S¹-Bott functions on manifolds
We study S¹-Bott functions on compact smooth manifolds. In particular, we investigate S¹-invariant Bott functions on manifolds with circle action. Вивчаються S¹-функцiї Ботта на компактних гладких многовидах. Зокрема, дослiджуються S¹-iнварiантнi функцiї Ботта на гладких многовидах з дiєю кола....
Gespeichert in:
| Veröffentlicht in: | Український математичний журнал |
|---|---|
| Datum: | 2012 |
| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Український математичний журнал
2012
|
| Schlagworte: | |
| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/165265 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | S¹-Bott functions on manifolds / D. Repovš, V. Shark // Український математичний журнал. — 2012. — Т. 64, № 12. — С. 1685-1698. — Бібліогр.: 15 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-165265 |
|---|---|
| record_format |
dspace |
| spelling |
Repovš, D. Sharko, V. 2020-02-12T19:10:04Z 2020-02-12T19:10:04Z 2012 S¹-Bott functions on manifolds / D. Repovš, V. Shark // Український математичний журнал. — 2012. — Т. 64, № 12. — С. 1685-1698. — Бібліогр.: 15 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165265 513.944 We study S¹-Bott functions on compact smooth manifolds. In particular, we investigate S¹-invariant Bott functions on manifolds with circle action. Вивчаються S¹-функцiї Ботта на компактних гладких многовидах. Зокрема, дослiджуються S¹-iнварiантнi функцiї Ботта на гладких многовидах з дiєю кола. Supported by the Slovenian-Ukranian research (Grant BI-UA/11-12-001) Supported by the National Agency for Science Innovation and Informatization of Ukraine (Grants 40/1-78-2012 and NM/337-2012) en Український математичний журнал Український математичний журнал Статті S¹-Bott functions on manifolds S¹-функцiї ботта на многовидах Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
S¹-Bott functions on manifolds |
| spellingShingle |
S¹-Bott functions on manifolds Repovš, D. Sharko, V. Статті |
| title_short |
S¹-Bott functions on manifolds |
| title_full |
S¹-Bott functions on manifolds |
| title_fullStr |
S¹-Bott functions on manifolds |
| title_full_unstemmed |
S¹-Bott functions on manifolds |
| title_sort |
s¹-bott functions on manifolds |
| author |
Repovš, D. Sharko, V. |
| author_facet |
Repovš, D. Sharko, V. |
| topic |
Статті |
| topic_facet |
Статті |
| publishDate |
2012 |
| language |
English |
| container_title |
Український математичний журнал |
| publisher |
Український математичний журнал |
| format |
Article |
| title_alt |
S¹-функцiї ботта на многовидах |
| description |
We study S¹-Bott functions on compact smooth manifolds. In particular, we investigate S¹-invariant Bott functions on manifolds with circle action.
Вивчаються S¹-функцiї Ботта на компактних гладких многовидах. Зокрема, дослiджуються S¹-iнварiантнi функцiї Ботта на гладких многовидах з дiєю кола.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165265 |
| citation_txt |
S¹-Bott functions on manifolds / D. Repovš, V. Shark // Український математичний журнал. — 2012. — Т. 64, № 12. — С. 1685-1698. — Бібліогр.: 15 назв. — англ. |
| work_keys_str_mv |
AT repovsd s1bottfunctionsonmanifolds AT sharkov s1bottfunctionsonmanifolds AT repovsd s1funkciíbottanamnogovidah AT sharkov s1funkciíbottanamnogovidah |
| first_indexed |
2025-11-25T22:47:35Z |
| last_indexed |
2025-11-25T22:47:35Z |
| _version_ |
1850573804446679040 |
| fulltext |
UDC 513.944
D. Repovš* (Ljubljana Univ., Slovenia),
V. Sharko** (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
S1-BOTT FUNCTIONS ON MANIFOLDS
S1-ФУНКЦIЇ БОТТА НА МНОГОВИДАХ
We study S1-Bott functions on compact smooth manifolds. In particular, we investigate S1-invariant Bott functions on
manifolds with circle action.
Вивчаються S1-функцiї Ботта на компактних гладких многовидах. Зокрема, дослiджуються S1-iнварiантнi функцiї
Ботта на гладких многовидах з дiєю кола.
1. Introduction. Let Mn be a compact closed manifold of dimension at least 3. We study the S1-Bott
functions on Mn. Separately we investigate S1-invariant Bott functions on M2n with a semi-free
circle action which have finitely many fixed points. The aim of this paper is to find exact values of
minimal numbers of singular circles of some indices of S1-invariant Bott functions on M2n.
Closely related to S1-Bott function on a manifoldMn is a more flexible object, the decomposition
of a round handle of Mn. In its turn, to study the round handles decomposition of Mn we use a
diagram, i.e., a graph which carries the information about the handles.
2. S1-Bott functions. Let Mn be a smooth manifold, f : Mn → [0, 1] a smooth function, and
x ∈ Mn one of its critical points. Consider the Hessian Γx(f) : Tx × Tx → R at this point. Recall
that the index of the Hessian is called the maximum dimension of Tx, where Γx(f) is negative
definite. The index of Γx(f) is called the index of the critical point x, and the corank of Γx(f) is
called the corank of x. Suppose that the set of critical points of f forms a disjoint union of smooth
submanifolds Ki
j whose dimensions do not exceed n − 1. A connected critical submanifold Ki0
j0
is called nondegenerate if the Hessian is nondegenerate on subspaces orthogonal to Ki0
j0
(i.e., has
corank equal to n− i0) at each point x ∈ Ki0
j0
.
Definition 2.1. A mapping f : Mn → [0, 1] is called a Bott function if all of its critical points
form nondegenerate critical submanifolds which do not intersect the boundary of Mn.
Consider the following important example of Bott functions:
Definition 2.2. A mapping f : Mn → [0, 1] is called an S1-Bott function if all of its critical
points form nondegenerate critical circles.
Note that S1-Bott functions do not exist on any smooth manifold [12]. S1-Bott functions have
been studied and used by many authors [1 – 7, 9, 11, 14].The following theorem can be found in
[8, 11].
Theorem 2.1. Let Mn be a smooth closed manifold, f : Mn → [0, 1] be a S1- Bott function,
and γ ⊂ Mn its critical circle. Then there is a system of coordinates in a neighborhood of γ of one
of the following types:
∗ Supported by the Slovenian-Ukranian research (Grant BI-UA/11-12-001).
