A simplicial (or combinatorial ) d- sphere is a simplicial complex homeomorphic to a d- dimensional sphere . Some simplicial spheres appear as the boundaries of a convex polyhedron ; however, in higher dimensions, most simplicial spheres cannot be obtained in this way.
The most important open problem in this area is the g-hypothesis formulated by , who asked about the possible number of faces of different dimensions of the simplicial sphere.
Content
Examples
- For any n ⩾ 3, the simple n- cycle C n is a simplicial circle , that is, a simplicial sphere of dimension 1. This construction gives all simplicial circles.
- The boundary of a convex polyhedron in R 3 with regular faces, such as an octahedron or an icosahedron , is a 2-sphere.
- More generally, the boundary of any ( d + 1) -dimensional compact (or bounded ) simplicial convex polyhedron in Euclidean space is a simplicial sphere.
Properties
It follows from the Euler formula that any simplicial 2-sphere with n vertices has 3 n - 6 edges and 2 n - 4 faces. The case n = 4 is realized as a tetrahedron. With the repeated implementation of the barycentric division, it is easy to construct simplicial spheres for any n ⩾ 4. However, Ernst Steinitz gave a description of 1-skeletons (edge graphs) of convex polyhedra in R 3 , from which it follows that any simplicial 2-sphere is the boundary of a convex polyhedron.
Branko Grünbaum constructed an example of a simplicial sphere that is not the boundary of a multidimensional polyhedron. proved that, in fact, “the majority” of simplicial spheres are not polyhedron boundaries. The smallest example exists in dimension d = 4 and has f 0 = 8 vertices.
gives upper bounds for the number of f i i -faces of any simplicial d- sphere with f 0 = n vertices. The hypothesis was proved for the polyhedral spheres in 1970 by 1] , and for the general simplicial spheres in 1975 - .
The g- hypothesis formulated by McMullen in 1970 raises the question of a complete description of f -vectors of simplicial d- spheres. In other words, what are the possible sets of the number of faces of each dimension of a simplicial d- sphere? For polyhedral spheres, the answer is given by the g- theorem , which was proved in 1979 by Biller and Lee (existence) and Stanley (necessity). It has been suggested that the same conditions are necessary for general simplicial spheres. For 2015, the hypothesis remained open for d = 5 and higher. In December 2018, Karim Adiprasito proved the hypothesis for all d [2] .
See also
- The Dehn - Somerville Equations
Notes
- ↑ McMullen, 1971 , p. 187-200.
- ↑ Adiprasito, 2018 .
Literature
- Karim Adiprasito. Combinatorial Lefschetz theorems beyond positivity . - 2018 .-- arXiv : 1812.10454v2 .
- Richard P. Stanley. Combinatorics and commutative algebra. - Second edition. - Boston, MA: Birkhäuser Boston, Inc., 1996 .-- T. 41 .-- C. x + 164. - (Progress in Mathematics). - ISBN 0-8176-3836-9 .
- P. McMullen. On the upper-bound conjecture for convex polytopes // J. Combinatorial Theory. - 1971. - Vol. 10 . - S. 187-200 .