Deformations of circle-valued Morse functions on surfaces
Let M be a smooth connected orientable compact surface and let Fcov(M,S1) be a space of all Morse functions f : M → S₁ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S₁ is either a constant map or a covering map. The space Fcov(M,S₁) is endowed...
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nasplib_isofts_kiev_ua-123456789-1663052025-02-09T18:18:29Z Deformations of circle-valued Morse functions on surfaces Деформації відображень Морса поверхонь у коло Maksymenko, S.I. Статті Let M be a smooth connected orientable compact surface and let Fcov(M,S1) be a space of all Morse functions f : M → S₁ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S₁ is either a constant map or a covering map. The space Fcov(M,S₁) is endowed with the C ∞-topology. We present the classification of connected components of the space Fcov(M,S₁). This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M. Нехай M — гладка зв'язна орієнтовна компактна поверхня. Позначимо через Fcov(M,S₁) простір усіх відображень Морса f:M→S₁, які не мають критичних точок на ∂M, а для кожної компоненти зв'язності V межі дМ обмеження f:V→S₁ є або постійним або накриваючим відображенням. Наділимо Fcov(M,S₁) топологією C∞. У статті наведено класифікацію компонент зв'язності простору Fcov(M,S₁). Цей результат узагальнює результати С. В. Матвєєва, В. В. Шарка та автора про функції Морса, що є локально постійними на ∂M. *This research is partially supported by grant of Ministry of Science and Education of Ukraine, No. M/150-2009. 2010 Article Deformations of circle-valued Morse functions on surfaces / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 10. — С. 1360–1366. — Бібліогр.: 6 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/166305 517.9 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Maksymenko, S.I. Deformations of circle-valued Morse functions on surfaces Український математичний журнал |
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Let M be a smooth connected orientable compact surface and let Fcov(M,S1) be a space of all Morse functions f : M → S₁ without critical points on ∂M such that, for any connected component V of ∂M, the restriction f : V → S₁ is either a constant map or a covering map. The space Fcov(M,S₁) is endowed with the C ∞-topology. We present the classification of connected components of the space Fcov(M,S₁). This result generalizes the results obtained by Matveev, Sharko, and the author for the case of Morse functions locally constant on ∂M. |
| format |
Article |
| author |
Maksymenko, S.I. |
| author_facet |
Maksymenko, S.I. |
| author_sort |
Maksymenko, S.I. |
| title |
Deformations of circle-valued Morse functions on surfaces |
| title_short |
Deformations of circle-valued Morse functions on surfaces |
| title_full |
Deformations of circle-valued Morse functions on surfaces |
| title_fullStr |
Deformations of circle-valued Morse functions on surfaces |
| title_full_unstemmed |
Deformations of circle-valued Morse functions on surfaces |
| title_sort |
deformations of circle-valued morse functions on surfaces |
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Інститут математики НАН України |
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2010 |
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Статті |
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https://nasplib.isofts.kiev.ua/handle/123456789/166305 |
| citation_txt |
Deformations of circle-valued Morse functions on surfaces / S.I. Maksymenko // Український математичний журнал. — 2010. — Т. 62, № 10. — С. 1360–1366. — Бібліогр.: 6 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT maksymenkosi deformationsofcirclevaluedmorsefunctionsonsurfaces AT maksymenkosi deformacíívídobraženʹmorsapoverhonʹukolo |
| first_indexed |
2025-11-29T13:33:51Z |
| last_indexed |
2025-11-29T13:33:51Z |
| _version_ |
1850131864352718848 |
| fulltext |
UDC 517.9
S. I. Maksymenko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
DEFORMATIONS OF CIRCLE-VALUED MORSE FUNCTIONS
ON SURFACES*
ДЕФОРМАЦIЇ ВIДОБРАЖЕНЬ МОРСА ПОВЕРХОНЬ У КОЛО
Let M be a smooth connected orientable compact surface. Denote by Fcov(M,S1) the space of all Morse
functions f : M → S1 having no critical points on ∂M and such that for every connected component V
of ∂M, the restriction f : V → S1 is either a constant map or a covering map. Endow Fcov(M,S1) with
C∞-topology. In this paper the connected components of Fcov(M,S1) are classified. This result extends the
results of S. V. Matveev, V. V. Sharko, and the author for the case of Morse functions being locally constant
on ∂M.
