Composing generic linearly perturbed mappings and immersions/injections

Abstract

Let 𝑁 (resp., 𝑈) be a manifold (resp., an open subset of ℝ^𝑚). Let 𝑓:𝑁 → 𝑈 and 𝐹:𝑈 → ℝ^𝓁 be an immersion and a C^∞ mapping, respectively. Generally, the composition 𝐹 ∘ 𝑓 does not necessarily yield a mapping transverse to a given subfiber-bundle of 𝐽¹(𝑁,ℝ^𝓁). Nevertheless, in this paper, for any 𝒜¹-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 𝐽¹(𝑁,ℝ^𝓁) 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 → ℝ^𝓁 and a given injection 𝑓:𝑁 → 𝑈. Furthermore, applications of the two main theorems are given.