∗∗ Supported by the National Agency for Science Innovation and Informatization of Ukraine (Grants 40/1-78-2012
and NM/337-2012).
c© D. REPOVŠ, V. SHARKO, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12 1685
1686 D. REPOVŠ, V. SHARKO
1. Trivial ν : S1 ×Dn−1(ε) → Mn, where Dn−1(ε) is a disc of radius ε, ν(S1 × 0) = γ, and
f
(
ν(θ, x)
)
= −x21 − . . .− x2λ + x2λ+1 + . . .+ x2n−1, for (θ, x) ∈ S1 ×Dn−1(ε).
2. Twisted τ : ([0, 1]×Dn−1(ε)/ ∼)→Mn, where τ is a smooth embedding such that (τ([0, 1])×
× 0/ ∼) = γ and f(τ(t, x)) = −x21 − . . . − x2λ + x2λ+1 + . . . + x2n−1, for (t, x) ∈ (τ : [0, 1] ×
× Dn−1(ε)/ ∼ ). Here ([0, 1] × Dn−1(ε)/ ∼) is diffeomorphic to S1 × Dn−1(ε) by identifying
0×Dn−1(ε) and 1×Dn−1(ε) by the mapping: (0, x1, . . . , xλ, xλ+1, . . . , xn−1)↔ (1,−x1, . . . , xλ,
−xλ+1, . . . , xn−1).
The number λ is called the index of the critical circle γ.
Let Mn be a smooth manifold, and f : Mn → [0, n] an S1-Bott function. We say that f is a
nice S1-Bott function if the submanifold Mi(f) = f−1
[
0, i+
1
2
]
contains all closed orbits of index
λ ≤ i. Each nice S1-Bott function defines a filtration on the manifold Mn : M0(f) ⊂ M1(f) ⊂ . . .
. . . ⊂ Mn−1(f) ⊂ Mn. It is well known [11] that the existence of a nice S1-Bott function on a
manifold is equivalent to existence of a decomposition of the manifold by round handles. We recall
some necessary definitions.
Definition 2.3. We define an n-dimensional round handle Rλ of index λ by Rλ = S1 ×Dλ ×
×Dn−λ−1, where Di is a disc of dimension i.
Define a twisted n-dimensional round handle TRλ of index λ, 0 < λ < n−1, by TRλ = [0, 1]×
× Dλ × Dn−λ−1/ ∼, where identification is given by the map: (0, x1, . . . , xλ, xλ+1, . . . , xn−1) ↔
↔ (1,−x1, . . . , xλ,−xλ+1, . . . , xn−1).
Apparently, Thurston [15] was the first to note that the existence of a S1-Bott function on mani-
fold is equivalent to existence of a decomposition of the manifold by handles. We shall describe this
fact in more details.
Definition 2.4. We say that the manifold Mn
λ is obtained from a smooth manifold Mn by
attaching a round handle of index λ if Mn
λ = Mn
⋃
ϕ S
1 ×Dλ ×Dn−λ−1, where ϕ : S1 × ∂Dλ ×
×Dn−λ−1 −→ ∂Mn is a smooth embedding.
Manifold Mn
λ is obtained from a smooth manifold Mn by gluing a twisted round handles of index
λ, if Mn
λ = Mn
⋃
ϕ[0, 1]×Dλ ×Dn−λ−1/ ∼, where ϕ : ([0, 1]× ∂Dλ ×Dn−λ−1/ ∼)→ Mn is a
smooth embedding.
Definition 2.5. Decomposition of a smooth manifold Mn by round handles is called a filtration
∂Mn × [0, 1] = Mn
0 (R) ⊂Mn
1 (R) ⊂ . . . ⊂Mn
n−1(R) = Mn, where the manifold Mn
i (R) obtained
from the manifold Mn
i−1(R) by gluing round and twisted round handles of index i. In the case when
Mn is a closed manifold, filtration begins with round handles of index 0.
In what follows we recall the relationship between S1 and the decomposition by round handles
[11].
Theorem 2.2. Let Mn be a smooth closed manifold. The following two conditions are equiva-
lent :
1. On the manifold Mn there is a nice S1-Bott function with the critical circles γ1, . . . , γk of
index λ1, . . . , λk with trivial coordinate systems and critical circles γ̃1, . . . , γ̃l of indices µ1, . . . , µl
with twisted coordinate systems.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1687
2. Manifold Mn admits a decomposition by round handles consisting of round handles Rλ1 , . . .
. . . , Rλk of index λ1, . . . , λk and of twisted round handles TRµ1 , . . . , TRµl of indices µ1, . . . , µl so
that the critical circle γi corresponds to a round handle Rλi , 1 ≤ i ≤ k, and the critical circle γ̃j
corresponds to a twisted round handle TRµj , 1 ≤ j ≤ l.
Thus each nice S1-Bott function on manifold Mn generates a round handle decomposition of
Mn and vice versa.
The following result belongs to D. Asimov [5].
Theorem 2.3. Let Mn be a smooth closed manifold (n > 3), and suppose that the Euler
characteristic χ(Mn) = 0. Then Mn admits a round handle decomposition.
For three-dimensional manifolds the situation is much more complicated [1, 12], there are closed
three-manifolds which do not admit a round handle decomposition. Of recent results in the three-
dimensional Poincare conjecture implies that a simply connected three-dimensional manifold admit
a round handle decomposition.
We are interested in conditions when an S1-Bott function on Mn has the property that all of its
critical circles have trivial coordinate system. We recall the necessary facts from [4].
By definition an n-dimensional handle Hλ of index λ is Hλ = Dλ × Dn−λ. We say that a
smooth manifold Mn
λ is obtained from a smooth manifold Mn by attaching handles of index λ if
Mn
λ = Mn
⋃
ϕD
λ × Dn−λ, where ϕ : ∂Dλ × Dn−λ −→ ∂Mn is a smooth embedding. ∂Dλ × 0
(Dλ × 0) is called the core (disc), and ∂Dn−λ × 0 (Dn−λ × 0) is called co-core sphere (disc) of the
handle Dλ ×Dn−λ.
A decomposition of a smooth manifold Mn by handles is a filtration ∂Mn × [0, 1] = Mn
0 ⊂
⊂ Mn
1 ⊂ . . . ⊂ Mn
n = Mn, where the manifold Mn
i is obtained from the manifold Mn
i−1 by
attaching handles of index i.
