Spin structures and codimension two embeddings of -manifolds up to regular homotopy

We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into -space inside the set of all regular homotopy classes of immersions with trivial normal bundles. As a consequence, we show that for a large class of -manifolds , the following phenomenon occurs: there exists a codimension two immersion of the -sphere whose double points cannot be eliminated by regular homotopy, but can be eliminated after taking the connected sum with a codimension two embedding of . This involves introducing and studying an equivalence relation on the set of spin structures on . Their associated -invariants also play an important role.