Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary
Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0...
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Інститут математики НАН України
2023
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Transactions of Institute of Mathematics of NAS of Ukraine| _version_ | 1872553040562618368 |
|---|---|
| author | Maksymenko, Sergiy Максименко, Сергій |
| author_facet | Maksymenko, Sergiy Максименко, Сергій |
| author_institution_txt_mv | [
{
"author": "Sergiy Maksymenko",
"institution": "Institute of Mathematics of the National Academy of Science of Ukraine"
}
] |
| author_sort | Maksymenko, Sergiy |
| baseUrl_str | https://trim.imath.kiev.ua/index.php/trim/oai |
| collection | OJS |
| datestamp_date | 2023-08-17T09:06:54Z |
| description | Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliation $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$. |
| doi_str_mv | 10.3842/trim.v20n1.532 |
| first_indexed | 2026-08-04T01:09:00Z |
| format | Article |
| fulltext |
Збірник праць Ін-ту математики НАН України (2023) т. 20, №1, 896–910
Diffeomorphism groups of Morse-Bott
foliation on the solid Klein bottle by
Klein bottles parallel to the boundary
Sergiy Maksymenko
Abstract. Let G be a Morse-Bott foliation on the solid Klein bottle K
into 2-dimensional Klein bottles parallel to the boundary and one singular
circle S1. Let also S1 r̂S2 be the twisted bundle over S1 which is a union
of two solid Klein bottles K0 and K1 with common boundary K. Then
the above foliation G on both K0 and K1 gives a foliation G1 on S1 r̂S2
into parallel Klein bottles and two singluar circles. The paper computes
the homotopy types of groups of foliated (sending leaves to leaves) and leaf
preserving diffeomorphisms for foliations G and G1.
Анотація. Нехай G – шарування Морса-Ботта на заповненій пляшці
Кляйна K на двовимірні пляшки Кляйна паралельні до межі BK та
центральне коло S1. Нехай також S1 r̂S2 – тотальний простір єдиного
нетривіального S2-розшарування над колом S1, яке є об’єднанням двох
копій K0 та K1 заповненої пляшки Кляйна зі спільною межею K. Тоді
шарування G на кожній копіїK0 таK1 визначає шарування G1 на S1 r̂S2
на паралельні пляшки Кляйна та два сингулярні кола. В роботі обчисле-
но гомотопічні типи груп дифеоморфізмів шарувань G та G1.
1. INTRODUCTiON
Let D2 = t|w| ď 1u be the unit disk in the complex plane, S1 = BD2 be
its boundary, and T = S1 ˆD2 be the solid torus. Define the following C8
function
f : T Ñ [0; 1], f(z, w) = |w|2,
and for every r P [0; 1] let Tr := f´1(r) be the inverse image of r. Evidently,
Tr is a 2-torus for r P (0; 1] and T0 is a circle. Let also
F = tTr | r P [0; 1]u
be the partition on T into the inverse images of f .
Notice that f is a Morse-Bott function (in some sense the most simplest
one), and the corresponding partition F is aMorse-Bott foliation, see e.g. [1,
2020 Mathematics Subject Classification: 57R30, 57T20
Keywords: foliation; diffeomorphism; homotopy type; solid Klein bottle
DOI : http://dx.doi.org/10.3842/trim.v20n1.532
896
Morse-Bott foliation on the solid Klein bottle 897
6,17]. Such foliations play and important role in Hamiltonian and Poisson
geometries, however in the present paper we will not use this interpretation.
Given a partition F on a manifold M we will say that a diffeomorphism
h : M Ñ M is F-leaf preserving if it leaves invaraint each leaf of F , i.e.
h(ω) = ω for all ω P F . Also, h is F-foliated whenever the image h(ω)
of each leaf ω P F is again a (perhaps some other) leaf of F . Then for a
subset X Ă M we will denote by Dlp(F , X) and Dfol(F , X) respectively
the groups of F-leaf preserving and F-foliated diffeomorphisms of M . If
X = ∅, we will omit it from notations. Evidently, Dlp(F , X) is a normal
subgroup of Dfol(F , X).
In a series of previous papers by the author and O. Khokhliuk [9–12,
16] there were computed homotopy types of groups of foliated and leaf
preserving diffeomorphisms of the above foliation F on T. Namely, the
following results are obtained:
Theorem 1.1 ([12, Theorem 1.1.1]). The group Dlp(F , BT) is weakly con-
tractible.
Theorem 1.2 ([16]). The pair
(
Dlp(F),Dlp(F , BT)) is a strong deforma-
tion retract of the pair
(
Dfol(F),Dfol(F , BT)).
