Hopf triangulations of spheres and equilibrium triangulations of projective spaces

TL;DR

This paper constructs and analyzes special triangulations of spheres and projective spaces, exploring their symmetries, minimality, and embeddings, with new results on equilibrium triangulations and polyhedral embeddings.

Contribution

It introduces Hopf triangulations of spheres, candidate minimal triangulations of projective spaces, and new polyhedral embeddings, expanding understanding of symmetric and minimal triangulations.

Findings

Constructed Hopf triangulations for spheres up to dimension 7

Identified non-existence of perfect equilibrium triangulations for certain complex projective spaces

Presented a new tight polyhedral embedding of P^3 into 6-space

Abstract

Following work by the first author and Banchoff, we investigate triangulations of real and complex projective spaces of real and complex dimension $k$ that are adapted to the decomposition into "zones of influence" around the points $[1,0,\ldots,0],$ $\ldots,$ $[0,\ldots,0,1]$ in homogeneous coordinates. The boundary of such a "zone of influence" must admit a simplicial version of the Hopf decomposition of a sphere into "solid tori" of various dimensions. We present such {\em Hopf triangulations} of $S^{2k-1}$ for $k \leq 4$, and give candidate triangulations for arbitrary $k$. In the complex case, a crucial role of this construction is the central $k$-torus as the intersection of all "zones of influence". Candidate triangulations of the $k$-torus with $2^{k+1}-1$, $k\geq 1$, vertices -- possibly the minimum numbers -- are well known. They admit an involution acting like complex conjugation and an automorphism of order $k+1$ realising the cyclic shift of coordinate directions in $\mathbb{C}P^k$. For $k=2$, this can be extended to what we call a {\em perfect equilibrium triangulation} of $\mathbb{C}P^2$, previously described in the literature. We prove that this is no longer possible for $k=3$, and no perfect equilibrium triangulation of $\mathbb{C}P^3$ exists. In the real case, the central torus is replaced by its fixed-point set under complex conjugation: the vertices of a $k$-dimensional cube. We revisit known equilibrium triangulations of $\mathbb{R}P^k$ for $k\leq 2$, and describe new equilibrium triangulations of $\mathbb{R}P^3$ and $\mathbb{R}P^4$. Finally, we discuss the most symmetric and vertex-minimal triangulation of $\mathbb{R}P^4$ and present a tight polyhedral embedding of $\mathbb{R}P^3$ into 6-space. No such embedding was known before.