Grassmannians and Pseudosphere Arrangements
Michael Gene Dobbins
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Grassmannians and Pseudosphere Arrangements




Michael Gene Dobbins





0.85


Department of Mathematical Sciences, Binghamton University (SUNY), Binghamton, New York, USA. 
 








We extend vector configurations to more general objects that have nicer combinatorial and topological properties, called weighted pseudosphere arrangements.
These are defined as a weighted variant of arrangements of pseudospheres, as in the Topological Representation Theorem for oriented matroids.
We show that in rank 3, the real Stiefel manifold, Grassmannian, and oriented Grassmannian are homotopy equivalent to the analogously defined spaces of weighted pseudosphere arrangements. 
As a consequence, this gives a new classifying space for rank 3 vector bundles and for rank 3 oriented vector bundles where the difficulties of real algebraic geometry that arrise in the Grassmannian can be avoided.
In particular, we show for all rank 3 oriented matroids, that the subspace of weighted pseudosphere arrangements realizing that oriented matroid is contractible. 
This is a sharp contrast with vector configurations where the space of realizations can have the homotopy type of any primary real semialgebraic set.




Math Subject Classification (2010)   52C30   52C40   14M15   57R22




§ INTRODUCTION



If we record all the possible ways a given vector configuration or affine point set can be partitioned by a hyperplane, the resulting combinatorial representation will be an oriented matroid <cit.>.
From this data, we can determine such information as, what points appear on the boundary of the convex hull of a point set, the faces of the resulting polytope, the solutions to a linear programming optimization problem, whether polytopes intersect, and the visibility between points around or on the boundary of a polytope.
Such representations can be used for proving theorems and developing algorithms for finite point sets in Euclidean space <cit.>.

Oriented matroids are more general objects, however, since not all oriented matroids arise from a vector configuration.  Indeed, it is ∃R-complete, which is at least NP-hard, to determine whether a given oriented matroid can be realized by a vector configuration.  
In contrast, the Topological Representation Theorem says that any oriented matroid can be realized by a pseudosphere arrangement <cit.>.






A pseudosphere arrangement is a topological analog to data representing the direction of each vector in a vector configuration; see Firgure <ref>.
More precisely, given a vector configuration in R^k, we can define an oriented hyperplane arrangement by associating each vector to its orthogonal complement, and then define an oriented sphere arrangement by intersecting this hyperplane arrangement with the unit sphere.  From this oriented sphere arrangement, we know the direction of each vector.  Also, each cell of the sphere arrangement corresponds to a way of partitioning the given vector configuration by a hyperplane, and so the sphere arrangement defines an oriented matroid.  
A pseudosphere arrangement is a collection of oriented topological embeddings of (k-2)-spheres in the (k-1)-sphere satisfying certain conditions, and the cells of a pseudosphere arrangement define an oriented matroid; see Sections <ref> and <ref>.
In this paper we deal with weighted pseudosphere arrangements, where each pseudosphere gets a weight analogous to the norm of a vector.


The hardness of deciding whether a given oriented matroid can be realized by a vector configuration is a consequence of Mnëv's Universality Theorem, which says that for any primary semialgebraic set X, there is a rank 3 oriented matroid M, such that the quotient by isometry of the subspace of vector configurations realizing M has the homotopy type of X  <cit.>.  
In other words, if you were hoping for the set of vector configurations that can be partitioned by a hyperplane in a certain fixed set of ways to define a `nice' topological space, then you may be disappointed to know that such a space can be as horrible as any space that you can define algebraically. 
The Topological Representation Theorem may be understood as saying that the space of realizations by pseudosphere arrangements is always at least non-empty.
Here we take this farther by showing that in contrast to Mnëv's theorem, the space of realizations by pseudosphere arrangements is always contractible up to isometry in rank 3; see Theorem <ref>[This is actually proved for weighted pseudosphere arrangements, but here the weights play no role.].  


We also consider the space of all n-element rank k weighted pseudosphere arrangements, which we call the pseudolinear Stiefel manifold _k,n.
One way to define the real Stiefel manifold _k,n is as the space of all n × k matrices with orthonormal columns, or equivalently up to diffeomorphism, as the space of all spanning configurations of n vectors in ℝ^k considered up to symmetric positive definite linear transformation. 
In rank 3, we show that each pseudolinear Stiefel manifold is homotopy equivalent to the corresponding real Stiefel manifold, _3,n≃_3,n.
Moreover, there is a natural embedding of _3,n in _3,n, and we provide a strong deformation retraction from _3,n to _3,n that is equivariant with respect to the orthogonal group _3; see Theorem <ref>.
This holds even in the case where n = ∞.
We also consider the quotient of the pseudolinear Stiefel manifold by the orthogonal and special orthogonal groups, which we call the pseudolinear Grassmannians and oriented pseudolinear Grassmannians.  We show that these are homotopy equivalent to the corresponding real Grassmannians and oriented Grassmannians. 
That is, we the have the homotopy equivalences, _k,n = _k,n / _k ≃_k,n = _k,n / _k and _k,n = _k,n / _k ≃_k,n = _k,n / _k.
This means that weighted pseudosphere arrangements effectively have the same global topology as vector configurations in rank 3. 





These homotopy equivalences motivate the association of weights to pseudosphere arrangements. 

As a vector configuration moves along on a path though a Stiefel manifold, some of the vectors may pass through the origin.  For a weighted pseudosphere arrangement, this would correspond to a pseudosphere vanishing from the arrangement and then reappearing somewhere else.  Since this must happen continuously, we include weights that go to zero as a pseudosphere vanishes. 