It is also known that the group D(T, BT) of diffeomorphisms of T fixed
on its boundary is contractible (N. Ivanov [8, Theorem 2]), while the group
of all diffeomorphisms D(T), contains as a deformation retract a certain
semidirect product A := (S1 ˆ S1) ¸ U , where
U = t(ε n
0 δ) | ε, δ P t˘1u, n P Zu Ă SL(2,Z)
(this is a classical result). In particular, π0D(T) – U . In fact, A is con-
tained even in Dlp(F , BT), and Theorems 1.1 and 1.2 imply that the fol-
lowing inclusions are weak homotopy equivalences:
tidTu � � // Dlp(F , BT) � � // Dfol(F , BT) � � // D(T, BT),
A � � // Dlp(F) �
� // Dfol(F) �
� // D(T).
Furthermore, gluing two copies T0 and T1 of the solid torus by some dif-
feomorphisms between their boundaries, one gets a lens space Lp,q. Then
the foliation F on each of those tori gives a foliation Fp,q on Lp,q into two
singluar circles and parallel 2-tori. In [12,16] there were also computed the
homotopy types of the groups Dlp(Fp,q) and Dfol(Fp,q).
In the present paper we will make similar computations for the non-
orientable counterpatrs of T and lens spaces: the solid Klein bottle K and
the twisted S2-bundle over the circle S1 r̂ S2.
898 S. Maksymenko
More precisely, consider the following orientation reversing diffeomor-
phism ξ : T Ñ T of order 2 given by ξ(w, z) = (w̄,´z). Then the quotient
space K := T/ξ is called the solid Klein bottle. It is a non-orientable 3-
manifold and the corresponding quotient map p : T Ñ K is its orientable
double covering. Moreover, it is evident that f ˝ξ = f , whence there exists a
C8 function g : K Ñ [0; 1] such that f = g ˝p. For t P [0; 1] let Kt = g´1(t)
be the corresponding level set of g, so K0 is a circle, while for t P (0; 1] the
set Kt is a (2-dimensional) Klein bottle. In particular, K1 = BK.
Note also that Tr = p´1(Kt) and the restriction maps p : Tr Ñ Kt
(oriented for r ą 0) double coverings.
Let G = tKtutP[0;1] be the partition of K into level sets of g. Then g is
a Morse-Bott function for which K0 is a non-degenerate critical manifold,
and all other points of K are regular for g. Therefore one can regard G as
a Morse-Bott foliation with the singluar leaf K0.
Our aim is to compute the homotopy types of the groups Dfol(G) and
Dlp(G) of G-foliated and G-leaf preserving diffeomorphism ofK respectively,
and their respective subgroups fixed on BK. In fact most of the preliminary
work is done in the mentioned above papers, and here we will just use
their results for explicit computations. We are also aimed here to illustrate
usefulness of the developed methods.
First we recall the following statement:
Lemma 1.3. D(K, BK) is contractible, while D(K) is homotopy equivalent
to S1 ˆ Z2 ˆ Z2, i.e. to the disjoint union of 4 circles.
History of proof. W. B. R. Likorish [14] shown that π0D(K) – Z2 ‘Z2 and
that each diffeomorphism of BK extends to some diffeomorphism ofK. The
latter can be rephrased so that the restriction map ρ : D(K) Ñ D(BK),
ρ(h) = h|BK, (being also a continuous homomorphism) is surjective. Evi-
dently, its kernel is the group D(K, BK) of diffeomorphisms of K fixed on
the boundary. Notice that the map ρ is known to be a locally trivial fibra-
tion which is a particular case of “local triviality for embeddings” statement
independently proved by J.Cerf [2], R. Palais [18], and E. Lima [15].
Also, N. Ivanov [8] obtianed a general result on Waldhausen manifold
which includes the statement that D(K, BK) is contractible.
This implies that ρ is a homotopy equivalence. Moreover, it is also proved
by C. Earle and J. Eeels [5] and A. Gramain [7] that the path components
of D(BK) are homotopy equivalent to the circle. Hence D(K) is homotopy
equivalent to S1 ˆ Z2 ˆ Z2, i.e. to the disjoint union of 4 circles. □
Morse-Bott foliation on the solid Klein bottle 899
Our first result shows that the corresponding G-foliated and G-leaf pre-
serving counterparts of the groups from Lemma 1.3 have the same homo-
topy types. So the situation here is literally the same as in Theorems 1.1
and 1.2.
Theorem 1.4. The following statements hold.
(1) The group Dlp(G, BK) is weakly contractible.
(2) The pair
(
Dlp(G),Dlp(G, BK)
)
is a strong deformation retract of the
pair
(
Dfol(G),Dlp(G, BK)
)
.
They imply that the following maps denoted by (w.)h.e. are (weak) homo-
topy equivalences:
tidKu � � w.h.e //
h.e
++
Dlp(G, BK)
� � h.e // Dfol(G, BK)
� � w.h.e // D(K, BK),
Dlp(G) � � h.e // Dfol(G) � � w.h.e. // D(K)
ρ
h.e.
