Functions and vector fields on C(CPⁿ) -singular manifolds
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irk-123456789-1653122020-02-14T01:26:08Z Functions and vector fields on C(CPⁿ) -singular manifolds Libardi Alice Kimie Miwa Sharko, V. Статті 2014 Article Functions and vector fields on C(CPⁿ) -singular manifolds / Libardi Alice Kimie Miwa, V. Sharko // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 311–315. — Бібліогр.: 7 назв. — англ. 1027-3190 http://dspace.nbuv.gov.ua/handle/123456789/165312 514.763.22 en Український математичний журнал Інститут математики НАН України |
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Статті Статті Libardi Alice Kimie Miwa Sharko, V. Functions and vector fields on C(CPⁿ) -singular manifolds Український математичний журнал |
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Libardi Alice Kimie Miwa Sharko, V. |
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Libardi Alice Kimie Miwa Sharko, V. |
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Libardi Alice Kimie Miwa |
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Functions and vector fields on C(CPⁿ) -singular manifolds |
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Functions and vector fields on C(CPⁿ) -singular manifolds |
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Functions and vector fields on C(CPⁿ) -singular manifolds |
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Functions and vector fields on C(CPⁿ) -singular manifolds |
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Functions and vector fields on C(CPⁿ) -singular manifolds |
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functions and vector fields on c(cpⁿ) -singular manifolds |
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Functions and vector fields on C(CPⁿ) -singular manifolds / Libardi Alice Kimie Miwa, V. Sharko // Український математичний журнал. — 2014. — Т. 66, № 3. — С. 311–315. — Бібліогр.: 7 назв. — англ. |
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Український математичний журнал |
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AT libardialicekimiemiwa functionsandvectorfieldsonccpnsingularmanifolds AT sharkov functionsandvectorfieldsonccpnsingularmanifolds |
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2025-07-14T18:18:57Z |
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UDC 514.763.22
Libardi Alice Kimie Miwa (I. G. C. E-Unesp-Univ. Estadual Paulista, Brazil),
Vladimir Sharko (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
FUNCTIONS AND VECTOR FIELDS ON C(CP n)-SINGULAR MANIFOLDS
ФУНКЦIЇ I ВЕКТОРНI ПОЛЯ НА C(CP n)-СИНГУЛЯРНИХ МНОГОВИДАХ
Let M2n+1 be a C(CPn)-singular manifold. We study functions and vector fields with isolated singularities on M2n+1. A
C(CPn)-singular manifold is obtained from a smooth manifoldM2n+1 with boundary which is a disjoint union of complex
projective spaces CPn ∪ CPn ∪ . . . ∪ CPn and subsequent capture of the cone over each component of the boundary.
Let M2n+1 be a compact C(CPn)-singular manifold with k singular points. The Euler characteristic of M2n+1 is equal
to χ(M2n+1) =
k(1− n)
2
. Let M2n+1 be a C(CPn)-singular manifold with singular points m1, . . . ,mk. Suppose that,
on M2n+1, there exists an almost smooth vector field V (x) with finite number of zeros m1, . . . ,mk, x1, . . . , xl. Then
χ(M2n+1) =
∑l
i=1
ind(xi) +
∑k
i=1
ind(mi).
Нехай M2n+1 — C(CPn)-сингулярний многовид. Ми вивчаємо функцiї i векторнi поля з iзольованими сингулярно-
стями на M2n+1. C(CPn)-сингулярний многовид виникає з гладкого многовиду M2n+1 з краєм, який є незв’язним
об’єднанням комплексного проективного простору CPn∪CPn∪ . . .∪CPn i послiдовностi конусiв над кожною ком-
понентою краю. Нехай M2n+1 — компактний C(CPn)-сингулярний многовид iз k сингулярними точками. Ейлерова
характеристика M2n+1 дорiвнює χ(M2n+1) =
k(1− n)
2
. Нехай M2n+1 — C(CPn)-сингулярний многовид iз син-
гулярними точками m1, . . . ,mk. Припустимо, що на M2n+1 iснує майже гладке векторне поле V (x) iз скiнченним
числом нулiв m1, . . . ,mk, x1, . . . , xl. Тодi χ(M2n+1) =
∑l
i=1
ind(xi) +
∑k
i=1
ind(mi).
