Semi-free $R^1$ action and Bott map
Let $M^{n}$ be a compact closed manifold of dimension at least 3. We study the $R^1$-Bott functions on $M^{n}$. Separately investigated $R^1$-invariant Bott functions on $M^{2n}$ with a semi-free circle action which has finitely many fixed points. The aim of this paper is to find exact...
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| author | Sharko, V. Gol’cov, D. Шарко, В. Гольцов, Д. |
| author_facet | Sharko, V. Gol’cov, D. Шарко, В. Гольцов, Д. |
| author_institution_txt_mv | [
{
"author": "V. Sharko",
"institution": "Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev"
},
{
"author": "D. Gol’cov",
"institution": "Institute of Mathematics of the National Academy of Sciences of Ukraine, Kiev"
}
] |
| author_sort | Sharko, V. |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2018-02-10T20:56:26Z |
| description | Let $M^{n}$ be a compact closed manifold of dimension at least 3. We study the $R^1$-Bott functions on $M^{n}$. Separately investigated $R^1$-invariant Bott functions on $M^{2n}$ with a semi-free circle action which has finitely many fixed points. The aim of this paper is to find exact values of minimal numbers of singular circles of some indices of $R^1$-invariant Bott functions on $M^{2n}$.
Closely related to $ R^1 $-Bott function on a manifold $ M^n $ is a more flexible object, the decomposition of round handle of $ M^n $. In its turn, to study the round handles decomposition of $M^n $ we use a diagram, i.e. a graph which carries the information about the handles. |
| first_indexed | 2026-08-04T01:05:27Z |
| format | Article |
| fulltext |
Збiрник праць Iн-ту математики НАН України 2013, том 6, N 6, 224–235
V. Sharko, D. Gol’cov
Semi-free R1 action and Bott map
1 Introduction
Let Mn be a compact closed manifold of dimension at least 3. We
study the R1-Bott functions onMn. Separately investigated R1-invariant
Bott functions on M2n with a semi-free circle action which has finitely
many fixed points. The aim of this paper is to find exact values of minimal
numbers of singular circles of some indices of R1-invariant Bott functions
on M2n.
Closely related to R1-Bott function on a manifold Mn is a more
flexible object, the decomposition of 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 R1-Bott maps
Let Mn be a smooth manifold and f : Mn → R1 smooth function
or f : Mn → R1 non-homotopy to zero a smooth map. Suppose that
x ∈ Mn one of its critical points of f . In neighborhood U of critical
point x in both cases the map f can be viewed as a function with values
in R. 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 their dimensions do not exceed n − 1.
A connected critical submanifold Ki0
j0
is called non-degenerate if the
c© V. Sharko, D. Gol’cov, 2013
V. Sharko, D. Gol’cov 225
Hessian is non-degenerate on subspaces orthogonal toKi0
j0
(i.e. has corank
equal to n− i0) at each point x ∈ Ki0
j0
.
Definition 2.1. A mapping f : Mn → R1 is called a Bott map 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 map:
Definition 2.2. A mapping f : Mn → R1 is called an R1-Bott map if
all of its critical points form nondegenerate critical circles.
Note that an R1-Bott map do not exist on any smooth manifold (see
Theorem 2.3).
Theorem 2.1. Let Mn be a smooth closed manifold and suppose that on
Mn there is R1- Bott map f : Mn → R1. Denote by γ ⊂Mn its critical
circle and let f(γ) = a. Then there is interval (a − ε, a + ε) ⊂ R1 and
a system of coordinates in a neighborhood of γ of one of the following
types:
1) Trivial ν : S1×Dn−1(ε)→Mn; where Dn−1(ε), a disc of radius ε,
ν(R1×0) = γ, and f(ν(θ, x)) = a−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)) = a−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 → R1 an R1-Bott map.
Each nice R1-Bott map defines a filtration on manifold Mn : M0(f) ⊂
M1(f) ⊂ ... ⊂ Mn−1(f) ⊂ Mn. The existence of a nice R1-Bott map
from manifoldMn into the circle is equivalent to existance of a R1-round
handle decomposition on the manifold Mn. We recall some necessary
definitions.