// // D(BK) » \
4
S1,
where the notations in bold denote new results and implications.
This theorem will be proved in Section 3.
As mentioned above a lens space is a 3-manifold obtianed by gluing two
solid tori T0 and T1 by some diffeomorphism between their boundaries,
and there are infinitely many mutually non-diffeomorphic lens spaces. On
the other hand, since every diffeomorphism of the Klein bottle K extends
to a diffeomorphism of the solid Klein bottle K, it follows that gluing two
copies of K by some diffeomorphism of their boundaries gives always rise
to the same manifold S1 r̂ S2 being a total space of a unique non-trivial
S2-bundle over S1 called the twisted S2-bundle over S1.
Therefore one can regard S1 r̂ S2 as the union of two Solid klein bot-
tles G0 and G1 glued by the indentity diffeomorphism of their boundaries.
On each Ki, i = 0, 1, we have defined above the foliation G into paralled
2-dimensional Klein bottles and one singluar circle. These foliations con-
stitute together a foliation on S1 r̂ S2 into parallel Klein bottles and two
singluar circles S1
i Ă Ki. We will denote that foliation by pG, and it will be
convenient to call it polar.
As a consequence of Theorem 1.4 we get the following description of
the homotopy types of pG-foliated and pG-leaf preserving diffeomorphisms of
S1 r̂ S2. Notice that each h P Dlp(pG) leaves invariant the common boun-
dary BK0 = BK1 which we will denote by K. Hence we have a well-defined
continuous restriction homomorphism ρ : Dlp(pG) Ñ D(K), ρ(h) = h|K .
900 S. Maksymenko
Its kernel is eveidently the group Dlp(pG,K) of pG-leaf preserving diffeomor-
phisms fixed on K.
Denote by Dfol
+ (pG) the (index 2) subgroup of Dfol(pG) consisting of diffe-
omorphisms leaving invaraint each singular circle S1
0 and S1
1 . Our second
result if the following Theorem 1.5 which will be proved in Section 4:
Theorem 1.5. The following statements hold.
(1) The group Dlp(pG) is a strong deformation retract of Dfol
+ (pG).
(2) The “restriction to K := BK0 = BK1 homomorphism”
ρ : Dlp(pG) Ñ D(K)
is a weak homotopy equivalence.
In particular, Dlp(pG) and Dfol
+ (pG) are weakly homotopy equivalent to the
disoint union of 4 circles, while Dfol(pG) is homotopy equivalent to the
disjoint union of 8 circles.
Note that M. Kim and F. Raymond [13], shown that
π0D(S1 r̂ S2) » Z2 ‘ Z2,
and the generators of that group can be chosen to be also the generators of
π0Dlp(pG). This gives the following
Corollary 1.6. The inclusions Dlp(pG) Ă Dfol
+ (pG) Ă D(S1 r̂ S2) induce
bijections on π0 groups:
π0Dlp(pG) = π0Dfol
+ (pG) = π0D(S1 r̂ S2) = Z2 ‘ Z2.
On the other hand, the homotopy type of D(S1 r̂ S2) is more compli-
cated. It was described in E. César and C. Rourke [4, Theorem 2], which
in turn extended the technnique from PhD theses by E. César [3].
2. TWiSTED BUNDLES OVER THE CiRCLE
In this sections we present an explicit model for the solid Klein bottle K
and the universal covers of K and KzK0. These notations will be used in
the proof of Theorem 1.4.
2.1. Universal cover of the solid Klein bottle K. Let
κ : R ˆ C Ñ R, κ(s, w) = s,
be the trivial vector bundle (of real dimension 2), and g1 : R ˆ C Ñ R be
a C8 function given by g1(s, w) = |w|2. It determines a norm (or scalar
product) on the fibers of κ.
Morse-Bott foliation on the solid Klein bottle 901
Consider the following C8 vector bundle isomorphism (ξ, η) reversing
orientation and having no fixed points:
R ˆ C
ξ: (s, w) ÞÑ (s+1, w̄) //
κ
��
R ˆ C
κ
��
R
η: s ÞÑ s+1 // R
(2.1)
It defines a free and properly discontinuous action of Z on R ˆ C. Denote
by S1 r̂ C = (RˆC)/Z the quotient space. Then R/η – S1 and the corres-
ponding quotient maps β : R ˆ C Ñ S1 r̂ C and σ : R Ñ S1, σ(s) = e2πis,
are universal coverings maps. Moreover, we get the well-defined quotient
vector bundle γ : S1 r̂ C ” (R ˆ C)/ξ Ñ R/η ” S1 such that the following
diagram is commutative:
R ˆ C
β //
κ
��
S1 r̂ C
γ
��
R σ // S1
Evidently, g1 ˝ ξ = g1, whence there exists a unique C8 function
g : S1 r̂ C Ñ R,
such that g1 = g ˝ β. Then K := g´1
(
[0; 1]
)
is the solid Klein bottle,
β : R ˆD2 Ñ K is the universal cover of K, and
G = tKr := g´1(r)urP[0;1]
consists of level sets of g. Note that we also have a norm on the fibers
| ¨ | : S1 r̂ C Ñ [0;+8], |x| = a
g(x),
for x P S1 r̂ C, so Kr consists of elements of S1 r̂ C of norm ?
r.
