Triangulations of grassmannians and flag manifolds

Abstract

MacPherson (in: Topological methods in modern mathematics: a symposium in Honor of John Milnor’s Sixtieth Birthday Stony Brook NY 1991, Perish, Houston, 1993. https://doi.org/10.2307/1970177) conjectured that the Grassmannian $\textrm{Gr}(2, \mathbb {R}^n)$ has the same homeomorphism type as the combinatorial Grassmannian $\text{ MacP }(2,n)$, while Babson (in: A combinatorial flag space, MIT, 1993. https://doi.org/10.2307/1970177) proved that the spaces $\text{ Gr }(2,\mathbb {R}^n)$ and $\text{ Gr }(1,2,\mathbb {R}^n)$ are homotopy equivalent to their combinatorial analogs, the simplicial complexes $\Vert \text{ MacP }(2,n)\Vert$ and $\Vert \text{ MacP }(1,2,n)\Vert$ respectively. We will prove that $\text{ Gr }(2, \mathbb {R}^n)$ and $\text{ Gr }(1,2, \mathbb {R}^n)$ are homeomorphic to $\Vert \text{ MacP }(2,n)\Vert$ and $\Vert \text{ MacP }(1,2,n)\Vert$ respectively.

1 Introduction

An oriented matroid can be thought of as a combinatorial abstraction of a vector space, or of a point configuration or of real hyperplane arrangements. The theory of oriented matroid comes with analogous notion to linear independence, convexity, general position, and subspaces.

Mnëv and Ziegler in [3] introduced the poset $G(k,\mathcal {M})$ of rank k strong map images of a rank n oriented matroid $\mathcal {M}$ called the oriented matroid Grassmannian. The poset was introduced to serve as a combinatorial model for $\textrm{Gr}(k, \mathbb {R}^n)$, the space of k dimensional subspaces of $\mathbb {R}^n.$ A special case is when $\mathcal {M}$ is the unique rank n oriented matroid on n elements. The resulting poset $\textrm{MacP}(k, n)$, is called the MacPhersonian. Similarly, the poset of flags $(\mathcal {N}_1, \mathcal {N}_2)$ of oriented matroids, where $\mathcal {N}_1$ is a rank p strong map image of $\mathcal {N}_2$ and $\mathcal {N}_2$ is a rank k strong map image of $\mathcal {M}$ is denoted by $G(p, k, \mathcal {M})$. The poset came up in the work of Babson in [2] and the work of Gelfand and MacPherson in [4].

Mnëv and Ziegler conjectured that $\Vert G(k, \mathcal {M})\Vert$, the geometric realization of the poset $G(k, \mathcal {M})$ has the homotopy type of $\textrm{Gr}(k, \mathbb {R}^n)$. For $k = 2$, it was proven by Babson in [2] that $\Vert G(2, \mathcal {M})\Vert$ has the same homotopy type as $\textrm{Gr}(2, \mathbb {R}^n)$. It was also proven in [2] that $\Vert \textrm{G}(1,2, \mathcal {M})\Vert$ has the same homotopy type as $\textrm{Gr}(1,2, \mathbb {R}^n)$.

We will show that the complexes $\Vert \text{ MacP }(2,n)\Vert$ and $\Vert \text{ MacP }(1,2,n)\Vert$ are homeomorphic to $\textrm{Gr}(2,\mathbb {R}^n)$ and $\textrm{Gr}(1,2,\mathbb {R}^n)$ respectively. It follows from Babson’s work in [2] that the complex $\Vert \text{ MacP }(2,n)\Vert$ and $\textrm{Gr}(2, \mathbb {R}^n)$ have the same homotopy type, and that the complex $\Vert \textrm{MacP}(1,2, n)\Vert$ has the same homotopy type as $\textrm{Gr}(1,2,\mathbb {R}^n)$. Also, it can easily be shown that for $k=1$, $\Vert \textrm{MacP}(1,n)\Vert$ is homeomorphic to $\mathbb {R}P^{n-1}$.

In Sect. 2, we will give basic background on oriented matroids, posets and regular cell complexes. There are canonical maps $\mu : \textrm{Gr}(k, \mathbb {R}^n) \rightarrow \textrm{MacP}(k,n)$ mapping a k dimensional subspace to the rank k oriented matroid it determines, and the map $\nu : \textrm{Gr}(p,k, \mathbb {R}^n) \rightarrow \textrm{MacP}(p, k, n)$ mapping a flag of subspaces to a flag of oriented matroids. To establish our main assertion about the topology of $\Vert \text{ MacP }(2,n)\Vert$ and $\Vert \text{ MacP }(1,2,n)\Vert$, we will prove in Theorem 34 that the stratification $\{\mu ^{-1}(M): M \in \textrm{MacP}(2,n)\}$ is a regular cell decomposition of $\textrm{Gr}(2, \mathbb {R}^n)$. Similarly, we will prove in Theorem 49 that the stratification $\{\nu ^{-1}(N, M): (N, M) \in \textrm{MacP}(1,2,n)\}$ is a regular cell decomposition for $\textrm{Gr}(1,2,\mathbb {R}^n)$.

In Proposition 36, we will show that $\mu ^{-1}(M)$ is homeomorphic to an open ball. Similarly, $\nu ^{-1}(N, M)$ will be shown in Proposition 50 to be homeomorphic to an open ball. In Proposition 41, the boundary $\partial \overline{\mu ^{-1}(M)}$ of $\mu ^{-1}(M)$ will be shown to be $\bigcup _{N< M} \mu ^{-1}(N)$ as the union of lower dimensional cells. We have similar result in Proposition 53 for the boundary of $\nu ^{-1}(N, M)$.

In Sects. 5 and 8, we will prove that augmented closed intervals in $\textrm{MacP}(2,n)$ and $\textrm{MacP}(1,2,n)$ are isomorphic to the augmented face posets of some regular cell decomposition of shellable spheres. To see this, we will prove that the augmented intervals have recursive atom ordering and are thin.

In Sects. 6 and 9, we will prove that the closures $\overline{\mu ^{-1}(M)}$ and $\overline{\nu ^{-1}(N, M)}$ are topological manifolds whose boundaries are spheres.

In rank $r \ge 3$, and for $M \in \textrm{MacP}(r,n)$, $\mu ^{-1}(M)$ is not necessarily connected, see [5] Page 368. Our argument will also make use of the fact that in rank 2, $\partial \overline{\mu ^{-1}(M)} = \bigcup _{N< M} \mu ^{-1}(N)$. This fact called normality, is also not necessarily true in rank r for $r \ge 3$. We will use throughout this part of the project, the realizability of rank 2 oriented matroids; that is every rank 2 oriented matroid can be obtained from an arrangement of vectors in $\mathbb {R}^2$. This fact is also in general not true for oriented matroids of rank at least 3, see [5] Proposition 1.5.1.. Detailed results on rank r oriented matroids for $r \ge 3$ can be found in [5].

2 Oriented matroids

Suppose $X \in \textrm{Gr}(p, \mathbb {R}^n)$. We will view elements of $\mathbb {R}^n$ as $1\times n$ row vectors, so that X is the rowspace of a $p \times n$ matrix.

The collection of all sign vectors

$$\begin{aligned} \{(\text{ sign }(x_1), \text{ sign }(x_2), \ldots , \text{ sign }(x_n)): (x_1, x_2, \ldots , x_n) \in X \} \end{aligned}$$

is a collection of sign vectors that are called the covectors of an oriented matroid. For a covector C, the set $\{i \in [n]: C(i) \ne 0\}$ is called the support of C. A covector of minimal support is called a cocircuit. A formal definition of the covector set of an oriented matroid will be given later in this section.

We can write in terms of column vectors as $X = \textrm{Rowspace}(v_1 \; v_2 \; v_3 \; \cdots \; v_n)$. Every element of X is of the form $(\alpha \cdot v_1, \alpha \cdot v_2, \ldots , \alpha \cdot v_n)$ for some $\alpha = (\alpha _1, \alpha _2, \ldots , \alpha _p) \in \mathbb {R}^p$. So, $(\textrm{sign}(\alpha \cdot v_1), \textrm{sign}(\alpha \cdot v_2), \ldots \textrm{sign}(\alpha \cdot v_n))$ is a covector of the oriented matroid $\mathcal {M}$ determined by X.

Let $\{v_{\alpha _i}\}_{\alpha _i}$ be the set of non-zero vectors in $\{v_1, v_2, \ldots v_n\}$. We consider the following arrangement $(v_{\alpha _i}^{\perp })_{\alpha _i}$ of oriented linear hyperplanes. The arrangement determines a cellular decomposition of $\mathbb {R}^p$. The intersection of the cellular decomposition with $S^{p-1}$ the unit sphere in $\mathbb {R}^p$ gives a cellular decomposition of $S^{p-1}$. A cell in $S^{p-1}$ corresponds to a non-zero covector of $\mathcal {M}$ and a non-zero covector of $\mathcal {M}$ corresponds to a cell in the cellular decomposition of $S^{p-1}$. The oriented matroid $\mathcal {M}$ with a covector set obtained this way is called a realizable oriented matroid. It should be noted that rank 1 and rank 2 oriented matroids are realizable, but oriented matroids of rank at least 3 are not necessarily realizable, details can be found in [5] Page 23.

Let $X = \textrm{Rowspace}(v_1\; v_2 \; v_3 \ldots \; v_n)$ as defined earlier and $\mathcal {M}$ the corresponding oriented matroid. We consider the following function $\chi : [n]^p \rightarrow \{+, -, 0\}$ associated to X

$$\begin{aligned} \chi (i_1, i_2, \ldots , i_p) = \textrm{sign}(\det (v_{i_1} \; v_{i_2} \cdots v_{i_p}) ) \end{aligned}$$

The collection $\{\pm \chi \}$ is independent of the choice of basis vectors for X. We will write the resulting oriented matroid as $\mathcal {M} = (\pm \chi )$.

In Fig. 1b, a rank 3 oriented matroid is obtained from an essential arrangement of equators in a 2-sphere $S^2$.

Fig. 1

An rank 3 oriented matroid from an arrangement of equators

Notation: Let E be a finite set and $X,Y\in \{0,+,-\}^E$ be sign vectors. The composition $X\circ Y$ is defined to be the element of $\{0,+,-\}^E$ with

$$\begin{aligned} X\circ Y(e)={\left\{ \begin{array}{ll} X(e)&{} \text{ if } X(e)\ne 0\\ Y(e)&{} \text{ otherwise } \end{array}\right. } \end{aligned}$$

Definition 1

([5], Page 159) Let E be a finite set and ${\mathcal {V}}^* \subseteq \{0,+,-\}^E$. ${\mathcal {V}}^*$ is the covector set of an oriented matroid on elements E if it satisfies all of the following.

• 1. ${\textbf{0}}\in {\mathcal {V}}^*$.

• 2. If $X\in {\mathcal {V}}^*$ then $-X\in {\mathcal {V}}^*$.

• 3. If $X,Y\in {\mathcal {V}}^*$ then $X\circ Y\in {\mathcal {V}}^*$.

• 4. If $X,Y\in {\mathcal {V}}^*$, $X\ne -Y$, and $e\in E$ such that $X(e)=-Y(e)\ne 0$, then there is a $Z\in {\mathcal {V}}^*$ such that $Z(e)=0$ and, for each $f\in E$,

  • if $X(f)=Y(f)=0$ then $Z(f)=0$,

  • If $+\in \{X(f),Y(f)\}\subseteq \{0,+\}$ then $Z(f)=+$, and

  • If $-\in \{X(f),Y(f)\}\subseteq \{0,-\}$ then $Z(f)=-$.

If ${\mathcal {V}}^*$ is the covector set of an oriented matroid, then the rank of the oriented matroid is the rank of ${\mathcal {V}}^*$ as a subposet of $\{0,+,-\}^E$.

