Nonorientable surfaces in some non-haken 3-manifolds

If a closed, irreducible, orientable 3-manifold M does not possess any 2-sided incompressible surfaces, then it can be very useful to investigate embedded one-sided surfaces in M of minimal genus. In this paper such 3-manifolds M are studied which admit embeddings of the nonorientable surface K of genus 3. We prove that a 3-manifold M of the above type has at most 3 different isotopy classes of embeddings of K representing a fixed element of H2(M, Z2). If M is either a binary octahedral space, an appropriate lens space or Seifert manifold, or if M has a particular type of fibered knot, then it is shown that the embedding of K in M realizing a specific homology class is unique up to isotopy.