Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold
This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the to...
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nasplib_isofts_kiev_ua-123456789-1485822025-02-09T21:55:36Z Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold Hladysh, B.I. Prishlyak, A.O. This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classification of above-mentioned functions in some neighborhood of their critical points. Secondly, we've constructed a chord diagram from the neighborhood of a critical level. Also the minimum number of critical points of such functions is being considered. And finally, the criterion of global topological equivalence of functions which belong to Ω(M) and have three critical points has been developed. This paper partially based on the talks of the first author given at the AUI’s seminars on Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold (March 2–15, 2017, AUI, Vienna, Austria) and partially supported by the project between the Austrian Academy of Sciences and the National Academy of Sciences of Ukraine on Modern Problems in Noncommutative Astroparticle Physics and Categorian Quantum Theory. 2017 Article Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold / B.I. Hladysh, A.O. Prishlyak // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 57R45; 57R70 DOI:10.3842/SIGMA.2017.050 https://nasplib.isofts.kiev.ua/handle/123456789/148582 en Symmetry, Integrability and Geometry: Methods and Applications application/pdf Інститут математики НАН України |
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This paper focuses on the problem of topological equivalence of functions with isolated critical points on the boundary of a compact surface M which are also isolated critical points of their restrictions to the boundary. This class of functions we denote by Ω(M). Firstly, we've obtained the topological classification of above-mentioned functions in some neighborhood of their critical points. Secondly, we've constructed a chord diagram from the neighborhood of a critical level. Also the minimum number of critical points of such functions is being considered. And finally, the criterion of global topological equivalence of functions which belong to Ω(M) and have three critical points has been developed. |
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Hladysh, B.I. Prishlyak, A.O. Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold Symmetry, Integrability and Geometry: Methods and Applications |
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Hladysh, B.I. Prishlyak, A.O. |
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Hladysh, B.I. |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold |
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topology of functions with isolated critical points on the boundary of a 2-dimensional manifold |
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Topology of Functions with Isolated Critical Points on the Boundary of a 2-Dimensional Manifold / B.I. Hladysh, A.O. Prishlyak // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 19 назв. — англ. |
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Symmetry, Integrability and Geometry: Methods and Applications |
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AT hladyshbi topologyoffunctionswithisolatedcriticalpointsontheboundaryofa2dimensionalmanifold AT prishlyakao topologyoffunctionswithisolatedcriticalpointsontheboundaryofa2dimensionalmanifold |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 050, 17 pages
Topology of Functions with Isolated Critical Points
on the Boundary of a 2-Dimensional Manifold
Bohdana I. HLADYSH and Aleksandr O. PRISHLYAK
Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv,
4-e Akademika Glushkova Ave., Kyiv, 03127, Ukraine
E-mail: bohdanahladysh@gmail.com, prishlyak@yahoo.com
Received November 18, 2016, in final form June 16, 2017; Published online July 01, 2017
https://doi.org/10.3842/SIGMA.2017.050
Abstract. This paper focuses on the problem of topological equivalence of functions with
isolated critical points on the boundary of a compact surface M which are also isolated cri-
tical points of their restrictions to the boundary. This class of functions we denote by Ω(M).
Firstly, we’ve obtained the topological classification of above-mentioned functions in some
neighborhood of their critical points. Secondly, we’ve constructed a chord diagram from
the neighborhood of a critical level. Also the minimum number of critical points of such
functions is being considered. And finally, the criterion of global topological equivalence of
functions which belong to Ω(M) and have three critical points has been developed.
Key words: topological classification; isolated boundary critical point; optimal function;
chord diagram
2010 Mathematics Subject Classification: 57R45; 57R70
1 Introduction
Topological classification of spaces with a defined structure on them is one of the main problems
of topology. The most fundamental works devoted to functions classification are papers by
G. Reeb [16], A.S. Kronrod [9], A.O. Prishlyak [14], V.V. Sharko [17], A.A. Kadubovskyi [7] and
I.A. Iurchuk [5]. A.S. Kronrod and G. Reeb have constructed the graph which allows to classify
simple functions on closed surfaces. O.V. Bolsinov and A.T. Fomenko [1] also investigate the
layer and layer equipped equivalence using the concept of atoms and f -atoms.
Currently, there are many papers which are devoted to exploration of topological properties
of functions on surfaces with the boundary, e.g., [4, 11, 12, 19]. In [4] we consider Morse
functions, whose critical points belong to the boundary and are non-degenerated critical points
of restriction of the function to the boundary of the manifold (called here mm-functions).
According to [15, Theorem 1.3] the function with isolated critical points on a closed surface
is locally topologically equivalent to the function f(x, y) = Re(x+ iy)k for some integer nonne-
gative k. In this paper we generalize this theorem up to the case when a critical point belongs
to the boundary of the surface.
Another kind of important problem is to find a minimum number of critical points of functions
defined on a fixed surface. This problem is considered in details in [4, 10]. We call the function,
having the minimum number of critical points as optimal. As well as in [4] the criterion of
optimality of a mm-function is proved.
The present paper examines the class of smooth functions with isolated critical points on
a manifold M belonging to the boundary and being also isolated critical points of their restric-
tions to ∂M . It is also focused on the problem of getting a local topological classification and
finding a criterion of optimality of previously described functions.
mailto:bohdanahladysh@gmail.com
mailto:prishlyak@yahoo.com
https://doi.org/10.3842/SIGMA.2017.050
2 B.I. Hladysh and A.O. Prishlyak
2 Local topological classif ication
Let M be a compact surface with the boundary ∂M and f : M → R be a smooth function on M
with a finite number of critical points all of which belong to ∂M . The finite number of critical
points is equivalent to their isolation due to the compactness of M .
