I think we can argue that Sg --the genus-g surface -- has only a trivial embedding in S4 , since Sg is topologically a wedge of g S1's, and there are no knotted S1's in S4 (meaning that any two embeddings of S1 in S4 are isotopic.) But I am not clear on why there are no non-trivial embeddings of Hg ----a 3D handlebody; a 3-sphere with g handles----in S4.
If the first argument about Sg works, can I use it somehow; specifically using the fact that Sg is the boundary of Hg, to argue that there are no non-trivial embeddings of an Hg in S4? Or do I need an additional assumption for this last to be true? Or can I use some sort of surgery argument ?
Thanks in Advance.