## 1 Oriented Matroids and Matrices

Throughout this section, we fix positive integers r and n.

Let X=(x 1,…,x n )∈ℝrn 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

$$\mathcal{X}_X:E^r\rightarrow \{-1,0,+1\},\quad \mathcal{X}_X(i_1, \ldots ,i_r):=\mathrm{sgn} \det (x_{i_1},\ldots ,x_{i_r}).$$

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. 1.

$$\mathcal{X}_{X}$$ is not identically zero.

2. 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. 3.

For all i 1,…,i r ,j 1,…,j r E such that $$\mathcal{X}_{X} (j_{k},i_{2},\ldots ,i_{r})\cdot \mathcal{X}_{X} (j_{1},\ldots, j_{k-1},i_{1}, j_{k+1},\ldots, j_{r})\geq 0$$ 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 non-realizable 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 AGL(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 ℝrn.

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 Richter-Gebert 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 ℝd is called an elementary semialgebraic set.

Let f 1,…,f p ∈ℤ[v 1,…,v d ] be polynomial functions on ℝd, and V⊂ℝd be an elementary semialgebraic set. For a p-tuple ε=(ε 1,…,ε p )∈{−,0,+}p, let

$$V_{\varepsilon}:=\bigl\{ v\in V\mid \mathrm{sgn} \bigl( f_i(v) \bigr)=\varepsilon_i \ \mbox{for}\ i=1,\ldots, p \bigr\}$$

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

$$\mathcal {R}(\mathcal {M},B):=\bigl\{ X \in \mathbb {R}^{3n}\mid x_i=e_1,\ x_j=e_2,\ x_k=e_3,\ \mathcal{X}_X=\chi \bigr\},$$

where e 1,e 2,e 3 are the fundamental vectors of ℝ3. 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 ℝd, 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 non-fixed, 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 non-irreducible. 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 non-irreducible.

### 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 non-uniform. 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 ℝ3. 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

$$\mathcal {R}^\ast \bigl(\chi^\varepsilon \bigr)\times (0,\infty )^{12}\cong \mathcal {R}\bigl(\mathcal {M}^\varepsilon\bigr),$$

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. non-irreducible) to show that the realization space $$\mathcal {R}(\mathcal {M}^{\varepsilon})$$ is disconnected (resp. non-irreducible).

The equation $$\mathcal{X}_{X(s,t,u)}=\chi^{\varepsilon}$$ means that

$$\mathrm{sgn} \det(x_i,x_j,x_k)=\chi^\varepsilon (i,j,k),\quad \mbox{for all}\ (i,j,k)\in E^3.$$
(1)

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

$$\chi^\varepsilon (i,j,k)=\mathrm{sgn} \det (x_i,x_j,x_k)|_{s=t=1/2,u=1/3}.$$

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,

$$0<s<1.$$
(2)

From the equations det(x 1,x 7,x 3)=t>0,det(x 1,x 4,x 7)=1−t>0, we get

$$0<t<1.$$
(3)

Moreover, we have the inequalities

(4)
(5)
(6)
(7)
(8)

From the equation det(x 9,x 12,x 13)=u(1−2s)(1−2t+tusu), we get

$$\mathrm{sgn} \bigl(u(1-2s)(1-2t+tu-su)\bigr)=\varepsilon.$$
(9)

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}⊂ℝ3. After we normalize the last coordinate for x i  (iE∖{1,2,9}), we can visualize the matrix on the affine plane {(x,y,1)∈ℝ3}≅ℝ2. 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

$$\mathcal {R}^\ast \bigl(\chi^- \bigr)= \left\{ (s,t,u)\in \mathbb {R}^3 \,\, \begin{array}{|c} 0<s<1,\,0<u<t<1-u, \\ (1-t)^2-su>0,\ t^2-(1-s)u>0, \\ (1-2s)(1-2t+tu-su)<0 \end{array} \right\},$$
$$\mathcal {R}^\ast \bigl(\chi^+ \bigr)= \left\{ (s,t,u)\in \mathbb {R}^3 \,\, \begin{array}{|c} 0<s<1,\ 0<u<t<1-u,\\ (1-t)^2-su>0,\ t^2-(1-s)u>0,\\ (1-2s)(1-2t+tu-su)>0 \end{array} \right\}.$$

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.

### Proof

There are two cases:

Note that

(10)
(11)
(12)

(⊂) For the case 1−2s>0 and 1−2t+tusu<0, the inequality 2t−1>0 follows from Eq. (10). Since we have 0<s<1/2<t<1, we get

(13)

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

(14)

(⊃) For the component 0<s<1/2<t<1, the inequalities 1−2t+tusu<0, (1−t)2su>0, 1−tu>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+tusu>0,t 2−(1−s)u>0,tu>0 from (14), and (1−t)2su>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

$$\mathcal {R}^\ast \bigl(\chi^0\bigr) = \left\{ (s,t,u)\in \mathbb {R}^3\, \begin{array}{|c} 0<s<1,\,0<u<t<1-u, \\ (1-t)^2-su>0,\ t^2-(1-s)u>0,\\ (1-2s)(1-2t+tu-su)=0 \end{array}\right\}.$$

Here we have the decomposition

The intersection of the two irreducible components is the set

$$\biggl\{(s,t,u)\in \mathbb {R}^3 \bigg| s=t=\frac{1}{2}, 0<u<\frac{1}{2}\biggr\}\cong \biggl\{ X \biggl(\frac{1}{2},\frac{1}{2},u\biggr)\bigg| 0<u<\frac{1}{2}\biggr\}.$$

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+tusu=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

$$\mathcal {R}^\ast :=\left\{(s,t,u)\in \mathbb {R}^3 \left| \begin{array}{c} 0<s<1,\,0<u<t<1-u,\\ (1-t)^2-su>0,\,t^2-(1-s)u>0 \end{array} \right. \right\}.$$

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) non-fixed. 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 3-space.