We denote the restriction of function f to the boundary ∂M of the surface M by f |∂M and
the set of smooth functions defined on M , whose critical points are isolated, belong to the
boundary and are also isolated critical points of restriction to ∂M by Ω(M). Thus, we consider
the following class of functions
Ω(M) =
{
f : M → R | f ∈ C∞(M), CP(f) = ICP(f) = ICP(f |∂M )
}
,
where CP(f) (ICP(f)) is the set of (isolated) critical points of function f .
Lemma 2.1. Let y be a regular value of a smooth function f : M → R, where M is a compact
surface with the boundary. Then the following statements hold true:
(i) the level f−1(y) consists of a finite number of circles and line segments;
(ii) for arbitrary open set U with smooth boundary being transversal to f−1(y), the intersection
f−1(y) ∩ U has a finite number of components.
Proof. Firstly, according to [13, Lemma 1], the preimage of a regular level is 1-dimensional
manifold. Thus, it can not include the isolated points. Also the limit of critical points is
a critical point. It follows from the continuity of partial derivatives and from the definition of
critical points.
Let y be a regular value of f . Firstly, we prove that level f−1(y) has a finite number of line
segments. Let us suppose that it doesn’t hold. Then there exists the component of the boundary
which intersects the level line of f in the infinite number of points. Whereas the function acquires
the same values, then (according to Rollya’s theorem) there exists the critical value of function
restriction to the boundary between every two such points. Thus, the restriction f |∂M has the
infinite number of critical points. It means that the sequence of critical points has the limit
point belonging to level f−1(y). The last statement contradicts to the condition as y is regular
value of f .
In the next part of this proof we are going to show that the level line f−1(y) contains a finite
number of the circles. In order to do this we glue all components of the boundary ∂M by 2-di-
mensional disks D2. Thus, we get a closed surface M ′. Let us continue the function f on these
disks to smooth function F with a finite number of critical points on each disk (we can do it by
arbitrary function and after that approximate it by Morse function). In such a way we get the
surface M ′ and the function F defined on M ′, whose critical points are also isolated. Then the
preimage F−1(y) is compact, because of the compactness of the surface M ′ and the closedness
of the set F−1(y). In what follows F−1(y) includes the finite number of circles. As a result, the
level line f−1(y) also includes the finite number of the circles for the initial surface M , because
the number of the circles can only decrease after rejection of glued disks. �
The function with isolated critical point, being not local extreme, on a closed surface is locally
topologically equivalent with the following function: f(x, y) = Re(x + iy)k for some integer k,
k ≥ 1 [15]. Therefore, firstly, we consider the level lines of function f(x, y) = Re(x+iy)k, defined
on a surface R2
+ = {(x, y) ∈ R2 | y ≥ 0} for k ∈ {1, 2, 3, 4} and in a general case. Also note
that p0 = (0, 0) is an isolated critical point of the function Re(x+ iy)k and 0 is a correspondent
critical value.
We say that a function f has a local topological presentation f(x, y) = Re(x+ iy)k, y ≥ 0, if
it is locally topologically equivalent to the function Re(x+ iy)k, y ≥ 0.
Topology of Functions with Isolated Critical Points on the Boundary 3
Figure 1.
If k = 1, then f has a local topological presentation f(x, y) = x, y ≥ 0. The level lines of f
are shown in Fig. 1.1. Evidently the level f(x, y) = 0 is the ray x = 0, y ≥ 0.
In case k = 2 we have f(x, y) = Re(x + iy)2 = x2 − y2, y ≥ 0 and level f(x, y) = 0 consists
of two rays y = x, y ≥ 0 and y = −x, y ≥ 0, see Fig. 1.2.
When k = 3, the function has a local presentation f(x, y) = Re(x+ iy)3 = x3 − 3xy2, y ≥ 0.
Its level lines are shown in Fig. 1.3. The critical level f(x, y) = 0 consists now of three rays
x = 0, y ≥ 0, y = x√
3
, y ≥ 0, and y = − x√
3
, y ≥ 0.
If k = 4, then f(x, y) = Re(x + iy)4 = x4 − 4x2y2 + y4, y ≥ 0 and its level set f(x, y) = 0
consists of four rays y = x
√
2 +
√
3, y ≥ 0, y = x
√
2−
√
3, y ≥ 0, y = −x
√
2 +
√
3, y ≥ 0, and
y = −x
√
2−
√
3, y ≥ 0, see Fig. 1.4.
Thus, in all the cases k ∈ {1, 2, 3, 4} the line y = 0 is the axis of symmetry and it intersects
with level f−1(0) of function f at only one point p0. Further we generalize this result to an odd
and even k.
If k = 2n+ 1 (for some integer n, n ≥ 1), then function f has the following local topological
presentation:
f(x, y) = Re(x+ iy)2n+1 =
n∑
j=0
(−1)jC2j
2n+1x
2n+1−2jy2j , y ≥ 0.
Finally, in case k = 2n (for some integer n, n ≥ 1) we get the function:
f(x, y) = Re(x+ iy)2n =
n∑
j=0
(−1)jC2j
2nx
2n−2jy2j , y ≥ 0.
In both cases the line y = 0 is the axis of symmetry, because if a point (x0, y0) belongs to
the level f−1(0), then the point (x0,−y0) also belongs to this level.
Lemma 2.2. Let p0 ∈ ∂M be a critical point of a function f ∈ Ω(M) with the correspondent
critical value f(p0) = 0. Then there exists a neighborhood U(p0) and a homeomorphism:
h : f−1(0) ∩ cl(U(p0))→ Con
(
k⋃
i=1
{xi}
)
for a finite set of points {xi, i = 1, k} and some k ∈ Z, k ≥ 1.