226 Semi-free R1 action and Bott map
Definition 2.3. We define an n-dimensional round handle Rλ of index
λ by Mλ = M1 ×Dλ ×Dn−λ−1, where Di is a disc of dimension i.
Define twisted n-dimensional round handle TMλ of index λ (0 < λ <
n−1) by TMλ = [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).
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 : R1×∂Dλ×Dn−λ−1 −→ ∂Nn
is a smooth embedding.
Manifold Mn
λ is obtained from a smooth manifold Mn by gluing a
twisted round handles of index λ, ifMn
λ = Nn
⋃
ϕ[0, 1]×Dλ×Dn−λ−1/ ∼,
where ϕ : ([0, 1]× ∂Dλ ×Dn−λ−1/ ∼)→Mn is a smooth embedding.
Definition 2.5. The M1- round handle decomposition on the closed
manifold Mn is called a filtration
Mn−1 × [0, ε]
⋃
Mn
0 (R) ⊂Mn
1 (R) ⊂ ... ⊂Mn
n−1(R) = Mn,
whereMn−1 is a closed submanifold ofMn, the manifoldMn
i (R) obtained
from the manifold Mn
i−1(R) by gluing round and twisted round handles of
index i .
In what follows we recall the relationship between S1 and the decom-
position by round handles ([11]).
Theorem 2.2. Let Mn be a smooth closed manifold. The following two
conditions are equivalent:
1) On the manifold Mn there is a nice R1-Bott map 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 coordi-
nate systems.
2) ManifoldMn 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 cor-
responds to a round handle Rλi
(1 ≤ i ≤ k), and the critical circle
γ̃j corresponds to a twisted round handle TRµj
(1 ≤ j ≤ l).
V. Sharko, D. Gol’cov 227
Thus each nice R1-Bott map from manifoldMn into the R1 generates
a round handle decomposition of Mn and vice versa.
We are interested in conditions when an R1-Bott map on Mn has the
property that all of its critical circles have trivial coordinate system. We
recall the necessary facts from an [4].
Lemma 2.1. Let Mn be a smooth closed manifold, f : Mn → R1 an
R1-Bott map, 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 in-
dices µ1, ..., µl with twisted coordinate systems, then the homology groups
H∗(f−1[c − ε, c + ε], f−1(c − ε),Z) is generated exactly by the handles
which correspond to the critical circles
γ1, ..., γk, γ̃1, ..., γ̃l. Each circle γi generates two subgroups that are iso-
morphic 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).
Corolary 2.1. Let Mn be a smooth closed manifold, f : Mn → R1 an
S1-Bott map, 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 groups H∗(f−1[ci − εi, ci +
εi], f
−1(ci − εi),Z) are free Abelian groups.
Thus we have a homological criterion when R1-Bott map do not have
critical circle with twisted coordinate systems.
In the next section, we give another class of R1-Bott map which do
not possess the critical circle with twisted coordinate systems.
Definition 2.6. Let Mn be a smooth closed manifold. The number
χi(M
n) = µ(Hi(M
n,Z))−µ(Hi−1(Mn,Z))+ . . .+(−1)i+1µ(H0(Mn,Z))
is called the i-th Euler characteristic of Mn,where µ(H) is a minimal
number of generators H.
Definition 2.7. 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(Mn) = χλ+1(Mn) = 0.
Definition 2.8. Let Mn be a smooth closed manifold. A round handle
decomposition is called quasiminimal, if one of the following holds:
228 Semi-free R1 action and Bott map
1) the number of round handles of index i equals to ρ(χi(M
n)) + εi,
where εi = 0,if dimension i+ 1 is nonsingular and εi = 1,if dimen-
sion 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(Mn)). 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 tech-
nique, we can easily prove the following fact [4].
Proposition 2.1. 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 dimensions, then Mn admits
a minimal decomposition into round handles.
Definition 2.9. Let the manifold Mn admits R1-Bott function, then
R1-Morse number MR1
i (Mn) of index i is the minimum number of
singular circles of index i taken over all R1-Bott functions on Mn.
