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Round fold maps on $3$–manifolds

Naoki Kitazawa and Osamu Saeki

Algebraic & Geometric Topology 23 (2023) 3745–3762

We show that a closed orientable 3–dimensional manifold admits a round fold map into the plane, ie a fold map whose critical value set consists of disjoint simple closed curves isotopic to concentric circles, if and only if it is a graph manifold, generalizing the characterization for simple stable maps into the plane. Furthermore, we also give a characterization of closed orientable graph manifolds that admit directed round fold maps into the plane, ie round fold maps such that the number of regular fiber components of a regular value increases toward the central region in the plane.

Dedicated to Professor Kazuhiro Sakuma on the occasion of his 60th birthday

round fold map, graph manifold, simple stable map, 3–manifold, cohomology ring, normal form plumbing graph
Mathematical Subject Classification
Primary: 57R45
Secondary: 57K30, 58K30
Received: 7 September 2021
Revised: 29 March 2022
Accepted: 14 July 2022
Published: 5 November 2023
Naoki Kitazawa
Institute of Mathematics for Industry
Kyushu University
Osamu Saeki
Institute of Mathematics for Industry
Kyushu University

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