In the case where Mn is a closed manifold, filtration begins by handles of index 0. There is
a close relationship between the expansion of manifold by round handles and handles, in [5] the
following lemma was proved.
Lemma 2.1. Let Mn = Mn
1 +Hλ +Hλ+1 be a smooth manifold obtained from manifold with
boundary Mn
1 by attaching handles of indices λ and λ + 1, which do not intersect (n > 2). Then
if λ > 0, the manifold Mn can be represented as Mn = Mn
1 + Rλ, where Rλ denotes the round
handle of index λ.
Lemma 2.2. Let Mn be a smooth manifold (n > 2) obtained from manifolds with boundary
Mn
1 by attaching a round (or twisted round) handles of index λ > 0. Then the manifold Mn can be
represented as Mn = Mn
1 + Hλ + Hλ+1. If the round handle Rλ was glued, then the intersection
index of Hλ and Hλ+1 is equal to 0.
If we glue to the twisted handle TRλ, then the intersection index Hλ and Hλ+1 is equal to ±2.
Proof. The case when the handle is attached was proved in [4] (Lemma VIII.2). If glue twisted
handle TRλ to Mn
1 , then the argument is the same. Let ϕ : ([0, 1] × ∂Dλ × Dn−λ−1/ ∼) −→
−→ ∂Mn
1 be a gluing map. Represent ϕ([0, 1] × 0 × 0/ ∼) as the sum of two segments I1 and I2
such that I1 ∩ I2 = ∂I1 = ∂I2 and I1 ∪ I2 = (ϕ
(
[0, 1] × 0 × 0/ ∼
)
. Consider the submanifold
Hλ = I1 ×Dλ ×Dn−λ−1. Obviously it can be regarded as a handle of index λ, which is attached
to ∂Mn
1 along the set ∂Dλ × Dn−λ−1 × I1 with the restriction of ϕ. It is clear that the manifold
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1688 D. REPOVŠ, V. SHARKO
Hλ+1 = TRλ \ (I1 ×Dλ ×Dn−λ−1) = I2×Dλ×Dn−λ−1) is the handle of the index λ+1, which
is attached to ∂(Mn
1 ∪Hλ) along the set (∂I2 ×Dλ ∪ I2 × ∂Dλ)×Dn−λ−1.
By construction, the intersection index of these two handles is ±2.
Lemma 2.2 is proved.
Lemma 2.3. Let Mn be a smooth closed manifold, f : Mn → [0, 1] an S1-Bott function, and c
its critical value. Suppose ε > 0, and that on the interval [c−ε, c+ε] there are no other critical values.
Assume that on the surface level f−1(c) there are critical circles γ1, . . . , γk of indices λ1, . . . , λk
with trivial coordinate systems and there are critical circles γ̃1, . . . , γ̃l of indices µ1, . . . , µl with
twisted coordinate systems. Then the homology groups H∗
(
f−1[c− ε, c+ ε], f−1(c− ε),Z
)
is gen-
erated exactly by the handles which correspond to the critical circles γ1, . . . , γk, γ̃1, . . . , γ̃l. Each
circle γi generates two subgroups that are isomorphic to Z, a direct product of the homology group
Hλi
(
f−1[c − ε, c + ε], f−1(c − ε),Z
)
, and the other in the homology group Hλi+1
(
f−1[c − ε, c +
+ ε], f−1(c − ε),Z
)
. Each circle γ̃j generates a subgroup Z2 which is direct product in a group
Hµj
(
f−1[c− ε, c+ ε], f−1(c− ε),Z
)
.
Proof. Consider a function f associated to the decomposition of the manifold f−1[c−ε, c+ε] by
the round and twisted handles. Thus the critical circles lie on the same level of the decomposition of
the round and twisted handles We can choose the handles so that they do not intersect each other. If
we replace the round handles by handles from the previous lemma it follows that each twisted round
handle of index λ generates the homology of a subgroup isomorphic to Z2 in dimension λ, and each
round handle of index λ generates the homology of two subgroups isomorphic to Z in dimensions λ
and λ+ 1.
Lemma 2.3 is proved.
Corollary 2.1. Let Mn be a smooth closed manifold, f : Mn → [0, 1] an S1-Bott function, and
c1, . . . , ck its critical values. Suppose εi > 0, 1 ≤ i ≤ k, such that the interval [ci − εi, ci + εi] has
no other critical values. Then on a level surface f−1(ci) there are only critical circles with trivial
coordinate systems if and only if the nonzero homology groupsH∗
(
f−1[ci−εi, ci+εi], f−1(ci−εi),Z
)
are free Abelian groups.
Thus we have a homological criterion when S1-Bott functions do not have critical circle with
twisted coordinate systems.
In the next section, we give another class of S1-Bott function which do not possess the critical
circle with twisted coordinate systems.
3. Diagrams of S1-Bott functions and their applications. In this section we explore S1-Bott
functions. We recall the definition of partitions of diagrams [4]. Partition diagrams represent the
construction of S1-Bott functions, especially for simply connected manifolds.
Consider the decomposition of a closed smooth manifold Mn by handles Mn
0 ⊂ Mn
1 ⊂ . . .
. . . ⊂Mn
n = Mn, where the manifold Mn
i is obtained from the manifold Mn
i−1 by attaching handles
of index i. Assume that Ci = Hi(M
n
i ,M
n
i−1,Z) ≈ Z⊕ . . .⊕ Z︸ ︷︷ ︸
ki
, where ki is the number of handles
of index i. Mean discs handles of index i form a basis for the homology groups Hi(M
n
i ,M
n
i−1,Z).
Using the exact homology sequence for the triple Mn
i−1 ⊂ Mn
i ⊂ Mn
i+1 we can construct a chain
complex of free Abelian groups: (C, ∂) : C0 ← . . .← Ci−1
∂i←− Ci
∂i+1←−−− Ci+1 ← . . .← Cn, whose
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1689
homology coincides with the homology of the manifold Mn. Suppose that manifold Mn is oriented.
The choice of orientation allows to orient medium and comedium sphere of the handle which allows
to determine the homology indices λ and λ+ 1 in manifolds ∂Mn
λ . Thus the homomorphism ∂λ
given by the matrix of indices of homologous intersections of right-hand and left-hand spheres of
handles in the submanifold ∂Mn
λ .