Нехай M — гладка зв’язна орiєнтовна компактна поверхня. Позначимо через Fcov(M,S1) простiр усiх
вiдображень Морса f : M → S1, якi не мають критичних точок на ∂M, а для кожної компоненти
зв’язностi V межi ∂M обмеження f : V → S1 є або постiйним або накриваючим вiдображенням.
Надiлимо Fcov(M,S1) топологiєю C∞. У статтi наведено класифiкацiю компонент зв’язностi простору
Fcov(M,S1). Цей результат узагальнює результати С. В. Матвєєва, В. В. Шарка та автора про функцiї
Морса, що є локально постiйними на ∂M.
1. Introduction. Let M be a compact surface and P be either the real line R or the
circle S1. Denote by F ′(M,P ) the subset of C∞(M,S1) consisting of maps f : M → P
such that
(1) all critical points of f are non-degenerate and belongs to the interior of M, so f
is a Morse function.
Let also Fl.c.(M,P ) be the subset of F ′(M,P ) consisting of maps f : M → P
such that
(2) f |∂M is a locally constant map, that is for every connected component W of
∂M the restriction of f to W is a constant map.
Moreover, for the case P = S1 let Fcov(M,S1) be another subset of F ′(M,S1)
consisting of maps f : M → S1 such that
(2′) for every connected component W of ∂M the restriction of f to W is either a
constant map or a covering map.
Thus
Fl.c.(M,S1) ⊂ Fcov(M,S1).
Endow all these spaces F ′(M,P ), Fl.c.(M,P ), and Fcov(M,S1) with the correspond-
ing C∞-topologies. The connected components of the spaces Fl.c.(M,P ) were de-
scribed in [1 – 4]. The aim of this note is to describe the connected components of the
space Fcov(M,S1) for the case when M is orientable.
To formulate the result fix an orientation of P and let f ∈ F ′(M,P ). Then for
each (non-degenerate) critical point of f we can define its index with respect to a given
orientation of S1. Denote by ci = ci(f), i = 0, 1, 2, the total number of critical points
of f of index i.
Moreover, suppose W is a connected component of ∂M such that the restriction of
f to W is a constant map. Then we associate to W the number εW (f) := +1 (resp.
*This research is partially supported by grant of Ministry of Science and Education of Ukraine, No. M/150-
2009.
c© S. I. MAKSYMENKO, 2010
1360 ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
DEFORMATIONS OF CIRCLE-VALUED MORSE FUNCTIONS ON SURFACES 1361
εk(f) := −1) whenever the value f(W ) is a local maximum (resp. minimum) with
respect to the orientation of P. If f |W is non-constant, then we put εW (f) = 0.
The following theorem describes the connected components of Fl.c.(M,P ).
Theorem 1 [1 – 4]. Let f, g ∈ Fl.c.(M,P ). Then they belong to the same path
component of Fl.c.(M,P ) iff the following three conditions hold true:
(i) f and g are homotopic as continuous maps ( for the case P = R this condition
is, of course, trivial );
(ii) ci(f) = ci(g) for i = 0, 1, 2;
(iii) εW (f) = εW (g) for every connected component W of ∂M.
If P = R and f = g on some neighbourhood of ∂M, then one can choose a homotopy
between f and g fixed near ∂M.
The case P = R was independently established by V. V. Sharko [1] and S. V. Matveev.
Matveev’s proof was generalized to the case of height functions and published in the
paper [2] by E. Kudryavtseva. The case P = S1 was proven by the author in [3].
Moreover, in [4] Theorem 1 was reproved by another methods.
The present notes establishes the following result.
Theorem 2. Suppose M is orientable. Let f, g ∈ Fcov(M,S1). Then they belong
to the same path component of Fcov(M,S1) iff the following three conditions hold true:
(i) f and g are homotopic as continuous maps;
(ii) ci(f) = ci(g) for i = 0, 1, 2;
(iii) εW (f) = εW (g) for every connected component W of ∂M such that f |W is a
constant map.