One source of interest in Grassmannians is as classifying spaces for vector bundles.  
Recall that a real rank k vector bundle is a space that locally has the structure of a product of R^k with a space B, called the base space; for a precise definition see <cit.>. 
The infinite Grassmannian _k,∞ is a classifying space for rank k vector bundles in the following sense. 
Every rank k vector bundle with base space B can be defined up to isomorphism by a map from B to _k,∞, and two maps define isomorphic vector bundles if and only if the maps are homotopy equivalent, provided that B is paracompact.
We show that the infinite pseudolinear Grassmannian _k,∞ is a classifying space for rank 3 vector bundles, the infinite oriented pseudolinear Grassmannian _k,∞ is a classifying space for rank 3 oriented vector bundles, and the pseudolinear Stiefel manifold is universal for _3 fiber bundles; see Corollary <ref>.




Oriented Matroids have a natural partial order, and the poset of all rank k oriented matroids on n elements is the MacPhersonian, _k,n.
This work is part of a project by the author to resolve the MacPhersonian conjecture in rank 3, which says that the Grassmannian is homotopy equivalent to the polyhedral chain complex of _k,n.  
One potential consequence of this conjecture is that we would have a representation of the homotopy type of the Grassmannian as a simplicial complex defined by purely combinatorial conditions.  
Another important consequence of the MacPhersonian conjecture is that we could represent a vector bundle over a simplicial complex as a poset map to the MacPhersonian in a way that gives a bijection between isomorphism classes of vector bundles and matroid bundles <cit.>.
This conjecture was motivated by the use of oriented matroids to compute Pontrjagin classes of triangulated manifolds <cit.>.  An erroneous proof of the conjecture has appeared and was subsequently retracted <cit.>. 
The present paper does not deal with the MacPhersonian, but may be regarded intuitively as evidence for this conjecture.





In the face of Mnëv's Universality Theorem, the MacPhersonian conjecture may seem overly optimistic for two reasons. 
First, we have a natural map  from the Grassmannian _3,n to the MacPhersonian _3,n sending each vector configuration to its associated oriented matroid, 
but this map is not surjective.  Second, the preimage ^-1(M) of an oriented matroid M can have the homotopy type of any primary semialgebraic set, even in rank 3.
That is, the Grassmannian may be decomposed into realizations of oriented matroids, but the resulting components have highly complicated topology, as do their intersections.  
The conjecture would suggest that we can simply ignore the topology of the components and their intersections, and this will have no bearing on the topology of the Grassmannian as a whole.
The MacPhersonian conjecture may already seem intuitively more reasonable in light of Theorem <ref> and Theorem <ref>, which factor  into maps ι and  through the pseudolinear Grassmannian _3,n such that the following diagram commutes,

    (m) [matrix of math nodes, row sep=4em,
    column sep=5em, text height=1.5ex, text depth=0.25ex]
       _3,n   
    _3,n      _3,n
    ;
    [->]
    (m-1-2) edge node[auto]  (m-2-3)
    (m-2-1) edge node[auto] ι (m-1-2)
    (m-2-1) edge node[auto]  (m-2-3)
    ;

the map ι is a homotopy equivalence, the map  is surjective, and for every rank 3 oriented matroid M∈_3,n, the preimage ^-1(M) ⊂_3,n is contractible.


Other obstacles may raise doubt on generalizing these results to rank higher than 3.  For example, the proofs of the main results here use the fact that the quotient by isometries of the space of self homeomorphisms of the 2-sphere is contractible <cit.>, which may fail in higher dimensions <cit.>.
Also, extension spaces of higher rank oriented matroids can be disconnected, even in the realizable case <cit.>. 
The fact that universality holds for realizations of oriented matroids even when restricted to rank 3, however, shows how crucial the rank 3 case is.


Several ideas used in this paper were developed based 
on discussions with Laura Anderson and 
on the author's prior work with Andreas Holmsen and Alfredo Hubard on generalizing Mnëv's Universality Theorem to arrangements of convex sets in the plane <cit.>, which in turn stemmed from work generalizing the Erdős-Szekeres thoerem from point sets to arrangements of convex sets <cit.>.
















§ MAIN DEFINITIONS





 §.§ Basic notation


Here we briefly give some basic notation which is mostly standard, but may not be universally consistent in the literature. 
We use round brackets 
(x_1,…,x_n) ∈ X^n to denote a sequence and curly brackets {x_1,…,x_n}⊂ X to denote a set.
To append an entry y to sequence I = (x_1,…,x_n), we write I · y = (x_1,…,x_n,y). 
We use the notation [n]_N{1,…,n} and [a,b]_R{x ∈R : a ≤ x ≤ b},
with round brackets for strict inequalities. 
The unit sphere in R^k is denoted by ^k-1{ x∈R^k : x^2 = 1 }, and the closed unit ball by ^k. 
We call the intersection of a pointed convex cone in R^k with the sphere ^k-1, a convex subset of the sphere, or we say a spherically convex set for emphasis.
For a convex subset C of the sphere or the projective plane and points x_i ∈ C, we use [x_1,…,x_n]_C to denote the polygonal path in C with vertices x_1,…,x_n in that order and geodesic edges. 


We denote the set of self homeomorphisms between topological spaces by 

    (X,Y) = {ϕ: X → Y : ϕ is a homeomorphism },
 
and (X) (X,X). 
Let ^+(^d) be 
the component of (^d) that contains the identity map,
i.e. positively oriented reparameterizations of the sphere.
_k denotes the orthogonal group of rank k, which we may regard as a subset of (^k-1) or as a set of (k× k)-matrices when convenient, and _k denotes the special orthogonal group. 




