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«Abstract. We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R3 (a family of 3-polytopes attached ...»

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Thus in every geometric realization Y of X, the points corresponding to p1,..., p9 form a realization ΛP of LP, which is irrational by Lemma 1. Thus Y must have an irrational vertex coordinate. Otherwise, the supporting planes of Y would be the defined by equations with rational coefficients, whence the points of ΛP would be the solutions of systems of linear equations with rational coefficients, hence rational, a contradiction.

4. An unrealizable polyhedral complex Proof of Theorem 1.2. Let A be a 3-simplex in R3, with vertices labeled x1, x2, x3, x4. Let ei,j = conv(xi, xj ), i = j, denote the edges of A. Attach a topological belt B consisting of 2 triangular prisms to the two edges x1 x2 and x3 x4 as shown in Figure 4. Call the resulting topological polyhedral configuration X.

Suppose that X has a geometric realization Y, with vertices yi corresponding to xi. Since the edges e1,2 and e3,4 both belong to the belt B, the corresponding edges of Y must be concurrent by Lemma 3.3. But then the vertices y1, y2, y3, y4 are all coplanar. Hence they do not determine a 3-simplex, a contradiction.

Proof of Theorem 1.3. Let X denote the polyhedral complex of Theorem 1.2. As shown in Figure 4, the vertices x1, x3, a1 and the edges between them determine a topological 2simplex, call it ∆1. Similarly, the vertices x2, x4, a2 and the edges between them determine a topological 2-simplex ∆2. That is, ∆1 and ∆2 are topological 2-polyhedral complexes.

We define S = X ∪ {∆1, ∆2 }.

GEOMETRIC REALIZATIONS OF POLYHEDRAL COMPLEXES 9

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Figure 4. The unrealizable topological polyhedral complex of Theorem 1.

2.

Note that S is not a 3-polyhedral complex by our definition, because it contains facets ∆1 and ∆2 which are not contained in any cell of S. However, we may create a polyhedral complex from S as follows. Let R denote the bounded component of the complement of S (i.e. R is the region surrounded by S). Let c ∈ R, and for each facet F in the boundary of R, add to S the topological cone with apex c and base F. In other words, cone from the point c. Let X ′ denote the resulting 3-polyhedral complex.

Then clearly X is a subcomplex of X ′, and |X ′ | ∼ B 3. Furthermore, X ′ has exactly 9 vertices, 24 edges, 25 facets, and 9 cells. The cells of X ′ are comprised of 5 tetrahedra, 2 triangular prisms, and 2 tetragonal pyramids.

Remark 4.1 It is worth noting that the unrealizable complex of Theorem 1.

2 is minimal, in the sense that any topological 3-polyhedral complex consisting of two polyhedra is geometrically realizable. To see this, let X be a topological 3-polyhedral complex consisting of two 3-polyhedra Q1 and Q2. If Q1 and Q2 share less than a 2-face, the result is immediate.

So suppose that Q1 and Q2 share a 2-face F. Let P1 and P2 be polytopes isomorphic to Q1 and Q2, respectively. Let F1 and F2 denote the facets of P1 and P2, respectively, that correspond to F. Barnette and Gr¨nbaum [BG] proved that the shape of one facet of a u 3-polytope may be arbitrarily prescribed. Therefore we may choose P1 and P2 so that F1 and F2 are congruent. Apply an affine transformation that identifies F1 with F2. The result is a geometric 3-polyhedral complex isomorphic to X.

5. Universality theorems Let us first note that the first irrational polytope result of Perles was later extended by Mn¨v to a general universality theorem [Mn], and then further extended to all 4e polytopes [R]. Similarly, Brehm’s result gives a universality theorem for self-intersecting 2-surfaces in R3 (see Subsection 7.2).

In what follows, we extend Theorem 1.1 to a similar universality result. Using belts, we can in fact mimic the constructions of Theorem 1.1 for any point and line configuration. In particular, for a point and line configuration L, we can construct a geometric 3-polyhedral complex X(L) such that every realization of X(L) generates a realization of L. The universality theorem for point and line configurations (see e.g. [P]) then implies, in particular, that 10 IGOR PAK, STEDMAN WILSON Figure 5. Left: A point and line configuration L that may have non-planar realizations in RP3. Right: The resulting planar configuration L as constructed in Lemma 5.1.

for any proper subfield K of the algebraic closure of Q, there is a geometric 3-polyhedral complex that cannot be realized with all vertex coordinates in K.

