Triangulations of grassmannians and flag manifolds

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 Gr(2,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textrm{Gr}(2, \mathbb {R}^n)$$\end{document} has the same homeomorphism type as the combinatorial Grassmannian MacP(2,n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ MacP }(2,n)$$\end{document}, while Babson (in: A combinatorial flag space, MIT, 1993. https://doi.org/10.2307/1970177) proved that the spaces Gr(2,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Gr }(2,\mathbb {R}^n)$$\end{document} and Gr(1,2,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Gr }(1,2,\mathbb {R}^n)$$\end{document} are homotopy equivalent to their combinatorial analogs, the simplicial complexes ‖MacP(2,n)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \text{ MacP }(2,n)\Vert $$\end{document} and ‖MacP(1,2,n)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \text{ MacP }(1,2,n)\Vert $$\end{document} respectively. We will prove that Gr(2,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Gr }(2, \mathbb {R}^n)$$\end{document} and Gr(1,2,Rn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ Gr }(1,2, \mathbb {R}^n)$$\end{document} are homeomorphic to ‖MacP(2,n)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \text{ MacP }(2,n)\Vert $$\end{document} and ‖MacP(1,2,n)‖\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \text{ MacP }(1,2,n)\Vert $$\end{document} respectively.


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, M) of rank k strong map images of a rank n oriented matroid M called the oriented matroid Grassmannian.The poset was introduced to serve as a combinatorial model for Gr(k, R n ), the space of k dimensional subspaces of R n .A special case is when M is the unique rank n oriented matroid on n elements.The resulting poset MacP(k, n), is called the MacPhersonian.Similarly, the poset of flags (N 1 , N 2 ) of oriented matroids, where N 1 is a rank p strong map image of N 2 and N 2 is

INTRODUCTION
a rank k strong map image of M is denoted by G(p, k, 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 G(k, M) , the geometric realization of the poset G(k, M) has the homotopy type of Gr(k, R n ).For k = 2, it was proven by Babson in [2] that G(2, M) has the same homotopy type as Gr(2, R n ).It was also proven in [2] that G(1, 2, M) has the same homotopy type as Gr(1, 2, R n ).
We will show that the complexes MacP(2, n) and MacP(1, 2, n) are homeomorphic to Gr(2, R n ) and Gr(1, 2, R n ) respectively.It follows from Babson's work in [2] that the complex MacP(2, n) and Gr(2, R n ) have the same homotopy type, and that the complex MacP(1, 2, n) has the same homotopy type as Gr(1, 2, R n ).Also, it can easily be shown that for k = 1, MacP(1, n) is homeomorphic to RP n−1 .
In Section 2, we will give basic background on oriented matroids, posets and regular cell complexes that is sufficient for this part of the project.We will also introduce the maps µ : Gr(k, R n ) → MacP(k, n) mapping a k dimensional subspace to the rank k oriented matroid it determines, and the map ν : Gr(p, k, R n ) → MacP(p, k, n) mapping a flag of subspaces to a flag of oriented matroids.To establish our main assertion about the topology of MacP(2, n) and MacP(1, 2, n) , we will prove in Theorem 32 that the stratification {µ −1 (M ) : M ∈ MacP(2, n)} is a regular cell decomposition of Gr(2, R n ).Similarly, we will prove in Theorem 46 that the stratification {ν −1 (N, M ) : (N, M ) ∈ MacP(1, 2, n)} is a regular cell decomposition for Gr(1, 2, R n ).
In Proposition 34, we will show that µ −1 (M ) is homeomorphic to an open ball.Similarly, ν −1 (N, M ) will be shown in Proposition 47 to be homeomorphic to an open ball.In Proposition 38, the boundary ∂µ −1 (M ) of µ −1 (M ) will be shown to be N <M µ −1 (N ) as the union of lower dimensional cells.We have similar result in Proposition 50 for the boundary of ν −1 (N, M ).
In Section 6 and Section 9, we will prove that the closures µ −1 (M ) and ν −1 (N, M ) are topological manifolds whose boundaries are spheres.
In rank r ≥ 3, and for M ∈ MacP(r, n), µ −1 (M ) is not necessarily connected.Our argument will also make use of the fact that in rank 2, ∂µ −1 (M ) = N <M µ −1 (N ).This fact called normality, is also not necessarily true in rank r for r ≥ 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 R 2 .This fact is also in general not true for oriented matroids of rank at least 3. Detailed results on rank r oriented matroids for r ≥ 3 can be found in [5].

