I am interested in singularity theory of differential mappings by topology. In knot theory, it is well-known thea two knots are equivalent if and only if one differ the other by some moves (i.e. Reidemeister moves). similar moves exists in the other mathematical objects. For example, surfaces in 4-space (Roseman moves) etc. Now I read a paper for moves for surfaces in 3-space. In the paper, by using stratified Morse theory (Cerf theory), complete moves are determined. I do not understand little for cerf theory. However, I am understanding to be determined complete Reidemeister type moves. The moves was determined by using singularities of a projection of proper isotopy and critical points. In the interval between critical points, not change the topology.
When we have known complete moves as above, how do we understand to make the moves? Maybe, after and before in a critical points change the topology.
Thank you for your considerations.