Definition 2

([5], Page 126) Let E be a finite set and r a positive integer. A rank r chirotope on elements E is a nonzero alternating function $\chi :E^r\rightarrow \{0,+,-\}$ satisfying the following Grassmann–Plücker relations: for each $x_2, x_3,\ldots , x_r, y_0, y_1, \ldots , y_r\in E$, the set

$$\begin{aligned} \{(-1)^i\chi (y_i, x_2, \ldots , x_r)\chi (y_0, \ldots , {\hat{y}}_i,\ldots , y_r)\} \end{aligned}$$

either is $\{0\}$ or contains both $+$ and −.

Definition 3

(Basis orientation) ([5], Page 138) A basis orientation of an oriented matroid $\mathcal {M}$ is a mapping $\chi$ of the set of ordered bases of $\mathcal {M}$ to $\{+1, -1\}$ satisfying the following two properties

• $\chi$ is alternating,

• For any ordered bases of $\mathcal {M}$ of the form $(e, x_2, x_3, \ldots , x_p)$ and $(f, x_2, x_3, \ldots , x_p)$, $e \ne f$, we have

  $$\begin{aligned} \chi (e, x_2, x_3, \ldots , x_p) = D(e)D(f), \end{aligned}$$

  where D is one of the two opposite cocircuits complementary to the hyperplane spanned by $\{x_2, x_3, \ldots , x_p\}$ in $\mathcal {M}.$

The following theorem establish the cryptomorphism between the definition of an oriented matroid using covectors and its definition using a chirotope.

Theorem 4

([5, 6], Page 128) Let $p\ge 1$ be an integer and E be a set. A mapping $\chi : E^p \rightarrow \{+1, 0, -1\}$ is a basis orientation of an oriented matroid of rank p on E if and only if it is a chirotope.

In general, an oriented matroid is obtained from an arrangement of pseudospheres. Figure 2 illustrates an arrangement of pseudospheres.

Theorem 5

([5, 6], Page 233)The Topological Representation Theorem (Folkman-Lawrence 1978) The rank r oriented matroids are exactly the sets $(E,\mathcal {V}^*)$ arising from essential arrangements of pseudospheres in $S^{r-1}$.

Fig. 2

Arrangement of Pseudospheres

Let $\{+, -, 0\}$ be a poset with the partial order $0 < -$ and $0< +$. The partial order on $\{+, -, 0\}^n$ is component-wise the partial order on $\{+, -, 0\}$.

Definition 6

([1, 5], Page 318) Let $\mathcal {M}$ and $\mathcal {N}$ be two rank r oriented matroids, and ${\mathcal {V}}^*(\mathcal {M})$ and ${\mathcal {V}}^*(\mathcal {N})$ the covector sets of $\mathcal {M}$ and $\mathcal {N}$ respectively. We say that $\mathcal {N} \le \mathcal {M}$ if and only if for every $X \in {\mathcal {V}}^*(\mathcal {N})$ there exist a $Y \in {\mathcal {V}}^*(\mathcal {M})$ such that $X \le Y$. The oriented matroid $\mathcal {M}$ is said to weak map to $\mathcal {N}$.

Definition 7

[1] $\textrm{MacP}(p,n)$ denotes the poset of all rank p oriented matroids on elements $\{1,2,\ldots , n\}$, with weak map as the partial order. The poset is called the MacPhersonian [1].

As discussed in Sect. 2, for every subspace $X \in \textrm{Gr}(p, \mathbb {R}^n)$, there is a corresponding rank p oriented matroid on n elements, say $\mu _X$. There is thus the canonical map $\mu : \textrm{Gr}(p, \mathbb {R}^n) \rightarrow \textrm{MacP}(p,n)$: $X \mapsto \mu _X$.

Let $\mathcal {M}$ be a rank p oriented matroid elements [n] and $\chi : [n]^p \rightarrow \{+, -,0\}$ its chirotope. We have the following abstraction of notions from vector spaces and convexity.

• 1. Loop: An element i is said to be a loop of $\mathcal {M}$ if $\chi (i, i_1, i_2, \ldots , i_{p-1}) = 0$ for any $p-1$ tuple $(i_1, i_2, \ldots , i_{p-1}).$

• 2. Basis: A set $\{i_1, i_2, \ldots , i_p\}$ of size p is said to be a basis of $\mathcal {M}$ if and only if $\chi (i_1, i_2, \ldots , i_p) \ne 0$.

• 3. Independence: A set $\{i_1, i_2, \ldots , i_k\}$ is said to be independent if it is contained in a basis of $\mathcal {M}$.

• 4. Parallel. A non-loop element i is said to be parallel to a non-loop element j if for every $p-1$ tuple $(i_1, i_2, \ldots , i_{p-1})$, we have that $\chi (i, i_1, i_2, \ldots , i_{p-1}) = \chi (j, i_1, i_2, \ldots , i_{p-1})$. Similarly, i is said to be anti-parallel to j if for every $p-1$ tuple $(i_1, i_2, \ldots , i_{p-1})$, we have that $\chi (i, i_1, i_2, \ldots , i_{p-1}) = -\chi (j, i_1, i_2, \ldots , i_{p-1}).$

• 5. Convex Hull: Let S be a subset of [n]. The convex hull of S is the set $\{i \in [n]: - \in C(S) \; \textrm{if}\; C(i) = - \; \text{ for } \text{ all } \; C \in {\mathcal {V}}^*(\mathcal {M}){\setminus } \{0\}\}.$

An oriented matroid also comes with an abstraction of the notion of subspaces of a vector space. Let V be a rank k subspace of $\mathbb {R}^n$ and W a rank p subspace of V. We have that the collection of sign vectors $\{(\text{ sign }(x_1), \text{ sign }(x_2), \ldots , \text{ sign }(x_n)): (x_1, x_2, \ldots , x_n) \in V\}$ is a subset of the collection $\{(\text{ sign }(y_1), \text{ sign }(y_2), \ldots , \text{ sign }(y_n)): (y_1, y_2, \ldots , y_n) \in W\}$. Let $\mathcal {M}$ be the rank k oriented matroid determined by V, and let $\mathcal {N}$ be the rank p oriented matroid determined by W. Then $\mathcal {V}^*(\mathcal {N}) \subseteq \mathcal {V}^*(\mathcal {M})$

Definition 8

[3] Let $\mathcal {M}$ be a rank k oriented matroid, and $\mathcal {N}$ a rank p oriented matroid. $\mathcal {N}$ is said to be a rank p strong map image of $\mathcal {M}$ if and only if ${\mathcal {V}}^*(\mathcal {N}) \subseteq {\mathcal {V}}^*(\mathcal {M})$.

Definition 9

[3] Let $\mathcal {M}$ be an oriented matroid. The poset of all rank p oriented matroids that are strong map image of $\mathcal {M}$ is denoted by $\textrm{G}(p, \mathcal {M})$.

In Fig. 3, the oriented matroid $\mathcal {N}$ is a rank 2 strong map image of $\mathcal {M}$.

Fig. 3

strong map image

We now define the combinatorial analog of the flag manifold $\textrm{Gr}(p,k, \mathbb {R}^n)$. The poset came up in the work of Gelfand and MacPherson in [4] and in the work of Babson in [2].

Definition 10

[2] We define $\textrm{MacP}(p,k,n)$ as the poset of pairs (N, M) of oriented matroids, where M is a rank k oriented matroid on n elements, and N is a rank p strong map image of M. The pair (N, M) is called a combinatorial flag. We say that $(N_1, M_1) \le (N_2, M_2)$ if and only if $M_1 \le M_2$ and $N_1 \le N_2$.

As in the case of the map $\mu : \textrm{Gr}(p,\mathbb {R}^n) \rightarrow \textrm{MacP}(p,n)$ discussed earlier, there is also the canonical map$\nu : \textrm{Gr}(k, p, \mathbb {R}^n) \rightarrow \textrm{MacP}(p,k,n)$ defined by $(W, V) \mapsto (N, M)$ where N and M are the oriented matroids determined by subspaces W and V respectively.

Proposition 11

Let $M_0\in \textrm{MacP}(2, n)$ be the rank 2 oriented matroid whose only basis is (1, 2), and let $N_0$ be a covector of $M_0$. Then $\bigcup _{(N, M) \ge (N_0, M_0)} \nu ^{-1}(N, M)$ can be embedded in $\textrm{Gr}(2, \mathbb {R}^{n+1})$

Proof

Suppose $(N, M) \in \textrm{MacP}(1,2,n)_{\ge (N_0, M_0)}$. Let $(Y, X) \in \nu ^{-1}(N, M)$. Then $X = \textrm{Rowspace}(e_1 \; e_2 \; v_3\; v_4\; \cdots v_n)$ for some vectors $v_i \in \mathbb {R}^2$. Also, $Y = \textrm{Rowspace}({\textbf {r}})$ for some ${\textbf {r}} \in X \subseteq \mathbb {R}^n$. Then ${\textbf {r}}$ is of the form $(\alpha \cdot e_1 \; \alpha \cdot e_2\; \alpha \cdot v_3 \cdots \alpha \cdot v_n )$ for some $\alpha \in \mathbb {R}^2$. There is a unique vector $v_{n+1} \in \mathbb {R}^2$ such that $v_{n+1} \cdot \alpha = 0$, $\Vert v_{n+1}\Vert = 1$ and $0 \le \textrm{Arg}(v_{n+1}) < \pi$.

The embedding $\varphi$ is then given by $(Y, X) \mapsto \textrm{Rowspace}(e_1\; e_2 \; v_3\; v_4 \cdots \; v_{n+1})$. $\square$

Let (Y, X) be any point in $\nu ^{-1}(N, M)$. We denote by $\iota (N, M)$ the image of (Y, X) under the map $\mu \circ \varphi$. Let $\chi '$ be a chirotope of $\iota (N, M)$. We observe that N is given by the functions $\pm \chi _{n+1}': [n] \rightarrow \{+, -,0\}$, where $\chi _{n+1}'(i) = \chi '(n+1, i).$

Proposition 12

Let $(N_0, M_0) \in \textrm{MacP}(1,2,n)$ such that $\{1,2\}$ is a basis of $M_0$. Then $\textrm{MacP}_{(N, M) \ge (N_0, M_0)}$ can embedded as a subposet $\textrm{MacP}(2, n+1)_{\ge \iota (N_0, M_0)}$ of $\textrm{MacP}(2, n+1)$.

Proof

It follows from the above identifications that $(N_1, M_1) \le (N_2, M_2)$ if and only if $\iota (N_1, M_1) \le \iota (N_2, M_2)$. $\square$

Remark 13

Let $(N_0, M_0)$ be in $\textrm{MacP}(1,2,n)$ and $\iota (N_0, M_0)$ be the corresponding rank 2 oriented matroid in $\textrm{MacP}(2,n+1)$ described above. Then the interval $\textrm{MacP}(1,2,n)_{\ge (N_0, M_0)}$ in $\textrm{MacP}(1,2,n)$ can be identified with the interval $\textrm{MacP}(2, n+1)_{\ge \iota (N_0, M_0)}$ in $\textrm{MacP}(2,n+1)$.

We can visualize an element of the flag manifold $\textrm{Gr}(1,2,\mathbb {R}^n)$ as in Fig. 4a. In Fig. 4b, the sign vectors $\{(+-++-), -(+-++-), 0 \}$ is the covector set of a rank 1 oriented matroid determined by the 1 dimensional subspace Y.

Fig. 4

A flag of subspaces, and the rank one oriented matroid

2.1 Reorientation of an oriented matroid

We have described a realizable rank p oriented matroid as determined by some arrangement of vectors in $\mathbb {R}^p$. We now discuss the reorientation of a rank p oriented matroid.

