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 c ...

and

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...

, a simplicial (or combinatorial) ''d''-sphere is a

simplicial complex In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set ...

homeomorphic In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...

to the ''d''-dimensional sphere. Some simplicial spheres arise as the boundaries of

convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...

s, 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 Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

, 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 Karim Alexander Adiprasito (born 1988) is a German mathematician working at the University of Copenhagen and the Hebrew University of Jerusalem who works in combinatorics. He completed his Ph.D. in 2013 at Free University Berlin under the supervi ...

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 th ...

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 ...

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...

, is a simplicial 2-sphere. * More generally, the boundary of any (''d''+1)-dimensional

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

(or

bounded Boundedness or bounded may refer to: Economics * Bounded rationality, the idea that human rationality in decision-making is bounded by the available information, the cognitive limitations, and the time available to make the decision * Bounded e ...

) simplicial

in the

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

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 for an ...

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 In mathematics, the barycentric subdivision is a standard way to subdivide a given simplex into smaller ones. Its extension on simplicial complexes is a canonical method to refine them. Therefore, the barycentric subdivision is an important tool i ...

, it is easy to construct a simplicial sphere for any ''n'' ≥ 4. Moreover,

Ernst Steinitz Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician. Biography Steinitz was born in Laurahütte (Siemianowice Śląskie), Silesia, Germany (now in Poland), the son of Sigismund Steinitz, a Jewish coal merchant, and ...

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 Gil Kalai (born 1955) is the Henry and Manya Noskwith Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, Israel, Professor of Computer Science at the Interdisciplinary Center, Herzliya, and adjunct Professor of mathematics a ...

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. ...

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

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

in December 2018.

References

*{{cite book , authorlink=Richard P. Stanley , first=Richard , last=Stanley , title=Combinatorics and commutative algebra , edition=Second , series=Progress in Mathematics , volume=41 , publisher=Birkhäuser , location=Boston , year=1996 , isbn=0-8176-3836-9 Algebraic combinatorics Topology

Examples

Properties

See also

References