We use x ↦φ for the function defined by substituting a value for x into a formula φ.
We may implicitly curry functions, and we separate arguments by `;' to emphasis when this is done.  That is, given a function f : X × Y → Z, we denote by f(c) : Y → Z the function y ↦ f(c;y).
We denote partial functions by f:X ↛Y.





 §.§ Vector configurations and Grassmannians


Let X be a k dimensional real vector space with an inner product ⟨,⟩, an orthonormal basis e_1,…,e_k, and unit sphere (X) = {u ∈ X: u=1}.  For the time being, we could regard X to be R^k, but we will mostly work in the vector space X = k defined later.

A vector configuration (a_1,…,a_n) in X is a Parseval frame when any of the following equivalent conditions are satisfied: 




  * For all u ∈(X), ⟨ a_1,u ⟩^2 + … +⟨ a_n,u ⟩^2 = 1. 

  * For all x ∈ X, ⟨ a_1,x ⟩^2 + … +⟨ a_n,x ⟩^2 = x^2.

  * The linear map x ↦ (⟨ a_1,x⟩,…,⟨ a_n,x⟩) from X to R^n is an isometry.

  * The (n× k)-matrix A with entries A_i,j = ⟨ a_i, e_j⟩ has orthonormal columns. 




Each class of vector configurations spanning X related by symmetric positive definite transformations contains a unique Parseval frame. 


The Stiefel manifold and Grassmannian can be defined in several ways that are homeomorphic to each other. Here, we define the Stiefel manifold _k,n = _k,n(X) for k ≤ n ∈N to be the space of Parseval frames of X indexed by [n]_N. 
We identify a Parseval frame that ends with a tail of zeros with the shorter frame where the trailing zero are removed so that _k,k⊂_k,k+1⊂_k,k+2⊂⋯,
and we define the infinite Stiefel manifold as the union of this ascending chain of spaces,

    _k,∞ = ⋃_n=k^∞_k,n.






We define the Grassmannian as the quotient by the orthogonal group _k,n = _k,n/_k.
We define the oriented Grassmannian as the quotient by the special orthogonal group _k,n = _k,n/_k. 
Similarly, we define the infinite Grassmannian _k,∞ and oriented Grassmannian _k,∞ as the quotient of _k,∞, which is also the union the ascending chain of Grassmannians.


We define a metric on _k,n for A = (a_1,…,a_n) and B = (b_1,…,b_n) by

    (A,B) max{a_i - b_i : i ∈ [n]_N},

and we define a metric on A,B∈_k,n by

    (A,B) inf{(A,B) : A ∈A, B ∈B}.





Since _k,n and _k,n are defined as the quotient of _k,n by a group of isometries,  is a metric and the metric topology is the same as the quotient topology.


On the infinite Stiefel manifold and infinite (oriented) Grassmannian, we use the direct limit topology, which is the finest topology such that the inclusion maps _k,n↪_k,∞ are continuous.
Equivalently, this topology is defined by the universal property that a function ϕ : _k,∞→ Y to any topological space Y is continuous if and only if the restriction of ϕ to _k,n is continuous for all n ≥ k.


While  defines a metric on _k,∞, the resulting metric topology is not the same as the direct limit topology on _k,∞, and likewise for the Grassmannians.  














There are various equivalent ways the Grassmannian could be defined, depending on how a point in the Grassmannian is represented, and likewise for the Stiefel manifold.  
A point in _k,n is commonly represented as an orthonormal sequence of k vectors in R^n, i.e. the columns of the matrix A in the last definition of a Parseval frame.
Likewise, a point in _k,n may alternatively be represented as a k-dimensional vector subspace of R^n, i.e. the column space of the matrix A.  Our choice of representation, the row vectors of A, is a matter of convenience.


For us it is more convenient to consider the rows of A rather than the columns so that later we will be able to extend these vector configurations to a larger space defined by weighted pseudospheres.
We choose to represent the elements of our vector configuration as row vectors of A so that the orthogonal group _k acts on the right.  Later _k will act by precomposition as a subspace of (^2), which makes this choice consistent with the convention that composition is written right to left.










 §.§ Polar representation and pseudosphere arrangements


We represent a vector a ∈R^k by the pair (a)  (a, (a)) where 

    (a) : ^k-1→{+,0,-},    (a; x) ⟨ a,x ⟩.

This is effectively a polar representation of a where the direction of a is represented by (a). 
We denote by k(R^k) the vector space isomorphic to R^k by , and inheriting the usual scalar multiplication, vector addition, inner product, norm, standard basis vectors, and the action of matrices as linear transformations.  Note that we do not have a nice formula for adding vectors a,b ∈k, other than a+b = (^-1(a)+^-1(b)).

We define weighted pseudospheres as an extension of k.  
A rank k oriented pseudosphere is a map θ : ^k-1→{+,0,-} such that there is some orientation preserving self-homeomorphism ϕ∈^+(^k-1) such that θ∘ϕ =  (e_k); see Figure <ref> Left.  
We may simply call θ a pseudosphere, with the understanding that it is oriented and has some rank.  






A non-trivial weighted pseudosphere is a pair α = (r,θ) consisting of a positive real number r>0 and a pseudosphere θ.  Additionally, there is the trivial weighted pseudosphere 0 =(0) = (0,  x ↦ 0), which is the origin of k.
We let 

    α r    and    (α) θ.