Lemma 2.2. 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 non-
degenerate 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 non-degenerate critical circle S1
j of index λ can be replaced by non-
degenerate critical points of idexes λ and λ+ 1 [1]. Therefore the contri-
bution in the formula of Euler characteristic this critical points will not
give and we obtain the desired formula. �
3 Manifolds with free R1-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 →
V. Sharko, D. Gol’cov 229
Mn/S1 is fibre bundle. Any smooth R1-invariant map f : Mn → R1
from the manifold Mn into the circle R1 is called an R1-invariant
round Bott map if each connected component of the singular set Σf is
non-degenerate critical circle.
It is clear that if f be a R1-invariant round Bott map from the man-
ifold Mn then it projection π∗(f) : Mn/S1 → R1, is a Morse map. And
conversaly, if g : Mn/S1 → R1 be a Morse map from the manifoldMn/S1
then π−1
∗ (g) = g ◦π : Mn → S1 is R1-invariant round Bott map from the
manifold Mn. The critical point of the index λ of the map g correspond
to critical circle of the index λ of the map π−1
∗ (g).
Definition 3.1. Let on smooth manifold Mn there are smooth free
circle action θ : Mn × S1 → Mn and R1-invariant round Bott map
f : Mn → S1. For the triple (Mn, θ, f) R1-equivariant round Morse-
Bott number of index i, MeqS1
i (Mn, θ, f) is the minimum number of
singular circles of index i taken over all homotopic to f R1-invariant
round Bott map from Mn into R1.
Definition 3.2. Let on smooth manifold Mn there is Morse maps f :
Mn → R1. For the couple (Mn, f) Morse-Novikov number of index
i, Mi(M
n, f) is the minimum number of critical points of index i taken
over all homotopic to f Morse maps from Mn into R1.
It is clear that there is following fact.
Corolary 3.1. Let on smooth manifold Mn there is smooth free circle
action θ : Mn×R1 →Mn and let p : Mn →Mn/R1 is natural projection.
Suppose that f : Mn/R1 → R1 be a Morse map. Then MeqR1
i (Mn, θ, f ·
p) = Mi(M
n/S1, f).
Definition 3.3. Let on smooth manifold Mn there is smooth free cir-
cle action θ : Mn × R1 → Mn. Then this circle action is minimal
if there exist R1-invariant round Bott map f : Mn → R1 such that
MeqR1
i (Mn, θ, f) = MS1
i (Mn, f) for all i.
Suppose that on smooth compact manifoldMn(n > 6) there is smooth
free circle action θ : Mn×R1 →Mn and let p : Mn →Mn/R1 is natural
projection. Suppose that π1(Mn) ≈ π1(Mn/R1) ≈ Z. Then from from
results of Novikov [2] it follows that
Mi(M
n/R1, f) = µ(Hi(M
n/R1, Z)) + µ(TorsHi−1(Mn/R1, Z))
230 Semi-free R1 action and Bott map
for any non-homotopy to zero Morse map f : Mn/R1 → R1. Therefore
corollary 3.1 implies that MeqR1
i (Mn, θ, f · p) = MR1
i (Mn, f)
Theorem 3.1. Let on smooth compact manifold Mn(n > 6) there is
smooth free circle action. Suppose that π1(Mn) ≈ π1(Mn/S1) ≈ Z.
Then this circle action is minimal if and only if
µ(Hi(M
n/S1, Z) + µ(TorsHi−1(Mn/S1, Z) = ρ(χi(M
n))
for all i.
Proof. Necessary. Suppose that on Mn there is minimal smooth
free circle action. If n > 6 from results of Novikov [2] it follows that
Morse number in dimension i of the manifold Mn/R1 is equal
Mi(M
n/S1) = µ(Hi(M
n/R1, Z)) + µ(TorsHi−1(Mn/R1, Z)). There is
equality
Mi(M
n/S1) = MeqR1
i (Mn). Because of the condition of minimal free
circle action there is equality Mi(M
n/R1) = MeqR1
i (Mn) = MR1
i (Mn) =
ρ(χi(M
n)).
