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A refined transversality theorem on crossings and its applications

Shunsuke Ichiki

TL;DR

This work advances singularity-theoretic transversality by introducing a refined, Hausdorff-measure-based transversality theorem for crossings of (g+π)∘f, where f is a C^r injection and g is a C^r map. By applying a parametric-transversality framework to the d-point cross-section, the authors derive sharp upper bounds on the Hausdorff dimension of the bad perturbation set for each 2 ≤ d ≤ d_f and show that almost all linear perturbations yield favorable transversality properties. A central application is a generalization of Mather's stability theorem for generic projections in the regime ell > 2 dim X, with explicit Hausdorff-dimension bounds on the exceptional set and extensions to embeddings, immersions, and normal crossings. The results unify and strengthen prior Lebesgue-measure outcomes and provide optimal exponents in key cases, underlining the practical impact for embedding and projection problems in smooth and lower-regularity settings.

Abstract

A transversality theorem is one of the most important tools in singularity theory, and it yields various applications. In this paper, we establish a refined transversality theorem on crossings from a new perspective of Hausdorff measures and give its various applications. Moreover, by using one of them, we generalize Mather's ``stability theorem for generic projections" in his celebrated paper ``Generic projections" under special dimension pairs.

A refined transversality theorem on crossings and its applications

TL;DR

This work advances singularity-theoretic transversality by introducing a refined, Hausdorff-measure-based transversality theorem for crossings of (g+π)∘f, where f is a C^r injection and g is a C^r map. By applying a parametric-transversality framework to the d-point cross-section, the authors derive sharp upper bounds on the Hausdorff dimension of the bad perturbation set for each 2 ≤ d ≤ d_f and show that almost all linear perturbations yield favorable transversality properties. A central application is a generalization of Mather's stability theorem for generic projections in the regime ell > 2 dim X, with explicit Hausdorff-dimension bounds on the exceptional set and extensions to embeddings, immersions, and normal crossings. The results unify and strengthen prior Lebesgue-measure outcomes and provide optimal exponents in key cases, underlining the practical impact for embedding and projection problems in smooth and lower-regularity settings.

Abstract

A transversality theorem is one of the most important tools in singularity theory, and it yields various applications. In this paper, we establish a refined transversality theorem on crossings from a new perspective of Hausdorff measures and give its various applications. Moreover, by using one of them, we generalize Mather's ``stability theorem for generic projections" in his celebrated paper ``Generic projections" under special dimension pairs.

Paper Structure

This paper contains 5 sections, 11 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Preliminaries for the proof of the main theorem
  4. Proof of the main theorem
  5. Applications of the main theorem

Key Result

Proposition 2.2

Let $f:X\to V$ be a $C^r$ injection and $g:V\to \mathbb{R}^\ell$ a $C^r$ mapping, where $r$ is a positive integer, $X$ is a $C^r$ manifold and $V$ is an open subset of $\mathbb{R}^m$. Suppose that Then, for any integer $d$ satisfying $2\le d\le d_f$, the set has Lebesgue measure zero in $\mathcal{L}(\mathbb{R}^{m},\mathbb{R}^{\ell})$.

Theorems & Definitions (28)

  • Definition 2.1
  • Proposition 2.2: Ichiki2018T
  • Theorem 2.3
  • Remark 2.4
  • Definition 3.1
  • Definition 3.2
  • Lemma 3.3: Ichiki2022a
  • Definition 5.1
  • Theorem 5.2
  • Remark 5.3
  • ...and 18 more