If each handle determines a vertex, and bridge the edges of those vertices for which the corre-
sponding handles have a non-zero intersection, then we obtain a graph. Note that the structure of this
graph can be complicated. However, it can be simplified.
It is known [4] that with the addition of handles all the matrices of homomorphisms ∂i, 0 ≤ i ≤ n,
can be made diagonal.
Suppose that Mn is a simply connected manifold, n > 5 and there are no handles of the indices 1
and n−1, then certainly the homology of the intersection indices of right-hand and left-hand spheres
coincide with their geometric intersection indices.
Thus a pair of adjacent handles with indices λ and λ+ 1 may either not intersect or have inter-
section ±1, ±2 or ±m, where |m| > 2. Since the Euler characteristic of a closed smooth manifold
Mn which admits a round handles decomposition is zero, it follows that the decomposition Mn by
handles, we can introduce the following object, a diagram. A diagram is a disconnected graph whose
vertices correspond to handles and whose edges connect vertices if and only if the intersection of the
handle is nonzero. More precisely:
Definition 3.1. Ωn is called a diagram of length n, if the plane is given by (n+1) set of points(
a10, . . . , a
1
k0
; a11, . . . , a
1
k1
; . . . ; an1 , . . . , a
n
kn
)
, which satisfy the following conditions:
1) for some i the set (ai1, . . . , a
i
ki
) may be empty,
2) k0 − k1 + k2 − . . .+ (−1)nkn = 0,
3) point of the set (ai1, . . . , a
i
ki
), 1 < i < n − 1, can be connected either with only one point
from the set
(
ai−1
1 , . . . , ai−1
ki−1
)
or with only one point from (ai+1
1 , . . . , ai+1
ki+1
) in one of three ways.
A set of points a01, . . . , a
1
k0
; . . . ; ai1, . . . , a
i
ki
is called an i-skeleton diagram Ωn.
A point at which the chart is not linked to some other point is called free. If the chart has a
fragment
,
then aii is called a semi-free point (intersection of the handle is ±2). If there is a fragment
,
then aii is called a dependent point (intersection of the handle is ±m). Fragment
is called inserted in dimension i (the index of intersection of corresponding handles is equal to ±1).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1690 D. REPOVŠ, V. SHARKO
Definition 3.2. A pair of points of dimension i and i + 1 are independent in the dimension i,
if there is no connection between them or if they form a fragment.
In what follows we divide a chart into disjoint pairs of independent points. Let us make a
restriction for the fragments of the diagram form
We do not allow breaking any of the fragments into a pair of the form (aij , a
i+1
t ), (aik, a
i+1
l ).
Definition 3.3. If a chart Ωn can be represented as the disjoint union of independent pairs of
points, then it admits a partition. A pair of points (aij , a
i+1
k ) of this partition is called the vertices of
the partition in dimension i.
Let us fix a partition diagram Ωn denoted by Ωn(σ). It is possible that the diagram Ωn(σ) does
not admit a partition since in some dimensions it may not have enough points for the formation of
independent pairs.
Definition 3.4. The base of the diagram Ωn is the diagram Ωn obtained from Ωn by eliminating
all inserts.
Definition 3.5. A stabilization of the diagram Ωn in dimension i is a diagram of the form
Ω
S(i)
n = Ωn ∪
i
Ai, where Ai is a new insert on dimension i.
Lemma 3.1. For each chart Ωn there exists its stabilization in dimensions i1, . . . , is, denoted
by Ω
S(i1,...,is)
n , such that the diagram Ω
S(i1,...,is)
n admits a partition.
Definition 3.6. The number χi(Ωn) = ki − ki−1 + . . .+ (−1)i+1k0 is called i-th Euler char-
acteristic of the diagram Ωn.
Obviously, the insertion of dimension i increases the i-th Euler characteristic of Ω
S(i)
n the unit
and it does not change the values of the remaining j-Euler characteristics χj
(
Ω
S(i)
n
)
= χj(Ωn) for
j 6= i.
Lemma 3.2. If the diagram Ωn admits a partition, then the number of vertices of the partition
Ωn in each dimension is the same for all of its possible partitions.
Suppose that the diagram Ωn admits a partition. Denote by mi(Ωn) the number of vertices in
dimension i of a partition Ωn and by M(Ωn) the number M(Ωn) =
∑i
j=0
mj(Ωn).
In light of the lemma this numbers does not depend on the choice of a particular partition of the
diagram Ωn.
Definition 3.7. Dimension λ of a chart Ωn is called singular if χλ−1(Ωn) = χλ+1(Ωn) = 0,
χλ(Ωn) = k > 0 and chart Ωn in dimensions λ does not consist of semi-free fragments.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1691
In the process of decomposition of the diagram Ωn into a pair of independent points is necessary
in this situation it to make one box of dimension λ−1 or in dimension λ+1 which leads to ambiguity.
The result will be different number of pairs in dimension λ+ 1 or in dimension λ+ 1, depending on
whether in any dimension we have made insertions.
Lemma 3.3. The diagram Ωn =
(
a10, . . . , a
1
k0
; a11, . . . , a
1
k1
; . . . ; an1 , . . . , a
n
kn
)
admit a partition
if and only if it has no negative i-th Euler characteristics and singular dimensions. If the diagram Ωn
admit partition then the number of vertices in dimension i of a partition is equal mi(Ωn) = χi(Ωn).
If the diagram Ωn not admit partition then there exists its stabilization ΩS
n , such that the dia-
gram ΩS
n admits a partition. Raises the question of the minimum possible number of the vertices in
dimension i among stabilized diagram Ω
Sj
n have a partition.
For diagram Ωn denote by ms
i (Ωn) the minimum possible number of the vertices in dimension
i among stabilized diagram Ω
Sj
n have a partition.
Let N be the set of integers, put ρ(n) =
1
2
(
n+ |n|
)
, where n ∈ N.
Theorem 3.1. Let Ωn be an arbitrary diagram, then ms
i (Ωn) of the diagram Ωn is equal
ms
i (Ωn) = ρ
(
χi(Ωn)
)
. If ΩS
n is a stabilization of diagram Ωn, then ms
i (Ω
S
n) ≥ ms
i (Ω).
Definition 3.8. For the diagram Ωn its i-th Morse number Mi(Ωn) is the the number ms
i (Ωn),
where Ωn is the base of the diagram Ωn.