Definition 14

We say that a rank p oriented matroid $\mathcal {M} = (\pm \chi _\mathcal {M})$ is obtained from a rank p oriented matroid $\mathcal {N} = (\pm \chi _\mathcal {N})$ by a reorientation of an element i if $\chi _\mathcal {M}(i_1, i_2, \ldots , i_p) = \chi _\mathcal {N}(i_1, i_2, \ldots , i_p)$ for all p-tuples $(i_1, i_2, \ldots , i_p)$ such that $i \notin \{i_1, i_2, \ldots , i_p\}$, and $\chi _\mathcal {M}(i,j_1, j_2,\ldots , j_{p-1}) = - \chi _\mathcal {N}(i,j_1, j_2, \ldots , j_{p-1})$ for all $p-1$ tuples $(j_1, j_2, \ldots j_{p-1}).$

In the language of vector arrangements, reorientation of an element i is simply that if $(w_1, w_2, w_3, \ldots ,w_i, \ldots , w_n)$ is a vector arrangement for $\mathcal {N}$, then $(w_1, w_2, w_3, \ldots , -w_i, \ldots , w_n)$ is a vector arrangement for $\mathcal {M}$.

It then follows that if $\mathcal {M}$ is obtained from $\mathcal {N}$ by a reorientation or a sequence of reorientations, then $\mu ^{-1}(\mathcal {M})$ is homeomorphic to $\mu ^{-1}(\mathcal {N})$. Another useful observation is the following lemma:

Lemma 15

Suppose $\mathcal {N}$ is obtained from $\mathcal {M}$ by reorientations of some elements in [n]. Then the posets $(\hat{0}, \mathcal {M})$ and $(\hat{0}, \mathcal {N})$ are isomorphic.

Proof

Suppose $\mathcal {N}$ is obtained from $\mathcal {M}$ by reorienting elements in $A \subseteq [n]$. Then the required poset isomorphism $R_A: (\hat{0}, \mathcal {M}) \rightarrow (\hat{0}, \mathcal {N})$ is obtained by taking $R_A(Y)$ as a reorientation of the rank 2 oriented matroid Y by elements in A. $\square$

It should also be noted that the homeomorphism type of $\mu ^{-1}(\mathcal {M})$ is unchanged by relabelling the elements of $\mathcal {M}$.

3 Posets, regular cell complexes and topological balls

3.1 Posets and Recursive atom ordering

Associated to every poset P is a simplicial complex $\Vert P\Vert$ whose n-simplices are chains $x_0< x_1< \cdots < x_n$ for some $x_i$ in P. $\Vert P\Vert$ is called the order complex of P. We will be studying the topology of the order complex of the posets $\textrm{MacP}(2,n)$ and $\textrm{MacP}(1,2,n)$.

Definition 16

([7], Page 8) A finite poset P is said to be semimodular if it is bounded, and whenever two distinct elements u, v both cover $x \in P$, there is a $z \in P$ that covers both u and v. The poset P is defined to be totally semimodular if it is bounded and every interval in P is semimodular.

Definition 17

([5], Page 213) Let P be a poset. P is said to be thin if every interval of length 2 in P has exactly four elements.

Notation 18

Let P be a poset. We denote its unique minimum element if it exists by $\hat{0}$. If the unique minimum does not exists, the poset $P \cup \hat{0}$ 4,, is a poset obtained from P by adjoining a unique minimum element to P. Similarly, the unique maximum element of P if it exists is denoted by $\hat{1}$. If it does not exists, then $P\cup \hat{1}$ is an augmented poset with a unique maximal element.

Definition 19

([7], Page 4) A graded poset P is said to admit a recursive atom ordering if the length of P is 1 or if the length of P is greater than 1 and there is an ordering $a_1, a_2, \ldots , a_t$ of the atoms of P which satisfies:

• 1. If $j \in \{1,2, \ldots , t\}$, then $[a_j, \hat{1}]$ admits a recursive atom ordering in which the atoms of $[a_j, \hat{1}]$ that comes first in the ordering are those that cover some $a_i$ where $i < j$.

• 2. For all $i < j$, if $a_i, a_j < y$, then there is a $k < j$ and an element $z \le y$ such that z covers $a_k$ and $a_j$.

The following theorem and proof appears in the work of Bjorner and Wachs ([7]). We will give below their proof of the only if direction.

Theorem 20

([7], Page 8) A graded poset P is totally semimodular if and only if for every interval [x, y] of P, every atom ordering of [x, y] is a recursive atom ordering.

Proof

Proof of the only if Let P be a totally semimodular poset with length greater than 1. If $[x,y] \ne P$, then [x, y] is totally semimodular, and by induction every atom ordering of [x, y] is a recursive atom ordering.

Let $a_1, a_2, \ldots , a_n$ be any atom ordering in P, since every atom ordering in $[a_j, \hat{1}]$ is recursive, order the atoms of $[a_j, \hat{1}]$ so that those that cover $a_i$ for some $i < j$ come first. Let $y \ge a_i, a_j$. Since P is totally semimodular, there is a $z \in P$ that covers both $a_i$ and $a_j$. $\square$

3.2 Regular cell complexes

Definition 21

A regular CW complex X is a collection of disjoint open cells $\{e_{\alpha }\}$ whose union is X such that:

• 1. X is Haursdorff

• 2. For each open m-cell $e_\alpha$, there exists a continuous map $f_\alpha : D^m \rightarrow X$ that is a homeomorphism onto $\overline{e_\alpha }$, maps the interior of $D^m$ onto $e_\alpha$, and maps the boundary $\partial D^m$ into a finite union of open cells, each of dimension less than m.

Let $\{e_{\alpha }\}$ be a regular cell decomposition of X and F(X) the face poset of the decomposition, with $e_\alpha \le e_\tau$ if and only if $e_\alpha \subseteq \partial \overline{e_\tau }$. The complex $\Vert F(X)\Vert$ is homeomorphic to X.

Theorem 22

([5], Page 213) If a poset P containing $\hat{0}$ and $\hat{1}$ is thin and admits a recursive atom ordering, then P is the augmented face poset of a regular cell decomposition of a PL sphere.

3.2.1 Cell collapse

Hersh in [8], introduced a general class of collapsing maps which may be performed sequentially on a polytope and preserving homeomorphism type. Each such map is defined by first covering a polytope face with a family of parallel lines or generally a family of parallel-like segments across which the face is collapsed. An example [8] of such a collapsing map is given below:

Example 23

([8], Page 7) Let $\Delta _2$ be the convex hull of (0, 0), (1, 0), (0, 1/2) in $\mathbb {R}^2$, and let $\Delta _1$ be the convex hull of (0, 0) and (1, 0) in $\mathbb {R}^2$. There is a continuous and surjective function $g: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ that acts homeomorphically on $\mathbb {R}^2\setminus \Delta _2$ sending it onto $\mathbb {R}^2 \setminus \Delta _1$. The simplex $\Delta _2$ is covered by vertical line segments with an end point in $\Delta _1$. The map g has the property that it maps each vertical line segment to its end point in $\Delta _1$.

Let $R = \{(x,y): -1 \le x \le 1, 0 \le y \le 1\}$. For $0 \le x \le 1$ and $0 \le y \le -x/2 + 1/2$, let $g(x,y) = (x,0)$. For $0 \le x \le 1$ and $-x/2 + 1/2 \le y \le 1$, let $g(x,y) = (x, \frac{y- 1/2+x/2}{1/2+x/2})$. For $-1 \le x \le 0$ and $0 \le y \le -x/2 + 1/2$, let $h(x,y) = (x, y\frac{-x}{-x/2+1/2})$. For $-1 \le x \le 0$ and $-x/2 + 1/2 \le y \le 1$, let $g(x,y) = (x, -1+2y)$ , and let g acts as the identity outside R. We note that $\Delta _2 \subseteq R$.

The map $g: \mathbb {R}^2 \rightarrow \mathbb {R}^2$ is such that $g(\Delta _2) = \Delta _1$ and g maps $\mathbb {R}^2{\setminus } \Delta _2$ homeomorphically onto $\mathbb {R}^2{\setminus } \Delta _1$. The map g is said to collapse the 2-simplex $\Delta _2$ onto its face $\Delta _1$.

Let $\sim$ be a relation on $\mathbb {R}^2$ defined as $(a,b) \sim (a', b')$ if and only if $g(a,b) = g(a',b')$. This induces the quotient map $g/\sim \;\;: \mathbb {R}^2/\sim \;\; \rightarrow \mathbb {R}^2$. The continuous map $g/\sim$ is one-to-one and onto and thus a homeomorphism. Hence, $\mathbb {R}^2/\sim$ is homeomorphic to $\mathbb {R}^2$.

Hersh gave a formal definition of a collapsing map below

Definition 24

[8] Let $g: X \rightarrow Y$ be a continuous, surjective function with the quotient topology on Y. That is, open sets in Y are the sets whose inverse images under g are open in X. We call such a map g an identification map.

Definition 25

([8], Page 23) Given a finite regular CW complex K on a set X an open cell L in K, define a face collapse or cell collapse of $\overline{L}$ onto $\overline{\tau }$ for $\tau$ an open cell contained in $\partial L$ to be an identification map $g: X \rightarrow X$ such that:

• 1. Each open cell of $\overline{L}$ is mapped surjectively onto an open cell of $\overline{\tau }$ with L mapped onto $\tau$.

• 2. g restricts to a homeomorphism from $K {\setminus } \overline{L}$ to $K {\setminus } \overline{\tau }$ and acts homeomorphically on $\overline{\tau }$.

• 3. The images under g of the cells of K form a regular CW complex with new characteristic maps obtained by composing on the left the original characterisitc maps of K with $g^{-1}: X \rightarrow X$ for those cells of K contained either in $\overline{\tau }$ or in $K \setminus \overline{L}$.

Definition 26

([8], Page 25) Let $K_0$ be a convex polytope, and let $\mathcal {C}_i^0$ be a family of parallel line segments covering a closed face $L_i^0$ in $\partial K_0$ with the elements of $\mathcal {C}_i^0$ given by linear functions $c: [0,1] \rightarrow L_i^0$. Suppose that there is a pair of closed faces $G_1, G_2$ in $\partial L_i^0$ with $c(0) \in G_1$ and $c(1) \in G_2$ for each $c \in \mathcal {C}_i^0$ and there is a composition $g_i \circ \cdots \circ g_1$ of face collapses on $K_0$ such that:

• 1. $g_i \circ \cdots \circ g_1$ acts homeomorphically on $\textrm{int}(L_i^0)$

• 2. For each $c \in \mathcal {C}_i^0$, $g_i \circ \cdots \circ g_1$ either sends c to a single point or acts homeomorphically on c.

• 3. Suppose $g_i \circ \cdots \circ g_1(c(t)) = g_i \circ \cdots \circ g_1(c'(t'))$ for $c\ne c' \in \mathcal {C}_i^0$ and some $(t,t') \ne (1,1)$. Then $t = t'$, and for each $t \in [0,1]$ we have $g_i \circ \cdots \circ g_1(c(t)) = g_i \circ \cdots \circ g_1(c'(t'))$.

Then call $\mathcal {C}_i = \{g_i \circ \cdots \circ g_1(c(t))| c \in \mathcal {C}_i^0\}$ a family of parallel-like curves on the closed cell $L_i = g_i \circ \cdots \circ g_1(L_i^0)$ of the finite regular CW complex $K_i = g_i \circ \cdots \circ g_1(K_0)$.

Theorem 27

([8], Theorem 4.21) Let $K_0$ be a convex polytope. Let $g_1, \ldots , g_i$ be collapsing maps with $g_j: X_{K_{j-1}} \rightarrow X_{K_j}$ for regular CW complexes $K_0, K_1, \ldots , K_i$ all having the underlying space X. Suppose that there is an open cell $L_i^0$ in $\partial K_0$ upon which $g_i \circ \cdots \circ g_1$ acts homeomorphically and a collection $\mathcal {C} = \{g_i \circ \cdots \circ g_1(c)| c \in \mathcal {C}_i^0\}$ of parallel-like curves covering $\overline{L_i}$ for $L_i = g_i \circ \cdots \circ g_1(L_i^0) \in K_i$. Then there is an identification map $g_{i+1}: X_{K_i} \rightarrow X_{K_{i+1}}$ specified by $\mathcal {C}$.

