On the embeddability of skeleta of manifold triangulations

TL;DR

This paper establishes a criterion for when the skeleton of a manifold triangulation can be embedded into Euclidean space, and demonstrates this with examples involving sphere bundles.

Contribution

It introduces a new criterion for embeddability of manifold skeletons based on submanifold complements and applies it to specific sphere bundle cases.

Findings

Criteria for embeddability of manifold skeletons in Euclidean space

Embeddability of (q-1)-skeletons of sphere bundle triangulations

Application to specific sphere bundle cases

Abstract

We show a criterion for a skeleton of a manifold triangulation being embeddable into Euclidean space in terms of the complement of a submanifold. As an application, we obtain embeddability of a $(q-1)$-skeleton of a triangulation of an $S^p$-bundle over $S^q$ into $\mathbb{R}^{p+q}$.