Notice that the formulations of both Theorems 1 and 2 look the same. The difference
is that in Theorem 1 every f ∈ Fl.c.(M,P ) takes constant values of connected com-
ponents of ∂W, while in Theorem 2 the restrictions of f ∈ Fcov(M,S1) to boundary
components W of M may also be covering maps and the degrees of such restrictions
f : W → S1 are encoded by homotopy condition (i).
I would like to thank A. Pajitnov for posing me question about connected compo-
nents of Fcov(M,S1) and useful discussions.
The proof of Theorem 2 follows the line of [3, 4]. First we prove R-variant of
Theorem 2 similarly to [4], see Theorem 3 below, and then deduce Theorem 2 from
Theorem 3 similarly to [3]. Therefore we mostly sketch the proofs indicating only the
principal differences.
2. R-variant of Theorem 2 for surfaces with corners. Let f ∈ Fcov(M,S1). Say
that v ∈ S1 is an exceptional value of f, if v is either a critical value of f or there exists
a connected component W of ∂M such that f(W ) = v.
Let v ∈ S1 be a non-exceptional value of f. Then its inverse image f−1(v) is
a proper 1-submanifold of M which does not contain connected components of ∂M.
Thus f−1(v) is a disjoint union of circles and arcs with ends on ∂M and transversal to
∂M at these points. Let M̂ be a surface obtained by cutting M along f−1(v).
Then M̂ can be regarded as a surface with corners and f induces a function f̂ : M̂ →
→ [0, 1] such that
(a) f̂ |
IntM̂
is Morse and has no critical points on ∂M̂ ;
(b) let W be a connected component of ∂M̂ ; then either f̂ |W is constant, or there
are 4kW points on W for some kW ≥ 1 dividing W into 4kW arcs
A1, B1, C1, D1, . . . , AkW
, BkW
, CkW
, DkW
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1362 S. I. MAKSYMENKO
such that f̂ strictly decreases on Ai, strictly increases on Ci, f̂(Bi) = 1, and f̂(Di) = 0
for each i = 1, . . . , kW .
We will now define the space of all such functions and describe its connected com-
ponents.
2.1. Space Fξ(M, I). Let M be a compact, possibly non-connected, surface. For
every connected component W of ∂M fix an orientation and a number kW ≥ 0, and
divide W into 4kW consecutive arcs
A1, B1, C1, D1, . . . , AkW
, BkW
, CkW
, DkW
directed along the orientation of W. If kW = 0 then we do not divide W at all.
Denote this subdivision of ∂M by ξ and the set of ends of these arcs by K = K(ξ).
We will regard K as
”
corners” of M.
Let also T+ (resp. T1, T−, and T0) be the union of all closed arcs Ai (resp. Bi, Ci,
Di) over all boundary components of M.
Let Fξ(M, I) be the space of all continuous functions f : M → I = [0, 1] satisfying
the following three conditions.
(a) The restriction of f to M \K is C∞, and all partial derivatives of f of all orders
continuously extend to all of M.
(b) All critical points of f are non-degenerate and belong to IntM,
f(IntM) ⊂ (0, 1), f−1(0) = T0, f−1(1) = T1,
and f |T+
(resp. f |T
−
) has strictly positive (resp. negative) derivative.
(c) Let W be a connected component of ∂M such that kW = 0. Then f |W is
constant and f̂(W ) ∈ (0, 1).
Notice that condition (a) means that f is a C∞-function on a surface with corners
and condition (b) implies that f strictly increases (decreases) on each arc Ai (Ci),
Again we associate to every f ∈ Fξ(M, I) the total number ci(f) of critical points
at each index i = 0, 1, 2. Moreover, to every connected component W of ∂M with
kW = 0 we associate the number εW (f) = ±1 as above.
The following theorem extends R-case of Theorem 1 to orientable surfaces with
corners.
Theorem 3. Suppose M is orientable and connected. Then f, g ∈ Fξ(M, I) be-
longs to the same path component of Fξ(M, I) iff
(i) ci(f) = ci(g) for i = 0, 1, 2;
(ii) εW (f) = εW (g) for every connected component W of ∂M with kW = 0.
Moreover, if f = g on some neighbourhood of T0 ∪ T1, then there exists a homotopy
relatively T0 ∪ T1 between these functions in Fξ(M, I).