The sequence of cell collapses needed in the proof of Theorem 46 will be of the form illustrated by the example below.

Example 28

Let $K_0 = \Delta ^3 \times [0,1] \times [1,2]\times [0,1]$, where $\Delta ^3 = \{(x_1, x_2, x_3): 0 \le x_1 \le x_2 \le x_3 \le \frac{\pi }{2} \}$ is a 3-simplex. Let $L_0 = \Delta ^3 \times \{0\} \times [1,2] \times [0,1]$ be a face of $K_0$. Let $G_1^0 = \{(0, x_2, x_3): 0= x_1 \le x_2 \le x_3 \le \frac{\pi }{2}\} \times \{0\} \times [1,2] \times [0,1]$ and $G_2^0 = \{(x_1, x_2, x_3): 0 \le x_1 = x_2 \le x_3 \le \frac{\pi }{2}\} \times \{0\} \times [1,2] \times [0,1]$. Then $G_1^0$ and $G_2^0$ are faces of $L_0$.

Let $(x,r) = ((0, x_2, x_3), 0, r_2, r_3) \in G_1^0$, and $c_{(x,r)}$ is a curve defined by:

$$\begin{aligned} c_{(x,r)}: [0,1] \rightarrow L_0: t \mapsto ((t\cdot x_2, x_2, x_3), (0, r_2, r_3)). \end{aligned}$$

The end point of the curve $c_{(x,r)}$ lies in $G_2^0$. Let $\mathcal {C}_0 = \{c_{(x,r)}: (x,r) \in G_1^0\}$. Then $\mathcal {C}_0$ is a collection of parallel-like line segments covering the face $L_0$. By Theorem 27, there is an identification map $g_1: K_0 \rightarrow K_0$ specified by $\mathcal {C}_0$. The map has the property that the curves in $\mathcal {C}_0$ are mapped to their end points in $G_2^0$.

Let $\sim _1$ be a relation on K defined by $(x,r) \sim _1 (x', r')$ if and only if $g_1(x, r) = g_1(x', r')$. Then the quotient $K_1 = K/\sim _1$ is homeomorphic to K.

Also, let $L_1 = \Delta ^3 \times [0,1] \times [1,2] \times \{0\}$ be a face of $K_0$, $G_1^1 = \{(x_1, x_2, x_3): x_1 \le x_2 \le x_3 \le \frac{\pi }{2}\} \times [0,1] \times [1,2] \times \{0\}$, and $G_2^1 = \{(x_1, x_2, \frac{\pi }{2}): x_1 \le x_2 \le x_3 = \frac{\pi }{2}\} \times [0,1] \times [1,2] \times \{0\}$. Then $G_1^1$ and $G_2^1$ are faces of $L_1$. Let $\mathcal {C}^0_1$ be a collection of parallel line segments covering $L_1$ and with endpoints in $G_1^1$ and $G_2^1$. Then $\mathcal {C}_1 = \{g_1(c)| c \in \mathcal {C}_1^0\}$ is a collection of parallel-like segments covering $g_1(L_1)$. By Theorem 27, there is an identification map $g_2: K_1 \rightarrow K_1$ specified by $\mathcal {C}_1$. Let $\sim _2$ be a relation on $K_1$ defined by $[(x,r)] \sim _2 [(x', r')]$ if and only if $g_2([(x,r)]) = g_2([(x', r')])$. Hence, $K_2 = K_1/\sim _2$ is homeomorphic to $K_1$.

In Theorem 46, we will apply homeomorphism-preserving cell collapses on the boundary of a closed ball of the form

$$\begin{aligned} B_M= & {} \left\{ (\theta _1, \theta _2, \ldots , \theta _{n_1}): 0 \le \theta _1 \le \theta _2 \cdots \le \theta _{n_1} \le \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (\eta _1, \eta _2, \ldots , \eta _{n_2}): \frac{\pi }{2} \le \eta _1 \le \eta _2 \le \cdots \le \eta _{n_2}\le \pi \right\} \\{} & {} \times [a,b]^{{l}_M-2}. \end{aligned}$$

The identification $\sim$ as in Example 28 is given here as in (Fig. 5).

Fig. 5

An identification on the boundary of $\mu ^{-1}(M)$

The identification as in Fig. 5 on the boundary $r_2 = 0, r_5 = 0$ gives the collapse of the 5-simplex $\{(\theta _2, \theta _3, \theta _4, \theta _5, \theta _6): 0 \le \theta _2 \le \theta _3 \le \cdots \theta _6 \le \frac{\pi }{2}\}$ onto the 3-simplex $\{(\theta _2, \theta _3, \theta _4, \theta _5, \theta _6): 0 = \theta _2 \le \theta _3 \le \theta _4 = \theta _5 \le \theta _6 \le \frac{\pi }{2}\}$.

3.2.2 Topological balls

Theorem 29

([9], Page 1) Let S be an $(n-1)$ sphere in $S^n$ and H a component of $S^n - S$. If $\overline{H}$ is a topological manifold with boundary, then $\overline{H}$ is homeomorphic to an n-ball.

Corollary 30

Suppose $\textrm{int}(D^n)$ is the interior of a closed unit ball centered at the origin. If $\overline{H} \subset \text{ int }(D^n)$ is a topological manifold whose boundary S is an $(n-1)$ sphere, then $\overline{H}$ is homeomorphic to a closed unit ball $D^n$.

Definition 31

Let M be a topological manifold and $S \subseteq M$. The subset S is said to be collared in M if there is a neighborhood $N(S) \subseteq M$ and a homeomorphism $h: S \times I \rightarrow N(S)$ satisfying $h(x,1) = x$.

Theorem 32

([10], Theorem 2) The boundary of a topological manifold M with boundary is collared in M.

Proposition 33

Suppose $G \subset X$ is homeomorphic to an open ball and $\overline{G}$ is a topological manifold whose boundary is a sphere. Then $\overline{G}$ is homeomorphic to a closed ball.

Proof

Let $h_1: G \rightarrow \text{ int }(D^n)$ be an homeomorphism to the open unit ball and $S = \partial \overline{G}$ the boundary of G. As $\overline{G}$ is a topological manifold whose boundary is S, by Theorem 32, there is a collared neighborhood N(S) of S in $\overline{G}$ and a homeomorphism $h_2: S \times I \rightarrow N(S) \subseteq \overline{G}$ with $h_2(S\times \{1\}) = S$. Let $A = h_2(S\times \{0\} ) \subset G$, and B the subset of G whose boundary is A.

So $A' = h_1({B}) \subset \text{ int }(D^n)$ is a topological manifold whose boundary is the sphere $h_1(A)$. By Corollary 30, $A'$ is homeomorphic to a closed ball , hence B is homeomorphic to a closed ball.

We will denote by $h_3: {B} \rightarrow D_{\frac{1}{2}}$ the homeomorphism of B to a closed ball, where $D_{\frac{1}{2}}$ is a closed ball of radius $\frac{1}{2}$ centered at the origin. Define:

$$\begin{aligned} h_4: S \times I \rightarrow D^n: \; (x,t) \mapsto (1 + t)\cdot h_3 \circ h_2(x,0). \end{aligned}$$

$h_4$ gives a homeomorphism from $N(S) \subseteq \overline{G}$ to the annulus in $D^n$ the unit ball. As $\overline{G} = B \cup N(S)$, the map $h: \overline{G} \rightarrow D^n$ given by $h = {h_3} \cup (h_4{\circ h_2^{-1}})$ is a homeomorphism by the Gluing Lemma [6]. $\square$

4 The stratification $\mu ^{-1}(M)$

Theorem 34

$\{\mu ^{-1}(M): M \in \; \text{ MacP }(2, n)\}$ is a regular cell decomposition of $Gr(2,\mathbb {R}^n)$.

Let M be a rank 2 oriented matroid. We will first determine the topology of $\mu ^{-1}(M) = \{X\in Gr(2, \mathbb {R}^n): (\pm \chi _X) = M\}$. We know from the background on oriented matroids in Sect. 2 that the topology of $\mu ^{-1}(M)$ is invariant under relabeling and reorientation of elements of the oriented matroid. We may thus assume that $\{1, 2\}$ is a basis of M for the rest of the proofs in this chapter. We may also assume that for any $X \in \mu ^{-1}(M)$, X can be uniquely expressed as $X = \text{ Rowspace }(e_1 \; e_2 \; v_3 \; {v_4} \; \cdots v_n)$ satisfying $0 \le \textrm{Arg}(v_k) < \pi$ for a non-zero vector $v_k$. For such a non-zero vector $v_k$, we let $\theta _k = \textrm{Arg}(v_k)$ and $r_k = |v_k|$, so that $v_k = r_ke^{i\theta _k}$.

Arranging the arguments of the non-zero vectors in an increasing order as $0< \theta _{l_1}< \theta _{l_2}< \cdots< \theta _{l_{n_1}} < \frac{\pi }{2}$, $\frac{\pi }{2}< \theta _{j_1}< \theta _{j_2}< \cdots< \theta _{j_{n_2}} < \pi$, we can thus uniquely represent X as $( (\theta _{l_1}, \theta _{l_2}, \ldots , \theta _{l_{n_1}}), (\theta _{j_1}, \theta _{j_2}, \ldots , \theta _{j_{n_2}}), (r_\lambda )_{\lambda } )$. An example of such an identification is given in Fig. 6.

Fig. 6

$X \; \text{ as } \text{ a } \text{ point }\; ((\theta _3, \theta _4, \theta _5), (\theta _6, \theta _8), (r_i)_{1\le i \le 9})$

The above representation of an element $X \in \mu ^{-1}(M)$ gives an identification

$$\begin{aligned} ({\textbf {ID1}}) \; \; \mu ^{-1}(M) \equiv A_M= & {} \left\{ (\theta _{l_1}, \theta _{l_2}, \ldots , \theta _{l_{n_1}}): 0< \theta _{l_1}< \theta _{l_2} \cdots< \theta _{l_{n_1}}< \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (\theta _{j_1}, \theta _{j_2}, \ldots , \theta _{j_{n_2}}): \frac{\pi }{2}< \theta _{j_1}< \theta _{j_2}< \cdots < \theta _{j_{n_2}}\le \pi \right\} \\{} & {} \times (0, \infty )^{l_M-2} \end{aligned}$$

Notation 35

Let $M \in \textrm{MacP}(2,n)$ and i a non-loop in M. $l_M$ denotes the number of non-loops element in M. $p_M$ denotes the number of distinct parallel classes of M. $P_M(i)$ denotes the set of elements that are parallel to i.

The following proposition then follows from the above identification of $\mu ^{-1}(M).$

Proposition 36

Let $M \in \text{ MacP }(2,n)$. Then $\mu ^{-1}(M)$ is homeomorphic to an open ball of dimension $h(M) = l_M + p_M -4$.

Notation 37

Let $M \in \textrm{MacP}(2,n)$. The dimension of $\mu ^{-1}(M)$ is given by $h(M) = l_M + p_M -4$

Proposition 38

Let $(\pm \chi _N, N), (\pm \chi _{M}, M)$ be two rank 2 oriented matroids. Suppose that M covers N. Then exactly one of the following possibilities holds:

CR1:

There is exactly one i such that $|P_M(i)|\ge 2$ in M, i is a loop in N and $\chi _N(j,k) = \chi _M(j,k)$ for $j,k \ne i$.

CR2:

There are exactly two distinct parallel/anti-parallel classes $P_M(i)$ and $P_M(j)$ in M such that the following holds:

$$\begin{aligned} \chi _N(a,b) = \left\{ \begin{array}{ll} 0 &{}\quad \text{ if } \, a, b \in P_M(i) \cup P_M(j)\\ \chi _M(a,b) &{} \quad \text{ otherwise } \end{array}\right\} \end{aligned}$$

Proof

We first check that if M and N satisfy CR1 or CR2 then M covers N. Suppose M and N satisfy CR1; that is $\chi _N(a,b) = \chi _M(a,b)$ for all $a, b \ne i$ and i is a loop of N. Let $N'$ be a rank 2 oriented matroid such that $N < N' \le M$. Then i is not a loop in $N'$ as otherwise $N' \le N$. We thus have that $\chi _{N'}(i, k) \ne 0$ if and only if $\chi _M(i,k) \ne 0$. Hence $N' = M$.

