On the homotopy type of the Acyclic MacPhersonian

Ricardo Strausz [email protected]

September 24, 2019

In 2003 Daniel K. Biss published, in the Annals of Mathematics [1], what he thought was a solution of a long standing problem, culminating a discovery due to Gel’fand and MacPher- son [4]; namely, he claimed that the MacPhersonian —a.k.a. the matroid Grassmannian— has the same homotoy type of the Grassmannian. Six years later he was encouraged to publish an “erratum” to his proof [2], observed by Nikolai Mn¨ev;up to now, the homotopy type of the MacPhersonian remains a mystery... d The Acyclic MacPhersonian AMn is to the MacPhersonian, as it is the class of all acyclic oriented matroids, of a given dimension d and order n, to the class of all oriented matroids, of the same dimension and order. The aim of this note is to study the homotopy type d of AMn in comparison with the Grassmannian G(n − d − 1, n − 1), which consists of all n − d − 1 linear subspaces of IRn−1, with the usual topology — recall that, due to the duality given by the “orthogonal complement”, G(n − d − 1, n − 1) and G(d, n − 1) are homeomorphic. For, we will define a series of spaces, and maps between them, which we summarise in the following diagram:

d ı d Ggn / AMgn π F S η ı ı

Ö m.c.f. ~ d t d π ı Gn o Sn

τ

arXiv:1504.06592v2 [math.AT] 21 Sep 2019  & d ı d Gcn / AMn In this diagram, the maps labeled ı represents inclusions and those labeled π are natural projections; the compositions of these will prove homotopical equivalences. The main d function is η which retracts the fat acyclic MacPhersonian AMf n —to be defined— to the d (Affine) Grassmannian Gn = G(n − d − 1, n − 1); this will be study through the mean

1 d curvature flow defined in the elements of Sn, the space of all spherical Radon separoids of dimension d and order n — see [12] for a compact introduction to separoid theory. A recent result of Zhao et al. (2018) allow us to prove that this flow “stretches” wiggled spheres, representing acyclic oriented matroids, onto geodesic spheres, representing affine configurations of points (cf. [6, 7]).

1 Preliminaries

We will represent the elements of all our spaces as “spherical” subspaces of the n-octahedron of radius 2 in IRn: n X Øn := {x = (xi) ∈ IR : |xi| = 2}. It is the combinatorial structure of these hyper-octahedra —or cross polytopes if you will— that allow us to encode the circuits of the oriented matroids we will be dealing with. The reader is encouraged to read [5, 8, 9] to better understand this approach; however, in order to be self-contained, we start by working out a simple example to have in mind during the rest of the exposition: Consider a configuration P of 4 points in the real line, say on the numbers (points) 1, 2, 3 and 4. From the affine point of view (modulus the affine group), this configuration is equivalent to the one on the numbers 1, 3, 5 and 7, but different from that on the numbers 1, 2, 3 and 5. The configuration P can be encoded with the matrix ( 1 2 3 4 ) which represents a linear function f: IR4 → IR. The minimal Radon partitions of this configuration are 2 † 13, 2 † 14, 3 † 14 and 3 † 24; these are the circuits of the associated oriented matroid. Here, and in the sequel, we denote by the symmetric relation † the fact that two subsets are disjoin but their convex hull do intersect (cf. [10]); that is,

A † B ⇐⇒ conv(A) ∩ conv(B) 6= ∅ and A ∩ B = ∅.

