Composing generic linearly perturbed mappings and immersions/injections

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.

19 pages. Accepted for publication in Journal of the Mathematical Society of Japan. The paper is a generalization of arXiv:1610.02880. As the appendix, the statement of the main theorem in arXiv:1607.03220 (the proof of it is in arXiv:1607.03220) is introduced in the final section of this paper

References in corpus (2)

• Recognizable classification of Lorentzian distance-squared mappings
• Generic linear perturbations