New perspectives on a classical embedding theorem
M. C. Crabb
TL;DR
This note unifies classical embedding criteria with modern triangulated-manifold results by casting embeddings as equivariant cohomology obstructions governed by the Euler class $e(\lambda)$. It shows that nonvanishing powers $e(\lambda)^m$ force obstructions to embedding via Stiefel-Whitney data and van Kampen–Flores type arguments, while vanishing higher obstructions permit embeddings, linking to complexified normal bundles and K-theory via $\gamma(\lambda)$. The simplicial analogue yields topological Radon-type results and generalizes to multi-obstruction settings, with connections to Frick–Harrison and Atiah’s K-theory perspective, as well as to Haefliger’s stable obstruction. Together, the results provide a cohesive framework spanning differential topology, combinatorial topology, and stable homotopy/K-theory obstructions for embeddings of manifolds into Euclidean spaces.
Abstract
In this expository note, recent results of Kishimoto and Matsushita on triangulated manifolds are linked to the classical criterion on the normal Stiefel-Whitney classes for existence of an embedding of a smooth closed manifold into Euclidean space of given dimension. We also look back at Atiyah's K-theoretic condition for the existence of a smooth embedding.