Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces 1 Oleksandra Khokhliuk and Sergiy Maksymenko

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Homotopy types of diﬀeomorphism groups of polar

Morse-Bott foliations on lens spaces, 1

Oleksandra Khokhliuk and Sergiy Maksymenko

Abstract. Let T=S1×D2be the solid torus, Fthe Morse-Bott foliation on Tinto 2-tori

parallel to the boundary and one singular circle S1×0, which is the central circle of the

torus T, and D(F, ∂T )the group of diﬀeomorphisms of Tﬁxed on ∂T and leaving each leaf

of the foliation Finvariant. We prove that D(F, ∂T )is contractible.

Gluing two copies of Tby some diﬀeomorphism between their boundaries, we will get

a lens space Lp,q with a Morse-Bott foliation Fp,q obtained from Fon each copy of T. We

also compute the homotopy type of the group D(Fp,q )of diﬀeomorphisms of Lp,q leaving

invariant each leaf of Fp,q .

1. Introduction

One of the main features and applications of algebraic topology and, especially, homotopy

theory, to real world problems is that very often homotopy invariants of “good” spaces are

discrete and can be described as certain properties “stable under small perturbations”.

On the other hand, the concrete computation of homotopy types can be done in not so

many cases. Actually, the homotopy type of a CW-complex Xis, in principle, determined

by Postnikov tower: one can associate to Xa certain “canonical” representative ¯

Xbeing

homotopy equivalent to X, see [25, §4.3, p. 410]. However, that construction is more theo-

retical, since ¯

Xis usually inﬁnite dimensional, and to construct ¯

Xone need to know all the

homotopy groups of X, which we currently do not completely know even for spheres. Real

computations of homotopy types usually are done by other methods, like theory of coverings

or ﬁbre bundles described in standard textbooks, e.g. [25].

For inﬁnite dimensional spaces, e.g. spaces of maps (inﬁnite dimensional manifolds), the

situation is more complicated. It is known that for a smooth compact manifold Xits group

of C∞diﬀeomorphisms D(X)endowed with C∞Whitney topology is a Fr´echet manifold,

and therefore has the homotopy type of a CW-complex, see [64] for discussion of homotopy

types of inﬁnite dimensional manifolds. Homotopy types of path components of D(X)are

completely computed for all compact manifolds with dim X≤2, [76,9,10,11,20]. There

is also a lot of results about the homotopy types of D(X)for dim X= 3, see e.g. [26,18,28]

and references therein. In higher dimensions it is known very little (just because that every

ﬁnitely presented group is the fundamental group of some manifold, so the classiﬁcation of

compact manifolds contains classiﬁcation of ﬁnitely presented groups), and the results are

2020 Mathematics Subject Classiﬁcation. 57R30, 57T20.

Key words and phrases. Foliation, diﬀeomorphism, homotopy type, lens space, solid torus.

arXiv:2210.11043v3 [math.AT] 1 Aug 2023

2 OLEKSANDRA KHOKHLIUK AND SERGIY MAKSYMENKO

mostly concern with identifying certain non-trivial elements of homotopy groups, e.g. [62,

75,21,8,43,4], see also good reviews by N. Smolentsev [77] and J. Milnor [61].

Another important class of inﬁnite dimensional spaces are groups of homeomorphisms and

diﬀeomorphisms preserving leaves of a certain foliation. Their homotopy types are studied

even worse. The most relevant results are obtained in the papers by and K. Fukui and

S. Ushiki [17] and K. Fukui [15]. They studied regular foliations on 3-manifolds with ﬁnitely

many Reeb components and proved that the identity path component of the group of foliated

(sending leaves to leaves) diﬀeomorphisms has the homotopy type of a product of circles, see

Remark 1.1.4 below.

Most of other results related with the structure of diﬀeomorphism groups of foliations

concern with extensions of results by M. Herman [27], W. Thurston [78], J. Mather [59,60],

D. B. A. Epstein [12] on proving perfectness groups of compactly supported diﬀeomorphisms

isotopic to the identity. Recall that a group is perfect, if it coincides with its commuta-

tor subgroup G= [G, G]. Then perfectness means triviality of the ﬁrst homology group

H1(K(G, 1),Z)of the corresponding Eilenberg-MacLane space, and this is where the ho-

motopy theory appears. The technique developed in the latter result by Epstein [12] was

extended by W. Ling [49] to a certain class of groups of homeomorphisms, see also [2].

That allowed in turn to extend the above results on perfectness of diﬀeomorphism groups,

to groups of leaf-preserving homeomorphisms and diﬀeomorphisms of nonsingular foliations,

see e.g. [68,70,69,72,73,22,1,47] and references therein.

However for singular foliations their groups of diﬀeomorphisms are less studied, e.g. [16,

71,55,46]. One of the important class of singular foliations constitute the so-called Morse-

Bott foliations, and in particular, foliations by level sets of Morse-Bott functions. They play

an important role in Hamiltonian dynamics and Poisson geometry, see e.g. [14,79,74,50,

81,57,58,13].

Also in [52] and [55, Theorem 3.7] the second author gave rather wide suﬃcient conditions

on a vector ﬁeld Fon a manifold Xunder which the identity path component Did(F)of the

group of orbit-preserving diﬀeomorphisms of Fis either contractible or homotopy equivalent

to the circle. Given h∈ Did(F)one can associate to each x∈Xthe time αh(x)between x

and h(x)along its orbit of F, see Section 2.6. Such a time αh(x)is not uniquely deﬁned,

however it is shown that certain assumptions on zeros of Fand the fact that h∈ Did(F)

allows to choose αhto be C∞and continuous in h.

That result as well as all the technique were applied further to study the homotopy types

of stabilizers and orbits of functions f∈ C∞(X, R)under the natural “left-right” action of the

product D(X)×D(R)on the space C∞(X, R)deﬁned by the following rule: (h, φ)·f=φ◦f◦h.

Notice that this action includes the natural “right” action of the subgroup D(X)×idRon

C∞(X, R). In [53] there were given wide conditions on funder which the inclusion of “right”

stabilizers and orbits into the corresponding “left-right” stabilizers and orbits are homotopy

equivalences. Moreover, for the case dim X= 2, the homotopy types of “right” stabilizers and

orbits were almost completely computed, see e.g. [54,51,56,45,44], and references therein.

However, the technique developed in the above papers is applicable only to 1-dimensional

foliations. Let us also mention that E. Kudryavtseva [38,39,40,41,42] studied the

homotopy types of spaces of Morse functions on surfaces, and partially rediscovering the

above results obtained a general structure of “right” orbits, see [56] for discussion.

In recent three papers [35,36,37] the present authors developed several techniques

for computations of homotopy types of groups of diﬀeomorphisms of Morse-Bott and more

DIFFEOMORPHISM GROUPS OF MORSE-BOTT FOLIATIONS ON LENS SPACES 3

general classes of “singular” foliations in higher dimensions. In this paper we present explicit

computations for some simplest Morse-Bott foliations on the solid torus and lens spaces,see

Theorems 1.1.2 and 7.1.2.

1.1. Main result. Let Sr={z∈C| |z|=r}r∈[0;1] be the family of concentric circles in

Ctogether with the origin S0= 0,

T={(w, z)∈C2| |z|= 1,|w| ≤ 1}∼

=S1×D2

be the solid torus in C2,T2:= S1×S1be its boundary, and F={S1×Sr}r∈[0,1] be the

partition of Tinto 2-tori parallel to the boundary and the central circle S1×0.

Equivalently, Fis a partition into the level sets of the following Morse-Bott function

f:T→R,f(w, z) = |z|2, for which the central circle S1×0is a non-degenerate critical

submanifold.

Say that a diﬀeomorphism h:T→Tis F-foliated if for each leaf ω∈ F its image h(ω)

is a (possibly distinct from ω) leaf of Fas well. Also his F-leaf preserving, if h(ω) = ωfor

each leaf ω∈ F. Deﬁne the following groups:

• D(T)is the group of diﬀeomorphisms of T;

• D(F)is the group of F-leaf preserving diﬀeomorphisms of T;

• Dﬁx(T, T 2)is the subgroup of D(T)consisting of diﬀeomorphisms ﬁxed on T2;

• Dﬁx(F, T 2) := D(F)∩ Dﬁx(T, T 2).

Endow all those groups with the corresponding C∞Whitney topologies. Our main result is

the following theorem proved in Section 5:

Theorem 1.1.1.The group Dﬁx(F, T 2)is weakly contractible, that is weakly homotopy

equivalent to a point (i.e. all its homotopy groups vanish).

Notice further that the homotopy types of the groups D(T)and Dﬁx(T, T 2)are well

known. In fact, Dﬁx(T, T 2)is contractible, see [29], while the homotopy type of D(T)can be

described as follows, see Section 6for details. Consider the following subgroup of GL(2,Z):

G:= ε0

m δ|m∈Z, ε, δ ∈ {±1}.