Here cl(U(p0)) is a closure of neighborhood U(p0), Con
(⋃k
i=1{xi}
)
is a union of k direct
lines having a single common point, being their common end (see Fig. 2).
Proof. We consider a neighborhood U of the point p0, that does not contain critical points of
function f with the exception of p0. Then the set K := (f−1(0)\{p0}) ∩ U does not contain
4 B.I. Hladysh and A.O. Prishlyak
Figure 2.
critical points of f and it is a 1-dimensional manifold. If K includes a closed curve, then within
this curve the function f will have a point of local extremum, which is impossible. In the same
way you can make sure that component K can not form the loop with the vertex p0.
Let us show that K has a finite number of components. Suppose there are infinitely many
such components. One can always choose a neighborhood U , that has smooth and transversal
to each component of K boundary. Then the set K ∩ ∂U has a limit point belonging to the
set K. The partial derivatives of f by the directions ∂U and K equal zero at this limit point.
It means that this point is critical. The last sentence contradicts to the conditions. Thus, we
proved the finiteness of components of the set K.
Afterwards, if it is necessary, again reduce the neighborhood U to one, that does not include
that components of K for which p0 is not a limit point. Then, the rest of components with
the point p0 form the union of k direct lines, which have a single point p0 being their common
end. �
Definition 2.3. Two given smooth functions f and g, defined on some surfaces M and N
respectively, are said to be layer equivalent if there exists a homeomorphism λ : M → N , which
maps the components of the level sets of f onto the components of the level sets of g.
Definition 2.4. Two given smooth functions f and g are said to be layer equipped equivalent in
some neighborhoods of their critical levels f−1(c1) and g−1(c2) if there exist ε1 > 0, ε2 > 0 and
a homeomorphism λ : f−1(c1 − ε1, c1 + ε1)→ g−1(c2 − ε2, c2 + ε2), which maps the components
of the level sets of f onto the components of the level sets of g and λ preserve the growing
directions of functions.
Definition 2.5. Two given smooth functions f and g, defined on some surfaces M and N
respectively, are said to be topologically equivalent if there exist homeomorphisms h1 : M → N ,
h2 : R→ R, such that h2 ◦ f = g ◦ h1 and h2 preserves the orientation of R.
Theorem 2.6. Function f ∈ Ω(M) is topologically equivalent to the function g(x, y) = x2 + y2,
y ≥ 0 (g(x, y) = −x2− y2, y ≥ 0) in some neighborhood of its local minimum (maximum) point.
Proof. Theorem statement has local nature, so we consider some small enough neighborhood
of a local minimum (maximum) point and all consideration will take place in this neighborhood.
Every level line of a function f ∈ Ω(M) is transversal to the boundary ∂M , because the
restriction of this function has no critical point with the exception of minimum (maximum)
point. We construct a grad-like vector field X, being tangent to the boundary. Let f takes
value c at p0. Consider the homeomorphism h which maps the level f−1(c+ ε) (for some ε > 0)
into level x2 + y2 = ε, y ≥ 0 of g.
Let us denote the trajectory of field X which passes though a point x by γ(x) and the
trajectory of field X which passes though a point y by β(y). Then, desired homeomorphism of
neighborhood is defined by the formula: H(x) = β(h(γ(x) ∩ f−1(c+ ε))) ∩ g−1(ε). �
Notice that a critical point is a saddle critical point if it is not the point of local minimum
or local maximum.
Topology of Functions with Isolated Critical Points on the Boundary 5
Figure 3.
Theorem 2.7. Let f ∈ Ω(M), p0 be a saddle critical point of function f . Then there exists
a neighborhood U(p0) of p0, such that the restriction f |U(p0) is topologically equivalent to the
function g(x, y) = Re(x + iy)k, y ≥ 0 which is defined in some neighborhood of (0, 0) for some
integer k ≥ 1.
Proof. Let U(p0) be the neighborhood with the properties described in Lemma 2.2. Then
critical level of f divides U(p0) into regions. We consider one of these regions and denote it
by V . Let us suppose that f(x) < 0 on V . Then a vector field grad(f) is directed inside the
region V at the points of the intersection ∂U(p0) ∩ ∂V .
We denote the trajectories of gradient vector field, passing through point x by φx. Let us
consider a map h : ∂U(p0)∩∂V → f−1(0)∩∂V being defined by the condition h(x) = φx∩f−1(0)
if φx ∩ f−1(0) 6= ∅ and h(x) = lim
t→∞
φx(t) = p0 if φx ∩ f−1(0) = ∅ (see Fig. 3). The map h
is continuous, because of the continuous dependence of the solution of differential equation on
initial condition. Then there exists a point being mapped into the point p0 and it means that
there exists the trajectory γ passing through ∂U(p0) ∩ ∂V and finishing at the point p0. If V
has common points with the boundary, then we change the field grad(f) in the neighborhood of
the boundary into a such one, which is tangent to the boundary in some neighborhood. Then,
the trajectory γ contains into the boundary.
The set K with such constructed trajectories γ divides the neighborhood U(p0) into regions.
Similarly to Theorem 2.6, we can construct a homeomorphism of each of these sectors (if it
is necessary, after some reduction of U(p0)) into correspondent sectors of function g(x, y) =
Re(x + iy)k, y ≥ 0 which maps the level lines into level lines. Also if critical point has only
one trajectory which enter in it, then the trajectories are mapped into trajectories by this
homeomorphism.
