New explicit construction of fold maps on general 7-dimensional closed and simply-connected spin manifolds

TL;DR

This paper introduces a new general method for explicitly constructing fold maps on 7-dimensional closed, simply-connected spin manifolds, advancing the understanding of their topology through concrete geometric tools.

Contribution

The paper presents a novel, general construction technique for fold maps on 7-dimensional spin manifolds, expanding the toolkit for studying their differential topology.

Findings

Constructed fold maps on a broad class of 7-dimensional spin manifolds.

Provided explicit examples illustrating the new construction method.

Enhanced the understanding of the topology of these manifolds through concrete fold maps.

Abstract

7-dimensional closed and simply-connected manifolds have been attractive as central and explicit objects in algebraic topology and differential topology of higher dimensional closed and simply-connected manifolds, which were studied actively especially in the 1950s--60s. Attractive studies of the class of these $7$-dimensional manifolds were started by the discovery of so-called exotic spheres by Milnor. It has influenced on the understanding of higher dimensional closed and simply-connected manifolds via algebraic and abstract objects. Recently this class is studied via more concrete notions from algebraic topology such as concrete bordism theory by Crowley, Kreck, and so on. As a new kind of fundamental and important studies, the author has been challenging understanding the class in constructive ways via construction of fold maps, which are higher dimensional versions of Morse functions. The present paper presents a new general method to construct ones on spin manifolds of the class.