For r P [0; 1] denote:
Kr := g´1
(
[0; r]
)
= tx P S1 r̂ C | |x| ď ?
ru, Cr := g´1
(
[r; 1]
)
.
Thus Kr is a “thinner” solid Klein bottle with boundary Kr, while Cr is a
collar of the boundary Klein bottle BK. In particular, K0 is a circle being
also the zero section of γ.
2.2. Universal cover of KzK0. Consider also another universal covering
map
α : R2 ˆ (0;+8) Ñ R ˆ (Czt0u), α(s, ϕ, r) = (s, re2πiϕ).
Then the composition:
ζ : R2 ˆ (0; 1]
αÝÝÑ R ˆ (D2zt0u) βÝÝÑ KzK0
902 S. Maksymenko
is the universal covering map for KzK0. Evidently, for each leaf Kr, r ą 0,
its inverse ζ´1(Kr) is the horizontal plane R2 ˆ r. In other words, the
composition
g2 := g1 ˝ α : R2 ˆ (0;+8) Ñ (0;+8)
is just the coordinate projection, g2(s, ϕ, r) = r.
One easily checks that the group of covering translations of ζ is generated
by the following diffeomorphisms
a,b : R2 ˆ (0 + 8) Ñ R2 ˆ (0 + 8),
a(s, ϕ, r) = (s, ϕ+ 1, r), b(s, ϕ, r) = (s+ 1,´ϕ, r), (2.2)
and that the following identities holds:
α ˝ a = α, α ˝ b = ξ ˝ α, b ˝ a = a´1 ˝ b.
Let us collect all those spaces and maps into the following commutative
diagram:
R2 ˆ (0; 1]
α
��
� � //
ζ
&&
R2 ˆ (0;+8)
α
�� g2
��
R ˆ (D2zt0u) � � //
β
��
R ˆD2 � � //
β
��
R ˆ C
β
��
g1
&&LL
LLL
LLL
LLL
L
KzK0
γ
--
� � // K = g´1([0; 1])
γ
��
� � // S1 r̂ C
γ
qq
g // R
S1
(2.3)
FiGURE 2.1. Universal coverings of K and KzK0
Morse-Bott foliation on the solid Klein bottle 903
2.3. Liftings of diffemorphisms. We will also use the following simple
statement concerning covering maps:
Lemma 2.4. Let β : X Ñ Y be a covering map between path connected
topological spaces, and ξ : X Ñ X be any covering transformation, i.e. a
homeomorphism satisfying β ˝ ξ = β. Let also h : Y Ñ Y be a continuous
map having a lifting h1 : X Ñ X, so β ˝ h1 = h ˝ β. Then each of the
following conditions implies that h1 commutes with ξ, i.e. ξ ˝ h1 = h1 ˝ ξ.
(1) There exists a point x P X such that ξ ˝ h1(x) = h1 ˝ ξ(x).
(2) There exists a subset A Ă Y being invariant under ξ, i.e. ξ(A) = A,
and such that h1 is fixed on A.
Proof. (1) Notice that ξ ˝ h1 and h1 ˝ ξ are also liftings of h, since
ξ ˝ h1 ˝ β = ξ ˝ β ˝ h = β ˝ h, h1 ˝ ξ ˝ β = h1 ˝ β = β ˝ h.
Moreover, by assumption, they coincide at x. Then by uniqueness of liftings
with one prescribed value, they should identically coincide on all of X.
(2) Let x P A be any point. Then, by assumption, ξ(x) P A. As h1 is fixed
on A we have that h1 ˝ ξ(x) = ξ(x) = ξ ˝ h1(x). Hence by (1), h1 ˝ ξ ” ξ ˝ h1
on all of X. □
Notice that each h P D(K, BK) lifts to a unique diffeomorphism
h1 : R ˆD2 Ñ R ˆD2
fixed on R ˆ S1, so β ˝ h1 = h ˝ β. Since R ˆ S1 is invariant under ξ, it
follows from Lemma 2.4 that h1 commutes with ξ.
Moreover, suppose in addition that h(K0) = K0, h1(Rˆ0) = Rˆ0, then
the restriction h1|Rˆ(D2zt0u) lifts in turn to a unique diffeomorphism
h2 : R2 ˆ (0; 1] Ñ R2 ˆ (0; 1]
fixed on R2 ˆ 1, so
α ˝ h2 = h1 ˝ α : R2 ˆ (0; 1] Ñ R ˆ (D2zt0u).