Homeomorphisms mentioned above are the same at the common boundary, that is why they
determine the searching one. �
Note that in case k = 1 the neighborhood U(p0) has two sectors having common points with
the boundary with the exception of p0, see Fig. 1.1.
3 Atoms of the function
Definition 3.1. A function which has at most one critical point at each level line will be called
simple.
Definition 3.2. An atom is a class of layer equivalence of function f restriction to the set
f−1([c− ε, c+ ε]), where c is a critical value of f , for small enough ε, such that the line segment
[c− ε, c+ ε] does not include critical values with the exception of c.
6 B.I. Hladysh and A.O. Prishlyak
Figure 4.
Figure 5.
Definition 3.3. A f -atom is a class of layer equipped equivalence of restriction of function f
to the set f−1([c− ε, c+ ε]), where c is critical value of f , for small enough ε as above.
Remark that every atom has corespondent two f-atoms, which can be obtained one from
another by changing the sing of the function. In what follows we consider only simple functions
and suppose that f(p0) = 0, where p0 is an isolated saddle critical point of functions f and f |∂M .
Let us consider the neighborhood of a critical point p0 bounded by f−1(−ε), f−1(ε) for some
small enough ε > 0, by some trajectories of a gradient field and by the boundary ∂M . The
parts of the surface where f > 0 and f < 0 will be called the positive and negative sectors of
function f . We depict these sectors by shaded and unshaded ones. The obtained surface has
the structure of (2k + 2)-gon. If we extend this neighborhood to the neighborhood of a critical
level, we get the neighborhood which is homeomorphic to a polygon with glued sides by linear
homeomorphism (e.g., in Fig. 4 the side CB is glued with the side DE).
Thus, atom has the structure of (2k + 2)-gon, which is presented in Fig. 5.
We put a circle with matched points corresponding to the previously described polygon. This
circle is the boundary of (2k+ 2)-gon and matched points are the points on the circle belonging
to the intersection of shaded and unshaded sectors (in other words, matched points belong to
the critical level).
We connect two matched points by a chord if and only if correspondent sides of polygon
become glued after extending of critical level neighborhood. In what follows we get the circle
with the matched points.
Further fix the orientation on the boundary to numerate the matched points on the circle.
If we change the orientation, we get the equivalent atom. Then we numerate matched points in
the following way: a point corresponding to a critical one p0 we denote by Q0, and the rest of
points we numerate according to the orientation of the boundary beginning with Q1 up to Qk
and point Q0 we consider as the point of reference. These points divide the circle into k + 1
black (thick) and grey (thin) arcs. These arcs correspond to positive and negative sections (see
Fig. 6).
Topology of Functions with Isolated Critical Points on the Boundary 7
Figure 6.
Then, every atom can be defined by the circle with k + 1 matched points and l chords (for
some l ∈ {0, 1, 2, . . . , [k2 ]}). Also one matched point corresponds to a critical point.
Definition 3.4. A chord diagram of a saddle critical level of the function defined on a smooth
compact surface with the boundary is the circle with the following elements:
(1) matched points, which are enumerated;
(2) chords, the ends of which are different matched points;
(3) coloration of arcs such that each two neighbor arcs with the exception of arcs near point
Q0, are of different colors.
Note that chord diagram defines f -atom and if we consider only elements (1) and (2) then
chord diagram defines atom.
Definition 3.5. Two chord diagrams are called equivalent if they can be obtained one from
another by turn or symmetry preserving the elements (1)–(3).
Definition 3.6. A free matched point on chord diagram is the one which is not connected with
another matched points by chord.
Chord diagrams are also considered in papers [7, 8, 18].
The circle of a chord diagram we denote by S1, matched points by 0, 1, . . . , k, and chord
which connects points i and j by lij , i, j ∈ {1, . . . , k}. We will say that j is a number of
a matched point Qj . Free points with the exception of Q0 we denote by Qi∗ , where i is the
number of correspondent matched points. Each matched point of chord diagram corresponds
to two vertices of the (previously described) (2k + 2)-gon and one of these points belongs to
a positive sector (f > 0) and another one to a negative sector (f < 0). That is why we denote
these points by Pi (positive) and Ni (negative), where i is the number of matched point Qi.
Lemma 3.7. A number of free points Nf.p can be calculated from the formula
Nf.p = k − 2 ·Nch + 1,
where Nch is a number of chords of chord diagram.
Proof. There exists a neighborhood of critical level which can be represented in the form of
polygon with (2k + 2) vertices and 2l glued sides (for some l). Then, the chord diagram of this
critical level includes k+ 1 matched points and l chords. Thus, after the renotation Nch := l we
get the formula of calculation of free points number. �
8 B.I. Hladysh and A.O. Prishlyak
In what follows we construct a substitution from the chord diagram. The substitution includes
the cycle (ij) (for some i, j ∈ {1, 2, . . . , k}) if and only if matched points Qi and Qj are connected
by a chord. This substitution includes only the cycles with 2 elements.
The substitutions constructed in previously described way will be called gluing substitution
and the atom with its gluing substitution τ (k) on the set {1, 2, . . . , k} will be denoted Aτ (k) .
Theorem 3.8. The following statements hold true:
1) each atom of saddle critical level coincides with atom Aτ (k) for some gluing substitution
τ (k) on the set {1, 2, . . . , k} and this substitution defines the gluing of atom sides;
2) a number Nk of atoms Aτ (k) can be calculated by the formula:
N1 = 1, N2 = 2, N3 = 4, Nk = 2
k−3∑
j=1
P
(k)
j + P
(k)
k−2,
where for each k the set of numbers P
(k)
j defined by a recurrent correlation
P
(k)
0 = 1, P
(k)
1 = k − 1, P
(k)
2 = k − 2,
P
(k)
j =
(
P
(k)
0 + P
(k)
1 + · · ·+ P
(k)
j−2
)
(k − j), j ∈ Z, j ≥ 3.
