On the number of components of folds of image simple fold maps

TL;DR

This work addresses when the parity of the number of connected components of the singular set $\Sigma(f)$ for image simple fold maps $f:M^m\to S$ (with $m\ge2$) to a surface is a homotopy invariant. It proves that parity is not invariant for odd $m\ge3$ into arbitrary surfaces, via two explicit constructive schemes (open book/round fold maps and allowable moves) that realize a parity change by $1$ through a generic homotopy, while also providing a new proof of invariance for the even-dimensional/orientable-target case and a counterexample for non-orientable targets. The paper further develops the toolkit of generic homotopies, local moves, and open-book techniques to produce explicit homotopies whose singular sets include a Möbius-band component, offering both a geometric and a combinatorial handle on parity changes. In addition, it refines the even-dimensional theory by giving a parity-invariance proof and derives corollaries involving Euler characteristics; it also exhibits high-dimensional non-invariance phenomena via Klein bottle and open Möbius-band constructions. Overall, the results delineate precise parity-invariance borders and supply constructive methods to alter parity, advancing Takase-related questions about fold-component parity in various dimensional and orientability contexts.

Abstract

A smooth map between manifolds is said to be \emph{image simple} if its restriction to its singular point set is a topological embedding. We study the parity of the number of connected components of the singular point set for image simple fold maps from a closed manifold of dimension $\ge 2$ to a surface. It is known that this parity is a homotopy invariant when the source manifold is of even dimension and the target surface is orientable. In this paper we show that for an arbitrary image simple fold map from a closed odd-dimensional manifold of dimension $\ge 3$ to a (possibly non-orientable) surface, this parity is not a homotopy invariant. We give two constructive proofs: one uses techniques from open book decompositions and round fold maps, and the other uses allowable moves. The constructed homotopies add one new component to the singular point set. We also give a new proof of the homotopy invariance of the parity of the number of connected components of the singular point set for image simple fold maps from a closed even-dimensional manifold to an orientable surface together with an example showing that this does not hold for maps into non-orientable surfaces in general.

Figures (12)

• Figure 1: Fold and cusp singularities for $m=2$
• Figure 2: Coorientation convention at a fold singularity
• Figure 3: Type II fold crossing and its reverse move
• Figure 4: Type III fold crossing
• Figure 5: Cusp-fold crossing and its reverse move

Theorems & Definitions (19)

• Example 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 3.1
• Example 4.1
• Lemma 4.2
• proof