Morse-Bott functions on manifolds with semi-free circle action
Let W²ⁿ be a closed manifold of dimension ≥ 6 with semi-free circle having finitely many fixed points. We study S¹-invariant Morse-Bott functions on W²ⁿ. The aim of this paper is to obtain exact values of minimal numbers of singular circles of some indexes of S¹-invariant Morse-Bott functions on W²ⁿ...
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| Cite this: | Morse-Bott functions on manifolds with semi-free circle action / V.V. Sharko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 518-523. — Бібліогр.: 4 назв. — англ. |
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| citation_txt | Morse-Bott functions on manifolds with semi-free circle action / V.V. Sharko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 518-523. — Бібліогр.: 4 назв. — англ. |
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| description | Let W²ⁿ be a closed manifold of dimension ≥ 6 with semi-free circle having finitely many fixed points. We study S¹-invariant Morse-Bott functions on W²ⁿ. The aim of this paper is to obtain exact values of minimal numbers of singular circles of some indexes of S¹-invariant Morse-Bott functions on W²ⁿ.
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Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 518-523
V. V. Sharko
Institute of Mathematics of NAS of Ukraine, Tereshchenkivs’ka
st. 3, Kyiv, 01601 Ukraine
E-mail: sharko@imath.kiev.ua
Morse-Bott functions on manifolds
with semi-free circle action
Let W 2n be a closed manifold of dimension ≥ 6 with semi-free circle having
finitely many fixed points. We study S1-invariant Morse-Bott functions on
W 2n. The aim of this paper is to obtain exact values of minimal numbers
of singular circles of some indexes of S1-invariant Morse-Bott functions on
W 2n.
Keywords: Semi-free circle action, manifold, Morse-Bott function, Morse
number.
1. S1-invariant Morse-Bott functions
Let W 2n be a closed smooth manifold. Suppose that W 2n
admits a smooth semi-free circle action with finitely many 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
conjugates 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 zi = xi+
√
−1yi.
The pair (U, h) will be called a standard chart at the point p.
Let f : W 2n → R be a smooth S1-invariant function on the mani-
fold W 2n. Denote by Σf the set of singular points of the function
c© V. V. Sharko, 2009
Morse-Bott functions on manifolds 519
f . It is clear that the set of isolated singular points Σf (pi) ⊂ Σf
of f coincides with the set of fixed points W S1
.
A point p ∈ W S1
is nondegenerate if the Hessian of the
function f at p is nondegenerate. For a nondegenerate fixed point
p there exist a standard chart (U, h) such that on U the function
f is given by the following formula:
f = f(p) − |z1|2 − . . .− |zλ|2 + |zλ+1|2 + . . .+ |zn|2.
Notice that the index of nondegenerate fixed point p is always
even.
Denote by Σf (S
1) the set singular points of the function f that
are disconnected union of circles. These circles will be called sin-
gular.
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 invari-
ant 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)))− |x1|2 − . . .− |xν |2 + |xν+1|2 + . . .+ |x2n−1|2.
By definition ν is the index of singular circle s.
Definition 1. A smooth S1-invariant function f : W 2n → R on
a manifold W 2n with a semi-free circle action which has isolated
fixed points is called an S1-invariant Morse-Bott function if each
connected component of the singular set Σf is either a nondegen-
erate fixed point or a nondegenerate critical circle, [3].
Definition 2. Assume that W 2n is the closed manifold with
a smooth semi-free circle action which has isolated fixed points
p1, . . . , pk. For any fixed point pi there exists a standard chart
(Ui, hi) such that each Ui is diffeomorphic to the unit disk D2n in
520 V. V. Sharko
Cn and that Ui are pairwise disjoint. Take any arbitrary integer
λi = 0, 1, . . . , n and define the following function on fi : Ui → R
by
fi = fi(pi) − |z1|2 − . . .− |zλi
|2 + |zλi+1|2 + . . . + |zn|2.
Theorem 1. Every smooth semi-free circle action on a closed
manifold with isolated fixed points p1, . . . , pk has an S1-invariant
Morse-Bott function f such that f = fi on Ui.
