Fold maps on small dimensional manifolds with prescribed singular set

Boldizsár Kalmár

Vol. 321 (2022), No. 2, 309–343

DOI: 10.2140/pjm.2022.321.309

Abstract

We give sufficient and necessary conditions for the existence of a fold map of a closed $(n+1)$-dimensional manifold with prescribed singular set into ${ℝ}^n$ for $4\le n\le 7$. To obtain the results about $8$-dimensional manifolds we establish formulas for stable $7$-framings on $7$-manifolds by using symplectic classes. Then we derive conditions about the cobordism classes of oriented $8$-manifolds which have fold maps into ${ℝ}^7$. We also obtain relations between global topological properties of the fold singular set like the self-intersection number and topological properties of the source manifold like the Euler characteristic and the signature. We use obstruction theory and the homotopy principle for fold maps.

Keywords

singular map, fold map, framing, obstruction, 8-manifold, signature, symplectic class

Mathematical Subject Classification

Primary: 57R25, 57R45

Secondary: 55R15, 57R20

Milestones

Received: 23 September 2021

Revised: 10 September 2022

Accepted: 13 September 2022

Published: 21 March 2023