Observe that, from the combinatorial point of view, the three configurations exemplified above define the same oriented matroid, while the affine group distinguish the last from the other two. The Radon partitions of the configuration P can be deduced from the solutions, in λi, to the following three equations: P λipi = 0, P λi = 0, P |λi| = 2, where the pi denote the 4 points of the configuration.  1/2   −1  For example, from the solution λ =   we deduce that 2 † 13.  1/2  0

2 ⊥ 4 Observe that the space of solutions S = Kf ∩ 1I ∩ Ø4 ⊂ IR is the intersection of three easy described spaces; namely, the kernel of the linear function, the hyperplane orthogonal to the vector 1I⊥ := (1, 1, 1, 1)>, and the 4-octahedron. Since the kernel is a flat of dimension 3, the hyperplane is also of dimension 3, and the 4-octahedron is an sphere of dimension 3, then S is homeomorphic to the sphere of dimension 1. ⊥ Conversely, given a geodesic 1-dimensional sphere contained in 1I ∩ Ø4, we can extend it to a 3-dimensional subspace, not contained in 1I⊥, which is the kernel of a linear function encoding a configuration of 4 points in the line. That is to say, all configurations of 4 points in the line are in 1-to-1 relation with the family of geodesic 1-dimensional spheres inside ⊥ 1I ∩ Ø4. Analogously, we can identify the configuration of 4 points in the line with the 2-subspaces of 1I⊥; therefore, the configurations of 4 points in the line are in 1-to-1 relation 1 3 with the points of the Grassmannian G4 of 2-subspaces of IR , the projective plane. Back to our example P , we can observe that S inherits from Ø4 the combinatorial structure of a cycle of length 8; namely, its vertices arise from the minimal Radon parti- tions, (1/2, −1, 1/2, 0)>, (2/3, −1, 0, 1/3)>, (1/3, 0, −1, 2/3)>, (0, 1/2, −1, 1/2)> and its an- tipodes, and the edges are the arcs that join these vertices along the circle S. Indeed, this is the so-called circuit graph of the associated oriented matroid (see [9]). However, if we are given just the labeled circuit graph of the oriented matroid, there is no canonic way to choose a single circle to represent it; in particular, there is no way to choose the configuration on the points 1, 2, 3 and 4 or the one on the points 1, 2, 3 and 5. We solve this “anomaly” by choosing a “wiggled” but canonic circle, namely we take the baricentres of each face of Ø4 representing the corresponding circuit; that is, we take the points (1/2, −1, 1/2, 0)>, (1/2, −1, 0, 1/2)>, (1/2, 0, −1, 1/2)>, (0, 1/2, −1, 1/2)>, with its antipodes, and connect them with geodesic arcs in the obvious way. ⊥ We can go one step further and consider “all those wiggled circles” contained in 1I ∩Ø4 such that its vertices are the circuits of an acyclic oriented matroid —this is what I call 1 the fat acyclic MacPhersonian AMf 4. 1 In a similar way, we can represent the acyclic MacPhersonian AM4 with “some wiggled ⊥ circles” contained in 1I ∩ Ø4; namely, for each acyclic oriented matroid, we take that ⊥ “circle” whose vertices are in the baricentres of the respective faces of 1I ∩ Ø4, and each 1 simplex of AM4 —each chain in the partial order of specialisation— is closed taking the 1 ⊥ “convex closure” in the space AMf 4. Since each face of 1I ∩ Ø4 is contractible, the natural map 1 AMf 4 π: ↓ 1 AM4 has contractible fibbers, and therefore it is a homotopical equivalence. 1 Finally, we can apply to each circle of AMf 4 a heat-type flow (we will use in the general 1 case the mean curvature flow; see [6, 7]) to stretch it and ending up with an element of G4 . 1 1 1 1 This exhibits a strong retraction η: AMf 4 → G4 showing that AMf 4, and therefore AM4, have

3 the homotopy type of the projective plane. In the appendix it can be fund a Python code that exhibits an implementation of such a flow for this case.

d 1.1 The Affine Grassmannian Gn

Given n points p1, . . . , pn in dimension d, we can build a d × n matrix M = (pi) consisting of these n column vectors. This matrix is naturally identified with the linear map which sends the cannonical base of IRn onto the points, and two such functions M,M 0 are affinely equivalent if, and only if, their respective kernels KM ,KM 0 intersect the hyperplane 1I⊥ := {x ∈ IRn : x · (1,..., 1)> = 0}

