Invariants of stable maps from the 3-sphere to the Euclidean 3-space

In the present work, we study the decompositions of codimension-one transitions that alter the singular set the of stable maps of $S^3$ into ${\mathbb{R}}^3,$ the topological behaviour of the singular set and the singularities in the branch set that involves cuspidal curves and swallowtails that alter the singular set. We also analyse the effects of these decompositions on the global invariants with prescribed branch sets.