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

Abstract

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.

Diffeomorphism groups of Morse-Bott foliation on the solid Klein bottle by Klein bottles parallel to the boundary

“…Statement (1) that the group D lp ( p G) is a strong deformation retract of D f ol + ( p G) is a direct consequence of results from [16]. (2) We should prove that the "restriction to K homomorphism"…”

“…In a series of previous papers by the author and O. Khokhliuk [9][10][11][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 D lp (F, BT) is weakly contractible.…”

“…Then the foliation F on each of those tori gives a foliation F p,q on L p,q into two singluar circles and parallel 2-tori. In [12,16] there were also computed the homotopy types of the groups D lp (F p,q ) and D f ol (F p,q ). In the present paper we will make similar computations for the nonorientable counterpatrs of T and lens spaces: the solid Klein bottle K and the twisted S 2 -bundle over the circle S 1 r ˆS2 .…”

Deformations of circle-valued Morse functions on 2-torus

“…I. Gelbukh classified Morse-Bott functions with the only 2 nondegenerate critical points on manifolds [4]. Topological properties foliations with Morse-Bott singularities of codimension-1 were investigated by B. Scárdua and J. Seade [19,20], and homotopy properties of diffeomorphisms preserving Morse-Bott foliations on lens spaces were studied by S. Maksymenko [14,15].…”