Homotopy types of diffeomorphism groups of polar Morse-Bott foliations on lens spaces, 1

Let $T=S^1\times D^2$ be the solid torus, $\mathcal{F}$ the Morse-Bott foliation on $T$ into $2$-tori parallel to the boundary and one singular circle $S^1\times 0$, which is the central circle of the torus $T$, and $\mathcal{D}(\mathcal{F},\mathrm{\partial }T)$ the group of diffeomorphisms of $T$ fixed on $\mathrm{\partial }T$ and leaving each leaf of the foliation $\mathcal{F}$ invariant. We prove that $\mathcal{D}(\mathcal{F},\mathrm{\partial }T)$ is contractible. Gluing two copies of $T$ by some diffeomorphism between their boundaries, we will get a lens space $L_{p,q}$ with a Morse-Bott foliation ${\mathcal{F}}_{p,q}$ obtained from $\mathcal{F}$ on each copy of $T$. We also compute the homotopy type of the group $\mathcal{D}({\mathcal{F}}_{p,q})$ of diffeomorphisms of $L_{p,q}$ leaving invariant each leaf of ${\mathcal{F}}_{p,q}$.