⊥ ⊥ in exactly the same subspace (cf. [8]): KM ∩ 1I = KM 0 ∩ 1I . If the points span the d-dimentional space, the kernel of such a function is of dimention ⊥ n − d, and we can identify the configuration with the n − d − 1 subspace KM ∩ 1I of the hyperplane 1I⊥ ≈ IRn−1; thus, we can identify each configuration with a point of the d n−1 Grassmannian Gn which consists of all n − d − 1 planes of IR , with the usual topology. ⊥ Furthermore, the intersection of each such a n − d − 1 plane KM ∩ 1I with the n- octahedron Øn define the Radon complex of the configuration; it is a CW-complex which is an (n − d − 2)-sphere (see [8]), whose 1-skeleton is the so-called circuit graph of the corresponding oriented matroid — or the cocircuit graph of its dual, if you will. The cocircuit graphs of oriented oriented matroids where characterised in [5, 9]. ⊥ Observe that 1I ∩ Øn corresponds to the space of all configurations of n = d + 2 d points in dimension d, modulus the affine group; it is the double cover of Gd+2, which is homeomorphic to the d-projective space.

1 Figure 1: An element of G3 .

d d We further endow Gn with a partition; namely, we relate two elements of Gn if their corresponding configurations represent the same oriented matroid; the realisation space d d 0 d R(M) ⊂ Gn of the configuration M ∈ Gn is the set of all those M ∈ Gn that represent the same (lebeled) oriented matroid (cf. [3] ch. 8). In particular, each realisation space d ⊥ in Gd+2 = 1I ∩ Øn/{−x, x} is contractible; for, observe that an affine configuration of n = d + 2 points in IRd is determined by a unique Radon partition (a unique circuit)

4 A † B, and its realisation space is then a cell which can be parametrised with the points of σ|A| × σ|B|, where σm represents the simplex of rank m (see Figure 2).

d 1.2 The Acyclic MacPhersonian AMn The family of all oriented matroids on n elements in dimension d can be partially ordered by specialisation; viz., we say that M ≤ M 0 if every minimal Radon partition of M 0 is a Radon partition of M. Given such a partial order, we can define the topological space of its chains, in the usual way, and if we considere only acyclic oriented matroids (of n elements d in dimension d) we get the acyclic MacPhersonian AMn. For example, consider all acyclic oriented matroids arising from 4 points spanning the affine plane; i.e., those acyclic oriented matroids whose circuits are the Radon partitions of 4 points in IR2. There are 3 labeled configurations in general position with the four points in the boundary of the convex hull, and 4 configurations where one of them is in the interior of the convex hull of the other three. These 7 uniform oriented matroids can be represented by 3 squares and 4 triangles that, when joining them together, form the hemicuboctahedron. The boundaries of these 7 “discs” represents those configurations which are not in general position (non-uniform acyclic oriented matroids); there are 12 edges coming from those where one element is in the interior of the segment of other two, ⊥ and 6 vertices representing those where two points coincide —this is exactly 1I ∩ Ø4 after 2 2 2 identifying antipodes. That is to say, AM4 is homeomorphic to G4 ; indeed, AM4 is the first 2 baricentric subdivision of G4 endowed with the partition described below.

2 Figure 2: AM4 is homeomorphic to the projective plane

d 1.3 The space of Spherical Radon Separoids Sn 2S
A separoid S = (S, †) is a set S endowed with a symmetric relation † ⊂ 2 on disjoin subsets of S (i.e., A † B ⇒ A ∩ B = ∅), closed as a filter; that is,

A † B and B ⊂ B0 ⊆ (S \ A) ⇒ A † B0.