The proof will be given in Section 4. Now we will deduce from this result Theo-
rem 2.
3. Proof of Theorem 2. Necessity is obvious, therefore we will prove only suffi-
ciency.
Let f, g ∈ Fcov(M,S1). Consider the following conditions (Pn), n ≥ 0, (Q), and
(R) for f and g.
(Pn) f (resp. g) is homotopic in Fcov(M,S1) to a map f̃ (resp. g̃) such that for some
common non-exceptional value v ∈ S1 of f̃ and g̃ the intersection f̃−1(v) ∩ g̃−1(v) is
transversal and consists of at most n points.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
DEFORMATIONS OF CIRCLE-VALUED MORSE FUNCTIONS ON SURFACES 1363
(Q) f (resp. g) is homotopic in Fcov(M,S1) to a map f̃ (resp. g̃) such that for some
common non-exceptional value v ∈ S1 of f̃ and g̃,
(i) f̃−1(v) = g̃−1(v),
(ii) f̃ = g̃ on some neighbourhood of f̃−1(v),
(iii) and for every connected component M1 of M \ f̃−1(v) the restrictions f̃ and g̃
onto M1 have the same numbers of critical points at each index.
(R) f is homotopic to g in Fcov(M,S1).
Notice that f and g always satisfy (Pn) for some n ≥ 0. We have to prove for
them condition (R). This is given by the following lemma, which completes the proof
of Theorem 2.
Lemma 1. Let f, g ∈ Fcov(M,S1). Suppose that f, g ∈ Fcov(M,S1) satisfy
conditions (i) – (iii) of Theorem 2. Then the following implications hold:
(Pn) ⇒ (Pn−1) ⇒ . . . ⇒ (P0) ⇒ (Q) ⇒ (R).
Proof. Implications (Pn) ⇒ (Pn−1) and (P0) ⇒ (Q) can be deduced from The-
orem 3 almost by the same arguments as [3] (Theorems 3, 4) were deduced from the
R-case of Theorem 1. The principal difference here is that one should work with 1-
submanifolds with boundary rather than with closed 1-submanifolds. The proof is left
for the reader.
(Q) ⇒ (R). Cut M along f−1(v) and denote the obtained surface with corners
by M̂. Then f (resp. g) induces on M̂ a function f̂ (resp. ĝ) belonging to Fξ(M
′, I).
Moreover, it follows from conditions (i) – (iii) of Theorem 1 for f and g and assumption
(iii) of (Q) that for every connected component M1 of M̂ the restrictions of f̂ and
ĝ to M1 satisfy conditions (i) and (ii) of Theorem 3. Hence they are homotopic in
Fξ(M
′, I) relatively some neighbourhood of the set T0 ∪ T1 corresponding to f−1(v).
This homotopy yields a desired homotopy between f and g in Fcov(M,S1).
Lemma 1 is proved.
4. Proof of Theorem 3. We will follow the line of the proof of Theorem 1, see [2, 4].
Suppose f, g ∈ Fξ(M, I) satisfy assumptions (i) and (ii) of Theorem 3. The idea is to
reduce the situation to the case when g = f ◦ h for some diffeomorphism h of M fixed
near ∂M, and then show that f ◦ h is homotopic in Fξ(M, I) to f, see Lemmas 4 – 6.
4.1. KR-graph. For f ∈ Fξ(M, I) define the Kronrod – Reeb graph (or simply KR-
graph) Γf of f as a topological space obtained by shrinking to a point every connected
component of f−1(v) for each v ∈ I. It easily follows from the assumptions on f that Γf
has a natural structure of a 1-dimensional CW-complex. The vertices of f corresponds
to the connected components of level sets f−1(v) containing critical points of f.
Notice that f can be represented as the following composite of maps:
f = fKR ◦ pf : M
pf
−−→ Γf
fKR
−−−−→ I,
where pf is a factor map and fKR is the induced function on Γf which we will call the
KR-function of f.
Say that f is generic if it takes distinct values at distinct critical points and connected
components W of ∂M with kW = 0. It is easy to show that every f ∈ Fξ(M, I) is
homotopic in Fξ(M, I) to a generic function.