Definition 3.9. A diagram Ωn is called exact if there exists a stabilization ΩS∗
n of Ωn, such
that ΩS∗
n admit partition with the number of vertices in dimension i is equal mi(Ω
S∗
n ) = Mi(Ωn)
simultaneously for all i .
Theorem 3.2. The diagram Ωn is exact if and only if it does not have singular dimensions.
A stabilization of diagram Ωn is called economical if
1) when χi(Ωn) = k < 0, perform k insert in the dimension i,
2) when i is singular dimension, then perform on insert in the dimension i − 1 or in dimension
i+ 1.
We now describe how we can construct a diagram Ωn on a decomposition of a smooth closed
manifold Mn on round handles.
Let Mn
0 (R) ⊂ Mn
1 (R) ⊂ . . . ⊂ Mn
n−1(R) = Mn, be a round handle decomposition Mn. By
Lemma 2.2, we replace each hand index λ on two ordinary handles of indices λ and λ + 1. As a
result, we obtain an expansion of the manifold Mn by handles: Mn
0 ⊂ Mn
1 ⊂ . . . ⊂ Mn
n = Mn.
Using this handle decomposition of Mn we can construct a chain complex of free abelian groups:
(C, ∂) : C0 ← . . . ← Ci−1
∂i←− Ci
∂i+1←−−− Ci+1 ← . . . ← Cn. Citing the matrix of differentials to the
diagonal form, we construct a diagram Ωn. The following fact holds:
Proposition 3.1. Let Mn
0 (R) ⊂ Mn
1 (R) ⊂ . . . ⊂ Mn
n−1(R) = Mn be a round handle decom-
position of manifold Mn and Ωn be a diagram associated to this decomposition. Assume that the
diagram Ωn has no semi-free vertices. If the diagram Ωn is the economical stabilization of its base
Ωn, then the original decomposition on the round handles had missing twisted round handles.
Proof. Indeed, in this case, the diagram Ωn does not allow the insertion of the round twisted
handle. All the points of insertion involved for the formation of vertices with other points of the
diagram. And by condition of the proposition semi-free vertices are not present.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1692 D. REPOVŠ, V. SHARKO
Remark 3.1. It is easy to construct a decomposition of a manifold Mn by the round handles,
among which are twisted round handles, but at the same time have associated with this decomposition
diagram no semi-free vertices.
Definition 3.10. LetMn be a smooth closed manifold. The number χi(Mn) = µ
(
Hi(M
n,Z)
)
−
− µ
(
Hi−1(M
n,Z)
)
+ . . . + (−1)i+1µ
(
H0(M
n,Z)
)
is called the i-th Euler characteristic of Mn,
where µ(H) is a minimal number of generators H.
Definition 3.11. A dimension λ of closed manifold Mn is called singular if Hλ(Mn,Z) is a
nonzero finite group distinct from Z2 ⊕ . . .⊕ Z2 and χλ−1(M
n) = χλ+1(M
n) = 0.
Definition 3.12. Let Mn be a smooth closed manifold. A round handle decomposition is called
quasiminimal, if one of the following holds:
1) the number of round handles of index i equals to ρ(χi
(
Mn)
)
+ εi, where εi = 0, if dimension
i+ 1 is nonsingular and εi = 1, if dimension i+ 1 is singular;
2) the number of round handles of index i equals to ρ
(
χi(M
n)
)
, if dimension i+ 1 is singular,
then there is only one handle of index i+ 2.
In both cases, the number of round handles of index i + 1 equals to ρ
(
χi+1(M
n)
)
. A round
handle decomposition is called minimal, if number of round handles of index i equals to ρ
(
χi(M
n)
)
for all i.
Using the decomposition of manifold on handles and the diagram technique, we can easily prove
the following fact [4].
Proposition 3.2. Let Mn be a smooth closed simply-connected manifold (n > 5). Then Mn
admits a quasiminimal decomposition into round handles. If manifold Mn have not singular dimen-
sions, then Mn admits a minimal decomposition into round handles.
Definition 3.13. Let the manifold Mn admits S1-Bott function, then S1-Morse number
MS1
i (Mn) of index i is the minimum number of singular circles of index i taken over all S1-Bott
functions on Mn.
Lemma 3.4. Let on a closed manifold Mn exist a smoth function f : Mn → R such that
each connected component of the singular set Σf of f is either a nondegenerate critical point pi,
i = 1, . . . , k, or a nondegenerate critical circle S1
j , j = 1, . . . , l. Then the Euler characteristic of the
manifold Mn is equal to χ(Mn) =
∑k
i=1
(−1)index(pi).
Proof. It is known that for any Morse function on the manifold Mn g : Mn → R with critical
points pi, i = 1, . . . , q, there is the formula χ(Mn) =
∑q
i=1
(−1)index(pi). By small perturbation of
the function f any nondegenerate critical circle S1
j of index λ can be replaced by nondegenerate crit-
ical points of idexes λ and λ+1 [1]. Therefore the contribution in the formula for Euler characteristic
of these critical points will be zero and we obtain the desired formula.
4. Manifolds with free S1-action. Let on smooth manifold Mn there is smooth free circle action.
Then of course the set Mn/S1 is a manifold and natural projection p : Mn →Mn/S1 is fibre bundle.
Any smooth S1-invariant function f : Mn → R on a manifold Mn is called an S1-invariant Bott
function if each connected component of the singular set Σf is nondegenerate critical circle.
It is clear that if f be a S1-invariant Bott function on the manifold Mn then it projection
π∗(f) : Mn/S1 → R, is a Morse function. And conversly, if g : Mn/S1 → R be a Morse function
on the manifold Mn/S1 then π−1
∗ (g) = g◦π : Mn → R is S1-invariant Bott function on the manifold
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1693
Mn. The critical point of the index λ of the function g correspond to critical circle of the index λ of
the function π−1
∗ (g).
In this situation for the manifold Mn S1-equivariant Morse number of index i, M eqS1
i (Mn) is
the minimum number of singular circles of index i taken over all S1-invariant Bott functions on Mn.
For the manifold Mn/S1 Morse number of index i, Mi(M
n/S1) is the minimum number of
critical points of index i taken over all Morse functions on Mn/S1.
