Round fold maps into the plane on some $7$-dimensional closed and simply-connected manifolds

TL;DR

This paper constructs explicit round fold maps into the plane on certain 7-dimensional simply-connected manifolds with specific cohomology rings, advancing understanding of their geometric and topological properties.

Contribution

It provides explicit examples of round fold maps on 7-manifolds with cohomology rings like that of complex projective space times a sphere, linking differential topology and algebraic topology.

Findings

Constructed round fold maps on specific 7-manifolds

Identified cohomology ring structures compatible with these maps

Enhanced understanding of fold maps on higher-dimensional manifolds

Abstract

Round fold maps are smooth maps on closed manifolds which are locally represented as the product maps of Morse functions and identity maps on open disks and whose singularity is realized as concentrically embedded spheres. The author previously introduced such maps. Our paper presents round fold maps on some $7$-dimensional simply-connected manifolds whose cohomology rings are isomorphic to that of the product of the $2$-dimensional complex projective space and a $3$-dimensional sphere. Such manifolds have been studied precisely by Wang and round fold maps on spin manifolds in these manifolds have been previously studied by the author. These manifolds form explicit classes of higehr dimensional closed and simply-connected manifolds, which are central objects in classical algeberic topology and differential topology. Understanding these manifolds in geometric and constructive ways is still attractive, which we think as pioneers. Fold maps are defined as smooth maps which are locally represented as the product maps of Morse functions and identity maps on open disks. They are fundamental and strong tools in generalizations of theory of Morse functions and applications to geometry of manifolds. Explicit construction of fold maps are difficult even on elementary or well-known manifolds whereas we can know the (non-)existence from Eliashberg's celebrating theory in the 1970s and related one in considerable cases.