5 A related pair A † B is called a Radon partition, and it is enough to know the minimal Radon partitions of a separoid to reconstruct it. The order of the separoid is the cardinal of its base set n = |S| and its dimension is the minimum d = d(S) that makes Radon’s lemma true; that is,

 S  d(S) := min d : ∀X ∈ ∃A ⊂ X : A † (X \ A). d + 2 We say that the separoid is in general position if every minimal Radon partition consist of d + 2 elements. Two disjoint subsets A, B ⊆ S which are not related (are not a Radon partition) are separated and we denote A | B. The separoid is acyclic if the empty set is separated from the base set ∅ | S; in other words, if no Radon partition involves the empty set:

A † B ⇒ |A||B| > 0.

All acyclic separoids can be represented with families of convex polytopes in IRn−1 and their separations by affine hyperplanes (cf. [?]). A Radon separoid is a separoid whose minimal Radon partition are unique on their supports; namely, if A † B and C † D are minimal Radon partitions, then

A ∪ B ⊆ C ∪ D ⇒ {A, B} = {C,D}.

Figure 3: Important clases of separoids.

The minimal Radon partitions of a Radon separoid are the circuits of an oriented matroid if they satisfy the so-called week elimination axiom: if A † B and C † D are

6 minimal Radon partitions, then

e ∈ B ∩ C ⇒ ∃E † F : E ⊆ (A ∪ C) \ e and F ⊆ (B ∪ D) \ e.

Oriented matroids are Steinitz separoids (cf. [?]); that is, their minimal Radon parti- tions satisfies the so-called Steinitz exchange axiom for circuits: if A † B is minimal and A ∪ B = d + 2, ∀x 6∈ (A ∪ B) ∃y ∈ (A ∪ B):(A \ y) † (B \ y) ∪ x. This property induces a cyclic order in the minimal Radon partitions of uniform oriented matroids of order n = d + 3, and therefore their Radon complex (to be defined below) is a cycle —see [8] for another application of this fact.

1.3.1 The Radon complex of a separoid

S Consider the n-cube Qn := (2 , ⊆) and identify its vertices with the subsets of the n-set S, in the usual way. Through this identification, it is easy to see that the faces of Qn are the intervals [A, B] := {X ∈ 2S : A ⊆ X ⊆ B} in the natural partial order induced by the inclusion. Since the n-octahedron Øn is the dual polytope of Qn, its faces can be labeled with such intervals too. Given a separoid S = (S, †), we will assign a subcomplex of R(S) ⊆ Øn made out of those faces [A, B¯] where A†B is a Radon partition and B¯ := S \B denotes the complement; R(S) is called the Radon complex of S (cf. [8]). We say that a separoid is spherical if its Radon complex is a sphere. In particular, oriented matroids are spherical Radon separoids.

2 Stretching spheres

To understand the acyclic MacPhersonian, we first identify each acyclic oriented matroid with its set of circuits. They are encoded in the vertices of the circuit graph, which is well know to be the 1-skeleton of a (n − d − 2)-sphere; indeed, we start with such a “wiggled sphere” and applying to it the mean curvature flow to stretch it, we end up with a “geodesical sphere” which represents an affine configuration of points. This way, we can assign to each acyclic oriented matroid an affine configuration of points, and in such a way that we are defining a homotopical retraction from the acyclic MacPhersonian to the affine Grassmannian.

n−2 2.1 Oriented Matroids as spheres inside Gn . In [9] we characterised the circuit graphs G(M) of oriented matroids as those graphs which can be embedded in what we called “the k-dual of the n-cube”, with some extra properties...

7 that was to say that its vertices can be labeled with the faces of

n−2 ⊥ n n X o n n X o Gn = 1I ∩ Øn := x ∈ IR : xi = 0 ∩ x ∈ IR : |xi| = 2 .

We will identify such circuits (and the corresponding vertices in the circuit graph) with n−2 the barycentre of such faces. The vertices of Gn are points of the form

eij = ei − ej,

n where i 6= j ∈ {1, . . . , n} and ei in the ith element of the canonical base of IR ; and each face of it can be determined in terms of the convex hull of the vertices that it contain. Therefore, each circuit A†B of a given oriented matroid M can (and will) be coordinatised in terms of differences of the canonical base vectors of IRn: X X X X cA†B = λaea − µbeb, where λa = − µb = 1.