Again, since R2 ˆ 1 is invariant under a and b, we get from Lemma 2.4
that h2 commutes with a and b. These liftings h1 and h2 of h will play an
important role for our proofs.
Let us also mention that the group Dlp(G, BK) can be defined as the
subgroup of D(K, BK) consisting of diffeomorphisms preserving the func-
tion g, i.e. satisfying the identity: g ˝ h = g. Moreover, if h P Dlp(G, BK),
both liftings h1 and h2 are defined and they satisfy the following idenitities:
g1 ˝ h1 = g1 and g2 ˝ h2 = g2.
904 S. Maksymenko
3. PROOF OF THEOREM 1.4
The second statement (2) that the pair
(
Dlp(G),Dlp(G, BK)
)
is a strong
deformation retract of
(
Dfol(G),Dlp(G, BK)
)
is a particular case of results
from [16].
For the proof of (1) we need to define an explicit model for the solid
Klein bottle K and the universal covers of K and KzK0.
(1) The proof of contractibility of the group Dlp(G, BK) almost literally
follows the proof of main result from [12] for similar foliation on the solid
torus, since the results of that paper are proved in a greater generality and
are applicable in the current situation. For the convenience of the reader
we will repeat certain arguments. Namely, we will define four subgroups of
Dlp(G, BK):
B4 Ă B3 Ă B2 Ă B1 Ă B0 = Dlp(G, BK),
and show1 that all the inclusions are homotopy equivalences, while the
smallest group B4 is weakly contractible.
To proceed with the proof it will be convenient to fix some C8 function
µ : [0; 1] Ñ [0; 1] such that µ = 0 on [0; 0.2] and µ = 1 on [0; 0.8].
1) Inclusion B1 Ă B0.
Let us define the group B1. Let h P B0 = Dlp(G, BK) be any element
and h1 : R ˆ D2 Ñ R ˆ D2 be its unique lifting fixed on R ˆ S1. Since
h(K0) = K0, it follows that h1(Rˆ0) = Rˆ0, so there exists an orientation
preserving diffeomorphism σ(h) : R Ñ R commuting with ξ and such that
h1(s, 0) =
(
σ(h)(s), 0
)
, s P R.
is R ˆ 0 is also invariant under ξ. In particular,
h1 ˝ ξ(s, 0) = h1(s+ 1, 0) =
(
σ(h)(s+ 1), 0
)
,
ξ ˝ h1(s, 0) = ξ(σ(h)(s), 0) =
(
σ(h)(s) + 1, 0
)
,
so σ(h)(s + 1) = σ(h)(s) + 1 for all s P R. Let D+
η (R) be the group of
all orientation preserving diffeomorphisms q of R satisfying the identity
q(s + 1) = q(s) + 1 for all s P R. Then the correspondence h ÞÑ σ(h) is a
well-defined map σ : B0 Ñ D+
η (R). One easily check that σ is a continuous
homomorphism.
Let B1 := ker(σ) be the kernel of σ. Thus B1 consists of G-leaf preserving
diffeomorphisms h such that their unique lifting h1 to the universal cover
R ˆD2 fixed on R ˆ S1 is also fixed on R ˆ 0.
1We also “reorder” here the groups Bi in comparison with their counterparts from [12].
Namely, we will make our diffeomorphisms fixed on the collar of BK at the fourth step,
while in [12] that was done from the beginning. This will slightly simplify the exposition.
Morse-Bott foliation on the solid Klein bottle 905
We claim that B1 is a strong deformation retract of B0. The proof it
easy and consists of two statements. Let us recall the arguments from [12].
a) σ admits a continuous section s : D+
η (R) Ñ Dlp(G, BK) = B0 satisfy-
ing s(idR) = idK. Thus s is a continuous map (not necessarily a homomor-
phism) such that σ ˝ s(g) = g for all g P D+
η (R).
Indeed, note that each q P D+
η (R) extends to a diffeomorphism
q̂ : R ˆD2 Ñ R ˆD2, q̂(w, s) =
(
w, µ(|w|)q(s) + (1 ´ µ(|w|)s))
fixed even on the set p´1(C0.8) = t(w, s) | |w| P [0.8; 1]u. One easily checks
that q̂ also commutes with ξ, and preserves the sets p´1(Kr), r P [0; 1], being
the inverses of the leaves of G. Hence q̂ yields a unique diffeomorphism
s(q) of K preserving the leaves of G and fixed on C0.8. In other words,
s(q) P Dlp(G, BK) = B0. Moreover, q̂ is in turn a unique lifting of s(q) fixed
on RˆS1, whence the correspondence q ÞÑ s(q) is the desired section of σ.
b) The group D+
η (R) is convex, and therefore contractible into the point
idR via the “stantard” homotopy:
H : D+
η (R) ˆ [0; 1] Ñ D+
η (R), H(q, t) = (1 ´ t)q + t idR.
