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July, 2018 Composing generic linearly perturbed mappings and immersions/injections
Shunsuke ICHIKI
J. Math. Soc. Japan 70(3): 1165-1184 (July, 2018). DOI: 10.2969/jmsj/77237723


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.

Funding Statement

The author was supported by JSPS KAKENHI Grant Number 16J06911.


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Shunsuke ICHIKI. "Composing generic linearly perturbed mappings and immersions/injections." J. Math. Soc. Japan 70 (3) 1165 - 1184, July, 2018.


Received: 30 January 2017; Published: July, 2018
First available in Project Euclid: 12 June 2018

zbMATH: 06966979
MathSciNet: MR3830804
Digital Object Identifier: 10.2969/jmsj/77237723

Primary: 57R45
Secondary: 57R42

Keywords: generalized distance-squared mapping , generic linear perturbation , immersion , injection , transversality

Rights: Copyright © 2018 Mathematical Society of Japan


Vol.70 • No. 3 • July, 2018
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