Also, suppose that M and N satisfy CR2. Then there are adjacent parallel classes $P_M(i)$, $P_M(j)$ with $\chi _N$ satsfying the definition in CR2. Suppose there is an $N'$ satsfying $N < N' \le M$. Then $\chi _{N'}(i,j) \ne 0$ as otherwise $N' \le N$. So, $\chi _{N'}(a,b) = \chi _M(a,b)$ for all $a \in P_M(i)$ and $b \in P_M(j)$. Hence $N' = M.$

Let $N_0 < M$, and let i, j be non-loops in $N_0$. $\chi _M(i, j) > 0$ implies that $\chi _{N_0}(i,j) \ge 0$. That is, for any subspace realization $\textrm{Rowspace}(e_1\; e_2 \; v_3 \; v_4 \cdots \; v_n ) \in \mu ^{-1}(M)$, $\textrm{Arg}(v_i) < \textrm{Arg}(v_j)$ implies that $\textrm{Arg}(w_i) \le \textrm{Arg}(w_j)$ for any vector realization $\textrm{Rowspace}(e_1\; e_2 \; w_3\; w_4 \cdots \; w_n) \in \mu ^{-1}(N_0)$. Also $\chi _M(i,j) = 0$ implies that $\chi _{N_0}(i, j) = 0$. In otherwords, $\textrm{Arg}(v_i) = \textrm{Arg}(v_j)$ implies that $\textrm{Arg}(w_i) = \textrm{Arg}(w_j)$ if i, j are non-loops in $N_0$ (Fig. 7).

Fig. 7

Covering relations in MacP(2, n)

Suppose $l_i$ is a non-loop in M such that all elements in $P_M(l_i)$ are loops in $N_0$. Let a be a non-loop in $N_0$ satisfying the condition that if for any element p such that $\chi _M(p, a) = +$, then $\chi _M(p, l_i) = +$. Similarly, let b be a non-loop in N satisfying the condition that if for any element p such that $\chi _M(b, p) = +$, then $\chi _M(l_i,p) = +$. The face of $\{(\theta _1, \theta _2, \ldots , \theta _n): 0 \le \theta _1 \le \theta _2 \le \cdots \le \theta _n \le \frac{\pi }{2}\}$ corresponding to $\theta _a = \theta _{l_1} = \theta _{l_2} = \cdot \theta _{l_i}< \cdots < \theta _b$ determines a rank 2 oriented matroid $N'$ such that $N_0< N' < M$. Repeating the same argument for all such $l_i$ with all elements in $P_M(l_i)$ loops in N, we can assume that $N'$ is a rank 2 oriented matroid such that $N' < M$ is obtained from M by sequences of CR2 and each parallel class $P_{N'}(i)$ contains a non-loop of N. If a, b are non-loops in N such that $\chi _{N}(a, b) = 0$ and $\chi _N'(a, b) = +$, this corresponds to the face of $\{(\theta _1, \theta _2, \ldots , \theta _{n_1}): 0 \le \theta _1 \le \theta _2 \le \cdots \le \theta _n \le \frac{\pi }{2}\}$ determined by $\theta _a = \cdots = \theta _b$. From $N'$ we can obtain by sequences of CR2 a rank 2 oriented $N''$ having the same number of distinct parallel class as N and the same number of non-loops as M. We can then obtain N from $N''$ by sequence(s) of ${\textbf {CR1}}$. In particular, if M covers $N_0$, then the proposition follows. $\square$

Corollary 39

$\text{ MacP }(2,n)$ is a ranked poset, with rank function $h(M) = \dim (\mu ^{-1}(M)) = l_M + P_M - 4$.

Lemma 40

If $N < M$ are rank 2 oriented matroids such that M covers N. Then $\mu ^{-1}(N) \subseteq \partial \overline{(\mu ^{-1}(M))}$.

Proof

Case 1: Suppose the covering relation $N < M$ is CR1. There is exactly one p in a parallel/anti-parallel class P of M such that $|P|\ge 2$ in M, p is a loop in N and $\chi _N(j,k) = \chi _M(j,k)$ for $j,k \ne i$.

Let $X_1 = \text{ Rowspace }(v_1 \; v_2 \; v_3 \; \cdots v_k) \in \mu ^{-1}(N)$. If p is parallel to j in M and p is a loop in N that is $v_p = 0$, then $\text{ Rowspace }(v_1 \; v_2 \; \cdots v_{p-1} \; \epsilon \cdot v_j \; v_{p+1} \ldots v_n) \in \mu ^{-1}(M)$ for every $\epsilon > 0$. Hence $\mu ^{-1}(N) \subseteq \partial \overline{\mu ^{-1}(M)}$

Case 2: Suppose the covering relation $N < M$ is CR2. There are two distinct parallel/anti-parallel classes $P_M(s)$ and $P_M(j)$ in M such that the following holds:

$$\begin{aligned} \chi _N(k,l) = \left\{ \begin{array}{ll} 0 &{} \quad \text{ if } \,k,l \in P_M(s) \cup P_M(j)\\ \chi _M(k,l) &{} \quad \text{ otherwise } \end{array}\right\} \end{aligned}$$

Let $X_1 = \text{ Rowspace }(e_1 \; e_2 \; \cdots \; v_n)$ be any vector arrangement in $\mu ^{-1}(N)$. Let $v_s = r_se^{i\theta _s}$, $v_j = r_je^{i\theta _s}$, and suppose without loss of generality that s and j are parallel in N. Let $v_j^\epsilon = r_se^{i(\theta _s + \epsilon )}$. If $\chi _{M}(s,j) = +$, ($v_j^{-\epsilon }$ if $\chi _{M}(p,j) = -$), then $\text{ Rowspace }(e_1 \; e_2 \; v_3\; \cdots v_{j-1}, v_j^\epsilon , v_{j+1}, \; \cdots \; v_n) \in \mu ^{-1}(M)$ for every sufficiently small value of $\epsilon$. Hence, $\mu ^{-1}(N) \subseteq \partial \overline{\mu ^{-1}(M)}.$ $\square$

Proposition 41

Let $M \in \text{ MacP }(2,n)$. Then $\partial \overline{\mu ^{-1}(M)} = \bigcup _{N < M} \mu ^{-1}(N)$

Proof

Let $\chi$ be a chirotope of M. Then $\overline{\mu ^{-1}(M)} \subseteq \{\text{ Rowspace }(v_1 \; v_2\; v_3\; \cdots \; v_n): \text{ sign }(\det (v_i \; v_j)) \in \{0, \chi (i,j)\}\}$. Let $X \in \overline{\mu ^{-1}(M)}$ and N the rank 2 oriented matroid determined by X. Then $N \le M$.

Conversely, suppose $N, M \in \text{ MacP }(2,n)$ such that $N < M$. We will show that $\mu ^{-1}(N) \subseteq \overline{\mu ^{-1}(M)}$.

Let $N = N_1< N_2< \cdots < N_k = M$ be a maximal chain from N to M. That $\mu ^{-1}(N_i) \subseteq \partial \overline{\mu ^{-1}(N_{i+1})}$ follows from Lemma 40.

From $\mu ^{-1}(N_i) \subset \partial \overline{(\mu ^{-1}(N_{i+1}))}$ for each i, we conclude that $\mu ^{-1}(N) \subseteq \partial \overline{(\mu ^{-1}(M))}$. $\square$

A useful observation from Propositions 38 and 41 is the following:

Suppose $N < M$ are rank 2 oriented matroids such that $P_M(l_1)$ and $P_M(l_2)$ are distinct parallel/anti-parallel classes in M, but $P_M(l_1) \cup P_M(l_2)$ is a parallel/anti-parallel class in N. Then we can obtain a rank 2 oriented matroid $N' \le M$ such that $N'$ covers N, $P_M(l_1)$ and $P_M(l_2)$ are in distinct parallel/anti-parallel class in $N'$.

An example of such a construction is the following:

Suppose $\text{ Rowspace }(v_1\; v_2 \; \ldots \; v_n)$ determines a vector realization for N and assume WLOG that $P_N(l_1) = P_M(l_1) \cup P_M(l_2)$. We obtain a rank 2 oriented matroid $N'$ as follows: let $v_{l_2} = r_{l_2}e^{i\theta _{l_2}}$ and, we denote by $\overline{P_M(l_1)}$ the subset of $P_M(l_1)$ that are anti-parallel to $l_2$ in N:

$$\begin{aligned} w_s = \left\{ \begin{array}{ccc} v_s &{} \text{ if } &{} s \notin P_M(l_1)\\ r_{l_2}e^{i(\theta _{l_2} + \epsilon )} &{} \text{ if } &{} s \in P_M(l_1)\setminus \overline{P_M(l_1)}, \chi _M(l_2, s) = +\\ r_{l_2}e^{i(\theta _{l_2} - \epsilon )} &{} \text{ if } &{} s \in P_M(l_1) \setminus \overline{P_M(l_1)}, \chi _M(l_2, s) = - \\ r_{l_2}e^{i(\theta _{l_2} + \pi - \epsilon )} &{} \text{ if } &{} s \in \overline{P_M(l_1)},\chi _M(l_2, s) = +\\ r_{l_2}e^{i(\theta _{l_2} + \pi + \epsilon )} &{} \text{ if } &{} s \in \overline{P_M(l_1)},\chi _M(l_2, s) = -\\ \end{array}\right\} . \end{aligned}$$

The rank 2 subspace $\text{ Rowspace }(w_1\; w_2\; \ldots \; w_n)$ determines a rank 2 oriented matroid $N' \le M$ that covers N.

5 Shellability of the interval $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$

Let $\textrm{MacP}(2,n) \cup \{\hat{0}\}$ be a poset obtained from $\textrm{MacP}(2,n)$ by adjoining a unique minimal element $\hat{0}$. Let $M \in \textrm{MacP}(2,n)$ and $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$ an interval in $\textrm{MacP}(2,n) \cup \{\hat{0}\}$. We will show that the poset $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$ is the augmented face poset of a regular cell decomposition of a $h(M)-1$ dimensional sphere , where h(M) is the dimension of $\mu ^{-1}(M)$ from Proposition 36.

Lemma 42

Let $W < T$ be rank 2 oriented matroids. Then the interval [W, T] is totally semimodular.

Proof

Let $N, N_1, N_2 \in [W, T]$ such that $N_2$ and $N_1$ cover N. In Proposition 38, we proved that there are two possible scenarios when $N_1$ covers N and similarly for $N_2$ covering N. So there are the following three distinct cases:

Case 1: If $N < N_1$ and $N < N_2$ are both case CR1 of Proposition 38. That is, there are $i\ne j$ such that i, j are loops in N but i is a non-loop in $N_1$ with parallel/anti-parallel class $P_1(i)$ in $N_1$ such that $|P_1(i)| \ge 2$ in $N_1$. Also, that j is a non-loop with parallel/anti-parallel class $P_2(j)$ in $N_2$ so that $|P_2(j)| \ge 2$ and j is a loop in N. Let $k_1 \in P_1(i) {\setminus } \{i,j\}$ and $k_2 \in P_2(j) {\setminus } \{i,j\}$. Then $k_1, k_2$ are non-loops in N. If $\text{ Rowspace }(v_1 \; v_2 \; \cdots \; v_n)$ determines a vector realization of N, then $\text{ Rowspace }(v_1\; v_2 \; \cdots v_{i-1} \; \epsilon _1 \cdot v_{k_1} \; v_{i+1}\; \cdots v_{j-1} \; \epsilon _2\cdot v_{k_2}\; \cdots v_n)$ (where $\epsilon _i = \pm 1$) gives a rank 2 oriented matroid M that covers both $N_1$ and $N_2$.

