Let $\mathcal{ G}$ be a Morse-Bott foliation on the solid Klein bottle $\mathbf{ K}$ into $2$-dimensional Klein bottles parallel to the boundary and one singular circle $S^1$. Let also $S^1\widetilde{ \times}S^2$ be the twisted bundle over $S^1$ which is a union of two solid Klein bottles $\mathbf{ K}_0$ and $\mathbf{ K}_1$ with common boundary $K$. Then the above foliations $\mathcal{ G}$ on both $\mathbf{ K}_0$ and $\mathbf{ K}_1$ gives a foliation $\mathcal{ G}'$ on $S^1\widetilde{ \times}S^2$ into parallel Klein bottles and two singluar circles. The paper computes the homotopy types of groups of foliated (sending leaves to leaves) and leaf preserving diffeomorphisms for foliations $\mathcal{ G}$ and $\mathcal{ G}'$.
Categories: math.GT math.AG math.AT math.DG math.DS