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 J 1 (N, R ℓ ). Nevertheless, in this paper, for any 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, 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.

Transversality theorems on generic linearly perturbed mappings

“…(3) The essential idea for the proofs of Theorems 1 and 2 is to apply Lemma 1, and it is similar to the idea of the proofs of [2, Theorems 1 and 2]. Note that in the special case r = ∞, from some results in [2], the results in this paper (Theorems 1 and 2 in this section and Corollaries 1 to 7 in Section 5) can be obtained.…”

“…Since max{2n − ℓ, 0} = 0, from Corollary 4, there exists a subset Σ of L(R m , R ℓ ) with Lebesgue measure zero such that for any π ∈ L(R m , R ℓ ) − Σ, the mapping (F π • f ) (2) : N (2) → (R ℓ ) 2 is transverse to ∆ 2 . For this proof, it is sufficient to prove that the mapping (F π • f ) (2) satisfies that (F π • f ) (2)…”

“…(2) As in [2], there is a case of s f = 3 as follows. If n + 1 ≤ m, N = S n and f : S n → R m is the inclusion f (x) = (x, 0, .…”

“…Moreover, in [2], for a given C ∞ immersion or a given C ∞ injection f : N → U , transversality theorems on a generic linearly perturbed mapping F π • f : N → R ℓ are given, where N is a C ∞ manifold, F : U → R ℓ is a C ∞ mapping and ℓ is an arbitrary positive integer which may possibly satisfy m ≤ ℓ.…”

“…On the other hand, in this paper, as improvements of some results in [2], two main transversality theorems (Theorems 1 and 2 in Section 2) and their applications on generic linearly perturbed mapping are given in the case where manifolds and mappings are not necessarily of class C ∞ .…”