We denote the kernel by S_α = S_θ = θ^-1(0).
We can scale weighted pseudospheres by s ∈R by s α = (|s|r,(s)θ) for s≠ 0 and 0α = 0.

A pseudosphere arrangement is a sequence of pseudospheres Θ = (θ_1,…,θ_n) that satisfies the following.
For all I ⊆ [n]_N, 
S_I = ⋂_i ∈ I S_θ_i is either empty or a topological sphere, meaning there is a homeomorphism ϕ_I : ^k_I→ S_I for some k_I ≤ k, 
and if S_I is non-empty then (θ_1 ∘ϕ_I,…, θ_1 ∘ϕ_I) is again a pseudosphere arrangement; see Figure <ref> Right.
A weighted pseudosphere arrangement is a sequence of rank k weighted pseudospheres A = (α_1,…,α_n) such that ((α_1),…,(α_n)) is a pseudosphere arrangement. 
We may simply write S_i = S_{i} = S_θ_i.
We say A and Θ are spanning when S_1∩…∩ S_n = ∅.  In other words, for every x ∈^k-1, there is some α_i that does not vanish at x.








 §.§ Pseudolinear Grassmannians


Throughout the rest of the paper, let the Stiefel manifold be _k,n = _k,n(k), i.e. the space of Parseval frames in k, and similarly let the Grassmannians and oriented Grassmannians consist of equivalence classes of Parseval frames in k as in Subsection <ref>.  We choose the vector space k for convenience in extending vector configurations to weighted pseudosphere arrangements.



We define the pseudolinear Stiefel manifold _k,n to be the set of all rank k spanning weighted pseudosphere arrangements indexed by [n]_N.
We define a (^k-1)-action on _k,n as follows.
For A = (α_1,…,α_n) ∈_k,n with α_i = (r_i,θ_i), let
α_i * ψ (r_i, θ_i ∘ψ), and let 
A*ψ (α_1*ψ,…,α_n*ψ).  This includes an extension of the _k-action on _k,n to _k,n as _k ⊆(^k-1) and for a ∈R^k, we have  

    (a)*Q = (a,  (a) ∘ Q) = (Q^* a).
  

We say that A is symmetric when -A = A*(-𝕀).


The pseudolinear Grassmannian _k,n is the quotient by the orthogonal group, _k,n = _k,n / _k, and the oriented pseudolinear Grassmannian _k,n is the quotient by the special orthogonal group, _k,n = _k,n / _k.

We extend the metrics on _k,n, _k,n, and _k,n to metrics on _k,n, _k,n, and _k,n as follows.
We first define distance between weighted pseudospheres by a weighted analog of Fréchet distance.
For weighted pseudospheres α_i = (r_i,θ_i) let
 
    (α_1,α_0) inf_ϕ_1, ϕ_0sup_x  r_1 ϕ_1(x) - r_0 ϕ_0(x)

where ϕ_i ∈^+(^k-1) such that θ_i ∘ϕ_i = (e_k)
and x ∈^k-1 such that ⟨ e_k, x⟩ = 0. 
Note that ϕ_i always exists by the definition of a pseudosphere, and also ϕ_i outside the equator of ^k-1 has no bearing on , so we may regard ϕ_i as a positively oriented parameterization of the kernel S_i.
For weighted pseudosphere arrangements A = (α_1,…,α_n) and B = (β_1,…,β_n) ∈, let

    (A,B) max_i∈ [n]_N(α_i,β_i).

For a pair A,B in _k,n or in _k,n, let 

    (A,B) inf{(A,B): A ∈A, B ∈B}.





Since _k,n and _k,n are defined as the quotient of _k,n by a group of isometries,  is a metric and the quotient topology is the same as the metric topology.



For a,b ∈k, we have (a,b) = a-b. Hence,  is an extension of the metrics on _k,n, _k,n, and _k,n, and the subspace topology is the same as the metric topology.



Again we identify weighted pseudosphere arrangements that only differ by a tail of all zeros, so that _k,k⊂_k,k+1⊂_k,k+2⊂⋯, and we define spaces _k,∞, _k,∞, _k,∞ as the union of the corresponding ascending chain of spaces with the direct limit topology.


We define the canonical bundles over the rank 3 pseudolinear Grassmannians to be the spaces

    _3,n (_3,n×R^3)/_3  = {{(A*Q, Q^* x) : Q ∈_3 } : A ∈_3,n, x ∈R^3 }

with the projection map ξ_3,n: _3,n→_3,n induced by (A,x) ↦ A.
Similarly, we define the canonical bundles over the oriented pseudolinear Grassmannians to be 
_3,n (_3,n×R^3)/_3 with projetion ξ_3,n: _3,n→_3,n.
We will prove that these are a fiber bundles, indeed vector bundles, in rank 3; see Lemma <ref>.













 §.§ Sign hyperfield-vector sets and chirotopes



For a weighted pseudosphere arrangement A = (α_1,…,α_n) = ((r_1,θ_1),…,(r_n,θ_n)), let

    (A)     (r_1,…,r_n) ∈R^n, 
     (A)    θ_1×…×θ_n : R^k →{+,0,-}^n,  
     (A)    { (A; v) : v ∈R^k}⊆{+,0,-}^n.

We call (A) the covector set of A. 
We call a set of sequences of signs X a sign hyperfield-vector set when X satisfies the vector axioms for oriented matroids,
or equivalently when X the set of covectors of an oriented matroid <cit.>. 




