Fold maps with singular value sets having no self-intersections and homological properties of their Reeb spaces
Naoki Kitazawa · arXiv (Cornell University) · 2015
In this paper, as a fundamental study on the theory of Morse functions and their higher dimensional versions or fold maps, and applications to geometric theory of manifolds, we study algebraic and differential topological properties of fold maps with singular value sets having no self-intersections and their source manifolds. As a specific case, round fold maps are defined as fold maps whose singular value sets are concentric spheres and they were introduced in 2012--13 and have been systematically studied by the author. Especially, in the present paper, we concentrate on such fold maps the inverse images of regular values by which are disjoint unions of spheres. The author previously studied homology and homotopy groups and the homeomorphism and diffeomorphism types of manifolds admitting round fold maps such that inverse images are always disjoint unions of spheres and here, we do such works for such fold maps which are not always round. For example, we show flexibility of (co)homology groups of Reeb spaces of such maps, which are essential tools in studying the manifolds.