Abstract
We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller pointsets than the known examples with this property. Moreover, we construct one on 13 points whose realization space is a connected but nonirreducible semialgebraic variety.
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1 Oriented Matroids and Matrices
Throughout this section, we fix positive integers r and n.
Let X=(x _{1},…,x _{ n })∈ℝ^{rn} be a real (r,n) matrix of rank r, and E={1,…,n} be the set of labels of the columns of X. For such a matrix X, a map \(\mathcal{X}_{X}\) can be defined as
The map \(\mathcal{X}_{X}\) is called the chirotope of X. The chirotope \(\mathcal{X}_{X}\) encodes the information regarding the combinatorial type, which is called the oriented matroid of X. In this case, the oriented matroid determined by \(\mathcal{X}_{X}\) is of rank r on E.
We note some properties which the chirotope \(\mathcal{X}_{X}\) of a matrix X satisfies.

1.
\(\mathcal{X}_{X}\) is not identically zero.

2.
\(\mathcal{X}_{X}\) is alternating, i.e. \(\mathcal{X}_{X}(i_{\sigma(1)},\ldots,i_{\sigma(r)}) =\mathrm{sgn} (\sigma) \mathcal{X}_{X}(i_{1},\ldots,i_{r})\) for all i _{1},…,i _{ r }∈E and all permutations σ.

3.
For all i _{1},…,i _{ r },j _{1},…,j _{ r }∈E such that \(\mathcal{X}_{X} (j_{k},i_{2},\ldots ,i_{r})\cdot \mathcal{X}_{X} (j_{1},\ldots, j_{k1},i_{1}, j_{k+1},\ldots, j_{r})\geq 0\) for k=1,…,r, we have \(\mathcal{X}_{X}(i_{1},\ldots,i_{r})\cdot \mathcal{X}_{X}(j_{1},\ldots,j_{r}) \geq 0\).
The third property follows from the identity
Generally, an oriented matroid of rank r on E (n points) is defined by a map χ:E ^{r}→{−1,0,+1}, which satisfies the above three properties ([1]). The map χ is also called the chirotope of an oriented matroid. We use the notation \(\mathcal {M}(E,\chi )\) for an oriented matroid which is on the set E and is defined by the chirotope χ.
An oriented matroid \(\mathcal {M}(E, \chi )\) is called realizable or constructible, if there exists a matrix X such that \(\chi=\mathcal{X}_{X}\). Not all oriented matroids are realizable, but we do not consider the nonrealizable case in this paper.
Definition 1.1
A realization of an oriented matroid \(\mathcal {M}=\mathcal {M}(E, \chi )\) is a matrix X such that \(\mathcal{X}_{X}=\chi\) or \(\mathcal{X}_{X}=\chi\).
Two realizations X,X′ of \(\mathcal {M}\) are called linearly equivalent, if there exists a linear transformation A∈GL(r,ℝ) such that X′=AX. Here we have the equation \(\mathcal{X}_{X'}=\mathrm{sgn} (\det A)\cdot \mathcal{X}_{X}\).
Definition 1.2
The realization space \(\mathcal {R}(\mathcal {M})\) of an oriented matroid \(\mathcal {M}\) is the set of all linearly equivalent classes of realizations of \(\mathcal {M}\), in the quotient topology induced from ℝ^{rn}.
Our motivation is as follows: In 1956, Ringel asked whether the realization spaces \(\mathcal {R}(\mathcal {M})\) are necessarily connected [6]. It is known that every oriented matroid on less than nine points has a contractible realization space. In 1988, Mnëv showed that \(\mathcal {R}(\mathcal {M})\) can be homotopy equivalent to an arbitrary semialgebraic variety [3]. His result implies that they can have arbitrary complicated topological types. In particular, there exist oriented matroids with disconnected realization spaces. Suvorov and RichterGebert constructed such examples of oriented matroids of rank 3 on 14 points, in 1988 and in 1996, respectively [5, 7]. However, it is unknown which is the smallest number of points on which oriented matroids can have disconnected realization spaces. See [1] for more historical comments.
One of the main results of this paper is the following.
Theorem 1.3
There exist oriented matroids of rank 3 on 13 points whose realization spaces are disconnected.
Let d and p be positive integers. The solution of a finite number of polynomial equations and polynomial strict inequalities with integer coefficients on ℝ^{d} is called an elementary semialgebraic set.
Let f _{1},…,f _{ p }∈ℤ[v _{1},…,v _{ d }] be polynomial functions on ℝ^{d}, and V⊂ℝ^{d} be an elementary semialgebraic set. For a ptuple ε=(ε _{1},…,ε _{ p })∈{−,0,+}^{p}, let
denote the corresponding subset of V. The collection of the elementary semialgebraic sets \((V_{\varepsilon})_{\varepsilon\in \{,0,+\}^{p}}\) is called a partition of V.
