Lips and swallow-tails of singularities of product maps

TL;DR

This paper demonstrates that certain local singularity moves, called lips and swallow-tails, in product maps of functions on 3-manifolds can be achieved through isotopies, advancing understanding of singularity manipulation.

Contribution

It proves that lips and swallow-tail singularity moves in product maps are realizable via isotopies of the functions, providing a new method for controlling singularities.

Findings

Lips and swallow-tail moves are generic local singularity moves.

These moves can be realized by isotopies of the functions.

The result applies to product maps of functions on 3-manifolds.

Abstract

Lips and swallow-tails are generic local moves of singularities of a smooth map to a 2-manifold. We prove that these moves of singularities of the product map of two functions on a 3-manifold can be realized by isotopies of the functions.