Sufficiently. Consider on manifold Mn/R1 Morse function with the
number of critical points of index i equal
Mi(M
n/R1) = µ(Hi(M
n/R1, Z)) + µ(TorsHi−1(Mn/R1, Z)). By the
construction and condition of the theorem we have the equalities
Mi(M
n/S1) = MeqR1
i (Mn) = ρ(χi(M
n)). But MR1
i (Mn) = ρ(χi(M
n))
and therefore free action of R1 is minimal. �
4 Manifolds with semi-free R1-action
Let M2n be a closed smooth manifold with semi-free R1-action which
has only isolated fixed points. It is known that every isolated fixed point p
of a semi-free R1-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)
on Cn = R2n with zj = xj +
√
−1yj . The pair (U, h) will be called a
standard chart at the point p. Let f : M2n → R1 be a smooth R1-
invariant map from the manifold M2n into the circle R1. Denote by Σf
V. Sharko, D. Gol’cov 231
the set of singular points of the map f . It is clear that the set of isolated
singular points Σf (pj) ⊂ Σf of f coincides with the set of fixed points
MR1
.
For a nondegenerate critical point pj there exist a standard chart
(Uj , hj) such that on Uj the map 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 (R1) the set singular points of the function f that are
disconnected union of circles. These circles will be called singular.
A circle s ∈ Σf(R
1) is called nondegenerate if there is an R1-invariant
neighborhood U of s on which R1 acts freely and such that the point
π(s) is nondegenerate for the function π∗(f) : U/R1 → R, induced on
U/R1 by the natural map π : U → U/R1. An invariant version of Morse
lemma says that there exist an R1-invariant neighborhood U of the circle
s and coordinates (x1, . . . , x2n−1) on U/R1 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 4.1. A smooth S1-invariant function f : M2n → R on a
manifoldM2n with a semi-free circle action which has isolated fixed points
is called : R1
∗-Bott function if each connected component of the singular
set Σf is either a nondegenerate fixed point or a nondegenerate critical
circle.
Theorem 4.1. 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 R1-invariant R1
∗-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/R
1 by the natural map π : Uj →
Uj/R
1. It is clear that function π∗(fj) is smooth on manifold (Uj \
232 Semi-free R1 action and Bott map
pj)/R
1. Denote by g smooth extension functions π∗(fj) on M2n/R1. By
small deformation of the function g, that is fixed on Uj/R
1, we shall
find function g1 on M2n/R1 such that g1 equal π∗(fj) on Uj/R1 and g1
have only non-degenerate critical points on M2n \
⋃
(Uj/R
1). Then the
function f = g1 ◦ p satisfy conditions of the theorem. �
Theorem 4.2. 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.
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 diffeo-
morfic to open disk D2n for any j. Consider manifold 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 bound-
ary of the manifold V 2n/S1 is disconnected union of complex projective
spaces CPn−1. If the number of the boundary components of the man-
ifold 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 4.2. Let f be an R1-invariant S1
∗-Bott function for smooth
semi-free circle action 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 4.1. It follows from Theorem 4.2 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 R1-invariant R1
∗-Bott functions f on
M2n with state Stf (Λ).
Definition 4.3. Let M2n 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.
V. Sharko, D. Gol’cov 233
The R1-Morse number MR1
i (M2n, St(Λ)) of index i is the minimum
numbers of singular circles of index i taken over all R1-invariant R1
∗-Bott
functions f on M2n with state Stf (Λ).
There is an unsolved problem: for a manifold M2n with a semi-free
circle action which has finitely many fixed points find exact values of
numbersMR1
i (M2n, St(Λ)) .
5 About R1-equivariant Morse numbers
MR1
i (M 2n, St(Λ))
Let M2n be a compact closed manifold of dimension with semi-
free circle action which has finite many fixed points p1, , ..., p2k. De-
note by π : M2n → M2n/R1 canonical map. The set M2n/R1 is
manifold with singular points π(p1), , ..., π(p2k). It is clear that neigh-
borhood of any singular point is cone over CPn−1. If f : M2n → R
be a smooth R1-invariant R1
∗-Bott function on the manifold M2n, then
π∗(f) : M2n/R1 → R is continuos function such that on smooth non-
compact manifold N2n−1 = M2n/R1 \
⋃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)/R1, its boundary 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 R1-invariant R1
∗-Bott func-
tions on the manifold M2n with states St(Λ) = (0, . . . , 0, 2n, . . . , 2n).
