therefore
the set $C_{i}\cup C_{j}=(Q_{i}\cup Q_{j})\setminus K$. As noted above in
(iv), one can assume that $f(C_{i})\subset\bigl{(}-\infty;f(K)\bigr{)}$ and
$f(C_{j})\subset\bigl{(}f(K);+\infty\bigr{)}$. Since $f\circ h=f$, so $h$
level sets of $f$, we see that $h$ can not interchange $C_{i}$ with $C_{j}$,
and thus $h(C_{i})=C_{i}$. Moreover, by (e), $h$ also preserves orientation of
$e$, whence it should also preserve orientation of $C_{i}$.

(b)$\Rightarrow$(g) Suppose $c^{+}_{2}(\mathbf{h})=c_{2}=n+1$ and thus
$c^{-}_{2}(\mathbf{h})=0$. Then by formula (6.1) we have that

$1=c^{+}_{0}(\mathbf{h})-\bigl{(}c^{+}_{1}(\mathbf{h})-c^{-}_{1}(\mathbf{h})\bigr{)}+n+1,$

whence
$c^{+}_{1}(\mathbf{h})=c^{+}_{0}(\mathbf{h})+c^{-}_{1}(\mathbf{h})+n>0$.

(c)$\Rightarrow$(g) Suppose $c^{+}_{2}(\mathbf{h})\geq 2$ but (g) fails, so
$c^{+}_{1}(\mathbf{h})=0$. Then by Lemma 6.5.1, we have that
$c^{+}_{2}(\widetilde{\mathbf{h}})=2c^{+}_{2}(\mathbf{h})\geq 4$,
$c^{+}_{1}(\widetilde{\mathbf{h}})=2c^{+}_{1}(\mathbf{h})=0$, whence by (6.2)

$2=c^{+}_{0}(\widetilde{\mathbf{h}})-c^{+}_{1}(\widetilde{\mathbf{h}})+c^{+}_{2}(\widetilde{\mathbf{h}})=c^{+}_{0}(\widetilde{\mathbf{h}})+c^{+}_{2}(\widetilde{\mathbf{h}})\geq
c^{+}_{2}(\widetilde{\mathbf{h}})\geq 4,$

which is impossible. Hence (g) should hold.

(b)&(e) $\Rightarrow$ (d) By assumption, all $1$\- and $2$-cells are
$h^{+}$-invaraint. As each $0$-cell $v$ of $K$ belongs to the closure of some
$1$-cell $e$, and $h$ preserves orientation of $e$, we see that $h$ also fixed
$v$. ∎

###  6.8. Disks of types (b)(T1) and (b)(T2)

By Lemma 6.7.1 we have an effective free action of the quotient
$\frac{\mathcal{S}(f,\partial\mathbb{M})}{\mathcal{S}(f;\hat{\mathbf{Y}})}\cong\frac{\pi_{0}\mathcal{S}(f,\partial\mathbb{M})}{\pi_{0}\mathcal{S}(f;\hat{\mathbf{Y}})}\cong\mathbb{Z}_{{\color[rgb]{1,0,0}b}}$
on $\hat{\mathbf{Y}}$, so we can regard it as a cyclic subgroup of the
permutation group of $\hat{\mathbf{Y}}$.

Let $g\in\pi_{0}\mathcal{S}(f,\partial\mathbb{M})$ be any element with
$\eta(g)=1$. Then its class in the quotient
$\mathbb{Z}_{{\color[rgb]{1,0,0}b}}$ is a generator of that group, so it is
represented by some bijection
$\widehat{g}:\hat{\mathbf{Y}}\to\hat{\mathbf{Y}}$.

Hence, $\widehat{g}^{{\color[rgb]{1,0,0}b}}=\mathrm{id}_{\hat{\mathbf{Y}}}$,
$\widehat{g}^{k}$ for $k=1,\ldots,{\color[rgb]{1,0,0}b}-1$ has no fixed
points, and every orbit of the action of $\mathbb{Z}_{{\color[rgb]{1,0,0}b}}$
consists of ${\color[rgb]{1,0,0}b}$ elements, so, in particular,
${\color[rgb]{1,0,0}b}$ divides $2n$ (the order of $\hat{\mathbf{Y}}$).

######  Lemma 6.8.1.

  1. (a)

If $\widehat{g}(Y_{i},1)=(Y_{k},\delta)$ for some $i,k\in\\{1,\ldots,n\\}$ and
$\delta\in\\{\pm 1\\}$, then $\widehat{g}(Y_{i},-1)=(Y_{k},-\delta)$.