Oriented Matroids
Suppose X ∈ Gr(p, R n ).We will view elements of R n as 1 × n row vectors, so that X is the rowspace of a p × n matrix.
The collection of all sign vectors {(sign(x 1 ), sign(x 2 ), . . ., sign(x n )) : (x 1 , x 2 , . . ., x n ) ∈ X} is a collection of sign vectors that are called the covectors of an oriented matroid.For a covector C, the set {i ∈ [n] : 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 Let {v αi } αi be the set of non-zero vectors in {v 1 , v 2 , . . .v n }.We consider the following arrangement (v ⊥ αi ) αi of oriented linear hyperplanes.The arrangement determines a cellular decomposition of R p .The intersection of the cellular decomposition with S p−1 the unit sphere in R p gives a cellular decomposition of S p−1 .A cell in S p−1 corresponds to a non-zero covector of M and a non-zero covector of M corresponds to a cell in the cellular decomposition of S p−1 .The oriented matroid 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].
Let X = Rowspace(v 1 v 2 v 3 . . .v n ) as defined earlier and M the corresponding oriented matroid.We consider the following function χ : The collection {±χ} is independent of the choice of basis vectors for X.We will write the resulting oriented matroid as M = (±χ).
In Figure 1(b), a rank 3 oriented matroid is obtained from an essential arrangement of equators in a 2-sphere S 2 .
Notation: Let E be a finite set and X, Y ∈ {0, +, −} E be sign vectors.The composition X • Y is defined to be the element of {0, ) Let E be a finite set and V * ⊆ {0, +, −} E .V * is the covector set of an oriented matroid on elements E if it satisfies all of the following.
and e ∈ E such that X(e) = −Y (e) = 0, then there is a Z ∈ V * such that Z(e) = 0 and, for each f ∈ E, If V * is the covector set of an oriented matroid, then the rank of the oriented matroid is the rank of V * as a subposet of {0, +, −} E .

Definition 3 (Basis orientation)([6]
) A basis orientation of an oriented matroid M is a mapping χ of the set of ordered bases of M to {+1, −1} satisfying the following two properties • χ is alternating, • For any ordered bases of M of the form (e, x 2 , x 3 , . . ., x p ) and (f, x 2 , x 3 , . . ., x p ), e = f , we have where D is one of the two opposite cocircuits complementary to the hyperplane spanned by {x 2 , x 3 , . . ., x p } in M.
The following theorem establish the cryptomorphism between the definition of an oriented matroid using covectors and its definition using a chirotope.Theorem 4 ( [7]) Let p ≥ 1 be an integer and E be a set.A mapping χ : E p → {+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 ( [7])The Topological Representation Theorem (Folkman-Lawrence 1978) The rank r oriented matroids are exactly the sets (E, V * ) arising from essential arrangements of pseudospheres in S r−1 .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]) Let M and N be two rank r oriented matroids, and V * (M) and V * (N ) the covector sets of M and N respectively.We say that N ≤ M if and only if for every X ∈ V * (N ) there exist a Y ∈ V * (M) such that X ≤ Y .The oriented matroid M is said to weak map to N .Definition 7 ([1]) MacP(p, n) denotes the poset of all rank p oriented matroids on elements {1, 2, . . ., n}, with weak map as the partial order.The poset is called the MacPhersonian [1].
Let M be a rank p oriented matroid elements [n] and χ : [n] p → {+, −, 0} its chirotope.We have the following abstraction of notions from vector spaces and convexity.

