Diffeomorphisms preserving Morse–Bott functions

Abstract

Let f : M → R be a Morse-Bott function on a closed manifold M , so the set Σ f of its critical points is a closed submanifold whose connected components may have distinct dimensions. Denote by S(f ) = {h ∈ D(M ) | f • h = f } the group of diffeomorphisms of M preserving f and let D(Σ f ) be the group of diffeomorphisms of Σ f . We prove that the "restriction to Σ f " map ρ : S(f ) → D(Σ f ), ρ(h) = h| Σ f , is a locally trivial fibration over its image ρ(S(f )).2010 Mathematics Subject Classification. 57R30, 57R45, 53B15,

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

“…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.…”

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

“…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.…”