A higher-dimensional version of Fáry's theorem

TL;DR

Given a PL embedding $\varphi:X\\to M$ of a finite simplicial complex $X$ into a PL $d$-manifold $M$, the paper proves there exists a triangulation $M'$ of $M$ containing $X$ as a subcomplex. The construction refines the embedding via subdivisions, employing a derived subdivision $\mathrm{D}$ and a biased derived subdivision $\mathrm{sd}$ to make $\mathrm{D}\widetilde{X}$ a strongly induced subcomplex of $\mathrm{sd}\widetilde{M}$, after which a sequence of edge subdivisions and valid edge contractions (per the Alexander-Newman framework) recovers $X$ inside $M'$. This yields a higher-dimensional PL strengthening of Fáry's theorem, clarifying when linear realizations can be achieved by triangulation refinements in dimensions beyond the plane. The result connects PL-topology techniques—subdivisions, strongly induced subcomplexes, and contraction/subdivision moves—to obtain a practical triangulation embedding, with implications for extending linear realizations within ambient manifolds.

Abstract

We prove a generalization of Istvan Fáry's celebrated theorem to higher dimension.

Figures (3)

• Figure 1.1: Derived subdivision $\text{\rm D}\Gamma$ as an induced subcomplex of the derived subdivision $\text{\rm D}\Delta$. Example shows a 4-cycle $\Gamma = \{12, 23, 34, 14\}$ (we list only the top dimensional faces) and $\Delta = \{12, 23, 34, 14, 24\}$ on the left, $\Delta = \{124, 23, 34\}$ on the right, respectively. The vertex $v$ certifies that the final complex is not strongly induced.
• Figure 1.2: On the left: Biased derived subdivision $\Delta'$ of the pair $\Gamma \subseteq \Delta$, where $\Delta$ is a 2-simplex and $\Gamma$ is the edge spanned by vertices 1 and 2. On the right: Let $\Gamma \subseteq \Delta$ be as on the left and let $\sigma = \{1, 123\}$ be the highlighted edge of $\Delta'$. Then $\hat{\sigma} = \{1\}, \sigma_1 = \{123\}$, $\mathrm{st}_\sigma\, \Delta'$ is the highlighted subcomplex and $\Gamma \cap \mathrm{st}_\sigma\, \Delta' = \Gamma \cap \sigma_1 = 12$.
• Figure 1.3: Example of an edge contraction and an edge subdivision. Edge $f$ is not valid as it is contained in a missing simplex determined by vertices $1,2$ and $6$. Edge $e$ is valid in $\Delta$ and $\text{\rm C}_e\Delta$ is a simplicial complex. Let $\Gamma$ be a subcomplex of $\Delta$ with top dimensional faces $25, 45$. Then $\text{\rm S}_e\Gamma$ is a subcomplex of $\text{\rm S}_e\Delta$, but it is not induced as certified by vertices $v, 4, 5$.

Theorems & Definitions (12)

• Theorem 1: fary
• Theorem 2: Kind-of Fáry's theorem
• Example 3
• Lemma 4: Section I 2Bing83
• Lemma 5: Chapter 1, Lemma 4 Zee63
• Lemma 6
• proof
• Lemma 7
• proof
• Lemma 8