Immersing n-spheres in 2n-space (Ex)
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Let denote the Stiefel manifold of orthogonal -frames in and consider the following fibration sequence
Complete the following:
- Use the Hirsch-Smale immersion theorem to give an interpretation of the boundary map in the homotopy exact sequence of the above fibration:
- Hence prove the following: if , any immersion is regularly homotopic to an embedding if and only if the normal bundle of is trivial.
- Given an example of an immersion with trivial normal bundle and which is not regularly homotopic to an embedding.