It naturally acts on R2by linear automorphisms and preserves the integral lattice Z2. Hence

it yields an action on 2-torus T2=R2/Z2by its automorphisms as a Lie group, and so one

can deﬁne the semidirect product T:= (R2/Z2)⋊Gcorresponding to that action. Moreover,

every element of Tnaturally extends to a certain F-leaf preserving diﬀeomorphism of T,

so we get an inclusion e:T ⊂ D(F). It is well known that the inclusion T ⊂ D(T)is a

homotopy equivalence, see Lemma 6.3.1 below. As a consequence of Theorem 1.1.1 we get

the following statement which will be proved in Section 6.4:

Theorem 1.1.2.Each of the following inclusions

Te

−−→ D(F)⊂ D(T),{idT} ⊂ Dﬁx(F, T 2)⊂ Dﬁx(T, T 2)

is a weak homotopy equivalence. In particular, every path component of D(F)is weakly

homotopy equivalent to 2-torus R,π0D(F)∼

=G, while πkDﬁx(F, T 2)=0for all k≥0.

Furthermore, gluing two copies of T(each equipped with the same foliation F) via some

diﬀeomorphism of T2, we will obtain a lens space Lp,q and a foliation Fp,q on it into two

singular circles and parallel 2-tori. We will call such a foliation polar. Then Theorem 1.1.1

will also allow us to compute the homotopy type of the group D(Fp,q)of Fp,q-leaf preserving

diﬀeomorphisms. Namely, the following statement holds:

4 OLEKSANDRA KHOKHLIUK AND SERGIY MAKSYMENKO

Theorem 1.1.3.The identity path component of D(Fp,q)is weakly homotopy equivalent

to 2-torus T2, and

π0D(Fp,q) = 









G,for L0,1=S1×S2,

Z2⊕Z2,for L1,0=S3, L2,1=RP3,

Z2for p > 2.

Moreover, if p > 2and q2̸=±1(mod p), then the inclusion D(Fp,q)⊂ D(Lp,q)is a weak

homotopy equivalence.

In fact, in Section 5we will obtain a more explicit description of the homotopy type of

D(Fp,q), see Theorem 7.1.2.

Remark 1.1.4.Notice that the above foliations Fand Fp,q are in some sense «dual» to

the ones considered by K. Fukui and S. Ushiki [17] and K. Fukui [15]. Namely, a solid torus

Tadmits a Reeb foliation Rhaving ∂Tas a leaf and being transversal to Fin the interior

of T. Further, let Lp,q be a lens space obtained by gluing two copies T0and T1of Tvia

some diﬀeomorphism between their boundaries. Then that Reeb foliation Rof each Tigives

a foliation Rp,q of the corresponding lens space Lp,q which has a joint leaf ∂T0=∂T1with

Fp,q and is transversal to other leaves of Fp,q.

Structure of the paper. For the proof of contractibility of G0=Dﬁx(F, T 2)in Theo-

rem 1.1.1 we will construct four nested subgroups G4⊂ G3⊂ G2⊂ G1⊂ G0and show that the

inclusions Gi+1 ⊂ Gi,i= 0,1,2,3, are homotopy equivalences, while the smallest group G4is

weakly contractible. Construction of a deformation of Giinto Gi+1,i= 0,1,2,3, requires a

speciﬁc technique, and in each case we will establish a more general result which holds for

foliations by level sets of arbitrary smooth and deﬁnite homogeneous functions.

Section 2contains several notations and constructions used throughout the paper and also

certain general results about ﬁberwise deﬁnite homogeneous functions on vector bundles. Let

p:E→Bbe a vector bundle over a smooth manifold Bwhich we identify with its image

in Eunder zero section. In Section 3we prove a “linearization” Theorem 3.1.2 allowing to

isotopy diﬀeomorphisms of the pair (E, B)to diﬀeomorphisms coinciding with some vector

bundle morphisms near B. That statement is a particular case of a recent preprint [37] by

the authors, and we present here another and independent proof. It will be used for the proof

of homotopy equivalence G3⊂ G2.

In Section 3.2 we also obtain several consequences of linearization theorem for vector

bundles of rank 1, and, in particular, Lemma 3.2.1 used for the inclusion G1⊂ G0, and

Lemma 3.2.2 used in the proof of Theorem 7.1.2. Also, in Section 3.3, we recall the results

from our paper [36] allowing to prove weak contractibility of G4, see Lemma 3.3.

In Section 4we prove Theorem 4.2.1 for the case B=S1which allows to “unloop diﬀeo-

morphisms from D(F)along the longitude” and prove homotopy equivalence G2⊂ G1.

The proofs of Theorem 1.1.1 and of the remained inclusion G4⊂ G3are given in Section 5.

