2023/09/22 by Kühnel, Wolfgang, Spreer, Jonathan
#52B70 #57Q91 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Primary: 57Q15 #Secondary: 52B15
paper · doi:10.48550/arxiv.2309.12728
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], …, [0,…,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 S2k-1 for k ≤ 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 2k+1-1, k≥ 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 ℂPk. For k=2, this can be extended to what we call a \em perfect equilibrium triangulation of ℂP2, previously described in the literature. We prove that this is no longer possible for k=3, and no perfect equilibrium triangulation of ℂP3 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 ℝPk for k≤ 2, and describe new equilibrium triangulations of ℝP3 and ℝP4. Finally, we discuss the most symmetric and vertex-minimal triangulation of ℝP4 and present a tight polyhedral embedding of ℝP3 into 6-space. No such embedding was known before.