Proof. 1) Let us consider a neighborhood of a saddle critical level correspondent to critical
point p0, being presented in the form of polygon with (2k + 2) vertices with glued (or unglued)
sides. We construct a critical level chord diagram, using this polygon. Thus each possible gluing
of sides of the polygon corresponds to some substitution τ , defined on the set {1, 2, . . . , k}. Also
it should be mentioned that if there is no side being connected with another one we get the
trivial substitution.
2) A number of atoms is equal to a number of non-equivalent chord diagram. That is why
the number of non-equivalent chord diagram with k + 1 matched points and “free” number of
chords (from 0 to [k2 ]) we also denote by Nk.
Hence, we can connect point Q1 with another one by a chord or not. The first action can be
made by k− 1 ways and the number of atoms is equal to Nk−2. In the second case we get k− 1
matched points and the number of possible gluings equals Nk−1. Thus, we have the recurrent
formula: Nk = Nk−1 + (k − 1)Nk−2, g ≥ 3.
Let us use the same arguments again and again to see the regularity:
Nk = Nk−1 + (k − 1)Nk = (1 + k − 1)Nk−2 + (k − 2)Nk−3
= (1 + k − 1 + k − 2)Nk−3 + (1 + k − 1)(k − 3)Nk−4
= (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3))Nk−4 + (1 + k − 1 + k − 2)(k − 4)Nk−5
= (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3) + (1 + k − 1 + k − 2)(k − 4))Nk−5
+ (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3))(k − 5)Nk−6 = · · ·
= ((1 + k − 1 + k − 2 + (1 + k − 1)(k − 3) + (1 + k − 1 + k − 2)(k − 4)
+ (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3))(k − 5) + · · ·
+ (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3) + · · ·
+ (1 + k − 1 + k − 2 + · · ·+ (k − (k − 2) + 2)(k − (k − 2) + 1)))Nk−(k−2)
+ ((1 + k − 1 + k − 2 + (1 + k − 1)(k − 3) + (1 + k − 1 + k − 2)(k − 4)
+ (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3))(k − 5) + · · ·
+ (1 + k − 1 + k − 2 + (1 + k − 1)(k − 3) + · · ·
Topology of Functions with Isolated Critical Points on the Boundary 9
+ (1 + k − 1 + k − 2 + · · ·+ (k − (k − 1) + 3))((k − (k − 1) + 1)))Nk−(k−1).
Consider the notations
P
(k)
0 = 1, P
(k)
1 = k − 1, P
(k)
2 = k − 2,
P
(k)
j =
(
P
(k)
0 + P
(k)
1 + · · ·+ P
(k)
j−2
)
(k − j)
then the previous sequence of equalities can be rewritten in the form
Nk = P
(k)
0 Nk−1 + P
(k)
1 Nk−1 =
(
P
(k)
0 + P
(k)
1
)
Nk−2 + P
(k)
0 P
(k)
2 Nk−3
=
(
P
(k)
0 + P
(k)
1 + P
(k)
2
)
Nk−3 + P
(k)
3 Nk−4
=
(
P
(k)
0 + P
(k)
1 + P
(k)
2 + P
(k)
3
)
Nk−4 + P
(k)
4 Nk−5
=
(
P
(k)
0 + P
(k)
1 + P
(k)
2 + P
(k)
3 + P
(k)
4
)
Nk−5 + P
(k)
5 Nk−6 = · · ·
=
(
P
(k)
0 + P
(k)
1 + · · ·+ P
(k)
k−2−1
)
Nk−(k−2) + P
(k)
k−1−1Nk−(k−1)
= N2
k−3∑
j=0
P
(k)
j +N1P
(k)
k−2 = 2
k−3∑
j=0
P
(k)
j + P
(k)
k−2. �
According to the formula from Theorem 3.8 we can calculate the number of atoms Aτ (k) . The
results of such calculations for k ≤ 20 are presented in Table 1.
Table 1.
k Nk k Nk k Nk k Nk
1 1 6 76 11 35696 16 46206736
2 2 7 232 12 140152 17 211799312
3 4 8 764 13 568504 18 997313824
4 10 9 2620 14 2390480 19 4809701440
5 26 10 9496 15 10349536 20 23758664096
4 Optimal functions
Further we consider a smooth surface M with a component of the boundary ∂M and a simple
smooth function f ∈ Ω(M).
Definition 4.1. We call a function f ∈ Ω(M) optimal on the surface M if it has the minimum
possible number of critical points on M among all functions from Ω(M).
4.1 Optimality criterion of a function
Theorem 4.2. Let f ∈ Ω(M) and M be a connected surface with the connected boundary, being
not homeomorphic to a 2-dimensional disk. Then the function f is optimal if and only if it has
exactly three critical points.
Proof. Firstly, remark that a function with three critical points has one minimum point, one
maximum point and the third one is a saddle critical point.
Necessity. To prove the theorem we show will the next statements: 1) existence of a smooth
function, that has three critical points on the surface M , being not 2-dimensional disk; 2) if
a function has two critical points on a surface N , where N is a 2-dimensional disk. Then from
10 B.I. Hladysh and A.O. Prishlyak
Figure 7.
1) follows that optimal function has no more than 3 critical points. It means that an optimal
function has exactly three critical points on the surface with boundary with the exception of
2-dimensional disk.
