293:, 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
68:, which asks about possible numbers of faces of different dimensions of a simplicial sphere. In December 2018, the g-conjecture was proven by
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constructed an example of a non-polytopal simplicial sphere (that is, a simplicial sphere that is not the boundary of a polytope).
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282:-spheres. In other words, what are the possible sequences of numbers of faces of each dimension for a simplicial
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proved that, in fact, "most" simplicial spheres are non-polytopal. The smallest example is of dimension
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Adiprasito, Karim (2019). "Combinatorial
Lefschetz theorems beyond positivity".
415:. Progress in Mathematics. Vol. 41 (Second ed.). Boston: Birkhäuser.
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implying that any simplicial 2-sphere is a boundary of a convex polytope.
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vertices. This conjecture was proved for simplicial convex polytopes by
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286:-sphere? In the case of polytopal spheres, the answer is given by the
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180:= 4 is realized by the tetrahedron. By repeatedly performing the
354:"Amazing: Karim Adiprasito proved the g-conjecture for spheres!"
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in the more general context of rational homology spheres.
184:, it is easy to construct a simplicial sphere for any
53:. Some simplicial spheres arise as the boundaries of
379:"On the upper-bound conjecture for convex polytopes"
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60:One important open problem in the field was the
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383:Journal of Combinatorial Theory, Series B
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264:for general simplicial spheres in 1975.
196:(or edge graphs) of convex polytopes in
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413:Combinatorics and commutative algebra
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127:More generally, the boundary of any (
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164:that any simplicial 2-sphere with
116:with triangular faces, such as an
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194:characterization of 1-skeleta
176:− 4 faces. The case of
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396:10.1016/0095-8956(71)90042-6
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124:, is a simplicial 2-sphere.
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307:Dehn–Sommerville equations
352:Kalai, Gil (2018-12-25).
241:-faces of any simplicial
108:The boundary of a convex
440:Algebraic combinatorics
278:-vectors of simplicial
182:barycentric subdivision
358:Combinatorics and more
377:McMullen, P. (1971).
172:− 6 edges and 2
226:upper bound theorem
51:-dimensional sphere
297:in December 2018.
41:simplicial complex
103:simplicial circle
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409:Stanley, Richard
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295:Karim Adiprasito
160:It follows from
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70:Karim Adiprasito
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188:≥ 4. Moreover,
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141:convex polytope
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361:. Retrieved
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389:: 187–200.
272:-conjecture
122:icosahedron
434:Categories
363:2018-12-25
335:1812.10454
313:References
156:Properties
118:octahedron
110:polyhedron
26:simplicial
208:Gil Kalai
85:≥ 3, the
445:Topology
411:(1996).
301:See also
291:-theorem
151:-sphere.
81:For any
76:Examples
18:geometry
192:gave a
143:in the
137:bounded
133:compact
87:simple
46:to the
37:-sphere
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91:-cycle
330:arXiv
101:is a
39:is a
417:ISBN
267:The
224:The
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