Notice that for each non-exceptional value v of f every connected component P of
f−1(v) is either an arc or a circle. We will distinguish the corresponding points on Γf
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1364 S. I. MAKSYMENKO
as follows: if P is an arc, then we denote the corresponding point on Γf in bold. Thus
on the KR-graph of f we will have two types of edges bold and thin.
Moreover, every vertex w of degree 1 of Γf corresponds either to a local extreme
of f or to a boundary component W of ∂M with kW = 0. In the first case w will be
called an e-vertex, and a ∂-vertex otherwise. ∂-vertexes will be denoted in bold.
Possible types of vertexes of Γf corresponding to saddle critical points together with
the corresponding critical level sets are shown in Fig. 1.
a b c
Fig. 1. Structure of f near saddle critical points.
Definition 1. Let f, g ∈ Fξ(M, I). Say that KR-functions of f and g are equiv-
alent if there exist a homeomorphism H : Γf → Γg between their KR-graphs and a
homeomorphism Φ: I → I which preserves orientation such that gKR = Φ−1◦fKR◦H
and H maps bold edges (resp. thin edges, ∂-vertexes) of Γg to bold edges (resp. thin
edges, ∂-vertexes) of Γf .
We will always draw a KR-graph so that the corresponding KR-function will be the
projection to the vertical line. This determines KR-function up to equivalence in the
sense of Definition 1.
The following statement can be proved similarly to [5, 6].
Lemma 2. Suppose M is orientable, and let f, g ∈ Fξ(M, I) be two generic
functions such that their KR-functions are KR-equivalent. Then there exist a diffeomor-
phism h : M → M and a preserving orientation diffeomorphism φ : I → I such that
g = φ−1 ◦ f ◦ h.
Since φ is isotopic to idI , it follows that g is homotopic in Fξ(M, I) to f ◦ h.
4.2. Canonical KR-graph. Consider the graphs shown in Fig. 2.
The graph X0(k), k ≥ 1, consists of a bold line “intersected” by another k− 1 bold
lines, the graph X±(k) is obtained from X0(k) by adding a thin edge directed either up
or down. The vertex of degree 1 on that thin edge can be either e- or ∂-one.
The graph Y is determined by five numbers: z, b−, b+, e−, e+, where z is the total
number of cycles in Y, b− (resp. e−) is the total number of ∂-vertexes (resp. e-vertexes)
being local minimums for the KR-function, and b+ and e+ correspond to local maxi-
mums.
We will assume that KR-function surjectively maps X∗(k) onto [0, 1], while Y is
mapped into interval (0, 1).
Definition 2. Let f ∈ Fξ(M, I). Say that f is canonical if it is generic and its
KR-graph Γf has one of the following forms:
(1) coincides either with one of X∗(k) for some k ≥ 1, or with Y for some e±, b±,
and z;
(2) is a union of X−(k) with X+(l) with common thin edge for some k, l ≥ 1, see
Fig. 3, a;
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
DEFORMATIONS OF CIRCLE-VALUED MORSE FUNCTIONS ON SURFACES 1365
} e+
}b+
}z
} e_
}
b_
}k
X k( ) 0
X 1( ) 0
X 1( ) +
X 1( )
_
X k( ) +
X k( )
_
Y
Fig. 2. Elementary blocks of canonical KR-graphs.
Fig. 3. Canonical KR-graph Γf .
(3) is a union of some X+(ki), i = 1, . . . , n, connected along their thin edges with
Y, see Fig. 3, b.
Every maximal bold connected subgraph of Γf will be called an X-block. Evidently,
such a block is isomorphic with X0(k) for some k.
Lemma 3. Let f ∈ Fξ(M, I) be a canonical function. Then the numbers ci(f),
kW , and εW (f) are completely determined by its KR-graph Γf and wise verse. More-
over, every X-block of Γf corresponds to a unique boundary component of M. In
particular, the collection of X-blocks in Γf is determined (up to order) by the partition
ξ of ∂M, and therefore does not depend on a canonical function f.
Proof. Since f is generic, c0(f) (resp. c2(f)) is equal to the total number of vertexes
of degree 1 being local minimums (resp. local maximums) of the restriction of fKR to
Y, while c1(f) is equal to the total number of vertexes of Γf of degrees 3 and 4.
