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.
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The most important open problem in the field is the g-conjecture, formulated by Peter McMullen, which asks about possible numbers of faces of different dimensions of a simplicial sphere.
Examples
Properties
It follows from Euler's formula that any simplicial 2-sphere with n vertices has 3n − 6 edges and 2n − 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 constructed an example of a non-polytopal simplicial sphere. Gil Kalai proved that, in fact, "most" simplicial spheres are non-polytopal. The smallest example is of dimension d = 4 and has f0 = 8 vertices.
The upper bound theorem gives upper bounds for the numbers fi of i-faces of any simplicial d-sphere with f0 = n vertices. This conjecture was proved for polytopal spheres 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 is open for d at least 5 (as of 2015).