Elimination of definite fold II

TL;DR

This paper provides a new constructive and algorithmic proof for eliminating definite folds in generic smooth maps of higher-dimensional manifolds into surfaces, and explores related stable maps and fibrations.

Contribution

It introduces a new simple, constructive, and algorithmic proof for the elimination theorem of definite folds, and studies stable maps and fibrations on 3-manifolds.

Findings

New proof for elimination of definite folds that is constructive and algorithmic

Explicit examples of maps without definite folds

Non-existence of singular Legendre fibrations on 3-manifolds

Abstract

In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algorithmic: the procedures enable us to construct various explicit examples. We also study simple stable maps of $3$--manifolds into the $2$--sphere without definite fold. Furthermore, we prove the non-existence of singular Legendre fibrations on $3$--manifolds, answering negatively to a question posed in our previous paper.