Geometric permutations of higher dimensional spheres

Geometric permutations of disjoint unit spheres

We show that a set of n disjoint unit spheres in R admits at most two distinct geometric permutations if n ≥ 9, and at most three if 3 ≤ n ≤ 8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R: if any subset of size 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.

fails for five and higher dimensional spheres

A construction of convex flag triangulations of five and higher dimensional spheres, whose h-polynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is provided. Let the f-polynomial fX of a simplicial complex X be defined by the formula fX(t): = ∑

Higher dimensional geometries related to Fuzzy odd-dimensional spheres

We study SO(m) covariant Matrix realizations of ∑m i=1 X 2 i = 1 for even m as candidate fuzzy odd spheres following hep-th/0101001. As for the fuzzy four sphere, these Matrix algebras contain more degrees of freedom than the sphere itself and the full set of variables has a geometrical description in terms of a higher dimensional coset. The fuzzy S is related to a higher dimensional coset SO(2...

Real Root Conjecture Fails for Five- and Higher-Dimensional Spheres

A construction of convex flag triangulations of five and higher dimensional spheres, whose h-polynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is provided. Let the f-polynomial fX of a simplicial complex X be defined by the formula fX(t) := ∑

On Neighbors in Geometric Permutations