Let ∂0V 2n−1 be a part of boundary of V 2n−1 consist from r compo-
nent 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. Us-
ing the function f we constructed on the manifold M2n R1-equivariant
R1
∗-Bott function F with the state St(0, . . . , 0, 2n, . . . , 2n), such that re-
striction π∗(F ) on V coinside with f . Therefore Morse number of in-
dex 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 5.1. Let M2n (2n > 8) be a closed smooth manifold admits a
smooth semi-free circle action with isolated fixed points p1, . . . , p2k. Then
234 Semi-free R1 action and Bott map
for the manifold M2n with the state St(Λ) = (0, . . . , 0, 2n, . . . , 2n)
MR1
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 diffeo-
morphic 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)/R1. 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 compo-
nent CP 2n−2, that corespondent Ui and ∂1V
2n be a part of boundary
consist from component CP 2n−2, that corespondent Uj . On manifold
V 2n = (M2n \ U)/R1 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 con-
structed on manifold M2n S1-equivariant S1
∗-Bott function F with the
state St(Λ) = (0, . . . , 0, 2n, . . . , 2n), such that restriction F on Ui coin-
side with fi, restriction F on Uj coinside with fj and restriction π∗(F )
on V coinside with f . Therefore Morse number of cobordism V equal
Mλ
R1(M2n, St(Λ)) In the paper [12] there is value of Morse number of a
cobordism. �
V. Sharko, D. Gol’cov 235
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|
| id | oai:trim.imath.kiev.ua:article-311 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:05:27Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/a6/5897356afa3170c249c756ab63130ea6.pdf |
| spelling | oai:trim.imath.kiev.ua:article-3112018-02-10T20:56:26Z Semi-free $R^1$ action and Bott map Напіввільна $R^1$ дія та відображення Ботта Sharko, V. Gol’cov, D. Шарко, В. Гольцов, Д. Let $M^{n}$ be a compact closed manifold of dimension at least 3. We study the&nbsp;$R^1$-Bott functions on $M^{n}$. Separately investigated $R^1$-invariant Bott functions on $M^{2n}$ with a semi-free circle action which has finitely many fixed points. The aim of this paper is to find exact values of minimal numbers of singular circles of some indices of $R^1$-invariant Bott functions on $M^{2n}$. Closely related to $ R^1 $-Bott function on a manifold $ M^n $ is a more flexible object, the decomposition of round handle of $ M^n $. In its turn, to study the round handles decomposition of $M^n $ we use a diagram, i.e. a graph which carries the information about the handles. Нехай $M^{n}$ - компактний замкнений многовид розмірності не менше 3.&nbsp; Ми вивчаємо&nbsp;&nbsp;$R^1$-функції Ботта на $M^{n}$. Окремо дослідженний $R^1$-інваріант функцій Ботта на $M^{2n}$ з напіввільною дією кола, яка має скінченне число фіксованих точок.&nbsp; Мета статті знайти точне значення мінімального числа сингулярних кіл для деяких індексів&nbsp; $R^1$-інваріанта функцій Ботта на $M^{2n}$. Тісно пов'язаним&nbsp; з $ R^1 $-функціями Ботта на многовиді $ M^n $ є більш гнучкіший об'єкт, розклад $ M^n $ на ручки. &nbsp; Для дослідження розкладу $M^n $ на ручки ми використовуємо діаграму, тобто граф який містить інформацію про ручки. Інститут математики НАН України 2013-06-26 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/311 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 10 No. 6 (2013): Brasilian-Polish Topology Workshop; 224-235 Сборник Трудов Института математики НАН Украины; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 224-235 Збірник Праць Інституту математики НАН України; Том 10 № 6 (2013): Бразильсько-польський симпозіум з топології; 224-235 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/311/290 Авторське право (c) 2013 Праці Інституту математики НАН України |
| spellingShingle | Sharko, V. Gol’cov, D. Шарко, В. Гольцов, Д. Semi-free $R^1$ action and Bott map |
| title | Semi-free $R^1$ action and Bott map |
| title_alt | Напіввільна $R^1$ дія та відображення Ботта |
| title_full | Semi-free $R^1$ action and Bott map |
| title_fullStr | Semi-free $R^1$ action and Bott map |
| title_full_unstemmed | Semi-free $R^1$ action and Bott map |
| title_short | Semi-free $R^1$ action and Bott map |
| title_sort | semi-free $r^1$ action and bott map |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/311 |
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