If we “fill up the faces” of the circuit graph, considering all its vectors, we end up with a PL-complex which is well known to be an sphere of dimension n − d − 2; that is, we can identify each oriented matroid on n elements in dimension d with a, not necesarly “flat”, n−2 (n − d − 2)-sphere inside Gn . An oriented matroid M is stretchable if and only if there is n−2 a flat (n − d − 2)-sphere inside Gn whose 1-skeleton is the circuit graph G(M) (cf., [8]).

d 2.2 The “fat” MacPhersonian AMf n We now consider a space which is much more “fatty” than the acyclic MacPhersonian and the affine Grassmannian, but contains both of them:

d n n−d−2 n−2 1 n−d−2 d o AMgn := S ,→ Gn : sk (S ) = G(M),M ∈ AMn .

That is, we take several copies of each sphere representing an oriented matroid, while n−2 considering all possible embeddings of such a sphere inside its natural ambiance space Gn , with the extra freedom of having its vertices in any point representig the corresponding circuit (not only the baricentre of it). n−2 d d Clearly, since all faces of Gn are contractible, AMgn and AMn have the same homotopy d d d type, and Gn is embedded in AMgn as it is in AMn (recall the diagram in the first page of this note). d d It remains to show that Gn is a homotopical retract of AMgn.

n−2 2.3 Mean Curvature Flow on Gn We need now to use some tools of differential geometry, so we need to “soft” our PL-spaces; for, we just change the PL-sphere Øn by an euclidian sphere, but keeping the partition

8 of such a sphere given by the faces of the octahedron. That is, in this section by Øn we will denote the sphere of radius 2 endowed with the partition induced by the intersection with the canonical subspaces of IRn, those generated by positive cones of the canonical base, and its negatives. So, we have the 2n vertices given by the n 1-dimensional subspaces spanned by the n canonic vectors e1, . . . , en; the edges between them are geodesic segments n
contained in the intersection of the 2 2-dimensional subspaces spanned by pairs of canonic vectors; geodesic triangles are the intersection of the positive cone of three independent canonic vectors, or its negatives; and so on... So, we preserve the combinatorics imposed by the octahedron, but our ambience space is now the differentiable euclidian sphere instead. d Analogously, when we consider S ∈ AMgn an element of the “fat” MacPhersonian, a n−2 “wiggly” (n−d−2)-sphere whose vertices lie on the “smoothed” faces of Gn , we consider d its embedding to be differentiable. That is, for each element S ∈ AMgn we have a smooth n−2 n−2 ⊥ immersion F0: S → Gn of an (n − d − 2)-sphere into the (n − 2)-sphere Gn = 1I ∩ Øn, endowed with the combinatorial decomposition induced by the n-octahedron. The one n−2 parameter family of immersions F : S × [0, ∞) → Gn satisfying ∂ ⊥ ( ∂t F (x, t)) = H(x, t), F (0, x) = F0(x), ∂ ⊥ ∂ where ( ∂t F (x, t)) denotes the normal component of ∂t F (x, t) and H(x, t) denotes the mean curvature vector of Ft(M), is known as the mean curvature flow with initial value F0. It can be proved that, if n − d − 2 ≥ 3 and d > 0 then Ft(S) converges uniformly d to a geodesic sphere F∞(S) ∈ Gn, which represents a configuration of n points in the d- d dimensional affine space. More generally, if S ∈ Sn is the Radon complex of a spherical d Radon separoid, then the limit F∞(S) ∈ Gn is a geodesic sphere; observe that for all d t ∈ [0, ∞) we have that Ft(S) ∈ Sn, therefore G is a strong retraction of S. d d It remains to prove that, if S ∈ AMgn then for all t ∈ [0, ∞) we have that Ft(S) ∈ AMgn.

Figure 4: Stretching spheres with Ricci-type flows

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