In Section 6we recall the description of the homotopy types of groups D(T2),D(T),

Dﬁx(T, T 2), see Lemmas 6.1.1,6.3.1, and in particular, prove deduce Theorem 1.1.2 from

Theorem 1.1.1, see Section 6.4.

Finally, in Section 7, we compute the homotopy type of D(Fp,q), and prove Theorem 7.1.2

being a detailed variant of Theorem 1.1.3.

DIFFEOMORPHISM GROUPS OF MORSE-BOTT FOLIATIONS ON LENS SPACES 5

2. Preliminaries

For a smooth compact manifold Mwe will denote by Did(M)the identity path component

of the group D(M)of its diﬀeomorphisms with respect to C∞Whitney topology. Also the

arrows →and →→ will mean monomorphism and epimorphism respectively.

2.1. Deformations. Let Xbe a topological space and H:X×[0; 1] →Xbe a homo-

topy. A subset A⊂Xwill be called invariant under Hif H(A×[0; 1]) ⊂A. Then is Ha

deformation of Xinto Aif H0= idX,H(A×[0; 1]) ⊂A, and H1(X)⊂A. It is well known

and is easy to see that in this case the corresponding inclusion map i:A⊂Xis a homotopy

equivalence and the map H1:X→Ais its homotopy inverse.

Moreover, if B⊂Xis another subset being invariant under H, then the restriction

H|B×[0;1] :B×[0; 1] →Bis a deformation of Binto A∩B, whence the inclusion A∩B⊂B

is a homotopy equivalence as well. In that case Hcan be regarded as a deformation of the

pair (X, B)into the pair (A, A ∩B), so the inclusion (A, A ∩B)⊂(X, B)is a homotopy

equivalence of pairs. We will use this observation several times.

2.2. Notation for several diﬀeomorphism groups. Throughout the paper the term

smooth means C∞and all manifolds and diﬀeomorphisms are assumed to be smooth if the

contrary is not said. Also all spaces of smooth maps, and in particular, diﬀeomorphism

groups, are endowed with C∞Whitney topology.

Let Mbe a manifold. Then for each subset B⊂Mwe will use the following notations1:

• D(M, B) = {h∈ D(M)|h(B) = B}is the group of diﬀeomorphisms of Mleaving B

invariant;

• Dﬁx(M, B)is the group of diﬀeomorphisms of Mﬁxed on B;

• Dnb(M, B)is the group of diﬀeomorphisms of Mﬁxed on some neighborhood of B

(which may vary for distinct diﬀeomorphisms).

Moreover, suppose Bis a proper submanifold of M, i.e. ∂B =B∩∂M, and p:E→Bis

a regular neighborhood of B, i.e. a retraction of an open neighborhood E⊂Mof Bendowed

with a structure of a vector bundle. Then, in addition,

• D(M, B, p)is the subgroup of D(M, B)consisting of diﬀeomorphisms hwhich coincide

with some vector bundle morphism ˆ

h:E→Eon some neighborhood of B(depending on

h); note that Dnb(M, B)⊂ Dﬁx(M, B, p)since every diﬀeomorphism hﬁxed near Bcoincides

thus with the identity automorphism idEof p;

• Dﬁx(M, B, p) := D(M, B, p)∩ Dﬁx(M, B);

• D(M, B, p, U)is the subset2of D(M, B, p)consisting of diﬀeomorphisms hwhich coin-

cide with some vector bundle morphism ˆ

h:E→Eon a given neighborhood Uof B;

• Dﬁx(M, B, p, U) := D(M, B, p, U)∩ Dﬁx(M, B).

Finally, let Q⊂Mbe a subset and η: [0; 1] →Ma path such that η(0), η(1) ∈Q.

•Then D0

ﬁx(M, Q, η)will denote the subgroup of Dﬁx(M, Q)consisting of diﬀeomorphisms

for which the paths ηand h◦η(having common ends) are homotopic relatively their ends.

1Those notations are used in diﬀerent parts of the paper, so the reader may skip this subsection and refer

to it on necessity.

2In general, it is not a group, since Umight not be not invariant under diﬀeomorphisms from D(M, B, p)

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HomotopytypesofdiffeomorphismgroupsofpolarMorse-Bottfoliationsonlensspaces,1OleksandraKhokhliukandSergiyMaksymenkoAbstract.LetT=S1×D2bethesolidtorus,FtheMorse-BottfoliationonTinto2-toriparalleltotheboundaryandonesingularcircleS1×0,whichisthecentralcircleofthetorusT,andD(F,∂T)thegroupofdiffeomorphism...

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Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces 1 Oleksandra Khokhliuk and Sergiy Maksymenko

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