Composing generic linearly perturbed mappings and immersions/injections

TL;DR

This paper establishes that generic linear perturbations of mappings and immersions can achieve transversality to certain fiber bundles, with applications in differential topology and singularity theory.

Contribution

It introduces new transversality results for compositions of perturbed mappings and immersions, extending classical theorems to broader contexts.

Findings

Generic linear perturbations achieve transversality to specified fiber bundles.

New transversality theorem for compositions of perturbed mappings and injections.

Applications demonstrate the utility of these transversality results.

Abstract

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given.