geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
and combinatorics, a simplicial (or combinatorial) ''d''-sphere is a simplicial complexhomeomorphic 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.
Examples
* For any ''n'' ≥ 3, the simple ''n''-cycle ''C''''n'' is a simplicial circle, i.e. a simplicial sphere of dimension 1. This construction produces all simplicial circles.
* The boundary of a convex
polyhedron
In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices.
A convex polyhedron is the convex hull of finitely many points, not all on ...
in R3 with triangular faces, such as an
octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
is a simplicial ''d''-sphere.
Properties
It follows from
Euler's formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that fo ...
that any simplicial 2-sphere with ''n'' vertices has 3''n'' − 6 edges and 2''n'' − 4 faces. The case of ''n'' = 4 is realized by the tetrahedron. By repeatedly performing the barycentric subdivision, it is easy to construct a simplicial sphere for any ''n'' ≥ 4. Moreover, Ernst Steinitz gave a characterization of 1-skeleta (or edge graphs) of convex polytopes in R3 implying that any simplicial 2-sphere is a boundary of a convex polytope.
Branko Grünbaum
Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentGil Kalai proved that, in fact, "most" simplicial spheres are non-polytopal. The smallest example is of dimension ''d'' = 4 and has ''f''0 = 8 vertices.
The
upper bound theorem In mathematics, the upper bound theorem states that cyclic polytopes have the largest possible number of faces among all convex polytopes with a given dimension and number of vertices. It is one of the central results of polyhedral combinatorics.
O ...
gives upper bounds for the numbers ''f''''i'' of ''i''-faces of any simplicial ''d''-sphere with ''f''0 = ''n'' vertices. This conjecture was proved for simplicial convex polytopes by Peter McMullen in 1970 and by Richard Stanley for general simplicial spheres in 1975.
The ''g''-conjecture, formulated by McMullen in 1970, asks for a complete characterization of ''f''-vectors of simplicial ''d''-spheres. In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial ''d''-sphere? In the case of polytopal spheres, the answer is given by the ''g''-theorem, proved in 1979 by Billera and Lee (existence) and Stanley (necessity). It has been conjectured that the same conditions are necessary for general simplicial spheres. The conjecture was proved by Karim Adiprasito in December 2018.