A refined transversality theorem on crossings and its applications

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.

💡 Research Summary

The paper revisits one of the most fundamental tools in singularity theory – transversality theorems – and upgrades the classical “measure‑zero” statements to precise Hausdorff‑measure estimates. The authors focus on multiple‑point crossings of smooth maps under generic linear perturbations, i.e. maps of the form (g + π) ∘ f where f is a C^r immersion (or injection) of a manifold X into an open set V⊂ℝ^m, g:V→ℝ^ℓ is an arbitrary C^r map, and π∈L(ℝ^m,ℝ^ℓ) is a linear map that plays the role of a generic perturbation. For each integer d≥2 up to the maximal crossing order d_f, they consider the d‑fold product map f^{(d)}:X(d)→(ℝ^ℓ)^d and the diagonal Δ_d⊂(ℝ^ℓ)^d (codimension ℓ(d‑1)). The set Σ_d of “bad” parameters π for which f^{(d)} fails to be transverse to Δ_d is the exceptional set whose size the paper quantifies.

The main result (Theorem 2.3) splits into two regimes. If dim X(d)−codim Δ_d≥0, then for any real s satisfying
 s ≥ mℓ − 1 + dim X(d) − codim Δ_d + 1/r,
the set Σ_d has s‑dimensional Hausdorff measure zero in the space L(ℝ^m,ℝ^ℓ). If dim X(d)−codim Δ_d<0, then for any s > mℓ + dim X(d) − codim Δ_d the same conclusion holds, and moreover every π∉Σ_d yields a genuine avoidance of the diagonal (i.e. f^{(d)}(π)(X(d))∩Δ_d=∅). In particular, choosing s=mℓ reproduces the classical Lebesgue‑measure‑zero statement, so the theorem strictly refines earlier results by providing explicit Hausdorff‑dimension bounds.

The proof rests on a Hausdorff‑measure version of Thom’s parametric transversality theorem (Lemma 3.4, originally from