The Topological Representation Theorem essentially says that for every weighted pseudosphere arrangement A, (A) is a sign hyperfield-vector set, and every sign hyperfield-vector set X is realized by a weighted pseudosphere arrangement A such that X = (A) <cit.>.  


A basis of X is a minimal subset I ⊂ [n]_N such that for every σ∈X∖ 0 there is i ∈ I such that σ(i) ≠ 0, and a subset of a basis is said to be independent. 
Equivalently, a set {i_1,…,i_m}⊂ [n]_N is an independent set of X when 

    { (σ(i_1),…,σ(i_m)) : σ∈X} = {+,0,-}^m,

and a maximal independent set is a basis of X.  
All bases have the same size, and this is called the rank of X.


The independent sets, bases, and rank of a sign hyperfield-vector set X are the same as those of the oriented matroid with covector set X.
There is, however, a dual oriented matroid M^* that has X as its vector set, and has an associated rank, bases, and independent sets that are different from what we use in this paper.




A hyperfield is a generalization of a field, and their study has lead to a vast generalization of matroids, oriented matroids, and vector spaces as an analog of vector spaces where the scalar field is replaced with a hyperfield.  One hyperfield is the sign hyperfield, and the analog of a vector space in this case is a sign hyperfield-vector set <cit.>.




The -realization space of a rank k sign hyperfield-vector set X on [n]_N is 

    (X) { A ∈_k,n :  (A) = X},

and the -realization space is 

    (X) { A ∈_k,n :  (A) = X}.

Similarly, we define the -realization and -realization spaces by

    (X) = (X)/_k ⊂_k,n,   (X) = (X)/_k ⊂_k,n.

We also regard these as realization spaces of the oriented matroid with covector set X.



We order the set of sign sequences {+,0,-}^n by the product of the relation (≤_) where 0 <_ (+), and 0 <_ (-), and the pair (+),(-) are incomparable. 
This makes each sign hyperfield-vector set X a graded poset and the dimension of σ∈X is its height minus 1.  In particular, 0 always has dimension -1.

For σ∈ (A), let 

    (A,σ) {u ∈^k-1:  (A; u) = σ},

and (A,σ) be the closure of (A,σ).



Edmonds and Mandel have shown that the subdivision of ^k-1 by a weighted pseudosphere arrangement is a regular cell decomposition.
In particular, the map σ↦(A,σ) is a poset isomorphism from (A) to the closed cells of A ordered by inclusion, with (A,0) = ∅ <cit.>.











For I ⊆ [n]_N, let _I(A)  (β_1,…,β_n) where 

    β_i = {[  α_i i∈ I;    0 i ∉I ]. .

We say I is a basis of A when |I| is the rank of A and _I(A) is spanning, or equivalently when I is a basis of (A).






We now come to realizations in the oriented (pseudolinear) Grassmannian.
We associate to each weighted pseudosphere arrangement a sign valued function on k-tuples of indices

    (A) : [n]_N^k →{+,0,-},

called the order type of A.  Unless I = {i_1,…,i_k} is a basis for A, (A;i_1,…,i_k) = 0.
If I is a basis, then C = (_I(A),(e_i_1+…+e_i_k)) is parameterized by a map s : Δ→ C from the standard (k-1)-simplex Δ such that the j-th facet of s(Δ) is contained in the i_j-th kernel S_i_j.
We define (A;i_1,…,i_k) ∈{+,-} to be the orientation of s. 
Given a rank k chirotope χ on [n]_N, the -realization and -realization spaces of χ are 

    (χ) { A ∈_k,n :  (A) = χ},   (χ) (χ)/_k ⊂_k,n,

and realization spaces (χ) and (χ) are defined similarly from _k,n.



Again by the Topological Representation Theorem, (A) is always a chirotope, and for every chirotope χ, (χ) is non-empty. 



Every sign hyperfield-vector set X corresponds to a pair of chirotopes {χ,-χ}, and we have (X)/_k = (χ) ∪(-χ).
On the other hand, (χ) is not closed under the action of _k, since (A*Q) = -(A) for Q ∈_k with (Q) = -1.
Hence, we use sign hyperfield-vector sets to represent elements of the Grassmannian, while we use chirotopes to represent elements of the oriented Grassmannian.
We deliberately prefer talking about “sign hyperfield-vector sets” to “oriented matroids” since these correspond to the case where we do not have an orientation.  




































§ RANK 3


The main goal of this section is to prove the following two theorems. 


The -realization space of every rank 3 oriented matroid (i.e. sign hyperfield-vector set) is contractible.
Also, the -realization space of every rank 3 chirotope is contractible.



For n ∈{3,…} or n = ∞, 
the pseudolinear Stiefel manifold   _3,n strongly and _3-equivariantly deformation retracts to the Stiefel manifold _3,n.  Hence the pseudolinear Grassmannian _3,n strongly deformation retracts to the Grassmannian _3,n, and the pseudolinear oriented Grassmannian _3,n strongly deformation retracts to the oriented Grassmannian _3,n.


grassmanpseudograssmansection

grassmanpseudograssman



_3,∞, _3,∞, and _3,∞ are respectively universal for principal _3-bundles, rank 3 vector bundles, and oriented rank 3 vector bundles.
Hence, _3,∞ and _3,∞ are classifying spaces.

















 §.§ Tools





We call pseudospheres in rank 3, pseudocircles.
We start by showing that for A∈_3,n  we can always fix A ∈A up to orthogonal transformation by fixing a coordinate system defined in terms of the pseudocircles in an _3 invariant way.  Essentially, we pick three independent elements of A to define a basis in R^3.
To this end we will define a partial function 

    : [n]_N^3 ×_3,n↛_3,

defined for pairs (I,A) where I is a basis of A, that satisfies the following lemma.