Case 2: If $N < N_1$ is case (a) of Proposition 38 and $N < N_2$ is case (b) of Proposition 38, that is, there is a non-loop i with $|P_1(i)| \ge 2$ in $N_1$ and i is a loop in N. Also, there are distinct parallel classes $P_2(j)$ and $P_2(k)$ in $N_2$ such that $P_2(j) \cup P_2(k)$ is a parallel class in N. Suppose WLOG that $t \in P_1(i)\setminus i$ is parallel to i in $N_1$.

We have that i is a loop in $N_2$ since $N_2$ covers N. Now, if $\text{ Rowspace }(v_1 \; v_2\;\ldots v_i \ldots \; v_n)$ determines a vector realization for $N_2$, then $\text{ Rowspace }(v_1\; v_2 \; \ldots v_{i-1}\; v_t\; v_{i+1}\ldots \; v_n)$ determines a vector realization for M that covers both $N_1$ and $N_2$.

Case 3:If $N < N_1$ and $N < N_2$ are both case (b) in Lemma 38. That is, there are distinct parallel classes $P_1(l_1), P_1(l_2)$ in $N_1$ such that $P_1(l_1) \cup P_1(l_2)$ is a parallel/anti-parallel class in N. Similarly, there are classes $P_2(l_3), P_2(l_4)$ in $N_2$ such that $P_2(l_3) \cup P_2(l_4)$ is a parallel class in N.

As in the observation after Proposition 41, we can obtain a rank 2 subspace $\text{ Rowspace }(w_1 w_2\; \ldots \; w_n)$ that determines a rank 2 oriented matroid M that covers both $N_1$ and $N_2$. $\square$

We have from the above lemma that for a rank 2 oriented matroid M, every bounded interval in $\text{ MacP }(2,n)_{\le M}$ is totally semimodular.

Let M be a rank 2 oriented matroid, and let N be a rank 2 oriented matroid in the interval $P = \text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$. Then the interval [N, M] is totally semimodular by Lemma 42. By Theorem 20, any atom ordering in [N, M] is a recursive atom ordering.

So to find a recursive atom ordering for $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$, we only need to order the atoms of $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$ as $X_1, X_2, \ldots , X_k$ so that the ordering satisfies conditions in Definition  19.

Proposition 43

Let M be a rank 2 oriented matroid. Then the interval $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$ has a recursive atom ordering.

Proof

We begin by ordering the atoms of M as follows (see Fig. 8 below): Fix a vector realization $A = \{v_1, v_2, v_3, \ldots , v_n\}$ that determines M. The vector realization A determines vector realizations for atoms of M. Let L be an affine line not parallel to any of the lines spanned by the non-zero vectors in $\{v_i\}$.

We label the vectors in the realization A in an increasing order from left to right in the order lines spanned by the vectors intersect the affine line L. We note that elements in the same parallel class $P_M(i)$ have their linear spans intersect L at the same point (in the Fig. 8 below, the pairs 1, 2 and 3, 4 labeled vectors are in a parallel class respectively).

Fig. 8

Recursive atom ordering

An atom of $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$ can be identified with (i, j) for some $i < j$ in the above ordering, and the vectors labeled by i, j form a basis for M. The unlabeled elements of the oriented matroid M are loops.

We will also let $(i, {\{j,r\}})$ denote a rank 2 oriented matroid in $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$ covering the atoms (i, j) and (i, r), with element labeled j parallel to element labeled r. Similarly, we have a notation of the form $({\{i,r\}}, j)$.

We consider a dictionary order for ordered pairs for the atoms of M as $X_1, X_2,\ldots , X_k$. We now verify the conditions in Definition 19. By Lemma 42, the interval $[X_i, M]$ is totally semimodular for each i, and so it has a recursive atom ordering by Theorem 20. In fact any ordering of the atoms in $[X_i, M]$ gives a recursive atom ordering of the poset $[X_i, M]$ by Theorem 20.

We now verify the second condition in Definition 43. Suppose $A_1 = (i_1, j_1) < A_2 = (i_2, j_2)$ in the dictionary order and $A_1, A_2 < Y$ where $Y \in \text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$. We will consider the following cases:

Case 1: Suppose $i_1 = i_2$. Then $j_1 < j_2$. Let j be the maximum label such that $j_1 \le j < j_2$ and the element labeled by j is a non-loop in Y. We obtain Z as $(i_1, {\{j, j_2\}})$, where j and $j_2$ are parallel if $\mathbb {R}_{\ge 0}(\epsilon _1 \cdot v_j), \mathbb {R}_{\ge 0}(\epsilon _2 \cdot v_{j_2})$ intersect L for $\epsilon _i \in \{\pm 1\}$, $\epsilon _1\cdot \epsilon _2 =1$, and anti-parallel if $\epsilon _1 \cdot \epsilon _2 = -1$. Such a Z covers both $(i_1, j)$ and $(i_1, j_2)$. We have $Z \le Y$ and $(i_1, j) < (i_2, j_2)$ in the dictionary order.

Case 2: Suppose $i_1 < i_2$. Let i be the maximum label such that $i_1 \le i < i_2$ and the element labeled i is not a loop in Y. Then Z is obtained as $({\{i, i_2\}}, j_2)$. Similarly, $i, i_2$ are parallel if $\mathbb {R}_{\ge 0}(\epsilon _1 \cdot v_i), \mathbb {R}_{\ge 0}(\epsilon _2 \cdot v_{i_2})$ intersect L for $\epsilon _i \in \{\pm 1\}$, $\epsilon _1\cdot \epsilon _2 =1$, and anti-parallel if $\epsilon _1 \cdot \epsilon _2 = -1$. Such a Z is a rank 2 oriented matroid covers the atoms $(i, j_2)$ and $(i_2, j_2)$. We have that $Z \le Y$ and $(i, j_2) < (i_2, j_2)$ in the dictionary order.

For each $j > 1$, let $Q_j = \{Y \in \text{ atom }[X_j, M]: Y \ge X_i \, \text{ for }\,\text{ some } i < j\}$. In the recursive atom ordering of $[X_j, M]$ for $j > 1$, we let elements of $Q_j$ come first. This determine a recursive atom ordering for the poset $[X_i, M]$ by Theorem 20, as the interval $[X_i, M]$ is totally semimodular for each i by Lemma 42.

Hence the poset $\text{ MacP }(2,n)_{\le M} \cup \{\hat{0}\}$ has a recursive atom ordering. $\square$

Lemma 44

Let N be a rank 2 oriented matroid. Then the interval $\text{ MacP }(2,n)_{\le N} \cup \{\hat{0}\}$ is thin.

Proof

In the interval $\text{ MacP }(2,n)_{\le N} \cup \{\hat{0}\}$ when $h(N) = 1$ in $\text{ MacP }(2,n)$, we know that $|\text{ MacP }(2,n)_{\le N} \cup \{\hat{0}\}| = 4$. So, we can consider intervals $[N_0, N_2]$ of length 2, where $N_0$ is a rank 2 oriented matroid. By the covering relations in Proposition  38, $3 \le |[N_0, N_2]| \le 4$. We will now prove that $|[N_0, N_2]| = 4$. Suppose $N_0< N_1 < N_2$ is a maximal chain in $[N_0, N_2]$, we will consider the covering relations $N_0 < N_1$ and $N_1 < N_2$ according to Proposition 38.

Case 1: If the coverings $N_0 < N_1$ and $N_1 < N_2$ are both CR1 of Proposition 38. That is, there are non-loops i, j so that the parallel/anti-parallel class $P_1(i)$ has size $|P_1(i)| \ge 2$ in $N_1$ and i is a loop in $N_0$. Also, the parallel/anti-parallel class $P_2(j)$ has size $|P_2(j)|\ge 2$ in $N_2$ and j is a loop in $N_1$. Now, suppose $\text{ Rowspace }(v_1\; v_2 \ldots \; v_n)$ is determines a vector realization for $N_2$. Then $\text{ Rowspace }(v_1 \; v_2 \; \ldots v_{i-1}, 0, v_{i+1}\; \ldots v_n)$ determines a vector realization for a rank 2 oriented matroid $N_1'\ne N_1$ so that $N_0< N_1' < N_2$.

Case 2: If the covering $N_0 < N_1$ is CR1 of Proposition 38 and the covering $N_1 < N_2$ CR2 of Proposition 38. That is, there is a non-loop i in $N_1$ with $|P_1(i)| \ge 2$ in $N_1$ so that i is a loop in $N_0$.

If the size of parallel/anti-parallel class $P_2(i)$ is of size $|P_2(i)|\ge 2$ in $N_2$, then we obtain $N_1'$ as in the above case from $N_2$.

If $|P_2(i)| = 1$ in $N_2$, then there are distinct classes $P_2(j), P_2(k) \ne P_2(i)$ in $N_2$ so that $P_2(i) \cup P_2(j)$ is a parallel/anti-parallel class in $N_1$; and if $\textrm{Rowspace}(v_1 \; v_2 \; v_3 \; \cdots v_k \cdots v_i \cdots v_n)$ is a vector realization for $N_2$, then $N_2$ covers the rank two oriented matroid $N_1'$ obtained from $\textrm{Rowspace}(v_1 \; v_2 \; v_3 \; \cdots v_k \cdots v_{i-1}\; v_k \; v_{i+1} \cdots \; v_n)$. So that $N_0< N_1' < N_2$ and $N_1 \ne N_1'$.

Case 3: If the covering $N_0 < N_1$ is CR2 of Proposition 38 and the covering $N_1 < N_2$ is CR1 of Proposition 38. That is, there is a non-loop k in $N_2$ so that $|P_2(k)| \ge 2$ and k is a loop in $N_1$. Let $j \in P_2(k){\setminus } \{k\}$ and $X_0= \text{ Rowspace }(w_1\; w_2 \ldots \; w_n)$ determines a vector realization for $N_0$. Then $w_k = 0$. We obtain the vector realization $X_1'$ that determines $N_1'$ by replacing $w_k = 0$ with $w_k' = w_j$. So that $N_0< N_1' < N_2$ and $N_1 \ne N_1'$.

Case 4: If the coverings $N_0 < N_1$ and $N_1 < N_2$ are both CR2 of Proposition 38. That is, there are classes $P_1(l), P_1(j)$ in $N_1$ so that $P_1(l) \cup P_1(j)$ is a parallel/anti-parallel class in $N_0$ and there are classes $P_2(k), P_2(r)$ in $N_2$ so that $P_2(k)\cup P_2(r)$ is a parallel/anti-parallel class in $N_1$. Let $X_0 = \text{ Rowspace }(v_1 \; v_2 \ldots \; v_n)$ determines a vector realization for $N_0$, and $v_k = r_ke^{i\theta _k}$.

We can similarly obtain a vector $X'_1 = \text{ Rowspace }(w_1\; w_2 \ldots \; w_n)$ as in the observation after Proposition 41 so that $N_1'$ the rank 2 oriented matroid determined by $X_1'$ is such that $N_0< N_1' < N_2$ and $N_1' \ne N_1$. $\square$

Proposition 45

Let M be a rank 2 oriented matroid. Then the interval $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$ is the augmented face poset of a regular cell decomposition of a PL sphere.

Proof of Proposition 45

The proposition now follows from Proposition 43, Lemma 44 and Theorem 22. $\square$

6 The topology of $\overline{\mu ^{-1}(M)}$

To prove that $\overline{\mu ^{-1}(M)}$ is homeomorphic to a closed ball, we will first prove that $\overline{\mu ^{-1}(M)}$ is a topological manifold with boundary $\bigcup _{N < M} \mu ^{-1}(N)$ as suggested by the following theorems and propositions.

Theorem 46

Let $M \in \text{ MacP }(2,n)$. The closure $\overline{\mu ^{-1}(M)}$ is a topological manifold whose boundary is $\bigcup _{N < M} \mu ^{-1}(N)$.

