Existence of 2-parameter crossings, with applications

TL;DR

This paper investigates the conditions under which the critical sets of Morse 2-functions on manifolds can be homotopically manipulated, providing examples and applications to 4-manifold theory.

Contribution

It characterizes when critical arcs of Morse 2-functions can be moved via homotopy and offers a library of examples for closed 4-manifolds, with applications to crown diagrams.

Findings

Criteria for moving critical arcs in Morse 2-functions

Examples of Morse 2-functions on closed 4-manifolds

Applications to crown diagram theory of 4-manifolds

Abstract

A Morse 2-function is a generic smooth map from a manifold M of arbitrary finite dimension to a surface B. Its critical set maps to an immersed collection of cusped arcs in B. The aim of this paper is to explain exactly when it is possible to move these arcs around in B by a homotopy and to give a library of examples when M is a closed 4-manifold. The last two sections give applications to the theory of crown diagrams of smooth 4-manifolds.