1. The functions on C(CPn)-manifolds. Let M2n+2 be a closed smooth manifold with semifree
S1-action
θ : S1 ×M2n+2 →M2n+2
which has only isolated fixed points. It is known that every isolated fixed point m of a semifree
S1-action has the following important property: near such a point the action is equivalent to a certain
linear S1 = SO(2)-action on R2n+2. More precisely, for every isolated fixed point m there exist
an open invariant neighborhood U of m and a diffeomorphism h from U to an open unit disk
D2n+2 in Cn+1 centered at origin such that h is conjugate to the given S1-action on U to the S1-
action on Cn with weight (1, . . . , 1). We will use both complex, (z1, . . . , zn+1), and real coordinates
(x1, y1, . . . , xn+1, yn+1) on Cn = R2n+2 with zj = xj +
√
−1yj . The pair (U, h) will be called a
standard chart at the point m.
Let M2n+2 be a manifold with finite many fixed points m1, . . . ,m2k. Denote by
π : M2n+2 →M2n+2/S1
the canonical map. The set of orbits N2n+1 = M2n+2/S1 is a manifold with singular points
π(m1), . . . , π(m2k). It is clear that a neighborhood of any singular point is a cone over CPn.
In general, a C(CPn)-singular manifold is obtained from a smooth manifold M2n+1 with bound-
ary which is a disjoint union of complex projective spaces CPn ∪CPn ∪ . . .∪CPn and subsequent
capture of the cone over each component of the boundary. For this type of C(CPn)-singular manifold
parity of the number of singular points depends on parity of the number n.
c© LIBARDI ALICE KIMIE MIWA, VLADIMIR SHARKO, 2014
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3 311
312 LIBARDI ALICE KIMIE MIWA, VLADIMIR SHARKO
Lemma 1.1. Let M2n+1 be a compact C(CPn)-singular manifold with k singular points. The
Euler characteristic of M2n+1 is equal χ(M2n+1) =
k(1− n)
2
.
Remark 1.1. For n even, the complex projective space CPn can not be the boundary of a smooth
compact manifold X2n+1.
Lemma 1.2. Let M2n+1 be a compact C(CPn)-singular manifold with k singular points. If n
is an odd number then the number k of singular points can be any number. If n is an even number
the number k of singular points is an even number.
Since C(CPn)-singular manifolds are topological spaces we can consider continuous functions
on them and because of the nature of C(CPn)-singular manifolds it is appropriate to consider
continuous functions which are smooth on the complement of the set of singular points. Also it
makes sense to study such functions on a C(CPn)-singular manifold whose singular points of the
manifold are critical points of these functions. More precisely, this means the following.
Let M2n+1 be a compact C(CPn)-singular manifold M2n+1 with singular points m1, . . . ,mk
and U(m1), . . . , U(mk) the respective closed neighborhood homeomorphic to the cone over CPn.
For any neighborhood U(mi) there is a disc D2n+2
i and a semifree action of the circle
θ : D2n+2
i × S1 → D2n+2
i
such that performed
D2n+2
i
π→ D2n+2
i /S1 ≈ U(mi).
We introduce in the disc D2n+2
i complex coordinates z1, . . . , zn and recall that the circle is the
set of complex numbers of modulus one. We assume that the action of the circle on the disc is defined
by the formula
θ(z1, . . . , zn) = eitz1, . . . , e
itzn.
Consider an arbitrary S1-invariant smooth function f : D2n+2
i → R with a single critical point in the
center of the disk. For example, let f be given by
f = −|z1|2 − . . .− |zλi |
2 + |zλi+1|2 + . . .+ |zn|2.
Notice that the index of the nondegenerate critical point 0 ∈ D2n+2
i of such function f is always
even.
Let π∗(f) : U(mi)→ R be the continuous function induced on U(mi) by the natural map
π : D2n+2
i → D2n+2
i /S1 ≈ U(mi).
It is clear that the function π∗(f) is smooth on the manifold U(mi) \mi.
Definition 1.1. The function π∗(f) : U(mi)→ R is called almost smooth function on the neigh-
borhood U(mi) with a singularity at the point mi.
