Hopf Triangulations of Spheres and Equilibrium Triangulations of Projective Spaces

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,… ,0], [ 1 , 0 , … , 0 ] , … , … , [0,… ,0,1] [ 0 , … , 0 , 1 ] in homogeneous coordinates. The boundary of such a “zone of influence” admits a simplicial version of the Hopf decomposition of a sphere into “solid tori” of various dimensions. We present such Hopf triangulations of S²ᵏ⁻¹ S 2 k - 1 for k ≤ 4 k ≤ 4 , and give candidate triangulations for arbitrary k . For equilibrium triangulations of CPᵏ C P k , the simultaneous intersection of all “zones of influence” is the central k -torus. For k≥ 1 k ≥ 1 , candidate triangulations of the k -torus with 2ᵏ⁺¹-1 2 k + 1 - 1 vertices—possibly the minimum numbers—are known. They admit an involution acting like complex conjugation, and an automorphism of order $$k+1$$ k + 1 realising the cyclic shift of coordinate directions in CPᵏ C P k . For $$k=2$$ k = 2 , this can be extended to what we call a perfect equilibrium triangulation of CP² C P 2 , previously described in the literature. We prove that this is not possible for