Proof. From results of paper [2] it follows that functions fi can be
extended from Ui to W 2n \⋃Ui. �
Theorem 2. The number of fixed points of any smooth semi-free
circle action on W 2n with isolated fixed points is always even and
equal to the Euler characteristic, χ(W 2n), of the manifold W 2n.
Proof. By Theorem 1 we construct on U1 the function
f1 = f1(p1) + |z1|2 + . . .+ |zn|2,
on Uj , (j ≥ 2) the function
fj = fj(pi) − |z1|2 − . . .− |zn|2
and extend such functions to S1-invariant Morse-Bott function f
on W 2n \⋃Ui. Since the manifold CPn is non-cobordant to zero
it follows that the number of fixed points of any smooth semi-free
circle action on W 2n with isolated fixed points is equal to the Euler
characteristic χ(W 2n) = 2k of W 2n. �
Definition 3. Let f be an S1-invariant Morse-Bott function for
smooth semi-free circle action with isolated fixed points p1, . . . , p2k
on a closed manifold W 2n. Suppose that the index of a critical
point pi of f is λi. The state of f is the collection of numbers
λ1, λ2, . . . , λ2k, which we will be denoted by Stf (λi).
Remark 1. From Theorem 1 it follows that for every smooth
semi-free circle action on a closed manifold W 2n with isolated fixed
points p1, . . . , p2k and any collection numbers λ1, λ2, . . . , λ2k, such
that 0 ≤ λi ≤ 2n there exists an S1-invariant Morse-Bott functions
Morse-Bott functions on manifolds 521
f on W 2n with state Stf (λi). Such a collection of numbers will be
denoted by St(λi) and called a state.
Definition 4. Let W 2n be a closed smooth manifold with smooth
semi-free circle action which has finitely many fixed points. The
S1-equivariant Morse number Mν
S1(W
2n, St(λi)) of index
ν of a state St(λi) of W 2n is the minimum number of singular
circles of index ν taken over all S1-invariant Morse-Bott functions
on W 2n with state St(λi).
The S1-equivariant Morse number Mν
S1(W
2n) of index ν
of W 2n is the minimum number of Mν
S1(W
2n, St(λi)) taken over
all states.
The S1-equivariant Morse number MS1(W 2n, St(λi)) of
a state St(λi) is the minimum number of singular circles of all
indices taken over all S1-invariant Morse-Bott functions on W 2n
with state St(λi).
The S1-equivariant Morse number MS1(W 2n) of W 2n is
the minimum number of MS1(W 2n, St(λi)) taken over all states.
There is an unsolved problem: for a manifold W 2n with a semi-
free circle action which has finitely many fixed points find ex-
act values of the numbers Mν
S1(W
2n, St(λi)), Mν
S1(W
2n),
MS1(W 2n, St(λi)), and MS1(W 2n).
Definition 5. An S1-invariant Morse-Bott function f on the
manifold W 2n with semi-free circle action which has finitely many
fixed points is
minimal for index ν of a state St(λi) if the number of
singular circles of f of index ν is equal to Mν
S1(W
2n, St(λi));
minimal for index ν if the number of singular circles of f of
index ν is equal to Mν
S1(W
2n);
minimal for state St(λi) if the number of all singular circles
of f is equal to MS1(W 2n, St(λi));
minimal if the number of all singular circles of f is equal to
MS1(W 2n).
522 V. V. Sharko
Theorem 3. Let W 2n (2n > 5) be a closed smooth simply-
connected manifold admits a smooth semi-free circle action with
isolated fixed points p1, . . . , p2k. Then on the manifold W 2n for the
state St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
) there exists a minimal (minimal for
index ν) S1-invariant Morse-Bott function g for the state
St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)
and
MS1(W 2n, St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)) =
=
n−1∑
i=1
µ(Hi((W
2n/S1) \ (pl+1 ∪ . . . ∪ p2k), p1, . . . , pl,Z)+
+
n−2∑
i=2
µ(Tors(Hi((W
2n/S1) \ (pl+1 ∪ . . . ∪ p2k), p1, . . . , pl,Z),
(
Mν
S1
(
W 2n, St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)
)
=
= µ
(
Hν((W
2n/S1) \ (pl+1 ∪ . . . ∪ p2k), p1, . . . , pl,Z)
)
+
+ µ
(
Tors(Hν−1((W
2n/S1) \ (pl+1 ∪ . . . ∪ p2k), p1, . . . , pl,Z)
))
,
where 0 ≤ l ≤ 2k (µ(H) – minimal number of generators of group
H).
