We clarify the structure of the set of regular homotopy classes containing embeddings of a 3-manifold into 5-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 3-manifolds M3, the following phenomenon occurs: there exists a codimension two immersion of the 3-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 M3. This involves introducing and studying an equivalence relation on the set of spin structures on M3. Their associated μ-invariants also play an important role.
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics