The universality theorem for neighborly polytopes
In this note, we prove that every open primary basic semialgebraic set is stably equivalent to the realization space of a neighborly simplicial polytope. This in particular provides the final step for Mnëv’s proof of the universality theorem for simplicial polytopes.
Key words and phrases:Realization space, universality theorem, simplicial polytope, neighborly polytope
2010 Mathematics Subject Classification:Primary 52B40; Secondary 52C40, 14P10
Mnëv’s Universality Theorem was a fundamental breakthrough in the theory of oriented matroids and convex polytopes. It states that the realization spaces of oriented matroids and polytopes, i.e. the spaces of point configurations with fixed oriented matroid/face lattice, can be arbitrarily complex. It comes in four flavours:
Theorem 1 (Universality Theorem [Mnë88]).
Let be a primary basic semialgebraic set defined over , then
there is an oriented matroid of rank whose realization space is stably equivalent to , and
there is a polytope whose realization space is stably equivalent to ;
if moreover is open, then
there is a uniform oriented matroid of rank whose realization space is stably equivalent to , and
there is a simplicial polytope whose realization space is stably equivalent to .
Mnëv announced this theorem in 1985 [Mnë85] and published a sketch of the proof in 1988 [Mnë88]. A more detailed reasoning can be found in his thesis [Mnë86] (in Russian). Shor [Sho91] simplified a key step in Mnëv’s line of reasoning for part (i) and (iii).
Moreover, part (i) of Theorem 1 was later elaborated upon by Richter-Gebert [RG95] and Günzel [Gün96], who proved the stronger Universal Partition Theorem for oriented matroids. Using Lawrence extensions to rigidify the face lattices, it is easy to prove part (ii) from part (i) [Mnë88][RG99]. Here, a face lattice is called rigid if it uniquely determines the oriented matroid defined by its vertices. Additionally, Theorem 1(ii) was generalized greatly by Richter-Gebert, who proved that already -dimensional polytopes are universal [RGZ95][RG96].
For a proof of part (iv), in contrast, only Mnëv’s original papers are available, apart of some preliminary results of Sturmfels [Stu88b] and Bokowski–Guedes de Oliveira [BG90]. Moreover, Mnëv’s elaborations for this case in [Mnë86][Mnë88] are specially concise and, in our opinion, incomplete. Hence, we think part (iv) of Theorem 1, although widely believed to be true, should be considered open until now. It is important to stress that, despite the wrong common belief, Lawrence extensions cannot be used to deduce the universality theorem for simplicial polytopes. We use a different approach to rigidify matroids, namely one based on a result of Shemer proving rigidity of neighborly polytopes [She82].
In particular, we establish here that neighborly polytopes, i.e. -polytopes with a complete -skeleton, are universal. Even more, we obtain this result with even-dimensional polytopes. Since all neighborly polytopes of even dimension are simplicial, this provides a proof of Theorem 1(iv).
Every open primary basic semialgebraic set defined over is stably equivalent to the realization space of some neighborly -dimensional polytope on vertices.
This is in contrast to the case of cyclic polytopes, who have trivial realization spaces [BS86, Example 5.1]. This holds more generally for all totally sewn and Gale sewn polytopes [She82][Pad13], of which cyclic polytopes are a particular case. These are large families of neighborly polytopes. Indeed, the number of Gale sewn polytopes with vertices in a fixed dimension is asymptotically at least , which is the current best lower bound for the number of all polytopes [Pad13].
The (homogenized) realization space of any even-dimensional neighborly -polytope on vertices constucted with the extended sewing or the Gale sewing construction is contractible, and in fact an open, piecewise smooth -ball.
Encouraged by the work of Richter-Gebert, we make the daring conjecture that universality holds even when we restrict to neighborly -dimensional polytopes.
Every open primary basic semialgebraic set defined over is stably equivalent to the realization space of some neighborly (and hence simplicial) -polytope.
A universality conjecture for simplicial -polytopes is supported by the existence of simplicial -polytopes without the isotopy property, that is, with disconnected realization space [BG90].
The following idea provides further motivation for our conjecture. Altshuler and Steinberg proved that vertex figures of neighborly -polytopes are always -dimensional stacked polytopes [AS73]. Despite the fact that realization spaces of stacked polytopes are trivial, their oriented matroids can be complicated: Notice that if is any planar point configuration, then there exists a stacked -polytope with a distinguished vertex such that the contraction of in coincides with .
Now, realization spaces of planar point configurations are universal, and it is conceivable that these realization spaces can be captured by constructing neighborly -polytopes having them as edge contractions, combined with the fact that all neighborly 4-polytopes are rigid.
2. Universality for simplicial neighborly polytopes
The realization space of an oriented matroid is the set of vector configurations that share the same oriented matroid, i.e. if is of rank , and is the ground set of , then
Similarly, if is a polytope in , and is its vertex set, then we define its (homogenized) realization space as
Hence, the realization space of a polytope is the union of the realization spaces of all oriented matroids whose (Las Vergnas) face lattice (i.e. the dual to the lattice of positive cocircuits) coincides with the face lattice of , see [BLSWZ93, Sec. 9.5].