Therefore for a calculation of the S1-equivariant Morse number of the index i there is possible
to use a Morse functions on the manifold Mn/S1. The following fact is clear:
Corollary 4.1. Let on smooth manifold Mn there is smooth free circle action. Then for the
manifold Mn S1-equivariant Morse number of index i is equal Morse number of index i for the
manifold Mn/S1.
Good example in this direction is the fibre bundle p : S2n+1 → CPn. For this S1-action a S1-
equivariant Morse number of even indexes equal 1 and for odd indexes equal 0.
The next example to show that S1-equivariant Morse number of the manifold Mn depends on
circle action.
Let p : S3 → S2 be the Hopf fibre bundle. Suppose that circle trivial act on S1. Using Hopf fibre
bundle and trivial circle action on S1 we shall construct new fibre bundle p× id : S3×S1 → S2×S1.
It is clear, that on manifold S2 × S1 there is a Morse function with one critical point of the indexes
0, 1, 2, 3. Therefore for such circle action on manifold S2 × S1 the S1-equivariant Morse number
for any index equal 1.
On the other side, let circle act trivially on S3 and q is rotation on S1. Consider fibre bundle
id× q : S3 × S1 → S3. On S3 there is a Morse function with one critical point of the indexes 0 and
3. Therefore in this situation for the manifold S3 × S1 the S1-equivariant Morse number for indexes
0 and 3 equal 1 and 0 for other indexes.
Remark 4.1. This example shows that for manifold S1-equivariant Morse number and S1-
Morse number of some index may be different.
Definition 4.14. Let on smooth manifold Mn there be a smooth free circle action. Then this
free circle action is minimal if for all indexes S1-equivariant Morse number is equal S1-Morse
number for the manifold Mn.
Corollary 4.2. Let on smooth simply-connected manifold Mn there be a smooth minimal free
circle action. Then manifold Mn have not singular dimensions.
Proof. Obviously, for dimension three corolary is valid.
A manifold, which allows free circle action has Euler characteristic zero. If n = 4, then free action
on simply-connected manifolds M4 non exist, since the Euler characteristic of a simply connected
four-dimensional manifold is always positive.
From the structure of the homology groups follows that simply-connected manifold Mn, 8 ≥
≥ n ≥ 5, have not singular dimensions.
Let n ≥ 9. Suppose that on Mn there be a minimal smooth free circle action. Obviosly, that
there be a equality Mi(M
n/S1) = M eqS1
i (Mn). By Smale theorem [10] on manifold Mi(M
n/S1)
there be a Morse function with the number of critical points of index i is equal Mi(M
n/S1) for all
i simultaneously. Therefore on the manifold Mn there exist S1-invariant Bott function f with the
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1694 D. REPOVŠ, V. SHARKO
number of critical circles of index i is equal Mi(M
n/S1) = M eqS1
i (Mn) for all i simultaneously.
Since free circle action is minimal, thereforeMS1
i (Mn) = M eqS1
i (Mn). If simply-connected manifold
Mn have singular dimension then S1-Bott function on Mn can not have the number of critical circles
of index i equal to i-th S1-Morse number MS1
i (Mn) for all i simultaneously. Consequently, the
manifold Mn has no singular dimensions.
Corollary 4.2 is proved.
Theorem 4.1. Let on smooth simply-connected manifold Mn there be a smooth free circle ac-
tion. Then this circle action is minimal if and only if µ(Hi(M
n/S1, Z)+µ(Tors Hi−1(M
n/S1, Z) =
= ρ
(
χi(M
n)
)
for all i.
Proof. From exact homotopy sequence of fibration follow that manifold Mn/S1 is simply-
connected. Let n = 3, using results about three dimension Poincare conjecture [13] can obtained that
M3 = S3, M3/S1 = S2 and we have Hopf fibre bundle p : S3 → S2. Therefore, Theorem 4.1 is
proved.
If n = 4, then free action on simply-connected manifolds M4 non exist.
Let n ≥ 5. Necessary. Suppose that on Mn there be a minimal smooth free circle action. If
n ≥ 5 from results of Smale and Barden [3, 10] it follows that Morse number in dimension i of the
manifold Mn/S1 is equal Mi(M
n/S1) = µ(Hi
(
Mn/S1, Z)
)
+ µ
(
Tors Hi−1(M
n/S1, Z)
)
. There
is equality Mi(M
n/S1) = M eqS1
i (Mn). Because of the condition of minimal free circle action there
is equality Mi(M
n/S1) = M eqS1
i (Mn) = MS1
i (Mn) = ρ
(
χi(M
n)
)
.
Sufficiently. Consider on manifold Mn/S1 Morse function with the number of critical points of
index i equal Mi(M
n/S1) = µ(Hi(M
n/S1, Z)) + µ(Tors Hi−1(M
n/S1, Z)). By the construction
and condition of the theorem we have the equalities Mi(M
n/S1) = M eqS1
i (Mn) = ρ(χi(M
n)). But
MS1
i (Mn) = ρ
(
χi(M
n)
)
and therefore free action of S1 is minimal.
Theorem 4.1 is proved.
Corollary 4.3. Let on smooth manifold Mn there be a smooth free circle action. Suppose that
manifold Mn/S1 is
a) π1(Mn/S1) ≈ Z, or π1(Mn/S1) ≈ Z⊕ Z, n > 6;
b) π1(Mn/S1) is infinite, n > 8.
Then S1-equivariant Morse number of index i for manifold Mn equal
a) Ŝi(2)(M
n/S1) + Ŝi+1
(2) (Mn/S1) + dimN(Z[π])
(
H i
(2)(M
n/S1,Z)
)
;
b) Di(Mn/S1) + Ŝi(2)(M
n/S1) + Ŝi+1
(2) (Mn/S1) + dimN(Z[π])
(
H i
(2)(M
n/S1,Z)
)
, for 3 < i <
< n− 3.
Proof. It follows from results of [4, 14] that on Mn there is Morse functions with the number of
critical points of index i equal Morse number of the manifold Mn/S1.
5. Manifolds with semi-free S1-action. Let M2n be a closed smooth manifold with semi-free
S1-action which has only isolated fixed points. It is known that every isolated fixed point p of a semi-
free 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. More precisely, for every isolated fixed point p there exist
an open invariant neighborhood U of p and a diffeomorphism h from U to an open unit disk D in Cn
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), and real coordinates (x1, y1, . . . , xn, yn)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1695
on Cn = R2n with zj = xj +
√
−1yj . The pair (U, h) will be called a standard chart at the point
p. Let f : M2n → R be a smooth S1-invariant function on the manifold M2n. Denote by Σf the set
of singular points of the function f. It is clear that the set of isolated singular points Σf (pj) ⊂ Σf of
f coincides with the set of fixed points MS1
.