For all I = (i_1,i_2,i_3) of distinct indices in [n]_N and Q ∈_3, we have 


  * (I) : {A ∈_3,n : I  is a basis of  A }→_3
is continuous,

  * (I;A*Q) =Q^*(I;A), 

  * ((1,2,3);(e_1,e_2,e_3)) = 𝕀,

  * (I;A*(I;A)) = 𝕀.




For A∈_3,n and an ordered basis I = (i_1,i_2,i_3) of A, 
let (I;A)∈_3 be given by the matrix with columns (u_1,u_2,u_3) defined as follows.
Let p_± k be the vertex (_I(A), ± (e_i_k)).
If p_- 1 = -p_1, then let 

    u_1    = p_1, 
     
    u_2    = ũ_2/ ũ_2 where ũ_2 = _u_1^⊥(p_2) = p_2 -⟨ u_1, p_2 ⟩ u_1, 
    
    u_3    = (A;i_1,i_2,i_3) ( u_1 × u_2).

Otherwise, we define a map φ∈(^2) that sends -p_1 to p_-1 and let (I;A) = (I;A * φ).

Choose a polar coordinate system for R^3 so that the 1st and 2nd coordinates are angle and radius in the plane spanned by p_1 =(1,0,0) and p_-1 = (1,ω,0) with ω∈ (0,π)_R and the 3rd coordinate is offset from this plane.  Define φ by

    φ(r,θ,h) =  (r,θω/π,h)    θ∈ [0,π]_R
     (r,θ (2-ω/π),h)    θ∈ [-π,0]_R.






The points p_1,p_-1 are where S_i_2 and S_i_3 meet, and assuming these are antipodal, Q = (I;A) is defined as the orthogonal transformation that sends e_1 to p_1, and sends e_2 into the half-plane extending from the line though p_1 in the direction of p_2, and orients the sphere so that S_i_1,S_i_2,S_i_3 appear counterclockwise in that order around the boundary of the triangular cell that is on the positive side of all three curves.
When p_1,p_-1 are not antipodal, we first deform the sphere to make these points antipodal to ensure that p_1 and p_2 are linearly independent.
Note also that p_3 may be in the plane spanned by p_1,p_2, so we do not use p_3 to find u_3.





If Θ_k = (θ_k,1,θ_k,2,θ_k,3) is a spanning pseudocircle arrangement for k∈{1,…,∞} and Θ_k →Θ_∞, then (Θ_k,e_1) →(Θ_∞,e_1).

Moreover, if {S_k,1, S_k,2} is an unoriented pseudocircle arrangement for k∈{1,…,∞}, and S_k,i→ S_∞,i in Fréchet distance, and S_∞,1≠ S_∞,2, then (S_k,1∩ S_k,2) → (S_∞,1∩ S_∞,2) in symmetric Hausdorff distance.



Since Θ_k is spanning, the point p_k = (Θ_k,e_1) ∈^2 is well defined.
Since ^2 is compact, there is a convergent subsequence p_k_j→ q.
Since θ_k,i→θ_∞,i for i ∈{2,3}, S_k,i→ S_∞,i in Fréchet distance, so there are maps ψ_k,i:S_k,i→ S_∞,i such that ∀ x  ∈^2, ψ_k,i(x)-x≤_k → 0.  In particular, this holds for x=p_k_j, so ψ_k_j,i(p_k_j) -q≤_k + p_k -q→ 0, 
so ψ_k_j,i(p_k_j) → q, which implies q ∈ (S_∞,2∩ S_∞,3).
We just have to show that θ_∞,1(q)=+.

Since p_∞ = (Θ_∞,e_1) is bounded away from θ_∞,1^-1{0,-}, and p^-_∞ = (Θ_∞,-e_1) is bounded away from θ_∞,1^-1{0,+}, and q ∈ (S_∞,2∩ S_∞,3) = {p_∞,p^-_∞}, there is some δ > 0 such that ∀ x ∈^2 if x-q≤δ then θ_∞,1(x) = θ_∞,1(q).
Also, since θ_k,1→θ_∞,1 there are maps ψ_k,1∈^+(^2) such that θ_k,1 = θ_∞,1∘ψ_k,1 and ∀ x ∈^2, ψ_k,1(x) -x≤'_k → 0.  Hence, if x-q≤δ -'_k then ψ_k,1(x)-q≤δ, so θ_k,1(x) = θ_∞,1(q).
For j sufficiently large, we have p_k_j -q≤δ -'_k_j, so θ_∞,1(q) = θ_k_j,1(p_k_j) = +.  Thus, q = p_∞.

Suppose that p_k does not converge to p_∞, then there is some other subsequence that is bounded away from p_∞ but has a subsubsequence that converges by compactness.  By the same argument above this subsubsequence must converge to p_∞, which is a contradiction.  Thus, p_k → p_∞

The second part of the lemma follows by a similar argument.



Since S_∞,1≠ S_∞,2, S_∞,1∩ S_∞,2 is a pair of points. 
For k large enough, we have S_k,1≠ S_k,2, so S_k,1∩ S_k,2 = {p_k,q_k} is a pair of points.
Since the space of pairs of points in ^2 is compact, there is a subsequence {p_k_j,q_k_j} that converges to some pair {p_∞,q_∞}. 
We may label these points so that p_k_j→ p_∞ and q_k_j→ q_∞.
Moreover, we label each pair {p_k,q_k} so that 
p_k -p_∞+q_k -q_∞≤q_k -p_∞+p_k -q_∞.


