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On the Acyclic Macphersonian

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On the Acyclic Macphersonian 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 { d gGn / AMgn π F S η { { Ö m:c:f: ~ d t d π { Gn o Sn τ arXiv:1504.06592v2 [math.AT] 21 Sep 2019 & d { d Gcn / AMn In this diagram, the maps labeled { 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 := fx = (xi) 2 IR : jxij = 2g: 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 y 13, 2 y 14, 3 y 14 and 3 y 24; these are the circuits of the associated oriented matroid. Here, and in the sequel, we denote by the symmetric relation y the fact that two subsets are disjoin but their convex hull do intersect (cf. [10]); that is, A y 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 jλij = 2; where the pi denote the 4 points of the configuration. 0 1=2 1 B −1 C For example, from the solution λ = B C we deduce that 2 y 13. @ 1=2 A 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? := fx 2 IRn : x · (1;:::; 1)> = 0g ? ? 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].
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