If f is given by
f = −|z1|2 − . . .− |zλi |
2 + |zλi+1|2 + . . .+ |zn|2
then the function π∗(f) : U(mi)→ R is called almost Morse function on the neighborhood U(mi).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
FUNCTIONS AND VECTOR FIELDS ON C(CPn)-SINGULAR MANIFOLDS 313
Definition 1.2. The function f : M2n+1 → R is called almost Morse function on the C(CPn)-
singular manifold M2n+1 if f is an almost Morse function in the neighborhoods U(mi) of singular
points mi of M2n+1 and f is a Morse function on a smooth manifold M2n+1 \
⋃
imi.
From Definition 1.1 follows that on any compact C(CPn)-singular manifold M2n+1 with singu-
lar points m1, . . . ,mk there exists an almost Morse function [1, 2].
The number of critical points of an almost Morse function is dependent of the structure of such
function in the neighborhood of singular points of the C(CPn)-singular manifold. Let us examine
this issue in more detail.
Definition 1.3. Let f be an almost Morse function on the C(CPn)-singular manifold M2n+1
with singular points m1, . . . ,mk. Denote by
π∗(fi) : U(mi)→ R
its almost Morse function in the neighborhood U(mi) of singular point mi of M2n+1. The state
of the almost Morse function f is the collection of all almost Morse functions in the neighborhood
U(mi) π∗(f1), π∗(f2), . . . , π∗(fk), which we will be denoted by St(f).
Consider the case where M2n+1, 2n ≥ 5, is a compact simply connected C(CPn)-singular
manifold.
Recall that for a simply connected smooth manifold we can calculate the Morse number via
its homology groups. More precisely, if we consider a closed manifold Nn and Morse functions
f : Nn → R then to count the Morse number for the class of such functions we can use the homology
group Hj(N
n,Z).
If we consider a compact manifold Nn with boundary ∂Nn = ∂1N
n
⋃
∂2N
n and Morse func-
tions f : (Nn, ∂1N
n, ∂2N
n) → R such that f−1(0) = ∂1N
n and f−1(1) = ∂2N
n then to calculate
the Morse numbers for this class of functions we use the group Hj(N
n, ∂1N
n,Z) [6].
Let M2n+1, 2n ≥ 5, be a compact simply connected C(CPn)-singular manifold with singular
points m1, . . . ,mk. Let σ be a permutation of (1, 2, . . . , k). We split the singular point m1, . . . ,mk
in two disjoint sets A and B consisting of s and k − s points, respectively:
A = mσ(1),mσ(2), . . . ,mσ(s), B = mσ(s+1),mσ(s+2), . . . ,mσ(k).
The case when A or B is empty set is not excluded. Consider the homology groups
Hj(M
2n+1 \B,A,Z).
Remark 1.2. If τ is another permutation of (1, 2, . . . , k) and
à = mτ(1),mτ(2), . . . ,mτ(s), B̃ = mτ(s+1),mτ(s+2), . . . ,mτ(k)
is another splitting of the singular points m1, . . . ,mk in two disjoint sets à and B̃ then, in general,
Hj(M
2n+1 \B,A,Z) 6= Hj(M
2n+1 \ B̃, Ã,Z).
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
314 LIBARDI ALICE KIMIE MIWA, VLADIMIR SHARKO
Theorem 1.1. Let M2n+1, 2n ≥ 5, be a compact simply connected C(CPn)-singular manifold
with singular points m1, . . . ,mk. Let σ be a permutation of (1, 2, . . . , k) and let A (with s points)
and B (with k − s points) be the split of the singular points m1, . . . ,mk in the two disjoint sets:
A = mσ(1),mσ(2), . . . ,mσ(s), B = mσ(s+1),mσ(s+2), . . . ,mσ(k).
We fix a collection of almost Morse functions
St = π∗(f1), π∗(f1), . . . , π∗(f1)︸ ︷︷ ︸
s
, π∗(f2), π∗(f2), . . . , π∗(f2)︸ ︷︷ ︸
k−s
in the neighborhoods U
(
mσ(1)
)
, U
(
mσ(2)
)
, . . . , U
(
mσ(s)
)
, U
(
mσ(s+1)
)
, . . . , U
(
mσ(k)
)
respectively,
where
f1 =
2n∑
i=1
|zi|2, f2 = 1−
2n∑
i=1
|zi|2.