Technically, the universality theorem for point and line configurations assumes that the realizations of a configuration are restricted to the projective plane RP2. If we allow realizations in RPd for d 2, some realizations may not lie entirely in a single plane. A point and line configuration L is said to be planar if every realization of L in RPd lies in a (projective) 2-plane. If a point and line configuration is not planar, there is a straightforward way to extend it to a planar configuration, which we describe in the following lemma.

Lemma 5.1 Let L = ([n], E) be a point and line configuration.

There is a point and line configuration L = ([3n + 1], E) such that E ⊆ E, and the points of L are coplanar in any realization of L in RPd. Furthermore, a planar realization of L contains a planar realization of L.





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resulting point and line configuration. Clearly, in every realization of L in RPd, each point p must lie in the plane determined by the two intersecting lines L and L′ corresponding to l and l′.

Finally, let Λ denote a realization of L. Since X ⊆ X and E ⊆ E, there is a subset Λ ⊆ Λ and a map f : X → Λ such that f is a bijection, and a collection of points {f (xi )}i∈I is collinear if I ⊆ E. To see that a collection of points {f (xi )}i∈I is collinear only if I ⊆ E, note ′ that each abstract line ei = i, wi, wi contains only one point of L, namely i. Furthermore, ′ contain no points of L. So the lines added to E to form E do not the two lines L and L enforce any new concurrencies among the points of L.

–  –  –

Figure 6. (Belts are shown schematically in purple) Left: A belt Bp which ensures that the three lines spanned by the red edges are concurrent in any realization of X.

Right: Two belts B and C, attached to the tetrahedron A along the green edges, which ensure that the three lines spanned by the red edges are not concurrent in any realization of X (so that p1 = p2 ). Note that B and C share the red edge of the tetrahedron T, but they do not share a 2-face.

Theorem 5.2 (Weak Universality Theorem) Let L be a point and line configuration, and let K be a proper subfield of the algebraic closure of Q.

There exists a geometric 3-polyhedral complex X(L) such that if X(L) has a realization over K then L has a planar realization over K. Moreover, the complex X(L) may be constructed using only triangular prisms.

Proof. We provide a sketch of the construction. By Lemma 5.1, we may assume that L is a planar configuration (if L is not planar, replace it with its planar extension). Let Λ denote a realization of L in RP3. For each line ℓ of Λ, place a tetrahedron with a marked edge e, such that ℓ = aff(e).

For each point p ∈ Λ, let Lp denote the set of lines of Λ containing p. For each such set Lp, add a belt Bp such that for each line ℓi ∈ Lp, the edge ei is identified with a lateral edge of Bp. See Figure 6 (left).

Finally, for each collection of 3 lines ℓi, ℓj, ℓk which are not concurrent in the realization Λ, place a tetrahedron Aijk with vertices labeled x1, x2, x3, x4. Add a belt Bijk such that each of the 3 edges ei, ej, and x1 x2 is identified with a lateral edge of Bijk. Add another belt Cijk such that each of the 3 edges ej, ek, and x3 x4 is attached along a lateral edge of Cijk. See Figure 6 (right). Call the resulting geometric 3-polyhedral complex X(L).

12 IGOR PAK, STEDMAN WILSON

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Theorem 5.2 is a weak universality theorem, in the sense that it does not imply that the realization spaces of X(L) and L are stably equivalent.

We would now like to investigate whether it is possible to obtain this latter type of result. We will find that by modifying our previous definitions slightly, we can in fact obtain a stronger and more general universality theorem. To this end, we adopt the definition of stable equivalence given in [R], and we define the realization spaces of X(L) and L as follows.

For a point and line configuration L = ([n], E), we define the (Euclidean) realization space of L (in R3 ) to be the set

R(L) = {(p1,..., pn ) ∈ R3n | Λ = {p1,..., pn } is a realization of L}.

In particular, we only allow realizations Λ in R3, rather than RP3 (that is, we do not allow points at infinity). This will be important for our final universality result. Notice that the coordinates of the realization space come with a particular order, induced by the natural order on [n]. That is, for each i ∈ [n], if (p1,..., pn ) ∈ R(L) then pi must be the point corresponding to i. For a geometric 3-polyhedral complex X with N vertices, then the realization space of X is the set R(X) = {(v 1,..., v N ) ∈ R3N | v 1,..., v N are the vertices of a realization X ′ of X}.