Proof

That the boundary of $\overline{\mu ^{-1}(M)} = \bigcup _{N < M} \mu ^{-1}(N)$ was obtained in Proposition 41.

For every $X \in \overline{\mu ^{-1}(M)}$, we will obtain a closed neighborhood $N_X$ of X that is homeomorphic to a closed ball of dimension h(M) and such that X is a point on the boundary of $N_X$. If $X \in \mu ^{-1}(M)$ the interior of $\overline{\mu ^{-1}(M)}$, then the above statement holds since $\mu ^{-1}(M)$ is homeomorphic to an open ball of dimension h(M) by Proposition 36.

Let $N = \mu (X)$. We may assume that $\{1,2\}$ is a basis of N. So, X can be obtained as $X = \text{ Rowspace }(e_1 \; e_2 \; v_1 \; v_2 \cdots \; v_{n-2}) \in \partial \overline{\mu ^{-1}(M)}$.

We recall from ${\textbf {ID1}}$ preceding Proposition 36 the following identification

$$\begin{aligned} \mu ^{-1}(M)\cong & {} \left\{ (x_1, x_2, \ldots , x_{n_1}): 0< x_1< x_2 \cdots< x_{n_1}< \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (y_1, y_2, \ldots , y_{n_2}): \frac{\pi }{2}< y_1< y_2< \cdots< y_{n_2} < \pi \right\} \times (0, \infty )^{l_M-2}. \end{aligned}$$

Let $r_j = |v_j|$ for $j \ge 3$. We define a closed ball of dimension $\dim (\mu ^{-1}(M))$ as follows:

$$\begin{aligned} B_M= & {} \left\{ (x_1, x_2, \ldots , x_{n_1}): 0 \le x_1 \le x_2 \cdots \le x_{n_1} \le \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (y_1, y_2, \ldots , y_{n_2}): \frac{\pi }{2} \le y_1 \le y_2 \le \cdots \le y_{n_2}\le \pi \right\} \times \prod _{j=1}^{L_M-2} \left[ \frac{r_j}{2}, r_j+1\right] . \end{aligned}$$

There is a map $T: B_M \rightarrow \overline{\mu ^{-1}(M)}$ defined as follows:

$\left( (\theta _1, \theta _2, \ldots \theta _{n_1}), (\theta _{n_1+1}, \theta _{n_1+2}, \ldots \theta _{n_1+n_2}), (r_j)_j\right) \mapsto \textrm{Rowspace}(e_1 \; e_2 \; w_1 \; w_2 \; \cdots w_{n-2})$, where $w_j= r_je^{i\theta _j}$. X is contained in the image of T.

The map T is not necessarily one-to-one on the boundary of $B_M$. Figure 9 illustrates two points on the boundary $\{r_3 = 0, r_6 = 0\}$ of $B_M$ with the same image under T.

Fig. 9

An identification on the boundary of $B_M$

Let $\sim$ be such identifications on the boundary of $B_M$ given by $x \sim y$ if and only if $T(x) = T(y)$. The identification is as discussed in Example 28.

The map $T' = T/\sim \;: B_M/\sim \; \rightarrow \overline{\mu ^{-1}(M)}$ is injective. By Theorem 27, $B_M/\sim$ is homeomorphic to $B_M$. The image $T'(B_M/\sim )$ is homeomorphic to $B_M/\sim$ by the compactness of $B_M/\sim$ and $T'(B_M/\sim ) \subset \overline{(\mu ^{-1}(M))}$ being Haursdorff.

So, the desired closed neighborhood of X in $\overline{\mu ^{-1}(M)}$ is $T'(B_M/\sim )\equiv B_M/\sim$ and it is homeomorphic to $B_M$ a closed ball of dimension $\dim (\mu ^{-1}(M))$. Hence, $\overline{\mu ^{-1}(M)}$ is a topological manifold with boundary. $\square$

The following theorem will now follow from Propositions 41, 43, Lemma 44, Theorems 22 and 46.

Theorem 47

Let M be a rank 2 oriented matroid. There is an homeomorphism from $\overline{\mu ^{-1}(M)}$ to an h(M)-dimensional closed ball.

Proof

The proof proceeds by induction on the height of M in $\text{ MacP }(2,n)$. The theorem is true for $h(M) = 0$ in which case $\overline{\mu ^{-1}(M)}$ is a 0-dimensional ball. Suppose $h(M) = l$, $l\ge 1$, and assume the theorem is true for every N in $\textrm{MacP}(2,n)$ with $h(N) < l$, that is, $\overline{\mu ^{-1}(N)}$ is a h(N)-dimensional closed ball. From Proposition 41, we know that $\partial \overline{(\mu ^{-1}(M))} = \bigcup _{N < M}(\mu ^{-1}(N))$, and by our inductive assumption $\bigcup _{N < M}(\mu ^{-1}(N))$ is a regular cell complex with cells $\{\mu ^{-1}(N): N < M\}$.

Proposition 43 says that the interval $[\hat{0},M]$ has a recursive atom ordering, and the interval is thin by Lemma 44. By Theorem 22, $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$ is an augmented poset of a regular cell decomposition of a $h(M)-1$-dimensional sphere. Hence, the regular cell complex $\partial \overline{(\mu ^{-1}(M))} = \bigcup _{N < M}(\mu ^{-1}(N))$ is homeomorphic to a sphere of dimension $h(M)-1$.

So, the boundary of $\mu ^{-1}(M)$ is homeomorphic to a $(h(M) -1)$ sphere. By Theorem 46, we now know that $\overline{\mu ^{-1}(M)}$ is a topological manifold; $\overline{\mu ^{-1}(M)}$ is a topological manifold whose boundary $\bigcup _{N < M}(\mu ^{-1}(N))$ is a sphere of dimension $h(M)-1$, and its interior $\mu ^{-1}(M)$ is an open ball by Proposition 36.

Hence, by Proposition 33, we obtain an homeomorphism from $\overline{\mu ^{-1}(M)}$ to a closed h(M)-dimensional ball. $\square$

Proof of Theorem 34

That $\{\mu ^{-1}(M): M \in \text{ MacP }(2,n)\}$ forms a regular cell decomposition of $Gr(2,\mathbb {R}^n)$ follows from Theorem 47. Hence, $\Vert \textrm{MacP}(2,n)\Vert$ is homeomorphic to $\textrm{Gr}(2, \mathbb {R}^n)$. $\square$

7 The stratification $\nu ^{-1}(N, M)$

As described in the background section, each point (Y, X) in $Gr(1,2,\mathbb {R}^n)$ can be identified with a point in $\textrm{Gr}(2,n)$ as illustrated in Fig. 10. In the Figure, we may assume that $\{1,2\}$ is a basis of $M = \mu (X)$, so that (Y, X) is identified with $\textrm{Rowspace}(e_1 \; e_2 \; v_3 \; v_4 \cdots \; v_{n+1})$

Fig. 10

Elements in $\textrm{Gr}(1,2, \mathbb {R}^n)$

Notation 48

Let M be a rank 2 oriented matroid and $N \in G(1, M)$ a rank 1 strong map image of M. The height of N in the poset G(1, M) is denoted by $h_M(N)$.

Theorem 49

$\{\nu ^{-1}(N,M): (N, M) \in \textrm{MacP}(1,2,n)\}$ forms a regular cell decomposition of $Gr(1,2,\mathbb {R}^n)$.

Let $(Y, X) \in \textrm{Gr}(1,2, \mathbb {R}^n)$, $N = \mu (Y)$ and $M = \mu (X)$. By the unique identification of a point $(Y, X) \in \textrm{Gr}(1,2, \mathbb {R}^n)$ with the point $\textrm{Rowspace}(e_1\; e_2 \; v_3 \; v_4 \; \cdots \; v_{n+1} )$ satisfying $0 \le \textrm{Arg}(v_j) < \pi$ for $j \ge 3$ and $\Vert v_{n+1}\Vert = 1$, we have the following identification of $\nu ^{-1}(N, M)$:

By (ID1), $\mu ^{-1}(M)$ is identified with

$$\begin{aligned} \mu ^{-1}(M) \equiv A_M= & {} \left\{ (\theta _{l_1}, \theta _{l_2}, \ldots , \theta _{l_{n_1}}): 0< \theta _{l_1}< \theta _{l_2} \cdots< \theta _{l_{n_1}}<\frac{\pi }{2}\right\} \\{} & {} \times \left\{ (\theta _{j_1}, \theta _{j_2}, \ldots , \theta _{j_{n_2}}): \frac{\pi }{2}< \theta _{j_1}< \theta _{j_2}< \cdots < \theta _{j_{n_2}}\le \pi \right\} \\{} & {} \times (0, \infty )^{l_M-2}. \end{aligned}$$

Then $\nu ^{-1}(N, M)$ can thus be identified with

$$\begin{aligned} ({\textbf {ID2}}) \; \; \nu ^{-1}(N, M)\equiv & {} \left\{ (\theta _{l_1}, \theta _{l_2}, \ldots , \Theta , \ldots , \theta _{l_{n_1}}):\right. \\{} & {} \left. 0< \theta _{l_1}< \theta _{l_2} \cdots< \Theta< \cdots< \theta _{l_{n_1}}< \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (\theta _{j_1}, \theta _{j_2}, \ldots , \theta _{j_{n_2}}): \frac{\pi }{2}< \theta _{j_1}< \theta _{j_2}< \cdots< \theta _{j_{n_2}}< \pi \right\} \times (0, \infty )^{l_M-2} \end{aligned}$$

The following proposition thus follows from the identification of $\nu ^{-1}(N, M)$ given in (ID2).

Proposition 50

Let $(N, M) \in \text{ MacP }(1,2,n)$. Then $\nu ^{-1}(N, M)$ is homeomorphic to an open ball of dimension $h(M) + h_M(N)$.

Lemma 51

([11], Lemma 3.8) Let $M_1, M_2 \in \text{ MacP }(2,n)$ and suppose that $M_1 < M_2$. Let $z_2 \in \mathcal {V}^*(M_2) \setminus \{0\}$. Then the set $A = \{z \in \mathcal {V}^*(M_1) {\setminus } \{0\}: z \le z_2 \}$ has a maximal element.

Notation 52

We will denote by $z_2^*$ the maximal non-zero covector in $\mathcal {V}^*(M_1)$ obtained above.

The following proposition follows as in Proposition 41 applied to elements $M' \in \textrm{MacP}(2,n+1)$, and by the identification of the interval $\textrm{MacP}(1,2,n)_{\ge (N_0, M_0)}$ with the interval $\textrm{MacP}(2, n+1)_{\ge M_0'}$ in Remark 13 .

Proposition 53

For any $(N_0, M_0) \in \text{ MacP }(1,2,n)$, we have

$$\begin{aligned} \partial \overline{(\nu ^{-1}(N_0,M_0))} = \bigcup _{(N, M) < (N_0,M_0)} \nu ^{-1}(N,M). \end{aligned}$$

Theorem 54

$\{\nu ^{-1}(N,M): (N, M) \in \textrm{MacP}(1,2,n) \}$ forms a regular cell decomposition for $Gr(1,2,\mathbb {R}^n)$, where $\nu : Gr(1,2, \mathbb {R}^n) \rightarrow \text{ MacP }(1,2,n)$ is the map described in Sect. 1.

8 Shellability of interval $\textrm{MacP}(1,2,n)_{\le (\pm z, M)}$

We prove the analogue of Lemma 42 here. We again use the identification of $\textrm{MacP}(1,2,n)_{(N_0, M_0)}$ with the interval $\textrm{MacP}(2, n+1)_{\ge M_0'}$. An interval $[(N_1, M_1), (N_2, M_2)]$ in $\textrm{MacP}(1,2,n)_{\ge (N_0, M_0)}$ can be identified with the interval $[M_1', M_2']$ in $\textrm{MacP}(2, n+1)_{\ge M_0'}$ where $M_1', M_2'$ are the rank 2 oriented matroid on $n+1$ elements corresponding to $M_1$ and $M_2$ respectively in Remark 13.

