On definition of a surface-knot

Question

A surface-knot is a closed surface embedded in 4-dimensianal space. Some authors define this embedding as a smooth embedding while others define it locally flatly. I don't know whether or not there is a difference between these two definitions?

Answer 1

Two references:

1. Wild branching surfaces and topological 4-manifolds by Piergallini and Zuddas (arXiv:1602.07459v1).

From this you conclude that there are tame closed surfaces in $S^4$ not isotopic to smooth ones (take a branched cover of a nonsmoothable 4-manifold over $S^4$). I am sure it was known earlier.

2. Exotic knottings of surfaces in the 4-sphere by Finashin, Kreck and Viro.

From this you conclude that there are smooth surfaces in $S^4$ which are topologically but not smoothly isotopic.