Then
Mλ(M
2n+1, St) = µ
(
Hλ(M
2n+1 \B,A,Z)
)
+ µ
(
TorsHλ−1(M
2n+1 \B,A,Z)
)
,
where µ(H) is the minimal number of generators of the group H.
2. Vector fields on C(CPn)-manifolds. Let V (x) be a smooth vector field on a smooth compact
manifold N2n+1 with boundary with a finite number of points C(CPn) in the interior of the manifold
N2n+1 where V (x) are zero. Suppose that the restriction of the field V (x) on the boundary ∂N2n+1
of the manifold N2n+1 is outwardly directed to the manifold N2n+1. Recall the definition of a index
zero of vector field V (x) on a smooth manifold N2n+1.
Definition 2.1. Let N2n+1 be a cone over C(CPn) and let V (x) be an almost smooth vector
field on N2n+1 such that the singular point n ∈ N2n+1 is an isolated zero point of V (x), the field
V (x) has finite number of zeros n1, . . . , nl and such zeros points belong to N2n+1 \ ∂N2n+1 and
V (x) on the boundary of the manifold N2n+1 is pointed out to the manifold N2n+1. The index
ind(n)V (x) of the vector field V (x) at the point n is defined as the
ind(n)V (x) = χ(N2n+1)−
l∑
i=1
ind(ni)V (x).
For definition of the index at a singular point of an arbitrary vector field on cone over C(CPn)
we will need to build additional. First we prove the following lemma.
Lemma 2.1. Let N2n+1 be a cone over C(CPn) and let V (x) be an almost smooth vector field
on N2n+1 such that the singular point n ∈ N2n+1 is an isolated zero point of V (x). Then on singular
manifold N2n+1 there exists an almost smooth vector field W (x) such that:
1) W (x) coincides with the field V (x) in some neighborhood of singular point n ∈ N2n+1;
2) W (x) has finite number of zero points and such zero points belong to N2n+1 \ ∂N2n+1;
3) W (x) on the boundary of the manifold N2n+1 is pointed out to the manifold N2n+1.
We must show that so defined index at the singular point n ∈ N2n+1 does not depend on the
choice of the vector field W (x).
We prove the following lemma.
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
FUNCTIONS AND VECTOR FIELDS ON C(CPn)-SINGULAR MANIFOLDS 315
Lemma 2.2. Let N2n+1 be a smooth closed manifold and let V (x) and W (x) be smooth vector
fields on N2n+1 × [0, 1) such that:
1) the vector field W (x) coincides with the vector field V (x) on N2n+1 × [1 − ε, 1), where
0 < ε < 1;
2) vector fields V (x) and W (x) have finite number of zeros;
3) vector fields V (x) and W (x) are not zero on the boundary N2n+1 × 0 of the manifold
N2n+1 × [0, 1) and pointed out to the manifold N2n+1 × [0, 1).
Let x1, . . . , xs and y1, . . . , yt be zeros of the vector fields V (x) and W (x) respectively. Then
s∑
i=1
ind(xi)V (x) =
t∑
i=1
ind(yi)W (x).
Proposition 2.1. Let N2n+1 be a cone over C(CPn) and V (x) is an almost smooth vector field
on N2n+1 such that the singular point n ∈ N2n+1 is an isolated zero of V (x). The index of the zero
n of the vector field V (x) in Definition 2.1 does not depend of the almost smooth vector field W (x).
Theorem 2.1. Let M2n+1 be a C(CPn)-singular manifold with singular points m1, . . . ,mk.
Suppose that on M2n+1 there exists an almost smooth vector field V (x) with finite number of zeros
m1, . . . ,mk, x1, . . . , xl. Then
χ(M2n+1) =
l∑
i=1
ind(xi) +
k∑
i=1
ind(mi).
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1, № 3. – P. 829 – 850.
4. Milnor D. Mathematics notes lectures on the h-cobordism theorem. – Princeton Univ. Press, 1965. – 122 p.
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Received 20.12.13
ISSN 1027-3190. Укр. мат. журн., 2014, т. 66, № 3
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