Paper detail

Diffeomorphisms preserving Morse-Bott functions

Let $f:M\to\mathbb{R}$ 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 $\mathcal{S}(f) = \{h \in \mathcal{D}(M) \mid f\circ h=h \}$ the group of diffeomorphisms of $M$ preserving $f$ and let $\mathcal{D}(Σ_f)$ be the group of diffeomorphisms of $Σ_f$. We prove that the "restriction to $Σ_f$" map $ρ:\mathcal{S}(f) \to \mathcal{D}(Σ_f)$, $ρ(h) = h|_{Σ_f}$, is a locally trivial fibration over its image $ρ(\mathcal{S}(f))$.