A differential topological study of compact manifolds having simple structures

TL;DR

This paper explores the classification of high-dimensional compact smooth manifolds with simple structures, focusing on special generic maps and their role in understanding manifold topology and differentiable structures.

Contribution

It provides explicit classifications and insights into manifolds with simple structures using differential topology and special generic maps, advancing understanding of high-dimensional manifold theory.

Findings

Classification of compact manifolds with boundary and simple structures.

Analysis of special generic maps and their topological implications.

Connections between Morse functions, special generic maps, and manifold structures.

Abstract

The present paper mainly presents, for example, explicit classifications of compact smooth manifolds having non-empty boundaries and simple structures where the dimensions are general. Studies of this type is fundamental and important. They also remain to be immature and difficult. This is due to the fact that the dimensions are high and this has prevented us from studying the manifolds in geometric and constructive ways. Moreover, most of the present work is motivated by explicit studies of higher dimensional variants of Morse functions: especially so-called special generic maps. The class of special generic maps is a natural class containing canonical projections of unit spheres and Morse functions on homotopy spheres with exactly two singular points. Their images are in general (compact) manifolds smoothly immersed to the targets and the dimensions of the images and the targets coincide. They know much about the topologies and the differentiable structures of the manifolds of the domains. The author has previously studied related problems and the present study is also a new study closely related to them. Last, we also present a dream for contribution to studies of special generic maps and higher dimensional variants of Morse functions and manifolds admitting them.