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

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

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,

$$ 0<s<1. $$

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

$$ 0<t<1. $$

Moreover, we have the inequalities


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

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

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 \(\mathcal {M}^{\varepsilon}\).

Fig. 1
figure 1

Column vectors of X 0

Fig. 2
figure 2

Realization of \(\mathcal {M}^{-}\) (on the left) and that of \(\mathcal {M}^{+}\) (on the right)

Fig. 3
figure 3

Realizations of \(\mathcal {M}^{0}\)

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.

Proposition 2.2


There are two cases:

Note that


(⊂) 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


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+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.

Fig. 4
figure 4

\(\mathcal {R}^{\ast}\) (on the top) and its partition \((\mathcal {R}^{\ast}(\chi^{\varepsilon}))_{\varepsilon \in \{-,0,+\}}\)