In the case r=3, a triple (i,j,k)∈E ^{3} is called a basis of χ if χ(i,j,k)≠0. Let B=(i,j,k) be a basis of χ such that χ(B)=+1. The realization space of an oriented matroid \(\mathcal {M}= \mathcal {M}(E,\chi)\) of rank 3 can be given by an elementary semialgebraic set
where e _{1},e _{2},e _{3} are the fundamental vectors of ℝ^{3}. For another choice of basis B′ of χ, we have a rational isomorphism between \(\mathcal {R}(\mathcal {M},B)\) and \(\mathcal {R}(\mathcal {M},B^{\prime})\). Therefore, realization spaces of oriented matroids are semialgebraic varieties.
The universal partition theorem states that, for every partition \((V_{\varepsilon})_{\varepsilon \in \{,0,+\}^{p}}\) of ℝ^{d}, there exists a family of oriented matroids \((\mathcal {M}^{\varepsilon})_{\varepsilon \in \{,0,+\}^{p}}\) such that the collection of their realization spaces with a common basis \((\mathcal {R}(\mathcal {M}^{\varepsilon},B))_{\varepsilon \in \{,0,+\}^{p}}\) is stably equivalent to the family \(( V_{\varepsilon})_{\varepsilon \in \{,0,+\}^{p}}\). See [2] or [4] for universal partition theorems.
We construct three oriented matroids \(\mathcal {M}^{\varepsilon}\) with ε∈{−,0,+} of rank 3 on 13 points, whose chirotopes differ by a sign on a certain triple. These oriented matroids present a partial oriented matroid with the sign of a single base nonfixed, whose realization space is partitioned by fixing the sign of this base. The two spaces \(\mathcal {R}(\mathcal {M}^{})\) and \(\mathcal {R}(\mathcal {M}^{+})\) are disconnected, and \(\mathcal {R}(\mathcal {M}^{0})\), which is a wall between the two, is connected but nonirreducible. So we also have the following.
Theorem 1.4
There exists an oriented matroid of rank 3 on 13 points whose realization space is connected but nonirreducible.
Remark 1.5
An oriented matroid \(\mathcal {M}(E,\chi)\) is called uniform if it satisfies χ(i _{1},…,i _{ r })≠0 for all i _{1}<⋯<i _{ r }∈E. Suvorov’s example on 14 points is uniform, and the examples which we construct are nonuniform. It is still unknown whether there exists a uniform oriented matroid on less than 14 points with a disconnected realization space.
2 Construction of the Examples
Throughout this section, we set E={1,…,13}.
Let X(s,t,u) be a real (3,13) matrix with three parameters s,t,u∈ℝ given by
This is a consequence of the computation of the following construction sequence. Both operations “∨” and “∧” can be computed in terms of the standard cross product “×” in ℝ^{3}. The whole construction depends only on the choice of the three parameters s,t,u∈ℝ. We have
We set \(X_{0}=X(\frac{1}{2},\frac{1}{2},\frac{1}{3})\). The chirotope χ ^{ε} is the alternating map such that
where ε∈{−,0,+}.
The oriented matroid which we will study is \(\mathcal {M}^{\varepsilon}:=\mathcal {M}(E,\chi^{\varepsilon})\).
Remark 2.1
We can replace X _{0} with \(X(\frac{1}{2},\frac{1}{2},u^{\prime})\) where u′ is chosen from \(\mathbb {R}\backslash\{1,0,\allowbreak \frac{1}{2},1,\frac{3}{2},2,3\}\). We will study the case \(0<u^{\prime}<\frac{1}{2}\). If we choose u′ otherwise, we can get other oriented matroids with disconnected realization spaces.
In the construction sequence, we need no assumption on the collinearity of x _{9},x _{12},x _{13}. Hence every realization of \(\mathcal {M}^{\varepsilon}\) is linearly equivalent to a matrix X(s,t,u) for certain s,t,u, up to multiplication on each column with positive scalar.
Moreover, we have the rational isomorphism
where \(\mathcal {R}^{\ast}(\chi^{\varepsilon}):= \{ (s,t,u)\in \mathbb {R}^{3}\mid \mathcal{X}_{X(s,t,u)} =\chi^{\varepsilon}\}\). Thus we have only to prove that the set \(\mathcal {R}^{\ast}(\chi^{\varepsilon})\) is disconnected (resp. nonirreducible) to show that the realization space \(\mathcal {R}(\mathcal {M}^{\varepsilon})\) is disconnected (resp. nonirreducible).