Lemma 55

Let $[(N_1, M_1), (N_2, M_2)]$ be an interval in $\textrm{MacP}(1,2,n)$. Then the interval $[(N_1, M_1), (N_2, M_2)]$ is totally semimodular.

Proposition 56

The interval $\textrm{MacP}(1,2,n)_\le (N, M) \cup \{\hat{0}\}$ has a recursive atom ordering for $(N,M) \in \textrm{Gr}(1, 2, \mathbb {R}^n)$.

Proof

We proceed as in Proposition 43. Let $(Y, X) \in \nu ^{-1}(N, M)$. We fix a vector arrangement $(e_1 \; e_2 \; v_3 \; v_4 \cdots v_{n+1})$ such that (Y, X) is identified with $\textrm{Rowspace}(e_1 \; e_2 \; v_3 \; v_4 \cdots v_{n+1})$. As in Proposition 43, we also fix an affine line L not parallel to any of the vectors in the above arrangement.

Let $A_1, A_2, \ldots A_k$ be the recursive atom ordering of atoms of $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$ as in Proposition  43. We denote by $L_z$ the labeling on the element $n+1$ as in Fig. 11.

Fig. 11

.

Let $A_t = (i,j)$ be an atom of $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}\}$. Let z be a non-zero covector of N. We denote by $z_t^* = \text{ max }\{w \in \mathcal {V}^*(A_t) {\setminus } 0: w \le z\}$. Let $N^*_t$ be a rank 1 oriented matroid with covectors $\{0, \pm z_t^*\}$. Then $(N_t^*, A_t) \le (N, M)$.

If $z_t^*$ is not a cocircuit of $A_t$, then there are cocircuits $z_t^0$, and $z_t^1$ satisfying $z_t^0(i) = 0$ and $z_t^1(j) = 0$ respectively. Let $N_t^0$ be a rank 1 oriented matroid with covectors $\{0, \pm z_t^0\}$, and let $N_t^1$ be a rank 1 oriented matroid with covectors $\{0, \pm z_t^1\}$. Then $(N_t^0, A_t)$ and $(N_t^1, A_t)$ are atoms in $\textrm{MacP}(1,2,n)_{\le (N, M)} \cup \{\hat{0}\}$ and are covered by $(N^*_t, A_t)$. If $z_t^*$ is a cocircuit of $A_t$, then $(N^*_t, A_t)$ is an atom in $\textrm{MacP}(1,2,n)_{\le (N, M)} \cup \{\hat{0}$. In this case, we say that $N^*_t = N_t^0 = N_t^1$.

In the case when when $z_t^*$ is not a cocircuit of $A_t$, we decide which of the atoms $(N_t^0, A_t)$ and $(N_t^1, A_t)$ is ordered just before the other. We consider the labeling obtained from the affine line L as in Fig. 11.

If $L_z \le i < j$ we order $(N_t^0, A_t)$ just before $(N_t^1, A_t)$ (see Fig. 12). If $i < L_z \le j$ we order $(N_t^1, A_t)$ just before $(N_t^0, A_t)$ (see Fig. 13). If $L_z > j$ we order $(N_t^1, A_t)$ just before $(N_t^0, A_t)$ (see Fig. 14). In the ordering of atoms of $\textrm{MacP}(1,2,n)_{\le (N_t^*, A_t)} \cup \{\hat{0}$, we denote the atom ordered first by $(N_t, A_t)$ while the second is denoted by $(N_t', A_t)$.

We then order the atoms of $\textrm{MacP}(1,2,n)_{\le (N, M)}$ as $(N_1, A_1), (N_1',A_1), (N_2, A_2), (N_2', A_2), \ldots (N_k, A_k), (N_k, A_k)$.

For each $(N_t, A_t)$ or $(N_t', A_t)$ in the above list, the intervals $[(N_t, A_t), (N, M)]$ and $[(N_t', A_t), (N, M)]$ are totally semimodular by Lemma 55. So by Theorem 20 the intervals $[(N_t, A_t), (N, M)]$, $[(N_t', A_t), (N, M)]$ have recursive atom ordering; in fact, any atom ordering gives a recursive atom ordering for the interval.

Fig. 12

$L_z \le L_4 < L_6$

Fig. 13

$L_1 < L_z \le L_4$

Fig. 14

$L_1< L_2 < L_z$

Let $(N_1, A_1), (N_1', A_1), (N_2, A_2), (N_2', A_2) \ldots , (N_k, A_k), (A_k', A_k)$ be the ordering of atoms in the interval $\textrm{MacP}(1,2,n)_{\le (N,M)} \cup \{\hat{0}\}$. We will now verify the second condition in Definition 19.

Let $(W, T) \in \textrm{MacP}(1,2,n)_{\le (N, M)}$, and $(N^1, A^1), (N^2, A^2)$ atoms in $\textrm{MacP}(1,2,n)_{\le (N,M)} \cup \{\hat{0}\}$. Let $A^1 = (i_1, j_1)$ and $A^2 = (i_2, j_2)$. Suppose $(N^1, A^1)$ precedes $(N^2, A^2)$ in the ordering of atoms in $\textrm{MacP}(1,2,n)$, and $(N^1, A^1), (N^2, A^2) < (W, T)$.

We recall from Proposition 43 that $(i, {\{j,k\}})$ denote a rank 2 oriented matroid covering (i, j), (i, k), and j is parallel/anti-parallel to k. Similarly, we have the notation ${\{i,k\}}, j)$. We will then consider the following cases:

Case 1: Suppose $A^1 = A^2 = (i_2, j_2)$ so that $N^1 \ne N^2$. Let w be a non-zero covector of W. We denote by $w^* = \max \{c\in \mathcal {V}^*(A^2){\setminus } 0: c \le w\}$. Let Z be a rank 1 oriented matroid with covectors $\{0, \pm w^*\}$. Then $(Z, A^2) \le (W, T)$ and covers both $(N^1, A^2)$ and $(N^2, A^2)$.

Case 2: Suppose $A^1 \ne A^2$ so that $A^1$ precedes $A^2$ in atom ordering in $\textrm{MacP}(2,n)_{\le M} \cup \{\hat{0}$. Suppose $i_1 = i_2$ so that $j_1 < j_2$. We obtained in Proposition 43 the rank 2 oriented matroid $L = (i_2, {\{j, j_2\}})$ such that $L \le T$ and L covers $A^2$ and $L' = (i_2, j)$ for some j satisfying $j_1 \le j < j_2$.

Suppose $L_z \le i_2$ as in Fig. 12. Let w be a cocircuit of L satisying $w(i_2) = 0$. Let $w_1^* = \max \{c\in \mathcal {V}^*(L'){\setminus } 0: c \le w\}$. Let $Z, Z_1$ be rank 1 oriented matroids with covectors $\{0, \pm w\} \; \textrm{and} \; \{0, \pm w_1^*\}$ respectively. Then $(Z, L) \le (W, T)$ and (Z, L) covers $(Z_1, L')$ and $(N^2, A^2)$.

Suppose $i_2 < L_z \le j_2$ or $L_z \ge j_2$. Let w be a cocircuit of L with $w(j) = w(j_2) = 0$. We obtain Z and $Z_1$ as described above, so that $(Z, L) \le (W, T)$ and (Z, L) covers $(Z_1, L')$ and $(N^2, A^2)$.

Case 3: Suppose $A^1 \ne A^2$ and $i_1 < i_2$. As in the proof of Proposition 43, we obtain $L = ({\{i, i_2\}},j_2) \le T$ such that L covers $A^2$ and $(i, j_2)$ for some $i_1 \le i < i_2$. Let $L' = (i, j_2)$. The flags (Z, L) and $(Z_1, L')$ can be obtain as in Case 2 by considering the cases when $L_z \le i$, $i < L_z \le j_2$ or $L_z > j_2$. $\square$

The following lemma also follows from the thinness of intervals of length 2 in $\textrm{MacP}(2, n+1)$, and the identification of an interval in $\textrm{MacP}(1,2,n)$ with an interval in $\textrm{MacP}(2,n+1)$.

Lemma 57

Every interval of length 2 in $\textrm{MacP}(1,2,n)_{\le (\pm z, M)} \cup \hat{0}$ is thin.

Using the above recursive atom ordering and Lemma 57, we have the following proposition.

Proposition 58

Let $(N, M) \in \text{ MacP }(1,2,n)$. The interval $\textrm{MacP}(1,2,n)_{\le (N, M)} \cup \{\hat{0}\}$ is an augmented face poset of a regular cell decomposition of a PL sphere.

Proof

The interval $\textrm{MacP}(1,2,n)_{\le (N, M)} \cup \{\hat{0}\}$ has a recursive atom ordering by Proposition 56, and the interval is thin by Lemma 57. Hence by Theorem 22, the interval $\textrm{MacP}(1,2,n)_{\le (N, M)} \cup \{\hat{0}\}$ is an augmented face poset of a regular cell decomposition of a PL sphere. $\square$

9 Topology of $\overline{\nu ^{-1}(N,M)}$

Theorem 59

Let $(N, M) \in \text{ MacP }(1,2,n)$. Then the closure $\overline{\nu ^{-1}(N,M)}$ is a topological manifold whose boundary is $\bigcup _{(N', M') < (N,M)} \nu ^{-1}(N', M').$

Proof

Let $(Y,X) \in \partial \overline{\nu ^{-1}(N,M)}$. The proof follows as in Theorem 46, with $B_M$ replaced with the ball

$$\begin{aligned} B_{(N,M)}= & {} \left\{ (x_1, x_2, \ldots \Theta , \ldots , x_{n_1}): 0 \le x_1 \le x_2 \cdots \le \Theta \cdots \le x_{n_1} \le \frac{\pi }{2}\right\} \\{} & {} \times \left\{ (y_1, y_2, \ldots , y_{n_2}): \frac{\pi }{2} \le y_1 \le y_2 \le \cdots \le y_{n_2}\le \pi \right\} \times \prod _{j=1}^{L_M-2} \left[ \frac{r_j}{2}, r_j+1\right] . \end{aligned}$$

To obtain a map $T': B_{(N, M)}/\sim \; \rightarrow \overline{\nu ^{-1}(N,M)}$. The image $T'(B_{(N, M)}/\sim )$ is the desired neighborhood of the point (Y, X), and $T'(B_{(N,M)}/\sim )$ is homeomorphic to a closed ball. $\square$

Theorem 60

Let $(N, M) \in \text{ MacP }(1,2,n)$. There is an homeomorphism from $\overline{\nu ^{-1}(N, M)}$ to an $h(M) + h_M(N)$-dimensional closed ball.

Proof

The proof follows as in Theorem 47. $\square$

Proof of Theorem 49

We conclude from Theorem that $\{\nu ^{-1}(N, M): (N, M) \in \text{ MacP }(1,2,n) \}$ is a regular cell decomposition of $\textrm{Gr}(1,2,\mathbb {R}^n)$.

Hence, $\Vert \textrm{MacP}(1,2,n)\Vert$ is homeomorphic to $\textrm{Gr}(1,2,\mathbb {R}^n)$. $\square$

10 Conclusion

We have proven that the complexes $\Vert \textrm{MacP}(2,n)\Vert$ and $\Vert \textrm{MacP}(1,2,n)\Vert$ associated to combinatorial Grassmannians have the same heomorphism type as the Grassmannian manifolds $\textrm{Gr}(2, \mathbb {R}^n)$ and $\textrm{Gr}(1,2, \mathbb {R}^n)$ respectively. Our argument relies majorly on the realizability of rank 2 oriented matroids, and the fact that stratas $\mu ^{-1}(M)$ and $\nu ^{-1}(\pm z, M)$ are homeomorphic to an open ball—the argument thus does not apply to oriented matroids of ranks at least 3.