  2. (b)

Let $(Y_{i_{0}},\delta_{i_{0}})\in\hat{\mathbf{Y}}$, and
$(Y_{i_{k}},\delta_{i_{k}})=\widehat{g}^{k}(Y_{i_{0}},\delta_{i_{0}})$,
$k=0,\ldots,{\color[rgb]{1,0,0}b}-1$, be all elements of its orbit. Then
exactly one of the following two possibilities holds:

    1. (T1)

either all disks $Y_{i_{0}},\ldots,Y_{i_{{\color[rgb]{1,0,0}b}-1}}$ are
mutually distinct,

    2. (T2)

${\color[rgb]{1,0,0}b}$ is even, the disks
$Y_{i_{0}},\ldots,Y_{i_{{\color[rgb]{0.64,0.08,0.98}m}-1}}$ are mutually
distinct, where ${\color[rgb]{0.64,0.08,0.98}m}={\color[rgb]{1,0,0}b}/2$, and
$(Y_{i_{{\color[rgb]{0.64,0.08,0.98}m}}},\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}}})=(Y_{i_{0}},-\delta_{i_{0}})$,
so, due to (a),

$\begin{matrix}(Y_{i_{0}},\delta_{i_{0}}),&(Y_{i_{1}},\delta_{i_{1}}),&\ldots,&(Y_{i_{{\color[rgb]{0.64,0.08,0.98}m}-1}},\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}-1}}),\\\
(Y_{i_{0}},-\delta_{i_{0}}),&(Y_{i_{1}},-\delta_{i_{1}}),&\ldots,&(Y_{i_{{\color[rgb]{0.64,0.08,0.98}m}-1}},-\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}-1}}),\end{matrix}$

are consecutive elements of the orbit of $(Y_{i_{0}},\delta_{i_{0}})$.

  3. (c)

Suppose $g(Y_{i})=Y_{j}$ for some $i,j$. Then we have the following
isomorphism:

$\pi_{0}\mathcal{S}(f|_{Y_{i}},\partial
Y_{i})\to\pi_{0}\mathcal{S}(f|_{Y_{j}},\partial Y_{j}),\qquad t\mapsto
gtg^{-1},$

which sends $\pi_{0}\Delta(f|_{Y_{i}},\partial Y_{i})$ onto
$\pi_{0}\Delta(f|_{Y_{j}},\partial Y_{j})$.

###### Proof.

Statement (a) is just the same as $(enumi)$, and (c) is evident.

(b) Suppose (b)(T1) fails, so
$g^{k}(Y_{i_{0}})=Y_{i_{k}}=Y_{i_{l}}=g^{l}(Y_{i_{0}})$ for some
$k<l\in\\{0,1,\ldots,{\color[rgb]{1,0,0}b}-1\\}$. Denote
${\color[rgb]{0.64,0.08,0.98}m}=l-k$. Then
$Y_{i_{0}}=g^{{\color[rgb]{0.64,0.08,0.98}m}}(Y_{i_{0}})=Y_{i_{u}}$. Since the
action of $\widehat{g}$ on $\hat{\mathbf{Y}}$ is free and
${\color[rgb]{0.64,0.08,0.98}m}<{\color[rgb]{1,0,0}b}$, we should have that
$(Y_{i_{0}},\delta_{i_{0}})\not=(Y_{i_{{\color[rgb]{0.64,0.08,0.98}m}}},\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}}})=(Y_{i_{0}},\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}}})$,
whence $\delta_{i_{0}}=-\delta_{i_{{\color[rgb]{0.64,0.08,0.98}m}}}$. But then

$\widehat{g}^{2{\color[rgb]{0.64,0.08,0.98}m}}(Y_{i_{0}},\delta_{i_{0}})=\widehat{g}^{{\color[rgb]{0.64,0.08,0.98}m}}(Y_{i_{0}},-\delta_{i_{0}})\stackrel{{\scriptstyle\ref{enum:lm:action_g_on_hY:star}}}{{=}}(Y_{i_{0}},\delta_{i_{0}}),$

i.e. $\widehat{g}^{2{\color[rgb]{0.64,0.08,0.98}m}}$ has a fixed element
$(Y_{i_{0}},\delta_{i_{0}})$, and therefore $2{\color[rgb]{0.64,0.08,0.98}m}$
must divide ${\color[rgb]{1,0,0}b}$. Since
${\color[rgb]{0.64,0.08,0.98}m}<{\color[rgb]{1,0,0}b}$, we should have that
$2{\color[rgb]{0.64,0.08,0.98}m}={\color[rgb]{1,0,0}b}$. ∎

Thus, in the case (b)(T1), if $h(Y_{i_{k}})=Y_{i_{k}}$ for some
$h\in\mathcal{S}(f,\partial\mathbb{M})$ and
$k=0,\ldots,{\color[rgb]{1,0,0}b}-1$, then $h$ must preserve orientation of
$Y_{i_{k}}$. On the other hand, in the case (b)(T2) for each $k$ there exists
$h\in\mathcal{S}(f,\partial\mathbb{M})$ such that $h(Y_{i_{k}})=Y_{i_{k}}$ and
$h$ reverses orientation of $Y_{i_{k}}$.

It will be convenient to say that a pair $(Y_{i_{0}},\delta_{0})$ as well as
its orbit is of type (b)(T1) or (b)(T2) depending on the corresponding cases.
Moreover, since $\widehat{g}(Y_{i_{0}},\delta_{0})$ and
$(Y_{i_{0}},-\delta_{0})$ have the same type, we can even say that the disk
$Y_{i_{0}}$ itself has the corresponding type.

######  Lemma 6.8.2.

Suppose all disks $Y_{1},\ldots,Y_{n}$ are of type (b)(T1). There exists a
$\Xi$-cellular homeomorphism $q:\mathbb{RP}^{2}\to\mathbb{RP}^{2}$ of order
${\color[rgb]{1,0,0}b}$ such that each of its iterations $q^{i}$,
$i=1,\ldots,{\color[rgb]{1,0,0}b}-1$, have a unique fixed point $x^{*}$.