The equation \(\mathcal{X}_{X(s,t,u)}=\chi^{\varepsilon}\) means that
We write some of them which give the equations on the parameters s,t,u. Note that for all (i,j,k)∈E ^{3}({i,j,k}≠{9,12,13}), the sign is given by
From the equation sgndet(x _{2},x _{3},x _{5})=sgn(s)=sgn(1/2)=+1, we get s>0. Similarly, we get det(x _{2},x _{5},x _{4})=1−s>0; therefore,
From the equations det(x _{1},x _{7},x _{3})=t>0,det(x _{1},x _{4},x _{7})=1−t>0, we get
Moreover, we have the inequalities
From the equation det(x _{9},x _{12},x _{13})=u(1−2s)(1−2t+tu−su), we get
Conversely, if we have Eqs. (2)–(9), then we get (1).
We can interpret a (3,13) matrix as the set of vectors {x _{1},…,x _{13}}⊂ℝ^{3}. After we normalize the last coordinate for x _{ i } (i∈E∖{1,2,9}), we can visualize the matrix on the affine plane {(x,y,1)∈ℝ^{3}}≅ℝ^{2}. Figure 1 shows the affine image of X _{0}. See Figs. 2, 3 for realizations of \(\mathcal {M}^{\varepsilon}\).
Proof of Theorem 1.3
We prove that \(\mathcal {R}^{\ast}(\chi^{})\) and \(\mathcal {R}^{\ast}(\chi^{+})\) are disconnected. From Eqs. (2)–(9), we obtain
First, we show that \(\mathcal {R}^{\ast}(\chi^{} )\) is disconnected; more precisely, that it consists of two connected components. We do this by proving the next proposition.
Proposition 2.2
Proof
There are two cases:
Note that
(⊂) For the case 1−2s>0 and 1−2t+tu−su<0, the inequality 2t−1>0 follows from Eq. (10). Since we have 0<s<1/2<t<1, we get
For the other case 1−2s<0, similarly, we get 1−2t>0 from Eq. (10). Since we have 0<t<1/2<s<1, we get
(⊃) For the component 0<s<1/2<t<1, the inequalities 1−2t+tu−su<0, (1−t)^{2}−su>0, 1−t−u>0 follow from (13). Thus we get t ^{2}−(1−s)u>0 from Eq. (11). The inequality u<t holds because t>1/2 and u<1−t.
For the other component 0<t<1/2<s<1, similarly, we get the inequalities 1−2t+tu−su>0,t ^{2}−(1−s)u>0,t−u>0 from (14), and (1−t)^{2}−su>0 from Eq. (12). Last, we get u<1−t from t<1/2 and u<t. □
For the set \(\mathcal {R}^{\ast}(\chi^{+})\), we have the following proposition.
Proposition 2.3
The proof is similar to that of Proposition 2.2 and is omitted.
Proof of Theorem 1.4
We show that \(\mathcal {R}^{\ast}(\chi^{0})\) consists of two irreducible components whose intersection is not empty. From Eqs. (2)–(9), we get
Here we have the decomposition
The intersection of the two irreducible components is the set
The proof is also similar to that of Proposition 2.2 and is omitted. □
Figure 3 shows two realizations of \(\mathcal {M}^{0}\). On the left, it shows the affine image of \(X(\frac{1}{2},\frac{3}{8},\frac{1}{4})\), on the irreducible component 1−2s=0. On the right, it shows the image of \(X( \frac{3}{4},\frac{11}{24},\frac{2}{7})\), on the other component 1−2t+tu−su=0. These images can be deformed continuously to each other via \(X(\frac{1}{2},\frac{1}{2},u)\) \((0<u<\frac{1}{2})\).
We set
The set \(\mathcal {R}^{\ast}\times (0,\infty)^{12}\) is rationally isomorphic to a realization space of a partial oriented matroid with the sign χ(9,12,13) nonfixed. The collection of semialgebraic sets \((\mathcal {R}^{\ast}(\chi^{\varepsilon}))_{\varepsilon \in\{,0,+\} }\) is a partition of \(\mathcal {R}^{\ast}\). Figure 4 illustrates this partition in 3space.
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Acknowledgements
I would like to thank Masahiko Yoshinaga for valuable discussions and comments. I also thank Yukiko Konishi for comments on the manuscript.
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Tsukamoto, Y. New Examples of Oriented Matroids with Disconnected Realization Spaces. Discrete Comput Geom 49, 287–295 (2013). https://doi.org/10.1007/s004540129456y
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DOI: https://doi.org/10.1007/s004540129456y