2.
Basis: A set {i 1 , i 2 , . . ., i p } of size p is said to be a basis of M if and only if χ(i 1 , i 2 , . . ., i p ) = 0. 3. Independence: A set {i 1 , i 2 , . . ., i k } is said to be independent if it is contained in a basis of M. 4. Parallel/Anti-parallel.An 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 , . . ., i p−1 ), we have that χ(i, i 1 , i 2 , . . ., i p−1 ) = χ(j, i 1 , i 2 , . . ., i p−1 ).Similarly, i is said to be anti-parallel to j if for every p − 1 tuple (i 1 , i 2 , . . ., i p−1 ), we have that χ(i, i 1 , i 2 , . . ., i p−1 ) = −χ(j, i 1 , i 2 , . . ., i p−1 ). 5. Convex Hull : Let S be a subset of [n].The convex hull of S is the set 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 R n and W a rank p subspace of V .We have that the collection of sign vectors {(sign(x 1 ), sign(x 2 ), . . ., sign(x n )) : (x 1 , x 2 , . . ., x n ) ∈ V } is a subset of the collection {(sign(y 1 ), sign(y 2 ), . . ., sign(y n )) : (y 1 , y 2 , . . ., y n ) ∈ W }. Let M be the rank k oriented matroid determined by V , and let N be the rank p oriented matroid determined by W . Then V * (N ) ⊆ V * (M) Definition 8 ([3]) Let M be a rank k oriented matroid, and N a rank p oriented matroid.N is said to be a rank p strong map image of M if and only if V * (N ) ⊆ V * (M).Definition 9 ([3]) Let M be an oriented matroid.The poset of all rank p oriented matroids that are strong map image of M is denoted by G(p, M).
In Figure 3, the oriented matroid N is a rank 2 strong map image of M. We now define the combinatorial analog of the flag manifold Gr(p, k, 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 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 As in the case of the map µ : Gr(p, R n ) → MacP(p, n) discussed earlier, we also have the map ν : Gr(k, p, R n ) → MacP(p, k, n) defined by (W, V ) → (N, M ) where N and M are the oriented matroids determined by subspaces W and V respectively.
We can visualize an element of the flag manifold Gr(1, 2, R n ) as in Figure 4 (a).In Figure 4

Reorientation of an oriented matroid
We have described a realizable rank p oriented matroid as determined by some arrangement of vectors in R p .We now discuss the reorientation of a rank p oriented matroid.
In the language of vector arrangements, reorientation of an element i is simply that if (w 1 , w 2 , w 3 , . . ., w i , . . ., w n ) is a vector arrangement for N , then (w 1 , w 2 , w 3 , . . ., −w i , . . ., w n ) is a vector arrangement for M.
It then follows that if M is obtained from N by a reorientation or a sequence of reorientations, then µ −1 (M) is homeomorphic to µ −1 (N ).Another useful observation is the following lemma: It should also be noted that the homeomorphism type of µ −1 (M) is unchanged by relabelling the elements of M.
3 Posets, Regular Cell Complexes and Topological Balls

Posets and Recursive atom ordering
Associated to every poset P is a simplicial complex P whose n-simplices are chains x 0 < x 1 < • • • < x n for some x i in P .P is called the order complex of P .We will be studying the topology of the order complex of the posets MacP(2, n) and MacP(1, 2, n).
Definition 15 ([8]) A finite poset P is said to be semimodular if it is bounded, and whenever two distinct elements u, v both cover x ∈ P , there is a z ∈ 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 16 ([8]) Let P be a poset.P is said to be thin if every interval of length 2 in P has exactly four elements.

