Elimination of cusps in dimension 4 and its applications

TL;DR

This paper studies how to eliminate cusps in 4-manifold maps, classifies the homotopies involved, and explores their effects on surface diagrams, providing new insights into 4-manifold representations.

Contribution

It introduces a classification of cusp merge homotopies and analyzes their impact on vanishing sets and surface diagrams in 4-manifold topology.

Findings

Classification of cusp merge homotopies via decorated curves

Description of vanishing set behavior under cusp merges

New examples of surface diagrams related to Lefschetz fibrations

Abstract

Several new combinatorial descriptions of closed 4-manifolds have recently been introduced in the study of smooth maps from 4-manifolds to surfaces. These descriptions consist of simple closed curves in a closed, orientable surface and these curves appear as so called vanishing sets of corresponding maps. In the present paper we focus on homotopies canceling pairs of cusps so called cusp merges. We first discuss the classification problem of such homotopies, showing that there is a one-to-one correspondence between the set of homotopy classes of cusp merges canceling a given pair of cusps and the set of homotopy classes of suitably decorated curves between the cusps. Using our classification, we further give a complete description of the behavior of vanishing sets under cusp merges in terms of mapping class groups of surfaces. As an application, we discuss the uniqueness of surface diagrams, which are combinatorial descriptions of 4-manifolds due to Williams, and give new examples of surface diagrams related with Lefschetz fibrations and Heegaard diagrams.