Since S_k,i→ S_∞,i in Fréchet distance, there are maps ψ_k,i:S_k,i→ S_∞,i such that ψ_k,i(x)-x→ 0 uniformly.  In particular this holds for x=p_k_j and for x=q_k_j, so ψ_k_j,i(p_k_j) → p_∞ and ψ_k_j,i(q_k_j) → q_∞, which implies that {p_∞,q_∞} = (S_∞,1∩ S_∞,2).

Suppose that {p_k,q_k} does converge.  Then, there is some other subsequence {p_k'_j,q_k'_j} that is bounded away from {p_∞,q_∞}, and again by compactness, we may assume that this subsequence converges, but by the same argument as above, {p_k'_j,q_k'_j} converges to {p_∞,q_∞}, which is a contradiction.








We have immediately that ((1,2,3);(e_1,e_3,e_3)) is the identity from the definition.

For any Q ∈_3 and any σ∈(A), we have 

    (A*Q, σ)    = {x : (A*Q;x) = σ}
       = {x : (A;Qx) = σ}
       = {Q^*y : (A;y) = σ} 
     = Q^*(A,σ).

In particular, A ↦ A*Q sends p_i to Q^*p_i, and so sends u_i to Q^*u_i.
Thus, (I;A*Q) = Q^*(I;A).
Furthermore, if Q = (I;A), then (I;A*Q) =Q^*Q=𝕀.

Finally, the vertices p_± 1 and p_2 depend continuously on A by Lemma <ref>, and u_1,u_2 depend continuously on these vertices, and u_3 depends continuously on u_1,u_2 up to change of sign, which is constant on connected components of the domain of (I).  Thus, (I) is continuous.





Let  be the sup metric on (^2), 
and let ψ = (ψ,𝕀).
For each rank 3 sign hyperfield-vector set X, this gives a topologically equivalent metric on (X) by 

    (A,B) = (A)-(B) + inf{ψ : ψ∈(^2) , A*ψ = B }.



For every rank 3 sign hyperfield-vector set X, the metrics  and  induce the same topology on (X).



 does not extend to a metric on _3,n that is topologically equivalent to , except for the case n=3.  Indeed, there are sign hyperfield-vector sets X≠Y and sequences A_k ∈(X) where A_k → A_∞∈(Y) with respect to , but A_k diverges with respect to .



To prove Theorems <ref> and <ref> will make use of the following.



There is a strong  _3-equivariant deformation retraction 

    : (^2) × [0,1]_R→(^2)
 
from homeomorphisms of the 2-sphere (^2) to the orthogonal group _3,
where Q ∈_3 acts by precomposition, i.e. fQ = f ∘ Q.



Although the deformation in <cit.> is not _3-equivariant, 
once we have a strong deformation retraction from (^2) to _3 the deformation can be modified to be equivariant using Lemma <ref>








Let κ be a strong deformation retraction from (^2) to (3).  
For f ∈(^2) let 

    A(f) = ((1,(e_1)∘ f), (1,(e_2)∘ f), (1,(e_3)∘ f)),
      γ(f) = ((1,2,3);A(f))



For Q ∈_3 we have γ(f∘ Q) = Q^-1∘γ(f) by Lemma <ref>.  Furthermore, if f ∈_3, then γ(f) = f^-1∘γ(𝕀) = f^-1.

Let

    (f,t) = κ(f ∘γ(f),t) ∘γ(f)^-1.

We have 

    (f ∘ Q,t)
       = κ(f ∘ Q ∘γ(f ∘ Q),t) ∘γ(f ∘ Q)^-1
       = κ(f ∘ Q ∘  Q^-1∘γ(f),t) ∘ (Q^-1∘γ(f))^-1
       = κ(f ∘γ(f),t) ∘γ(f)^-1∘ Q 
       = (f ,t) ∘ Q.

Thus,  is equivariant with respect to the action of _3 by precomposition.
To check  is a strong deformation retraction, observe 

    (f,0) = f ∘γ(f) ∘γ(f)^-1 = f,


    (f,1) = κ(f ∘γ(f),1) ∘γ(f)^-1∈_3,

and if f ∈_3, then

    (f,t) = κ(f ∘ f^-1,t) ∘ f = f.




Note that we could alternatively make a deformation that is _3-equivariant with respect to postcomposition in Theorem <ref> by considering the inverse of the map being deformed, but we cannot do this on both sides at the same time.  Is there a deformation that is equivariant with respect to both postcomposition and precomposition?




Fix u,v,w ∈^1. 
Let S_k be a simple closed curve in ^2 and a_k,b_k,c_k ∈ S_k distinct for each k ∈{1,2,…,∞} such that S_k → S_∞ in Fréchet distance and a_k → a_∞, b_k → b_∞, c_k → c_∞. 
Then, there is a unique homeomorphism h_k for each k from the closed unit disk to the closed region bounded by S_k that is conformal on the interior of the disk and sends u,v,w 
respectively to a_k,b_k,c_k.
Furthermore, h_k converges uniformly to h_∞.


The above theorem is just useful to the present paper as a variation of the Canonical Schoenflies Theorem in dimension 2 <cit.>.  We do not make use of conformality.  An important distinction is that the Canonical Schoenflies Theorem provides a parameterization of a Jourdan region that depends on an embedding of a cylinder, which is to say that it depends on a parameterization of a pseudocircle along with a collar neighborhood of that pseudocircle.  In higher dimensions this may be necessary, but since we are working only in ^2, we can make use of a parameterization that depends only on a pseudocircle and the image of 3 points on the pseudocircle.
Another important feature of this variation is that it is _3-equivariant.