For a nondegenerate critical point pj there exist a standard chart (Uj , hj) such that on Uj the
function f is given by the following formula:
f = f(p)− |z1|2 − . . .− |zλj |
2 + |zλj+1|2 + . . .+ |zn|2.
Notice that the index of nondegenerate critical point pj is always even.
Denote by Σf (S1) the set singular points of the function f that are disconnected union of circles.
These circles will be called singular.
A circle s ∈ Σf(S
1) is called nondegenerate if there is an S1-invariant neighborhood U of s on
which S1 acts freely and such that the point π(s) is nondegenerate for the function π∗(f) : U/S1 →
→ R, induced on U/S1 by the natural map π : U → U/S1. An invariant version of Morse lemma
says that there exist an S1-invariant neighborhood U of the circle s and coordinates (x1, . . . , x2n−1)
on U/S1 such that the function π∗(f) has the following presentation:
π∗(f) = π∗
(
f(π
(
s)
))
− x21 − . . .− x2λ + x2λ+1 + . . .+ x22n−1.
By definition λ is the index of singular circle s.
Definition 5.1. A smooth S1-invariant function f : M2n → R on a manifold M2n with a
semi-free circle action which has isolated fixed points is called: S1
∗ -Bott function if each connected
component of the singular set Σf is either a nondegenerate fixed point or a nondegenerate critical
circle.
Theorem 5.1. Assume that M2n is the closed manifold with a smooth semi-free circle action
which has isolated fixed points p1, . . . , pk. Let for any fixed point pj consider standard chart (Uj , hj)
and function
fj = fj(pi)− |z1|2 − . . .− |zλj |
2 + |zλj+1|2 + . . .+ |zn|2
on Uj , where λj is an arbitrary integer from 0, 1, . . . , n.
Then there exist an S1-invariant S1
∗ -Bott function f on M2n such that f = fj on Uj .
Proof. Consider on Uj the function fj . Let π∗(fj) : Uj/S
1 → R, continuos function induced on
Uj/S
1 by the natural map π : Uj → Uj/S
1. It is clear that function π∗(fj) is smooth on manifold
(Uj \ pj)/S1. Denote by g smooth extension functions π∗(fj) on M2n/S1. By small deformation
of the function g, that is fixed on Uj/S1, we shall find function g1 on M2n/S1 such that g1 equal
π∗(fj) on Uj/S
1 and g1 have only nondegenerate critical points on M2n \
⋃
(Uj/S
1). Then the
function f = g1 ◦ p satisfics conditions of the theorem.
Theorem 5.2. The number of fixed points of any smooth semi-free circle action on M2n with
isolated fixed points is always even and equal to the Euler characteristic of the manifold M2n.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1696 D. REPOVŠ, V. SHARKO
Proof. In first we consider following functions:
f1 = f1(p1) + |z1|2 + . . .+ |zn|2 on U1 and fj = fj(pi)− |z1|2 − . . .− |zn|2
on Uj , 2 ≤ j ≤ l, and extend such functions to S1-invariant Bott function f on manifold M2n \
U1
⋃
U2
⋃
. . .
⋃
Ul. We suppose that Uj is diffeomorfic to open disk D2n for any j. Consider mani-
fold V 2n = W 2n \
⋃
Uj . The boudary of manifod V 2n is disconnected union of spheres S2n−1. By
construction of manifold V 2n there is free cirle action. The boundary of the manifold V 2n/S1 is
disconnected union of complex projective spaces CPn−1. If the number of the boundary components
of the manifold V 2n/S1 is odd then we glue pairwise boundary components and obtain compact
smoth manifold with with boundary CPn−1. From the well known fact that the manifold CPn−1 is
non-cobordant to zero it follows that the number of fixed points of any smooth semi-free circle action
on M2n with isolated fixed points is even. The value of the Euler characteristic χ(M2n) = 2k is
follow from Lemma 3.4.
Definition 5.2. Let f be an S1-invariant S1
∗ -Bott function for smooth semi-free circle ac-
tion with isolated fixed points p1, . . . , p2k on a closed manifold M2n. Denote by λj the index
of a critical point pj of the function f. The state of the function f is the collection of numbers
Λ = (λ1, λ2, . . . , λ2k), which we will be denoted by Stf (Λ). It is clear that all numbers λj are even
and (0 ≤ λj ≤ 2n).
Remark 5.1. It follows from Theorem 5.1 that for every smooth semi-free circle action on
a closed manifold M2n with isolated fixed points p1, . . . , p2k and any collection even numbers
Λ = (λ1, λ2, . . . , λ2k), such that 0 ≤ λj ≤ 2n there exists an S1-invariant S1
∗ -Bott functions f on
M2n with state Stf (Λ).
Definition 5.3. LetM2n be a closed smooth manifold with smooth semi-free circle action which
has finitely many fixed points p1, . . . , p2k. Fix any collection even numbers Λ = (λ1, λ2, . . . , λ2k),
such that 0 ≤ λj ≤ 2n.
The S1-Morse numberMS1
i
(
M2n, St(Λ)
)
of index i is the minimum numbers of singular circles
of index i taken over all S1-invariant S1
∗ -Bott functions f on M2n with state Stf (Λ).
The following is an unsolved problem: for a manifold M2n with a semi-free circle action which
has finitely many fixed points find exact values of numbersMS1
i
(
M2n, St(Λ)
)
.
6. About S1-equivariant Morse numbers MS1
i
(
M2n, St(Λ)
)
. Let M2n be a compact closed
manifold of dimension with semi-free circle action which has finite many fixed points p1, , . . . , p2k.
Denote by π : M2n → M2n/S1 the canonical map. The set M2n/S1 is manifold with singu-
lar points π(p1), . . . , π(p2k). It is clear that neighborhood of any singular point is a cone over
CPn−1. If f : M2n → R is a smooth S1-invariant S1
∗ -Bott function on the manifold M2n, then
π∗(f): M2n/S1 → R is a continuos function such that on smooth non-compact manifold N2n−1 =
= M2n/S1 \
⋃2k
j=1 π(pj) it is Morse function.