1) Firstly, we consider the case of oriented surface. Let M be an oriented surface by genus g
with one component of the boundary, which is obtained by gluing of atom Aτ (4g+1) with the
substitution
(1, 4g)(2, 4g + 1)(3, 4g − 2)(4, 4g − 1) · · · (2i− 1, 4g − 2i+ 2)
× (2i, 4g − 2i+ 3) · · · (2g − 1, 2g + 2)(2g, 2g + 3).
The surface M can be defined by a chord diagram shown in Fig. 7.1.
On the other hand, the atom of critical level of function Re(x+ iy)4g+1, defined on some non-
oriented surface by genus 2g+1, can be represented in the form of (4g+2)-gon and correspondent
chord diagram coincides with already described (see Fig. 7.2) chord diagram.
Let us consider half-disks
D2
+ =
{
(x, y) ∈ R2 |x2 + y2 ≤ 1, y ≥ 0
}
, D2
− =
{
(x, y) ∈ R2 |x2 + y2 ≤ 1, y ≥ 0
}
,
homeomorphisms
ĥ1,2 : [0, 4g + 1]→ ∂±D
2
±,
where
∂+D
2
+ =
{
(x, y) ∈ ∂D2
+ | y > 0
}
, ∂−D
2
− =
{
(x, y) ∈ ∂D2
− | y < 0},
which map
ĥ1,2(j) =
(
−1 +
j
4g + 1
,±
√
1− (−1 +
j
4g + 1
)2
)
, j = 0, 4g + 1,
and embeddings
h1,2 : [0, 4g + 1]→ ∂M,
Topology of Functions with Isolated Critical Points on the Boundary 11
such that
h1(0) = P0, h1(1) = P1, h1(2) = P4g, h1(3) = P4g+1, . . . ,
h1(4g) = P2g, h1(4g + 1) = P2g+1, h2(0) = N0, h2(1) = N4g+1,
h2(2) = N2, h2(3) = N1, . . . , h2(4g) = N2g+2, h2(4g + 1) = N2g+1.
Then we clue the obtained atom by half-disks D2
± ∪ [−1, 1]× {0} according to the map
ĥ1,2 ◦ h1,2 : D2
± → ∂M.
After this gluing we get a surface with one component of the boundary. Also function
Re(x + iy)4g+1 can be continued to the half-disks by its values on the boundary, being used
to clue, and, if it is necessary, we can smooth the function (similarly, for example, to work [3,
Section 5]).
A non-oriented surface and a smooth function on it with three isolated critical points can be
obtained by using previous considerations with the atom Aτ (4g+3) and the substitution
(1, 4g + 2)(2, 4g + 3)(3, 4g)(4, 4g + 1) · · · (2i− 1, 4g − 2i+ 4)(2i, 4g − 2i+ 5) · · ·
× (2g − 1, 2g + 4)(2g, 2g + 5)(2g + 1, 2g + 3)
(see Fig. 6.2), which coincide with the atom of saddle critical level of function Re(x + iy)4g+3,
and gluing by half-disks D2
± according to the embeddings
ĥ1,2 ◦ h1,2 : D2
± → ∂M,
where
ĥ1,2 : [0, 4g + 1]→ D2
±,
ĥ1,2(j) =
(
−1 +
j
4g + 3
,±
√
1− (−1 +
j
4g + 3
)2
)
, j = 0, 4g + 3,
and
h1,2 : [0, 4g + 3]→ ∂M, h1(0) = P0, h1(1) = P1, h1(2) = P4g+2,
h1(3) = P4g+3, . . . , h1(4g + 1) = P2g+1, h1(4g + 2) = P2g+3,
h1(4g + 3) = P2g+2, h2(0) = N0, h2(1) = N4g+3, h2(2) = N2,
h2(3) = N1, . . . , h2(4g + 2) = N2g+1, h2(4g + 3) = N2g+2.
Also it should be mentioned that the atom Aτ (4g+3) contains the twisted rectangle which corre-
sponds to chord (2g + 1, 2g + 3).
Thus, we constructed the smooth function with three isolated critical points on the boundary
and also are critical points of function restriction to the boundary of the surface.
2) Let function f has two critical points on a surface N . Then we consider a gradient-like
vector field for f , being tangent toN (similar to [2]). After this we consider the function mapping
levels of function f on N into line segments y = const on D2, and trajectories of gradient-
like vector field are mapped into curves γt = {c cos t, t(1 − c) + sin t | c ∈ [0, 1]}, t ∈ [t−c , t
+
c ],
where t−c , t+c are the smallest modulo roots of equations t(1−c)+sin t = −1 and t(1−c)+sin t = 1
accordingly. This function defines the homeomorphism between N and 2-dimensional disk D2,
because only one level line and one trajectory path though each point of D2.
Sufficiency. Suppose that it doesn’t hold. Then there exists a function, having exactly three
critical points on a defined surface but being not optimal. It means that optimal function on
this surface has two critical points (because a smooth function on every compact surface has, at
least, two critical points). Thus, this surface is a 2-dimensional disk because of the item 2) in
necessity part of the proof. In such a way we get the contradiction. The theorem is proved. �
12 B.I. Hladysh and A.O. Prishlyak
Note that in case of 2-dimensional disk an optimal function has two critical point and can be
realized by a height function.
Further we suppose that optimal function has critical values equal −1, 0, 1 (we can do it,
because there exists a homeomorphism of straight line, mapping three critical values of optimal
function into points −1, 0, 1).
