On real algebraic realization of round fold maps of codimension $-1$

TL;DR

This paper explores the real algebraic realization of round fold maps of codimension -1, linking singularity theory, differential topology, and real algebraic geometry to understand manifold structures.

Contribution

It introduces new methods for realizing round fold maps of codimension -1 within real algebraic geometry, expanding the understanding of their topological and algebraic properties.

Findings

Real algebraic realization of round fold maps achieved for codimension -1

Connections established between singularity theory and real algebraic geometry

New techniques developed for representing these maps algebraically

Abstract

The canonical projections of the unit spheres are generalized to special generic maps and round fold maps, for example. They are generalizations from the viewpoint of singularity theory of differentiable maps and these maps restrict the topologies and the differentiable structures of the manifolds. We are concerned with round fold maps, defined as smooth maps locally represented as the product map of a Morse function and the identity map on a smooth manifold, and maps with singular value sets being concentric spheres. A bit different from differential topology, we are concerned with real algebraic geometric aspects of these maps. We discuss real algebraic realization of round fold maps of codimension $-1$ as our new work. Real algebraic realization of these maps is of fundamental and important studies in real algebraic geometry and a new study recently developing mainly due to the author.