Then a) allows to construct the following homeomorphism
η : B1 ˆ D+
η (R) ” ker(σ) ˆ D+
η (R) Ñ Dlp(G,C0.8),
η(h, q) = h ˝ s(q).
which is “fixed on B1” in the sense that η(h, idR) = h for all h P B1. As
D+
η (R) is contractible into the point idR, it now follows that B1 is a strong
deformation retract of B0 = Dlp(G,C0.8). We refer the reader to [12] for
the details.
2) Inclusion B2 Ă B1.
Define B2 to be the subgroup of B1 consisting of diffeomorphisms h co-
inciding with some vector bundle morphism q : S1 r̂ C Ñ S1 r̂ C on K0.2.
To see what this means consider the standard disk bundle γ : K Ñ S1
from (2.3), and for every y P S1 and r P (0; 1] denote by
Dr(y) = γ´1(y) XKr
the closed 2-disk of radius r in the fibre over y. Notice that the intersec-
tions of Dr(y) with the leaves of G constitute the partition of Dr(y) into
concentric circles. The following lemma is easy and directly follows from
definitions of the above covering maps.
Lemma 3.1. Let h P B1 then the following conditions are equivalent:
(1) h P B2;
906 S. Maksymenko
(2) h(D0.2(y)) = D0.2(y) for each y P S1 and the restriction
h : Dr(y) Ñ Dr(y)
is a rotation (i.e. a linear isomorphism preserving concentric circles);
(3) there exists a C8 function λh : R Ñ R such that
h1(w, s) = (we2πiλh(s), s), λh(s+ 1) = ´λh(s),
for all s P R and w P D2 with |w| ď 0.2;
(4) there exists a C8 function λh : R Ñ R such that
h2(s, ϕ, r) = (s, ϕ+ λh(s), r), λh(s+ 1) = ´λh(s),
for all s P R and r P (0; 0.2];
In this case such a function λh in (3) and (4) is the same, and it is also
unique. □
Proof. Equivalence of conditions (1)-(4) is easy. It also follows from (4)
that λh is uniquely determined by h2. □
Now, by “linearization theorem” [11], the inclusion B2 Ă B1 is a homo-
topy equivalence. Notice that in our situation G consists of level sets of a
positive definite 2-homogeneous on fibers function g. In this case the proof
of that “linearization theorem” can be simplified as it was shown in [12, The-
orem 3.1.2].
More precisely, the deformation of B1 into B2 can be defined as follows.
Let U be a neighborhood of the central circle K0 (i.e. the zero section of
p : S1 r̂ C Ñ S1) and h : U Ñ S1 r̂ C a smooth embedding such that
h(K0) = K0, but it is not necessarily fixed on K0. Then one can define the
following vector bundle isomorphism
Tfib(h) : S
1 r̂ C Ñ S1 r̂ C, Tfib(h)(x) = lim
tÑ0
1
th(tx),
which can be regarded as a “partial derivative” of h at points of K0 in
the direction of fibers of the vector bundle p : S1 r̂ C Ñ S1, see [11]. In
particular, Tfib(h) is well defined for all h P B1. It is easy to see that if
h P B2, so it coincides with some vector bundle morphism q on K0.2, then
Tfib(h) ” q on all of S1 r̂ C.
Define also the following function
ϕ : [0; 1] ˆ (S1 r̂ C) Ñ R, ϕ(t, x) = t+ (1 ´ t)µ(|x|).
Evidently, ϕ(t, x) = 0 exactly on the set 0 ˆ K0.2. Now a deformation
H : B1 ˆ [0; 1] Ñ B1 of B1 into B2 can be given by the following formula:
H(h, t)(x) =
#
h(ϕ(t,x)x)
ϕ(t,x) , (t, x) P (
[0; 1] ˆK)z(0 ˆK0.2
)
,
Tfib(h)(x), (t, x) P 0 ˆK0.2.
Morse-Bott foliation on the solid Klein bottle 907
Inbdeed, one can show (using Hadamard lemma) that H(h, t) is a diffeo-
morphism of K belonging to B1 for all (h, t) P B1 ˆ [0; 1], and the map H
is continuous with respect to the corrresponding C8 Whitney topologies,
see [12, Theorem 3.1.2]. Moreover, it is evident, that
H1(h) = h, H0(h)|K0.2 = Tfib(h)|K0.2 .
The latter means that H0(h) coincides with the vector bundle morphism
Tfib(h) on K0.2, and thus H0(h) belongs to B2. Finally, if h is already in B2,
then it easily follows from the formulas for H, that Ht(h) = h = Tfib(h) on
K0.2. Thus H is a homotopy of B1 which deforms B1 into B2 and leaves B2
invariant. This means that H is a deformation of B1 into B2, and therefore
the inclusion B2 Ă B1 is a homotopy equivalence whose homotopy inverse
is the map H0 : B1 Ñ B2.
