Related papers: Many neighborly spheres

We have identified some necessary conditions for the existence of rigid sphere designs. In particular, we have successfully resolved the conjecture proposed by [Ban87]; Given fixed positive integers t and d, we show that there exist only…

A beautiful result of Br\"ocker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most…

We show that there are simple 4-dimensional polytopes with n vertices such that all separators of the graph have size at least $\Omega(n/\log n)$. This establishes a strong form of a claim by Thurston, for which the construction and proof…

We prove that the Podles spheres $S_q^2$ converge in quantum Gromov-Hausdorff distance to the classical 2-sphere as the deformation parameter $q$ tends to 1. Moreover, we construct a $q$-deformed analogue of the fuzzy spheres, and prove…

It is known that there are finitely many simplicial complexes (up to isomorphism) with a given number of vertices. Translating to the language of $h$-vectors, there are finitely many simplicial complexes of bounded dimension with $h_1=k$…

For each $d\geq 3$, $n \geq 10$, and $k_1, k_2, \ldots, k_{d-1}\geq 2$ with $k_1+k_2+\ldots+k_{d-1}\leq n-1$, we construct a regular $d$-polytope whose automorphism group is of order $2^n$ and whose Schl\"afli type is $\{2^{k_1},2^{k_2},…

The dodecahedral conjecture states that the volume of the Voronoi polyhedron of a sphere in a packing of equal spheres is at least the volume of a regular dodecahedron with inradius 1. The authors prove the conjecture following the…

The diameter of a strongly connected $d$-dimensional simplicial complex is the diameter of its dual graph. We provide a probabilistic proof of the existence of $d$-dimensional simplicial complexes with diameter $ (\frac{1}{d \cdot d!} -…

J.-P. Roudneff conjectured in 1991 that every arrangement of $n \ge 2d+1\ge 5$ pseudohyperplanes in the real projective space $\mathbb{P}^d$ has at most $\sum_{i=0}^{d-2} \binom{n-1}{i}$ complete cells (i.e., cells bounded by each…

Let $\mathbb{R}^n$ be the n-dimensional Euclidean space with $O$ as the origin. Let $\wedge$ be a lattice of determinant $1$ such that there is a sphere $|X|<R$ which contains no point of $\wedge$ other than $O$ and has $n$ linearly…

Several recent papers have addressed the problem of characterizing the $f$-vectors of cubical polytopes. This is largely motivated by the complete characterization of the $f$-vectors of simplicial polytopes given by Stanley, Billera, and…

We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured…

Let $Q(X)$ be any integral primitive positive definite quadratic form with discriminant $D$ and in $k$ variables where $k\geq4$. We give an upper bound on the number of integral solutions of $Q(X)=n$ for any integer $n$ in terms of $n$, $k$…

A discrete d-manifold is a finite simple graph G=(V,E) where all unit spheres are (d-1)-spheres. A d-sphere is a d-manifold for which one can remove a vertex to make it contractible. A graph is contractible if one can remove a vertex with…

For $d \geq 2$ and $G$ a finite abelian group, define $T_d(G)$ to be the minimum number of vertices $n$ so that there exists a simplicial complex $X$ on $n$ vertices which has the torsion part of $H_{d - 1}(X)$ isomorphic to $G$. Here we…

The structure of on-shell and off-shell 2D, (4,4) supersymmetric scalar multiplets is investigated, in components and in superspace. We reach the surprising result that there exist eight {\underline {distinct}} on-shell versions and an even…

We prove that if a pure simplicial complex of dimension d with n facets has the least possible number of (d-1)-dimensional faces among all complexes with n faces of dimension d, then it is vertex decomposable. This answers a question of J.…

We study relations among characteristic classes of smooth manifold bundles with highly-connected fibers. For bundles with fiber the connected sum of $g$ copies of a product of spheres $S^d \times S^d$ and an odd $d$, we find numerous…

We present an infinite sequence of smooth embeddings of a connected sum of 6 projective planes in the 4-sphere, which are all ambient homeomorphic, but pairwise ambient non-diffeomorphic. The double covers of the 4-sphere ramified along…

The degree of a map between orientable manifolds is a fundamental concept in topology, providing deep insights into the structure of manifolds and the behavior of maps between them. Recently, this notion has been extensively studied,…