Year: 2026
Type: article
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 $$[1,0,\ldots ,0],$$ [ 1 , 0 , … , 0 ] , $$\ldots ,$$ … , $$[0,\ldots ,0,1]$$ [ 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 $$S^{2k-1}$$ S 2 k - 1 for $$k \le 4$$ k ≤ 4 , and give candidate triangulations for arbitrary k . For equilibrium triangulations of $$\mathbb {C}P^k$$ C P k , the simultaneous intersection of all “zones of influence” is the central k -torus. For $$k\ge 1$$ k ≥ 1 , candidate triangulations of the k -torus with $$2^{k+1}-1$$ 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 $$\mathbb {C}P^k$$ C P k . For $$k=2$$ k = 2 , this can be extended to what we call a perfect equilibrium triangulation of $$\mathbb {C}P^2$$ C P 2 , previously described in the literature. We prove that this is not possible for
Authors: Wolfgang Kühnel, Jonathan Spreer
Institutions: University of Stuttgart, The University of Sydney
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,090
OpenAlex ID: W4387031460