4.2 The case of oriented surface with one component of the boundary
Definition 4.3. A homeomorphism h : [0, k] → S1 ∪ Int{lmn |m,n ∈ {1, . . . , k}} will be called
a full way between free points Q0 and Qi∗ (for some i ∈ 1, k) if it satisfies the conditions:
1) h(0) = Q0, h(k) = Qi∗ ; 2) ∀ t ∈ {1, 2, . . . , k − 1}: h(t) ∈ {Q1, Q2, . . . , Qi∗−1, Qi∗+1, . . . , Qk};
3) ∀ j ∈ {1, 2, . . . , k − 1} ∀ t ∈ (j, j + 1): f(t) belongs either to the interior of arc, or to the
interior of chord; 4) the direction can not be changed during the moving on fixed chord diagram.
Theorem 4.4. A chord diagram of saddle critical level of optimal function on oriented surface
with one component of the boundary satisfies the following conditions:
1) every chord divides the circle into two arcs, each of which contains an even number of
matched points;
2) the chord diagram has k+1 = 4n+2 matched points (for some integer n, n ≥ 1) and there
exist exactly two free points, one of which is Q0;
3) there exist two full ways between free points.
Proof. 1) The first item means that we can connect only such matched points of chord diagram
which have odd and even numbers. It follows from the orientation of the surface, because in
another case corresponding atom includes the Möbius strip.
2) According to the previous notations, the number of marked points is equal to k + 1 and
the number of chords is equal to l. Let us show that: a) k is odd (⇔ k + 1 is even); b) the
chord diagram includes two free points; c) l = 2n for some integer nonnegative n. Then from b)
and c) follows that k ≡ 1 (mod 4) (it holds if and only if k + 1 = 4n+ 2).
a) The function f changes the sign when it passes thought critical point because f(p0) = 0.
It means that arcs, being adjacent with point Q0, are of deferent color. Thus, the number of
matched points is even, because the color of arcs alternates by passing the circle S1 of chord
diagram.
b) Suppose that it doesn’t hold. Then the chord diagram has less than 2n chords and, at
least, 3 free points. It means that appropriate 3 sides of atom belong to the boundary because
after extend of the neighborhood of critical level these 3 sides don’t clue with another sides.
That is why, at least, 4 half-disks D2
± become glued to the surface with further increase of the
neighborhood. The last sentence infers that this function has at least 5 critical points and as
a result isn’t optimal.
Thus, chord diagram of optimal function on the oriented surface with one component of the
boundary has 2 free points, one of which Q0 corresponds to a critical point.
c) Gluing of each pair of matched points (with odd and even number) is equivalent to attaching
of 1-handle (rectangle). Also this gluing increases the number of components of the boundary
by 1 (if 1-handle is glued to one component of the boundary) or decreases by 1 (if 1-handle
is glued to two components of the boundary) and at the same time the genus of the surface
becomes increased after each gluing.
We start from a half-disk D2
+ = {(x, y) ∈ R2 |x2 + y2 ≤ 1, y ≥ 0} (without glued 1-handles),
having one component of the boundary and every glued 1-handle changes the number of com-
ponents of the boundary by 1. It means that we can get the surface with one component of the
boundary only in case of even number of gluings.
Topology of Functions with Isolated Critical Points on the Boundary 13
3) Let us suppose that there does not exist a full way on a defined chord diagram. Then we
get, at least, two components of the boundary of the surface, which contradicts to the conditions
of the theorem. �
Corollary 4.5. A gluing substitution doesn’t include the cycle (i, j + 1)(i+ 1, j) for all possible
i, j ∈ {0, k}.
Proof. If the gluing substitution includes the cycle (i, j + 1)(i+ 1, j) for some i, j, then chord
diagram doesn’t have a full way, because every path from point Q0 to arbitrary point Qp,
p ∈ {0, k}\{i, i + 1, j, j + 1} doesn’t pass through the arcs î, i+ 1, ̂j, j + 1 and chords li,j+1,
li+1,j . �
Corollary 4.6. Every free point of a chord diagram of saddle critical level, except of Q0, has
an odd number.
Proof. From 2) follows that, with the exception of marked point Q0, we have matched points
Q1, . . . , Q4n+1, among which exactly 2n have an even number and 2n+ 1 have an odd number.
Then another free point has an odd number (the second item of Theorem 4.4). �
Theorem 4.7 (criterion of topological equivalence). Optimal functions are topologically equi-
valent if and only if their chord diagrams of saddle critical levels are equivalent.
Proof. Necessity. The topological equivalence of optimal functions induces the equivalence of
chord diagrams, because of the construction of chord diagrams. Then the equivalence of their
chord diagram, what follows from topological equivalence of optimal functions.
Sufficiency. Suppose that chord diagrams of optimal functions f and h are equivalent. In
what follows the existence of homeomorphisms from level lines of functions f and h to level
lines of function Re(x+ iy)k (Theorem 4.2). Let ϕ and ψ be these homeomorphisms. Then the
homeomorphism from some neighborhood of a saddle critical point of function f to correspondent
neighborhood of h can be defined by the formula ϕ−1 ◦ ψ.
Let us consider the atom of critical level without a critical point. Then a chord or a free point
(with the exception of Q0) conforms to each connected component. The equivalence of chord
diagrams defines the bijection between these level components, that is why, we can continue the
homeomorphism between neighborhoods of critical point of functions f and g to homeomorphism
of neighborhood of critical level of these functions.