Definition 17 ([8]
) 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 , . . ., a t of the atoms of P which satisfies: 1.For all j = 1, 2, . . ., t [a j , 1] admits a recursive atom ordering in which the atoms of [a j , 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 ≤ y such that z covers a k and a j .
The following theorem and proof appears in the work of Bjorner and Wachs ( [8]).We will give below their proof of the only if direction.

Theorem 18 ([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] = 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 , . . ., an be any atom ordering in P , since every atom ordering in [a j , 1] is recursive, order the atoms of [a j , 1] so that those that cover a i for some i < j come first.Let y ≥ a i , a j .Since P is totally semimodular, there is a z ∈ P that covers both a i and a j .

Regular cell complexes
For each open m-cell e α , there exists a continuous map f α : D m → X that is a homeomorphism onto e α , maps the interior of D m onto e α , and maps the boundary ∂D m into a finite union of open cells, each of dimension less than m.
Let {e α } be a regular cell decomposition of X and F (X) the face poset of the decomposition, with e α ≤ e τ if and only if e α ⊆ ∂e τ .The complex F (X) is homeomorphic to X.
Theorem 20 ( [8]) If a poset P containing 0 and 1 is thin and admits a recursive atom ordering, then P is the augmented face poset of a regular cell decomposition of a P L sphere.

Cell Collapse
Hersh in [9] 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 [9] of such a collapsing map is given below: Example 21 ( [9]) Let ∆ 2 be the convex hull of (0, 0), (1, 0), (0, 1/2) in R 2 , and let ∆ 1 be the convex hull of (0, 0) and (1, 0) in R 2 .There is a continous and surjective function g : R 2 → R 2 that acts homeomorphically on R 2 \∆ 2 sending it onto R 2 \∆ 1 .The simplex ∆ 2 is covered by vertical line segments with an end point in ∆ 1 .The map g has the property that it maps each vertical line segment to its end point in ∆ 1 .
Let R = {(x, y) : Let g acts as the identity outside R. Definition 23 ( [9]) Given a finite regular CW complex K on a set X an an open cell L in K, define a face collapse or cell collapse of L onto τ for τ an open cell contained in ∂L to be an identification map g : X → X such that:

Hersh gave a formal definition of a collapsing map below
1.Each open cell of L is mapped surjectively onto an open cell of τ with L mapped onto τ .2. g restricts to a homeomorphism from K \ L to K \ τ and acts homeomorphically on τ .3. The images under g of the cells of K form a regular CW complex with new characteristic maps obtained by composing the original characterisitc maps of K with g −1 : X → X for those cells of K contained either in τ or in K \ L.
Definition 24 ([9]) Let K 0 be a convex polytope, and let C 0 i be a family of parallel line segments covering a closed face L 0 i in ∂K 0 with the elements of C 0 i given by linear functions c : [0, 1] → L 0 i .Suppose that there is a pair of closed faces G 1 , G 2 in ∂L 0 i with c(0) ∈ G 1 and c(1) ∈ G 2 for each c ∈ C 0 i and there is a composition i and some (t, t ′ ) = (1, 1).Then t = t ′ , and for each t ∈ [0, 1] we have Theorem 25 ( [9]) Let K 0 be a convex polytope.Let g 1 , . . ., g i be collapsing maps with g j : X Kj−1 → X Kj for regular CW complexes K 0 , K 1 , . . ., K i all having the underlying space X.Suppose that there is an open cell L 0 i in ∂K 0 upon which The sequence of cell collapses needed in the proof of Theorem 43 will be of the form illustrated by the example below.