A basic semialgebraic set in is the set of solutions to a finite number of rational polynomial equalities and inequalities; it is called primary if all the inequalities in its definition are strict. A basic semialgebraic set is a stable projection of a basic semialgebraic set if, for the projection , we have that and that for every , the fiber is the relative interior of a non-empty polyhedron defined by equalities and strict inequalities that depend polynomially on . Two basic semialgebraic sets and are rationally equivalent if there is a homeomorphism such that and are rational functions. Two basic semialgebraic sets and are stably equivalent if they belong to the same equivalence class generated by stable projections and rational equivalences. We denote the stable equivalence of and as . We refer to [RG96][RG99] for more detailed definitions of these concepts.
A technique that pervades our proofs is the use of lexicographic extensions (see [BLSWZ93, Section 7.2]). When is realized by a vector configuration , then its lexicographic extension by , where are elements of and are signs, is realized by adjoining to the vector for any small enough (cf. Figure 2.1). The following lemma is straightforward, see also [BLSWZ93, Lemma 8.2.1 and Proposition 8.2.2].
Let denote any oriented matroid, and let denote a lexicographic extension of . Then the projection induced by deletion
is surjective, and its fibers are (polynomially parametrized) polyhedra of dimension .
We start with an observation of Mnëv/Shor concerning universality of uniform oriented matroids [Mnë85][Sho91], which is proven using constructible oriented matroids and a substitution technique from [Mnë88][JMLSW89].
For every open primary basic semialgebraic set defined over , there exists a rank uniform oriented matroid such that
The key step of the proof of Theorem 2 is to use a construction of Kortenkamp, who proved that every -dimensional point configuration of at most points appears as a face figure of a neighborly polytope [Kor97]. For larger point configurations, this is still an open problem, first asked by Perles.
For the proof, recall that in oriented matroid theory a polytope is called rigid if the face lattice of determines the oriented matroid spanned by the vertices of (see [Zie95, Section 6.6]). In particular, for all rigid polytopes, we have .
For every uniform oriented matroid of rank on elements, there exists a neighborly polytope with vertices in dimension such that
By a theorem of Kortenkamp [Kor97, Theorem 1.2], every realizable oriented matroid of rank can be extended to the Gale dual of an even-dimensional neighborly polytope by performing lexicographic extensions, obtaining a rank matroid on elements. By Lemma 5 we obtain
Let be the face lattice of the Gale dual of . Now, oriented matroid duality preserves realization spaces and, by [She82, Theorem 2.10] and [Stu88a, Theorem 4.2], every neighborly polytope of even dimension is rigid. Hence,
Let be a polytope with a flag of faces , and define . A point is said to be sewn onto through if it realizes the lexicographic extension of by , where these sets represent their elements in any order. Shemer [She82] proved that if is even dimensional and neighborly and is sewn through a universal flag then is also neighborly. Here, a universal flag of is a flag consisting of faces in every odd dimension, and such that the quotients of by these faces are still neighborly. This is extended in [Pad13] by relaxing the condition of being a universal flag to containing a universal subflag. This technique of generating neighborly polytopes — iteratively sewing starting from a cyclic polytope — is called the (extended) sewing construction (see [She82] and [Pad13, Section 3] for details).
Similarly, let be the oriented matroid of a neighborly -polytope , with dual , and let be an oriented matroid whose dual is obtained by doing first a lexicographic extension in general position of by followed by a lexicographic extension by . Then is also the oriented matroid of a neighborly polytope (of dimension ). This operation is called Gale sewing, and the neighborly polytopes obtained by repeating this procedure from a polygon or a -polytope are called Gale sewn (cf. [Pad13]).
Cyclic polytopes arise as a special case of these constructions: they can be obtained by repeatedly extended sewing from a simplex, as well as by Gale sewing from a polygon or certain stacked -polytope [Pad13].
Proof of Theorem 3.
Observe first that, since these are even-dimensional neighborly polytopes, they are rigid and therefore it suffices to argue at the level of realization spaces of oriented matroids (instead of polytopes). Moreover, up to a change to the dual in Gale sewing that does not affect the realization space, both constructions are based on performing a sequence of lexicographic extensions starting on a configuration with trivial realization space.
Hence, at the oriented matroid level, their realizations spaces are open and contractible. Indeed, in each step the fibers of the deletion map are open polyhedra (Lemma 5). Even more, since these fibers are piecewise polynomially (and hence piecewise smoothly) parametrized, then the whole realization space is a piecewise smooth ball. ∎
Acknowledgements We want to thank Günter Ziegler for his insightful comments on a previous version of this manuscript. Also, we wish to thank Nikolai Mnëv, Jürgen Richter-Gebert and Bernd Sturmfels for helpful discussions concerning the history of the universality theorem, and the state of the universality theorem for simplicial polytopes in particular. Finally, we want to thank Ivan Izmestiev for translating part of N. Mnëv’s doctoral thesis, and Hiroyuki Miyata for sparking our interest in this problem.
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