Choose an invariant neighborhood Ui of the point pj diffeomorphic to the open unit disc
D2n ⊂ Cn and set U =
⋃2k
j=1 Uj . Consider compact manifold V 2n−1 = (M2n \ U)/S1, its bound-
ary is a disconnected union of complex projective spaces ∂V 2n−1 = CPn−1
1 ∪ . . . ∪ CPn−1
2k . It is
clear that manifold V 2n−1 \ ∂V 2n−1 and manifold N2n−1 are diffeomorphic. We use a manifold
V 2n−1 for the study of S1-invariant S1
∗ -Bott functions on the manifold M2n with state St(Λ) =
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
S1-BOTT FUNCTIONS ON MANIFOLDS 1697
= (0, . . . , 0, 2n, . . . , 2n). Let ∂0V 2n−1 be a part of boundary of V 2n−1 consist from r component
CP 2n−2, 2k−1 ≥ r ≥ 1, and ∂1V 2n−1 = ∂V 2n−1\∂0V 2n−1. On the manifold with boundary V 2n−1
constructed Morse function f : V → [0, 1], such that f−1(0) = ∂0V
2n−1 and f−1(1) = ∂1V
2n.
Using the function f we constructed on the manifold M2n S1-equivariant S1
∗ -Bott function F
with the state St(0, . . . , 0, 2n, . . . , 2n), such that restriction π∗(F ) on V coinside with f. There-
fore Morse number of index i Mi(V
2n−1, ∂0V
2n−1) of manifold with boundary V 2n−1 is equal
MS1
i (M2n, St(0, . . . , 0, 2n, . . . , 2n).
Theorem 6.1. Let M2n (2n > 8) be a closed smooth manifold admits a smooth semi-free
circle action with isolated fixed points p1, . . . , p2k. Then for the manifold M2n with the state St(Λ) =
= (0, . . . , 0, 2n, . . . , 2n)
MS1
i (M2n, St(Λ) = Di(V 2n−1, ∂0V
2n−1) + Ŝi(2)(V
2n−1, ∂0V
2n−1)+
+Ŝi+1
(2) (V 2n−1, ∂0V
2n−1) + dimN(Z[π])
(
H i
(2)(V
2n−1, ∂0V
2n−1)
)
for 3 ≤ i ≤ 2n− 4.
Proof. Choose an invariant neighborhood Ui of the point pi diffeomorphic to the unit disc
D2n ⊂ Cn and set U =
⋃
i Ui. Let fi be a function on Ui equal
fi = |z1|2 + . . .+ |zn|2 and fj on Uj equal fj = 1− |z1|2 − . . .− |zn|2,
for i = 1, . . . , r, j = r+ 1, . . . , 2k− r. Consider the manifold V 2n = (M2n \U)/S1. It is clear that
its boundary is a disconnected union of complex projective spaces ∂V 2n = CP 2n−2
1 ∪ . . .∪CP 2n−2
2k .
Let ∂0V 2n be a part of boundary of V 2n consist from r component CP 2n−2, that correspondent Ui
and ∂1V 2n be a part of boundary consist from component CP 2n−2, that correspondent Uj . On mani-
fold V 2n = (M2n \ U)/S1 constructed Morse function f : V → [0, 1], such that f−1(0) = ∂0V
2n
and f−1(1) = ∂1V
2n. Using the function f we constructed on manifold M2n S1-equivariant S1
∗ -Bott
function F with the state St(Λ) = (0, . . . , 0, 2n, . . . , 2n), such that restriction F on Ui coinside with
fi, restriction F on Uj coinside with fj and restriction π∗(F ) on V coinside with f. Therefore Morse
number of cobordism V equalMλ
S1
(
M2n, St(Λ)
)
. In the paper [14] there is value of Morse number
of a cobordism.
Theorem 6.1 is proved.
1. Матвев С. В., Фоменко А. Т., Шарко В. В. Круглые функции Морса и изоэнергетические поверхности интег-
рируемых гамильтоновых систем // Мат. сб. – 1988. – 135, № 3. – С. 325 – 345.
2. Фоменко А. Т., Цишанг Х. О топологии трехмерных многообразий, возникающих в гамильтоновой механи-
ке // Докл. АН СССР. – 1987. – 294, № 2. – С. 283 – 287.
3. Barden D. Simply-connected five manifolds // Ann. Math. – 1965. – 82, № 3. – P. 365 – 385.
4. Sharko V. V. Functions on manifolds: algebraic and topological aspects // Transl. Math. Monogr. – 1993. – 131.
5. Asimov D. Round handle and non-singular Morse – Smale flows // Ann. Math. – 1975. – 102, № 1. – P. 41 – 54.
6. Bott R. Lecture on Morse theory, old and new // Bull. Amer. Math. Soc. – 1982. – 7, № 2. – P. 331 – 358.
7. Franks J. Morse – Smale flows and homotopy theory // Topology. – 1979. – 18, № 2. – P. 199 – 215.
8. Franks J. Homology and Dynamical systems // CMBS Regional Conf. Ser. Math. – Providence, R. I.: Amer. Math.
Soc., 1982. – 49. – 120 p.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
1698 D. REPOVŠ, V. SHARKO
9. Kogan M. Existence of perfect Morse functions on spaces with semi-free circle action // J. Symplectic Geometry. –
2003. – 1, № 3. – P. 829 – 850.
10. Smale S. On the structure of manifolds // Amer. J. Math. – 1962. – 84, № 3. – P. 387 – 399.
11. Miyoshi S. Foliated round surgery of codimension-one foliated manifolds // Topology. – 1983. – 21, № 3. – P. 245 – 262.
12. Morgan J. W. Non-singular Morse – Smale flows on 3-dimensional manifolds // Topology. – 1979. – 18, № 1. –
P. 41 – 53.
13. Morgan J. W., Tian T. G. Ricci flowand the Poincaré conjecture // Amer. Math. Soc. – 2007. – 3. – 570 p.
14. Sharko V. V. New L2-invariants of chain complexes and applications, C∗-algebra and elliptic theory // Trends Math.
– Basel, Switzerland: Birkhäuser, 2006. – P. 291 – 321.
15. Thurston W. Existence of codimension-one foliation // Ann. Math. – 1976. – 104, № 2. – P. 249 – 268.
Received 12.10.12
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 12
|