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

TL;DR

This paper investigates how certain topological changes in stable maps from the 3-sphere to 3-space affect their singular sets and global invariants, focusing on decompositions involving cuspidal curves and swallowtails.

Contribution

It introduces a detailed analysis of decompositions of codimension-one transitions and their impact on the singular set and global invariants of stable maps from $S^3$ to $ ^3$.

Findings

Classified types of singularities involving cuspidal curves and swallowtails.

Analyzed the effects of topological decompositions on global invariants.

Provided new insights into the structure of stable maps from 3-sphere to 3-space.

Abstract

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.