POSETS, REGULAR CELL COMPLEXES AND TOPOLOGICAL BALLS
The end point of the curve c (x,r) lies in G 0 2 .Let C 0 = {c (x,r) : (x, r) ∈ G 0 1 }.Then C 0 is a collection of parallel-like line segments covering the face L 0 .By Theorem 25, there is an identification map g 1 : K 0 → K 0 specified by C 0 .The map has the property that the curves in C 0 are mapped to their end points in G 0 2 .Let ∼ 1 be a relation on K defined by (x, r) ∼ 1 (x ′ , r ′ ) if and only if and G 1 2 are faces of L 1 .Let C 0 1 be a collection of parallel line segments covering L 1 and with endpoints in G 1  1 and } is a collection of parallel-like segments covering g 1 (L 1 ).By Theorem 25, there is an identification map g 2 : In Theorem 43, we will apply homeomorphism-preserving cell collapses on the boundary of a closed ball of the form The identification ∼ as in Example 26 is given here as in Figure 9.

Topological Balls
Theorem 27 ( [10]) Let S be an (n − 1) sphere in S n and H a component of S n − S.
If H is a topological manifold with boundary, then H is homeomorphic to an n-ball.
Corollary 28 Suppose int(D n ) is the interior of a closed unit ball centered at the origin.
sphere, then H is homeomorphic to a closed unit ball D n .
Definition 29 Let M be a topological manifold and S ⊆ M .The subset S is said to be collared in M if there is a neighborhood N (S) ⊆ M and a homeomorphism h : S × I → N (S) satisfying h(x, 1) = x.
Theorem 30 ( [11]) The boundary of a topological manifold M with boundary is collared in M .
Proposition 31 Suppose G ⊂ X is homeomorphic to an open ball and G is a topological manifold whose boundary is a sphere.Then G is homeomorphic to a closed ball.Proof Let h 1 : G → int(D n ) be an homeomorphism to the open unit ball and S = ∂G the boundary of G.As G is a topological manifold whose boundary is S, by Theorem 30, there is a collared neighborhood N (S) of S in G and a homeomorphism ) is a topological manifold whose boundary is the sphere h 1 (A).By Corollary 28, A ′ is homeomorphic to a closed ball.
We will denote by h 3 : where D 1 2 is a closed ball of radius 1 2 centered at the origin.Define: h 4 gives a homeomorphism from N (S) ⊆ G to the annulus in D n the unit ball.The map h : G → D n given by h = (h 3 • h 1 ) ∪ h 4 is a homeomorphism by the Gluing Lemma.[7] 4 The stratification µ −1 (M ) Let M be a rank 2 oriented matroid.We will first determine the topology of µ −1 (M ) = {X ∈ Gr(2, R n ) : (±χ X ) = M }.We know from the background on oriented matroids in Section 2 that the topology of µ −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 ∈ µ −1 (M ), X can be uniquely expressed as Arranging the arguments of the non-zero vectors in an increasing order as 0 we can thus uniquely represent X as ((θ l1 , θ l2 , . . ., θ ln 1 ), (θ j1 , θ j2 , . . ., θ jn 2 ), (r λ ) λ ).An example of such an identification is given in Figure 6.The above representation Notation 33 Let M ∈ 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/anti-parallel classes of M .P M (i) denotes the set of elements that are parallel/anti-parallel to i.
The following proposition then follows from the above identification of µ −1 (M ).
Proposition 35 Let (±χ N , N ), (±χ 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)| ≥ 2 in M , i is a loop in N and χ N (j, k) = χ M (j, k) for j, k = 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: Also, suppose that M and N satisfy CR2.Then there are adjacent parallel classes P M (i), P M (j) with χ N satsfying the definition in CR2.Suppose there is an Let N 0 < M , and let i, j be non-loops in N 0 .χ M (i, j) > 0 implies that χ N0 (i, j) ≥ 0. That is, for any subspace realization Rowspace(e implies that Arg(w i ) ≤ Arg(w j ) for any vector realization Rowspace(e 1 e 2 w 3 w 4 • • • wn) ∈ µ −1 (N 0 ).Also χ M (i, j) = 0 implies that χ N0 (i, j) = 0.In otherwords, Arg(v i ) = Arg(v j ) implies that Arg(w i ) = Arg(w j ) if i, j are non-loops in N 0 .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 χ M (p, a) = +, then χ M (p, l i ) = +.Similarly, let b be a non-loop in N satisfying the condition that if for any element p such that χ M (b, p) = +, then χ M (l i , p) = +.
The face of {(θ 1 , θ 2 , . . ., θn) 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 χ N (a, b) = 0 and χ ′ N (a, b) = +, this corresponds to the face of {(θ 1 , θ 2 , . . ., θn 1 ) : 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 CR1.In particular, if M covers N 0 , then the proposition follows.
