Twist maps and codimension-1 spun embeddings

We study codimension $1$ embeddings preserving open book structures. In particular, we prove that every closed orientable 3-manifold admits a codimension-1 spun embedding in a finite connected sum of $S^2 \times S^2$s and $S^2 \tilde{\times} S^2$s. We discuss some explicit constructions of planar open books on 3-manifolds and their codimension $1$ spun embeddings. To construct these embeddings, we use sphere twist maps and push maps. We also give a simple proof for nontriviality of the twist map along a nonseparating $S^n$ in the group of orientation preserving diffeomorphisms of $S^1 \times S^n \setminus D^{n+1}$, relative to the boundary.