3) Inclusion B3 Ă B2.
Denote by C8̊(R,R) the subset of the space C8(R,R) consisting of func-
tions δ : R Ñ R satisfying the identity δ(s+ 1) + δ(s) ” 0 for all s P R.
Recall that, by Lemma 3.1, to each h P B2 one associates a unique
C8 function λh : R Ñ R satisfying conditions of Lemma 3.1. In particular,
λh P C8̊(R,R), and thus we get a well-defined map λ : B2 Ñ C8̊(R,R). One
easily checks λh1˝h2 = λh1 + λh2 for all h1, h2 P B2, so the correspondence
h ÞÑ λh is a continuous homomorphism of topological groups.
Let B3 := ker(λ) be the kernel of λ, so it consists of elements h P B2 for
which λh ” 0, i.e. h2 is fixed on R2 ˆ (0; 0.2]. One easily checks that the
following two statements hold.
a) C8̊(R,R) is convex and therefore contractible.
b) The homomorphism λ : B2 Ñ C8̊(R,R) admits a continuous section
s : C8̊(R,R) Ñ B2. Actually, for each δ P C8̊(R,R) we have the
following diffeomorphism
h1
δ : R ˆD2 Ñ R ˆD2, h1
δ(s, w) =
(
s, we2πi(1´µ(r))δ(s)
)
.
Evidently, it is fixed near R ˆ S1, fixed on R ˆ 0, commutes with the
covering translation ξ : R ˆ D2 Ñ R ˆ D2 and preserves the function
g1. Hence h1
δ yields a unique diffeomorphism hδ : K Ñ K, which is
fixed near BK and preserves g. In turn, h1
δ is lifting of hδ and is of the
form (3) of Lemma 3.1. Hence hδ P B2.
Now, similarly to the proof for the inclusion B1 Ă B0, these conditions
imply that B3 is a strong deformation retract of B2.
4) Inclusion B4 Ă B3.
Let B4 be the subgroup of B3 consisting of G-leaf preserving diffeomor-
phisms fixed on the collar C0.8. Then the inclusion B4 Ă B3 is a homotopy
equivalence. Indeed, the deformation H : B3 ˆ [0; 1] Ñ B3 of B3 into B4 can
908 S. Maksymenko
be given by:
H(h, t)(x) =
#
h
(
(t|x| + (1 ´ t)µ(|x|)) x
|x|
)
, |x| ą 0,
h(x), |x| ď 0.2.
(3.1)
Thus H1(h) = h, H0 is fixed on C0.8.
5) Weak contractibility of B4.
Similarly to the last step of the proof of [12, Theorem 1.1.1] the group
B4 can be embedded into the loop space of the group of diffeomorphisms
of K, and that inclusion is a weak homotopy equivalence.
Namely, for each h P B4 define the following path γh : [0; 1] Ñ D(K)
given by
γh(t)(x) =
#
x, t = 0,
1
th(tx), t P (0; 1].
Since h is fixed near K0 and preserves “parallel” Klein bottles Kt, t P (0; 1],
it follows that γh is a well-defined continuous loop such that γ(t) = idK for
t P [0; 0.2] Y [0.8; 1].
Moreover, the additional assumptions that for each h P B4
´ its lifting h1 is fixed near R ˆ 0
´ while the lifting h2 of h|KzK0
is fixed on R2 ˆ (0; 0.2],
imply that h actually represents a null-homotopic loops in Ω(Did(K)). Thus
the correspondence h ÞÑ γh is an embedded of B4 into the path component
Ω0(Did(K), idK) of Ω(Did(K), idK) consisting of null-homotopic loops.
It is shown in [10, Corollary 1.10] that the latter inclusion
B4 Ă Ω0(Did(K), idK)
is a weak homotopy equivalence. In particular, for i ě 1 we have the
following isomorphisms:
πiB4 = πiΩ0(Did(K), idK) = πi+1Did(K) = πi+1S
1 = 0.
Hence all homotopy groups of B4 vanish.
This completes the proof of Theorem 3. □
Remark 3.2. Formula (3.1) for the homotopy in the case 4) is also appli-
cable for all h P Dlp(G) not only belonging to B3, and it gives a deformation
of Dlp(G) into Dlp(G,C0.8). In particular, Dlp(G,C0.8) is also weakly con-
tractible.
4. PROOF OF THEOREM 1.5
Let S1 r̂ S2 be the twisted S2-bundle over S1 glued from two copies ofK0
and K1 by some diffeomorphism of their boundaries, and K := BK0 = BK1
Morse-Bott foliation on the solid Klein bottle 909
be their common boundary. Let also pG be the foliation into two circles and
Klein bottles parallel ot K.