Lemma 37 If N < M are rank 2 oriented matroids such that M covers N .Then Proof Case 1: Suppose the covering relation N < M is CR1.There is exactly one p in a parallel/anti-parallel class 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: Let X 1 = Rowspace(e 1 e 2 • • • vn) be any vector arrangement in µ −1 (N ).Let vs = rse iθs , v j = r j e iθs , and suppose WLOG that s and j are parallel in for every sufficiently small value of ǫ.Hence, and N the rank 2 oriented matroid determined by X.Then N ≤ M .
Conversely, suppose N, M ∈ MacP(2, n) such that N < M .We will show that A useful observation from Proposition 35 and Proposition 38 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 ) ∪ P M (l 2 ) is a parallel/anti-parallel class in N .Then we can obtain a rank 2 oriented matroid N ′ ≤ 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 Rowspace(v 1 v 2 . . .v n ) determines a vector realization for N and assume WLOG that P N (l 1 ) = P M (l 1 ) ∪ P M (l 2 ).We obtain a rank 2 oriented matroid N ′ as follows: let v l2 = r l2 e iθ l 2 and, we denote by P M (l 1 ) the subset of P M (l 1 ) that are anti-parallel to l 2 in N : The rank 2 subspace Rowspace(w 1 w 2 . . .w n ) determines a rank 2 oriented matroid N ′ ≤ M that covers N .
5 Shellability of the interval MacP(2, n) ≤M ∪ { 0} We will show that the poset MacP(2, n) ≤M ∪ { 0} is the augmented face poset of a regular cell decomposition of a h(M ) − 1 dimensional sphere.
Lemma 39 Let W < T be rank 2 oriented matroids.Then the interval [W, T ] is totally semimodular.
Proof Let N, N 1 , N 2 ∈ [W, T ] such that N 2 and N 1 cover N .In Proposition 35, 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 35.That is, there are i = j such that i, j are loops in N but i is a non-loop in N 1 with parallel/anti-parallel class We consider a dictionary order for ordered pairs for the atoms of M as X 1 , X 2 , . . ., X k .We now verify the conditions in Definition 40.By Lemma 39, the interval [X i , M ] is totally semimodular for each i, and so it has a recursive atom ordering by Theorem 18.In fact any ordering of the atoms in [X i , M ] gives a recursive atom ordering of the poset [X i , M ] by Theorem 18.
We now verify the second condition in Definition 40.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 ∈ MacP(2, n) ≤M ∪{ 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 ≤ j < j 2 and the element labeled by j is a non-loop in Y .We obtain Z as (i 1 , jj 2 ), where j and j 2 are parallel if and anti-parallel if ǫ 1 • ǫ 2 = −1.Such a Z covers both (i 1 , j) and (i 1 , j 2 ).We have Z ≤ 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 ≤ i < i 2 and the element labeled i is not a loop in Y .Then Z is obtained as (ii 2 , j 2 ).Similarly, Such a Z is a rank 2 oriented matroid covers the atoms (i, j 2 ) and (i 2 , j 2 ).We have that Z ≤ Y and (i, j 2 ) < (i 2 , j 2 ) in the dictionary order.
For each j > 1, let Q j = {Y ∈ atom[X j , M ] : Y ≥ X i for 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 18, as the interval [X i , M ] is totally semimodular for each i by Lemma 39.
Lemma 41 Let N be a rank 2 oriented matroid.
we will consider the covering relations N 0 < N 1 and N 1 < N 2 according to Proposition 35.Case 1: If the coverings N 0 < N 1 and N 1 < N 2 are both CR1 of Proposition 35.That is, there are non-loops i, j so that the parallel/anti-parallel class P 1 (i) has size |P 1 (i)| ≥ 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)| ≥ 2 in N 2 and j is a loop in N 1 .Now, suppose Rowspace(v 1 v 2 . . .vn) is determines a vector realization for N 2 .Then Rowspace(v 1 v 2 . . .v i−1 , 0, v i+1 . . .vn) determines a vector realization for a rank 2 oriented matroid If the size of parallel/anti-parallel class P 2 (i) is of size |P 2 (i)| ≥ 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) = P 2 (i) in N 2 so that P 2 (i)∪P 2 (j) is a parallel/anti-parallel class in N 1 ; and if Rowspace(v and k is a loop in N 1 .Let j ∈ P 2 (k) \ {k} and X 0 = Rowspace(w 1 w 2 . . .wn) 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 = N ′ 1 .Case 4: If the coverings N 0 < N 1 and N 1 < N 2 are both CR2 of Proposition 35.That is, there are classes P 1 (l), P 1 (j) in N 1 so that P 1 (l)∪P 1 (j) is a parallel/antiparallel class in N 0 and there are classes P 2 (k), P 2 (r) in N 2 so that P 2 (k) ∪ P 2 (r) is a parallel/anti-parallel class in N 1 .Let X 0 = Rowspace(v 1 v 2 . . .vn) determines a vector realization for N 0 , and v k = r k e iθ k .
We can similarly obtain a vector X ′ 1 = Rowspace(w 1 w 2 . . .wn) as in the observation after Proposition 38 so that N ′ 1 the rank 2 oriented matroid determined by Proposition 42 Let M be a rank 2 oriented matroid.Then the interval MacP(2, n) ≤M ∪ { 0} is the augmented face poset of a regular cell decomposition of a P L sphere.

