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On the embeddability of skeleta of manifold triangulations

Daisuke Kishimoto, Takahiro Matsushita

TL;DR

The paper introduces a criterion linking the embeddability of a $d$-skeleton to the embeddability of the complement of a submanifold, enabling new positive examples and a constructive framework. It shows that for an $S^p$-bundle over $S^q$, the $(q-1)$-skeleton embeds into $\\mathbb{R}^{p+q}$, and, when $p+q=2d$ with $p<q$, the $d$-skeleton embeds into $\\mathbb{R}^{2d}$. Via connected-sum techniques, infinitely many closed $2d$-manifolds admit triangulations whose $d$-skeleta are embeddable, providing a broad contrast to nonembeddability results tied to Stiefel-Whitney classes. The proofs rely on smooth triangulations, transversality, isotopy extension, and a Schoenflies-type argument for connected sums.

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 $\mathbb{R}^{p+q}$.

On the embeddability of skeleta of manifold triangulations

TL;DR

The paper introduces a criterion linking the embeddability of a -skeleton to the embeddability of the complement of a submanifold, enabling new positive examples and a constructive framework. It shows that for an -bundle over , the -skeleton embeds into , and, when with , the -skeleton embeds into . Via connected-sum techniques, infinitely many closed -manifolds admit triangulations whose -skeleta are embeddable, providing a broad contrast to nonembeddability results tied to Stiefel-Whitney classes. The proofs rely on smooth triangulations, transversality, isotopy extension, and a Schoenflies-type argument for connected sums.

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 -skeleton of a triangulation of an -bundle over into .

Paper Structure

This paper contains 2 sections, 8 theorems, 7 equations.

Table of Contents

  1. Introduction
  2. Proof

Key Result

Theorem 1.2

Let $M$ be an $n$-manifold. If there is a $p$-dimensional compact submanifold $L$ of $M$ such that $M-L$ is embeddable into $\mathbb{R}^m$, then for any smooth triangulation $K$ of $M$, the $(n-p-1)$-skeleton $K_{n-p-1}$ is embeddable into $\mathbb{R}^m$.

Theorems & Definitions (13)

  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Conjecture 1.4
  • Proposition 1.5
  • Corollary 1.6
  • Theorem 2.1: see H
  • Theorem 2.2: see H
  • Lemma 2.3
  • proof
  • ...and 3 more