Furthermore, it easily follows from Fig. 1, c, that every X-block N of Γf corre-
sponds to a collar of some boundary component W of M such that kW is equal to the
total number of local minimums (= local maximums) of the restriction of fKR to N.
Finally, every connected component W of ∂M with kW = 0 corresponds to a ∂-
vertex w on Y. Moreover, εW = −1 (resp. εW = +1) iff w is a local minimum (resp.
local maximum) of the restriction of fKR to Y.
Lemma 3 is proved.
Lemma 4. Let f ∈ Fξ(M, I). Then f is homotopic in Fξ(M, I) to some canoni-
cal function.
Proof. Consider the following elementary surgeries of a KR-graph shown in Fig. 4.
It is easy to see that each of them can be realized by a deformation of f in Fξ(M, I).
Then similarly to [2] (Lemma 11) one can reduce any KR-graph of f ∈ Fξ(M, I) to a
canonical form using these surgeries. We leave the details for the reader.
Lemma 4 is proved.
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
1366 S. I. MAKSYMENKO
Fig. 4. Elementary surgeries of KR-graph.
Lemma 5. Let f, g ∈ Fξ(M, I) be two canonical functions satisfying assumptions
(i) and (ii) of Theorem 3. Then f (resp. g) is homotopic in Fξ(M, I) to another canonical
function f̃ (resp. g̃) such that g̃ = f̃ ◦ h for some diffeomorphism h : M → M fixed
near ∂M.
Proof. It follows from Lemma 3 and assumptions on f and g that their KR-graphs
have the same Y -blocks and the same (up to order) X±(k)-blocks. Then, using surgeries
of Figure 4 applied to Γg, we can reduce the situation to the case when KR-functions of
f of g are KR-equivalent. Whence by Lemma 2 we can also assume that there exists a
diffeomorphism h : M → M such that g = f ◦h. Moreover, changing g similarly to [2]
or [4] one can choose h so that it preserves orientation of M, maps every connected
component W of ∂M onto itself, and preserves subdivision ξ on W. Then using the
assumptions on f and g near ∂M, one can show that h is isotopic to the identity near
∂M.
Lemma 5 is proved.
Lemma 6. Let h : M → M be a diffeomorphism fixed near ∂M and f ∈
∈ Fξ(M, I) be a canonical function. Then f ◦h is homotopic in Fξ(M, I) to f relatively
some neighbourhood of ∂M.
Proof. Since every X-block of Γf corresponds to a collar N(W ) of some boundary
component W of ∂M, we can assume that h is fixed on some neighbourhood of N(W ).
Therefore we may cut off N(W ) from M and assume that f takes constant values
at each boundary component of ∂M. Then f is homotopic to f ◦ h relatively some
neighbourhood of ∂M by the arguments similar to the proof of Theorem 1, see [4].
Lemma 6 is proved.
Theorem 3 now follows from Lemmas 4 – 6.
1. Sharko V. V. Functions on surfaces. I // Some Problems in Contemporary Mathematics (in Russian) //
Pr. Inst. Mat. Nats. Akad. Nauk Ukrajiny. Mat. Zastos. – 1998. – 25. – P. 408 – 434.
2. Kudryavtseva E. A. Realization of smooth functions on surfaces as height functions // Mat. Sb. – 1999.
— 190, № 3. – P. 29 – 88.
3. Maksymenko S. I. Components of spaces of Morse mappings // Some Problems in Contemporary
Mathematics (in Russian) // Pr. Inst. Mat. Nats. Akad. Nauk Ukrajiny. Mat. Zastos. – 1998. – 25. –
P. 135 – 153.
4. Maksymenko S. Path-components of Morse mappings spaces of surfaces // Comment. math. helv. – 2005.
— 80, № 3. – P. 655 – 690.
5. Kulinich E. V. On topologically equivalent Morse functions on surfaces // Meth. Funct. Anal. and Top.
– 1998. – 4, № 1. – P. 59 – 64.
6. Bolsinov A. V., Fomenko A. T. Introduction to the topology of integrable Hamiltonian systems (in
Russian). – Moskow: Nauka, 1997.
Received 08.06.10
ISSN 1027-3190. Укр. мат. журн., 2010, т. 62, № 10
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