Proof of Proposition 42
The proposition now follows from Proposition 40, Lemma 41 and Theorem 20.
To prove that µ −1 (M ) is homeomorphic to a closed ball, we will first prove that µ −1 (M ) is a topological manifold with boundary N <M µ −1 (N ) as suggested by the following theorems and propositions.
For every X ∈ ∂µ −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 .
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 .
Let ∼ be such identifications on the boundary of B M given by x ∼ y if and only if T (x) = T (y).The identification is as discussed in Example 26.
The map The following theorem will now follow from Proposition 38, Proposition 40, Lemma 41, Theorem 20 and Theorem 43.Theorem 44 Let M be a rank 2 oriented matroid.There is an homeomorphism from µ −1 (M ) to an h(M )-dimensional closed ball.

Conclusion
We have proven that the complexes MacP(2, n) and MacP(1, 2, n) associated to combinatorial Grassmannians have the same heomorphism type as the Grassmannian manifolds Gr(2, R n ) and Gr(1, 2, R n ) respectively.Our argument relies majorly on the realizability of rank 2 oriented matroids, and the fact that stratas µ −1 (M ) and ν −1 (±z, M ) are homeomorphic to an open ball -the argument thus does not apply to oriented matroids of ranks at least 3.

Fig. 1
Fig. 1 An rank 3 oriented matroid from an arrangement of equators

6 NFig. 3
Fig. 3 strong map image Fig. 4 A flag of subspaces, and the rank one oriented matroid

Lemma 14
Suppose N is obtained from M by reorientations of some elements in [n].Then the posets ( 0, M) and ( 0, N ) are isomorphic.Proof Suppose N is obtained from M by reorienting elements in A ⊆ [n].Then the required poset isomorphism R A : ( 0, M) → ( 0, N ) is obtained by taking R A (Y ) as a reorientation of the rank 2 oriented matroid Y by elements in A.
Definition 22 ([9]) Let g : X → 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.