Essential embeddings of annuli and Möbius bands in $3$-manifolds

Abstract:

In this paper we give conditions when the existence of an “essential” map of an annulus or Möbius band into a 3-manifold implies the existence of an “essential” embedding of an annulus or Möbius band into that 3-manifold. Let ${\lambda _1}$ and ${\lambda _2}$ be disjoint simple “orientation reversing” loops in the boundary of a 3-manifold M and A an annulus. Let $f:(A,\partial A) \to (M,\partial M)$ be a map such that ${f_\ast }:{\pi _1}(A) \to {\pi _1}(M)$ is monic and $f(\partial A) = {\lambda _1} \cup {\lambda _2}$. Then we show that there is an embedding $g:(A,\partial A) \to (M,\partial M)$ such that $g(\partial A) = {\lambda _1} \cup {\lambda _2}$.