Thus, we continue this homeomorphism to the chords. And we have the point either on chord
or on the boundary of (8n+4)-gon for each trajectory of gradient-like vector field. It means that
we get the bijection between these trajectories. Thus, the homeomorphism between appropriate
trajectories can be constructed and it satisfies the condition that the points with one and the
same values map one into another (in other words, this homeomorphism preserves the value of
the function). We obtain the homeomorphism, because on every trajectory there exits a single
point with a fixed value from (0, 1). These homeomorphisms define the homeomorphism of the
surface and it is a topological equivalence. �
Theorem 4.8 (realization). If a chord diagram satisfies the conditions 1)–3) of Theorem 4.4,
then there exists an optimal function, chord diagram of saddle critical level of which coincides
with the first one.
Proof. A chord diagram can be defined by a number k and by a gluing substitution. That
is why, we consider the function f(x, y) = Re(x + iy)4n+1, y ≥ 0, its isolated critical point
p0 = (0, 0) and ε-neighborhood of critical level f(p0) = 0 (for some ε > 0), which is presented in
the form of (4n + 2)-gon with colored sections (see Fig. 5.1). Let us suppose that sides of this
14 B.I. Hladysh and A.O. Prishlyak
Figure 8.
polygon are line segments and have the length which equals 2ε. Then enumerate the sides of
this polygon from S0 up to S4n+1 as it is shown in Fig. 8.
After this we glue the rectangle [0, 1] × [−ε, ε] to sides Si and Sj (according to the gluing
substitution) by the following image: if gluing substitution includes the cycle (i, j), then
p : {0, 1} × [−ε, ε]→ {Si, Sj}, p((0,−ε)) = Q2i, p((0, ε)) = Q2i−1,
p((1,−ε)) = Q2j−1, p((1,−ε)) = Q2j−1, p((1, ε)) = Q2j .
Define the function f(x, y) on glued rectangle by its second coordinate f(x, y) = y. As a result,
we get an atom of saddle critical level. Whereafter we glue this atom by half-disks
D+ =
{
(x, y) ∈ R2 |x2 + (y − ε)2 ≤ (1− ε)2, y ≥ ε
}
,
D− =
{
(x, y) ∈ R2 |x2 + (y + ε)2 ≤ (1− ε)2, y ≤ −ε
}
,
as it was done in Theorem 4.2 in oriented case and continue function f on this half-disks by the
formula f(x, y) = y. If it is necessary we can smooth the obtained function (similar to work [3]).
Thus, we get the surface and the smooth function which is defined on it, such that chord diagram
of saddle critical level of this function coincides with the initial chord diagram. �
4.3 Examples of calculations
From Theorems 4.4, 4.7 and 4.8 follows that every chord diagram defines (up to topological
equivalence) an optimal function on oriented surface with one component of the boundary if
and only it satisfies the conditions 1)–3) of Theorem 4.4.
Using this properties of chord diagrams, there was calculated the number of optimal layer
(topological) non-equivalent functions defined on oriented surface with one component of the
boundary in cases:
1) surface by genus 1: 1 (1):
Topology of Functions with Isolated Critical Points on the Boundary 15
2) surface by genus 2: 5 (8):
3) surface by genus 3: 94 (182).
Also in case of oriented surface by genus 3 we used chord diagrams with 7 chords of closed
surface (chords 1–25), which are described in detail in paper [6].
We take into account possible symmetries and rotations of these chord diagrams and in such
a way we count the number of nonequivalent atoms and f -atoms of saddle critical level. We use
the abbreviation c.d. for chord diagrams.
16 B.I. Hladysh and A.O. Prishlyak
Thus, for example, we describe how many atoms we get from each c.d. 1–25: from c.d. 1 we
get 1 atom, from 2 – 7, 3 – 4, 4 – 1, from 5, 6, 7 and 8 we get 7 atoms, but 9 c.d. coincides with
the 6 one and 10 with the 1 c.d., from 11 we get 4 atoms, 12 c.d. coincides with the 4, 13 – with
the 8 and 14 – with the 5. We get 7 atoms from 15 c.d., but 16 c.d. coincides with the 15 c.d.
and 17 – with the 7. We get 1 atom from 18 c.d., from 19 – 4 atoms. The 20 c.d. coincides with
the 2 c.d., but 21 c.d. corresponds to 4 atoms, 22 – to 7 atoms, each c.d. 23 and 25 corresponds
to 8 atoms. The 24 c.d. coincides with the 21 c.d.
As a result we get 94 optimal functions of class Ω(M) up to layer equivalence. In the same
way you can make sure that there exist 182 functions from Ω(M) with three critical points up
to topological equivalence.
5 Conclusion
We’ve got the following results for the function with isolated critical point on the boundary
of the surface which are also isolated critical points of the restriction of this function to the
boundary of the surface:
• local topological presentation about saddle critical point and also about maximum and
minimum critical points;
• the recurrent formula of a number of saddle critical level atoms;
• criterion of optimality;
• criterion of topological equivalence of optimal functions in terms of chord diagrams in case
of oriented surface with one component of the boundary.
Problems of topological equivalence of optimal functions defined on surface with more than
one component of the boundary and on non-oriented surface are still open.
Acknowledgements
This paper partially based on the talks of the first author given at the AUI’s seminars on
Topology of functions with isolated critical points on the boundary of a 2-dimensional manifold
(March 2–15, 2017, AUI, Vienna, Austria) and partially supported by the project between the
Austrian Academy of Sciences and the National Academy of Sciences of Ukraine on Modern
Problems in Noncommutative Astroparticle Physics and Categorian Quantum Theory.
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https://doi.org/10.1016/S0012-365X(99)00347-7
1 Introduction
2 Local topological classification
3 Atoms of the function
4 Optimal functions
4.1 Optimality criterion of a function
4.2 The case of oriented surface with one component of the boundary
4.3 Examples of calculations
5 Conclusion
References
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