Proof. Choose an invariant neighborhood Ui of the point pi diffeo-
morphic to the unit disc D2n ⊂ Cn and set U =
⋃
i Ui. Consider
the manifold V 2n = (W 2n \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
k .
Morse-Bott functions on manifolds 523
The set W 2n/S1 is simply-connected. It is easy to see using van
Kampen theorem that (W 2n \ U)/S1 is simply-connected as well.
From S. Smale’s theorem [4] is follows that on (W 2n \U)/S1 there
exists a minimal Morse function which we used to construct an
S1-invariant Morse-Bott function g for state
St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)
on the manifold W 2n. The values of
MS1
(
W 2n, St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)
)
and
Mν
S1
(
W 2n, St(0, . . . , 0︸ ︷︷ ︸
l
, 2n, . . . , 2n︸ ︷︷ ︸
2k−l
)
)
follow from S. Smale’s theorem and simple homology calcu-
lation. �
Remark 2. Using diagrams technique, [1], one can give estimates
for equivariant Morse number for other states. This will be made
in forthcoming paper.
References
[1] V. V. Sharko. Functions on Manifolds: Algebraic and topological
aspects, Translations of Mathematical Monographs, 131, American
Mathematical Society, 1993.
[2] A. Jankowski, R. Rubinsztein. Functions with non-degenerate critical
points on manifolds with boundary // Comment. Math. Prace Mat. –
1972. – V. 16. – PP. 99–112.
[3] M. Kogan. Existence of perfect Morse functions on spaces with semi-
free circle action // Journal of Symplectic Geometry. – 2003. – V. 1. –
PP. 829–850.
[4] S. Smale. On the structure of manifolds // Amer. Journal of Math. –
1962. – V. 84. – PP. 387-399.
|
| id | nasplib_isofts_kiev_ua-123456789-6331 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-2910 |
| language | English |
| last_indexed | 2025-12-07T17:38:50Z |
| publishDate | 2009 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Sharko, V.V. 2010-02-23T14:53:18Z 2010-02-23T14:53:18Z 2009 Morse-Bott functions on manifolds with semi-free circle action / V.V. Sharko // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 518-523. — Бібліогр.: 4 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6331 Let W²ⁿ be a closed manifold of dimension ≥ 6 with semi-free circle having finitely many fixed points. We study S¹-invariant Morse-Bott functions on W²ⁿ. The aim of this paper is to obtain exact values of minimal numbers of singular circles of some indexes of S¹-invariant Morse-Bott functions on W²ⁿ. en Інститут математики НАН України Геометрія, топологія та їх застосування Morse-Bott functions on manifolds with semi-free circle action Article published earlier |
| spellingShingle | Morse-Bott functions on manifolds with semi-free circle action Sharko, V.V. Геометрія, топологія та їх застосування |
| title | Morse-Bott functions on manifolds with semi-free circle action |
| title_full | Morse-Bott functions on manifolds with semi-free circle action |
| title_fullStr | Morse-Bott functions on manifolds with semi-free circle action |
| title_full_unstemmed | Morse-Bott functions on manifolds with semi-free circle action |
| title_short | Morse-Bott functions on manifolds with semi-free circle action |
| title_sort | morse-bott functions on manifolds with semi-free circle action |
| topic | Геометрія, топологія та їх застосування |
| topic_facet | Геометрія, топологія та їх застосування |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/6331 |
| work_keys_str_mv | AT sharkovv morsebottfunctionsonmanifoldswithsemifreecircleaction |