In geometry and combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicial spheres arise as the boundaries of convex polytopes, however, in higher dimensions most simplicial spheres cannot be obtained in this way.
One important open problem in the field was the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by Karim Adiprasito in the more general context of rational homology spheres.[1][2]
combinatorics, a simplicial (or combinatorial) d-sphere is a simplicial complex homeomorphic to the d-dimensional sphere. Some simplicialspheres arise as the...
simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial...
In mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed...
5-sphere, but its triangulation (induced by some triangulation of A) is not a PL manifold. In other words, this gives an example of a finite simplicial...
g-conjecture on the possible numbers of faces of different dimensions in a simplicialsphere (also Grünbaum conjecture, several conjectures of Kühnel) (Karim Adiprasito...
The simplicial complex recognition problem is a computational problem in algebraic topology. Given a simplicial complex, the problem is to decide whether...
upper bound theorem states that if Δ {\displaystyle \Delta } is a simplicialsphere of dimension d − 1 {\displaystyle d-1} with n {\displaystyle n} vertices...
polytope C(n,d) maximizes the number fi of i-dimensional faces among all simplicialspheres of dimension d − 1 with n vertices. The moment curve in R d {\displaystyle...
Hopf elements. If X is any finite simplicial complex with finite fundamental group, in particular if X is a sphere of dimension at least 2, then its homotopy...
manifolds. In December 2018, he proved Peter McMullen's g-conjecture for simplicialspheres. For his work, he won the 2020 EMS Prize of the European Mathematical...
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...
illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory...
to a simplicial complex, its fundamental group can be described explicitly in terms of generators and relations. If X is a connected simplicial complex...
building blocks of discretizations of spacetime; that is, to build simplicial manifolds. 3-sphere Aitchison geometry Causal dynamical triangulation Complete graph...
discretization of a manifold. It usually means a piecewise linear manifold made by simplicial complexes. A digital manifold is a special kind of combinatorial manifold...
definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations...
question was the extension of this characterization from simplicial polytopes to simplicialspheres, the g-conjecture, which was resolved in 2018 by Karim...
(When only triangular faces are used, they are two-dimensional finite simplicial complexes.) In general, for any finite CW-complex, the Euler characteristic...
topological data analysis is to: Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter. Analyse these topological...
In geometry, the simplicial honeycomb (or n-simplex honeycomb) is a dimensional infinite series of honeycombs, based on the A ~ n {\displaystyle {\tilde...
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