# New Examples of Oriented Matroids with Disconnected Realization Spaces

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## Abstract

We construct oriented matroids of rank 3 on 13 points whose realization spaces are disconnected. They are defined on smaller point-sets than the known examples with this property. Moreover, we construct one on 13 points whose realization space is a connected but non-irreducible semialgebraic variety.

## Keywords

Oriented matroids Realization space## 1 Oriented Matroids and Matrices

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

*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

*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*.

*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 \(\mathcal{X}_{X} (j_{k},i_{2},\ldots ,i_{r})\cdot \mathcal{X}_{X} (j_{1},\ldots, j_{k-1},i_{1}, j_{k+1},\ldots, j_{r})\geq 0\) for*k*=1,…,*r*, we have \(\mathcal{X}_{X}(i_{1},\ldots,i_{r})\cdot \mathcal{X}_{X}(j_{1},\ldots,j_{r}) \geq 0\).

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 *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 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 ℝ^{ d } is called an elementary semialgebraic set.

*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

*p*-tuple

*ε*=(

*ε*

_{1},…,

*ε*

_{ p })∈{−,0,+}

^{ p }, let

*V*. The collection of the elementary semialgebraic sets \((V_{\varepsilon})_{\varepsilon\in \{-,0,+\}^{p}}\) is called a

*partition*of

*V*.

*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

*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 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}.

*X*(

*s*,

*t*,

*u*) be a real (3,13) matrix with three parameters

*s*,

*t*,

*u*∈ℝ given by

^{3}. The whole construction depends only on the choice of the three parameters

*s*,

*t*,

*u*∈ℝ. We have

*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.

*s*,

*t*,

*u*. Note that for all (

*i*,

*j*,

*k*)∈

*E*

^{3}({

*i*,

*j*,

*k*}≠{9,12,13}), the sign is given by

*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,

*x*

_{1},

*x*

_{7},

*x*

_{3})=

*t*>0,det(

*x*

_{1},

*x*

_{4},

*x*

_{7})=1−

*t*>0, we get

*x*

_{9},

*x*

_{12},

*x*

_{13})=

*u*(1−2

*s*)(1−2

*t*+

*tu*−

*su*), we get

*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

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

*s*>0 and 1−2

*t*+

*tu*−

*su*<0, the inequality 2

*t*−1>0 follows from Eq. (10). Since we have 0<

*s*<1/2<

*t*<1, we get

*s*<0, similarly, we get 1−2

*t*>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 1−2

*t*+

*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 1−2*t*+*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

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 1−2*s*=0. On the right, it shows the image of \(X( \frac{3}{4},\frac{11}{24},\frac{2}{7})\), on the other component 1−2*t*+*tu*−*su*=0. These images can be deformed continuously to each other via \(X(\frac{1}{2},\frac{1}{2},u)\) \((0<u<\frac{1}{2})\).

*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.

## Notes

### 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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