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 \(\mathcal {D}(\mathcal {F},\partial T)\) the group of diffeomorphisms of T fixed on \(\partial T\) and leaving each leaf of the foliation \(\mathcal {F}\) invariant. We prove that \(\mathcal {D}(\mathcal {F},\partial T)\) 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 \(\mathcal {F}_{p,q}\).