Statement (1) that the group Dlp(pG) is a strong deformation retract of
Dfol
+ (pG) is a direct consequence of results from [16].
(2) We should prove that the “restriction to K homomorphism”
ρ : Dlp(pG) Ñ D(K)
is a weak homotopy equivalence.
As noted above, due to [2,15,18], this homomorphism is a locally trivial
fibration whose fiber, Dlp(pG,K), is the group of diffeomorphisms fixed on
K. Since ρ is surjective, it suffices to show weak contractibility of Dlp(pG,K).
Let C be a neighborhood of K being a union of collars C0.8 of K in
K0 and K1. Then the inclusion Dlp(pG, C) Ă Dlp(pG,K) is a homotopy
equivalence. The proof is similar to the formula (3.1) in the proof of the
case 4) of Theorem 1.4.
On the other hand, Dlp(pG, C) is homeomorphic to the product
Dlp(pG,C0.8) ˆ Dlp(pG,C0.8)
of two copies of Dlp(pG,C0.8) being weakly contractible by Remark 3.2.
Hence Dlp(pG, C) is weakly contractible as well. Therefore Dlp(pG,K) is also
weakly contractible. Theorem 1.5 is completed.
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Sergiy Maksymenko
ALGEBRA AND TOPOLOGY DEPARTMENT
INSTiTUTE OF MATHEMATiCS OF NATiONAL ACADEMY OF SCiENCES OF UKRAiNE
TERESHCHENKiVSKA STR., 3, KYiV, 01024, UKRAiNE
Email: maks@imath.kiev.ua
ORCID: 0000-0002-0062-5188
|
| id | oai:trim.imath.kiev.ua:article-532 |
| institution | Transactions of Institute of Mathematics of NAS of Ukraine |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-08-04T01:09:00Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | ojs |
| resource_txt_mv | trimimathkievua/00/434228c387279ae7e7c30f707e037700.pdf |
| spelling | oai:trim.imath.kiev.ua:article-5322023-08-17T09:06:54Z Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary Групи дифеоморфізмів шарування Морса-Ботта на заповненій пляшці Клейна пляшками Клейна паралельними межі Maksymenko, Sergiy Максименко, Сергій Let $\mathcal{G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{\times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{K}_0$ and $\mathbf{K}_1$ with common boundary $K$. Then the above foliation $\mathcal{G}$ on both $\mathbf{K}_0$ and $\mathbf{K}_1$ gives a foliation $\mathcal{G}'$ on $S^1\widetilde{\times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{G}$ and $\mathcal{G}'$. Нехай $\mathcal{G}$ -- шарування Морса-Ботта на заповненій пляшці Кляйна $\mathbf{K}$ на двовимірні пляшки Кляйна паралельні до межі $\partial\mathbf{K}$ та центральне коло $S^1$. Нехай також $S^1\widetilde{\times}S^2$ -- тотальний простір єдиного нетривіального $S^2$-розшарування над колом $S^1$, яке є об'єднанням двох копій $\mathbf{K}_0$ та $\mathbf{K}_1$ заповненої пляшки Кляйна зі спільною межею $K$. Тоді шарування $\mathcal{G}$ на кожній копії $\mathbf{K}_0$ та $\mathbf{K}_1$ визначає шарування $\mathcal{G}'$ на $S^1\widetilde{\times}S^2$ на паралельні пляшки Кляйна та два сингулярні кола. В роботі обчислено гомотопічні типи груп дифеоморфізмів шарувань $\mathcal{G}$ та $\mathcal{G}'$. Інститут математики НАН України 2023-08-17 Article Article application/pdf https://trim.imath.kiev.ua/index.php/trim/article/view/532 10.3842/trim.v20n1.532 Transactions of Institute of Mathematics, the NAS of Ukraine; Vol. 20 No. 1 (2023): Modern problems of mathematics and its applications, III; 896-910 Сборник Трудов Института математики НАН Украины; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 896-910 Збірник Праць Інституту математики НАН України; Том 20 № 1 (2023): Сучасні проблеми математики та її застосувань III; 896-910 3083-7529 1815-2910 en https://trim.imath.kiev.ua/index.php/trim/article/view/532/517 Авторське право (c) 2023 Sergiy Maksymenko http://creativecommons.org/licenses/by/4.0 |
| spellingShingle | Maksymenko, Sergiy Максименко, Сергій Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title | Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title_alt | Групи дифеоморфізмів шарування Морса-Ботта на заповненій пляшці Клейна пляшками Клейна паралельними межі |
| title_full | Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title_fullStr | Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title_full_unstemmed | Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title_short | Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary |
| title_sort | diffeomorphism groups of morse-bott foliation on the solid klein bottle by klein bottles parallel to the boundary |
| url | https://trim.imath.kiev.ua/index.php/trim/article/view/532 |
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