Hopf triangulations of spheres and equilibrium triangulations of projective spaces

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.