Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces

Khokhliuk, Oleksandra; Maksymenko, Sergiy

doi:10.48550/arxiv.2210.11043

Cited by 1 publication

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“…Ivanov [19] proved that the group D(T, ∂T) of all diffeomorphism of T fixed on the boundary is contractible. Also in a recent joint paper [21] of the author with O. Khokhliuk it is computed the weak homotopy type of D lp (F ) and shown that all homotopy groups of D lp (F , ∂T) vanish. One of the main results of this paper accomplishes the results of [21] with the following theorem:…”

Section: Solid Torusmentioning

confidence: 99%

“…As a consequence of [19,21] we will get a description of homotopy types of the mentioned above groups, see Theorem 2.2 below. To formulate the corresponding statement we need to define certain subgroups of D lp (F ) which will also be used later in the paper.…”

Section: Solid Torusmentioning

confidence: 99%

“…contractibility of D(T, ∂T), is given in [19,Theorem 2]. The fact, that the inclusion T ⊂ D(T) is a homotopy equivalence, is classical and follows from contractibility of D(T, ∂T), see [21] for exposition of those results. Also for the proof that the corresponding induced map π 0 T = G → π 0 D(T) is an isomorphism see e.g.…”

Section: Solid Torusmentioning

confidence: 99%

“…[40,Theorem 14]. The upper horizontal weak homotopy equivalences are established in [21], and the right vertical homotopy equivalence are contained in Theorem 2.1. This implies that the lower arrows are weak homotopy equivalences as well.…”

Section: Solid Torusmentioning

confidence: 99%

“…In a recent series of joint papers by the author with O. Khokhliuk [23,24,22,21] there were developed several techniques for computations of homotopy types of diffeomorphisms groups of Morse-Bott and more general classes of singular foliations in higher dimensions. In particular, [21] computes the homotopy types of diffeomorphisms groups of leaf preserving diffeomorphisms for mentioned above foliations of the solid torus and lens spaces. In this paper we find the relationship between those groups and the groups of diffeomorphisms sending leaves to leaves, see Theorems 2.1, 2.2, 3.1, 3.4 below.…”

Section: Introductionmentioning

confidence: 99%

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Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2

Maksymenko¹

2023

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Let F be a Morse-Bott foliation on the solid torus T = S 1 × D 2 into 2-tori parallel to the boundary and one singular circle S 1 × 0. A diffeomorphism h : T → T is called foliated (resp. leaf preserving) if for each leaf ω ∈ F its image h(ω) is also leaf of F (resp. h(ω) = ω). Gluing two copies of T by some diffeomorphism between their boundaries, one gets a lens space L p,q with a Morse-Bott foliation F p,q obtained from F on each copy of T . Denote by D f ol (T, ∂T ) and D lp (T, ∂T ) respectively the groups of foliated and leaf preserving diffeomorphisms of T fixed on ∂T . Similarly, let D f ol (L p,q ) and D lp (L p,q ) be respectively the groups of foliated and leaf preserving diffeomorphisms of F p,q . In a recent joint paper of the author it is shown that D lp (T, ∂T ) is weakly contractible (all homotopy groups vanish), which allowed also to compute the weak homotopy type of D lp (L p,q ). In the present paper it is proved that D lp (T, ∂T ) is a strong deformation retract of D f ol (T, ∂T ). As a consequence the weak homotopy type of D f ol (L p,q ) for all possible pairs (p, q) is computed.

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Section: Solid Torusmentioning

confidence: 99%

Section: Solid Torusmentioning

confidence: 99%