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Torus fibrations on 33-dimensional manifolds which are smooth functions with finitely many critical values and stable fold maps

Primarily human-written textHuman understanding: all partsmath.GN — General Topologymath.GT — Geometric Topology

Contributed by Naok Kitazawa ↗

Version 1 / Nov 04, 2026 / CC BY 4.0

Abstract

We present smooth functions which are torus fibrations with finitely many critical values on 33-dimensional closed manifolds, from (stable) fold maps into the plane. Fold maps are higher dimensional versions of Morse functions and introduced and studied first in the 1950s by Thom and Whitney. Saeki, Sakuma, and so on, are important researchers, studying related singularity theory and differential topology mainly, since the 1990s. 33-dimensional closed manifolds always admit ones into the plane. Surface fibrations are important tools and objects in complex geometry, symplectic geometry, and 44-dimensional geometry. It is important to construct important examples of the fibrations. In our paper, we present a 33-dimensional related new study. Related to this, the author has succeeded in explicit construction of explicit functions with prescribed level sets and fold maps whose singular values sets are concentric on manifolds such as connected sums of sphere bundles over spheres.

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Version history

  1. v1Submitted by Naok KitazawaInitial depositCurrentNov 03, 2026