GEOMETRIC REALIZATIONS OF POLYHEDRAL COMPLEXES 13

Consider the natural map f : R(X(L)) → R(L) that assigns to each realization X ′ of X(L) the realization of L generated by X ′. The following informal argument shows that f will not be a stable equivalence in general. Suppose that X(L) is constructed so that its belts consist of a very large number of prisms. Let Λ be a realization of L, and consider the fiber A = f −1 (Λ). Since the belts of X(L) consist of a large number of prisms, for a given pair of belts B1 and B2 of X(L), we may construct a realization X ′ ∈ A in which the corresponding belts are knotted, and a realization Y ′ ∈ A in which they are not knotted. Since we forbid the possibility that the belts B1 and B2 may intersect one another arbitrarily, the knot in X ′ is non-trivial. That is, there is no continuous path from X ′ to Y ′ in R(X(L)) ⊂ R3N. Therefore A is not path-connected. But all fibers of a stable equivalence must be path-connected.

By modifying our definition of polyhedral complex slightly, we can eliminate the problem encountered in the previous paragraph. The idea is to preserve the face identifications in the complex, but allow polytopes to self-intersect (so in particular we will allow belts to intersect one another arbitrarily).

We define a geometric d-polyhedral arrangement X = (X, A) to consist of a set X = n i=1 Xi, where each Xi is a (face lattice of a) polytope, together with a set A ⊂ X of distinguished common faces, such that any face F ∈ A belongs to at least two polytopes, and any two polytopes have at most one common face that is contained in A. That is, if F ∈ A and F ∈ Xi ∩ Xj, then G ∈ A for all other G ∈ Xi ∩ Xj.

/ A geometric polyhedral arrangement Y = (Y, B) is a realization of X = (X, A) if there is a bijection f : X → Y such that Xi ≃ f (Xi ), and the face poset isomorphisms gi : Xi → f (Xi ) satisfy F ∈ A if and only if gi (F ) ∈ B.

The realization space R(X ) of a polyhedral arrangement X is defined in the obvious way.

Note that any two polytopes in X may intersect in more than a common face of both, but they can only have one common face distinguished by membership in A. That is, only the common face F ∈ A is required to be a common face of both polytopes in every realization, although the intersection of the polytopes may consist of much more.

Given a geometric d-polyhedral complex X = n Xi, we may construct a corresponding i=1 geometric d-polyhedral arrangement X by taking A = {F ∈ X | F ∈ Xi ∩ Xj for some i = j} and X = (X, A). The only difference between X and X is that in realizations of X, we allow the polytopes to self-intersect arbitrarily. However, the intersections corresponding to the faces in A are required to hold in all realizations of X. For this reason, Theorem 5.2 holds if we replace “polyhedral complex” with “polyhedral arrangement”, and the proof is identical. With this understanding, we may prove the desired universality results.

Theorem 5.3 Let L be a point and line configuration.

Then there is a polyhedral arrangement X (L) such that R(X (L)) is stably equivalent to R(L).

Proof. Let L be a point and line configuration, with realization Λ ⊂ R3. Let Z(L) denote the corresponding geometric 3-polyhedral complex constructed from Λ as in Theorem 5.2.

Let Z(L) be the polyhedral arrangement corresponding to Z(L). We begin by constructing a polyhedral arrangement X = X (L) by adding additional polytopes to Z(L). The purpose of adding these polytopes is simply to force the realizations of L generated by realizations of X to lie in R3 (rather than RP3 ), hence in R(L).

14 IGOR PAK, STEDMAN WILSON

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Figure 7. A belt, shown schematically in purple, attached to the four indicated edges of the tetrahedra.

This belt forces the point p to be a vertex of the tetrahedron T in every realization of X.

Let pi ∈ Λ, and place a tetrahedron (or triangular prism) Ti such that pi is a vertex of Ti. Let e1 and e2 denote two of the edges of Ti containing pi. Let B denote a belt of Z(L) whose lateral edges are concurrent at pi. Let e3 and e4 denote two of the lateral edges of B.

Now construct a new belt Ci that is attached along four of its lateral edges to the four edges e1, e2, e3, e4. See Figure 7. Adding Ti and Ci to Z(L) for each i ∈ [n] yields the polyhedral arrangement X = X (L).



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