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

Let T = S 1 × D 2 be the solid torus, F the Morse–Bott foliation on T into 2-tori parallel to the boundary and one singular circle S 1 × 0 , which is the central circle of the torus T , and D ( F , ∂ T ) the group of diffeomorphisms of T fixed on ∂ T and leaving each leaf of the foliation F invariant. We prove that D ( F , ∂ 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 F p , q obtained from F on each copy of T . We also compute the homotopy type of the group D ( F p , q ) of diffeomorphisms of L p , q leaving invariant each leaf of F p , q .