Traversally Generic & Versal Vector Flows: Semi-Algebraic Models of Tangency to the Boundary

TL;DR

This paper introduces boundary generic and traversally generic vector fields on compact manifolds with boundary, providing universal algebraic models for their boundary interactions, inspired by singularity theory and polynomial deformations.

Contribution

It develops local versal algebraic models for boundary interactions of generic vector fields, establishing universality across all equidimensional manifolds based on dimension.

Findings

Universal models depend only on the manifold's dimension.

Boundary and flow descriptions use universal polynomial deformations.

The models generalize classical singularity notions to boundary vector fields.

Abstract

Let $X$ be a compact smooth manifold with boundary. In this article, we study the spaces $\mathcal V^\dagger(X)$ and $\mathcal V^\ddagger(X)$ of so called boundary generic and traversally generic vector fields on $X$ and the place they occupy in the space $\mathcal V(X)$ of all fields (see Theorems \ref{th3.4} and Theorem \ref{th3.5}). The definitions of boundary generic and traversally generic vector fields $v$ are inspired by some classical notions from the singularity theory of smooth Bordman maps \cite{Bo}. Like in that theory (cf. \cite{Morin}), we establish local versal algebraic models for the way a sheaf of $v$-trajectories interacts with the boundary $\d X$. For fields from the space $\mathcal V^\ddagger(X)$, the finite list of such models depends only on $\dim(X)$; as a result, it is universal for all equidimensional manifolds. In specially adjusted coordinates, the boundary and the $v$-flow acquire descriptions in terms of universal deformations of real polynomials whose degrees do not exceed $2\cdot \dim(X)$.