Hopf triangulations of spheres and equilibrium triangulations of projective spaces (2309.12728v1)

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.