Theorem <ref> is essentially a reformulation of a theorem of Tibor Radó <cit.>, see also <cit.>.  Briefly, this reformulation is as follows.  
The Riemann Mapping Theorem implies that every simply connected open region in ^2 is conformally equivalent to the open unit disk. Carathéodory showed that if the region is bounded by a simple closed curve S ⊂^2, then this conformal map extends to a homeomorphism h from the closed unit disk to the closure of the region <cit.>.  Radó showed that if simple closed curves S_k have parameterizations that converge uniformly, then there exist such maps h_k that converge uniformly <cit.>.  Finally, the image of 3 distinct points on the unit circle determines a conformal automorphism of the unit disk depending uniformly continuously on the 3 points.  One way to see this is as follows.  Conformal automorphisms of the unit disk correspond bijectively via conjugation by the map z ↦z+i/iz+1 to conformal automorphisms of the upper half plane of C, 
which are precisely the complex extensions of projective automorphisms of the real projective line.  Projective automorphisms of the line are determined bijectively by the image of any 3 points.  
Specifically, the automorphism φ that respectively sends 0,1,∞ to the distinct values x_0,x_1,x_∞∈R∪{∞} is given by the linear fractional transformation 
φ(z) = (z-x_0)(x_1-x_∞)/(z-x_∞)(x_1-x_0),  which is also the cross-ratio (z,x_1;x_0,x_∞). 
































 §.§ Interpolation


Given a rank 3 sign hyperfield-vector set X, we define a map 

    : (X) ×(X) →(^2)

such that A*(A,B) = B. 
Equivalently, for each σ∈X, (A,B;(B,σ)) = (A,σ). 


To define , we use Theorem <ref> to parameterize the 2-cells of a given pseudocircle arrangement. 
We define (A,B) first on 0-cells then 1-cells then 2-cells; see Figure <ref>.  Let V_d be the set of all d-dimensional sign hyperfield-vectors of X.  Equivalently, V_d is the set of all σ such that (A;σ) is a d-dimensional cell of A.
Note that as a subset of the sphere, the Hausdorff dimension of (A;σ) might be higher than d. 





For each σ∈V_2, fix a choice of 3 distinct elements υ_1,υ_2,υ_3 ∈V_0 with υ_i <_σ; lets say the 3 lexicographically smallest elements.  These are the sign hyperfield-vectors of 3 vertices of the boundary of (A,σ) for A ∈(X).
Let p_i = (cos (2π i/3), sin (2π i/3)) ∈^1 for i ∈{1,2,3} be the 3 points spaced uniformly about the unit circle so p_3 = e_1.
Let

    _A,σ: ^2 →(A,σ)

be the unique homeomorphism as in Theorem <ref> that is conformal on the interior and respectively sends p_i to the vertex with sign hyperfield-vector υ_i, so (A,υ_i) = {_A,σ(p_i)}.



For the 0-cells there is only one possibility, since a 0-cell is just a single point, so 
for υ∈V_0, 
let (A,B;(B,υ)) = (A,υ).

For each τ∈V_1, fix a sign hyperfield-vector σ∈V_2 such that σ >_τ and let 

    ϕ_A,τ : [0,1]_R→∂^2
 be the positively oriented constant speed parameterization of the arc _A,σ^ -1((A,τ)).
Let 

    (A,B) = _A,σ^1∘ϕ_A,τ^1∘ϕ_B,τ^-1∘_B,σ^-1  on (B,τ).


For each σ∈V_2, let 
ϕ_A,B,σ : ^2 →^2 by 

    ϕ_A,B,σ(x) = x(_A,σ^-1∘(A,B)^1∘_B,σ^1(x/x) )

for x ≠ 0, let ϕ_A,B,σ(0)=0, 
and let 

    (A,B) = _A,σ^1∘ϕ_A,B,σ^1∘_B,σ^-1  on (B,σ).








For any rank 3 sign hyperfield-vector set X,  is continuous on (X) ×(X) and A* (A,B) = B.




The restriction of  to the diagonal is the identity, (A,A;x)=x.  





    (A*Q_1,B*Q_2) = Q_1^* ∘(A,B) ∘ Q_2

since A*Q_1* (Q_1^* ∘(A,B) ∘ Q_2) = B*Q_2.  





Let S_k be a sequence of oriented simple closed curves in ^2 that converge to an oriented simple closed curve S_∞ in Fréchet distance, and let a_k,b_k ∈ S_k such that a_k → a_∞ and b_k → b_∞.  Then, the oriented paths P_k ⊂ S_k from a_k to b_k converge to the oriented path P_∞⊂ S_∞ from a_∞ to b_∞ in Fréchet distance.



For x,y ∈^1, let [x,y]_^1 be the counter-clockwise arc from x to y.

Since S_k → S_∞, there is a sequence of embeddings f_k : ^1 →^2 with f_k(^1) = S_k 
and _k → 0 
such that for all u ∈^1, f_k(u) - f_∞(u) < _k.  
Fix w ∈^1 such that f_∞(w) ∈ P_∞∖{a_∞,b_∞}.

By choosing k sufficiently large, _k is sufficiently small, so that we may assume following,

    f_∞(w) -a_∞ > 2_k,   f_∞(w) -b_∞ > 2_k,   and  b_∞ - a_∞ > 4_k.
 
Let x_k ∈^1 be the first point clockwise of w such that 
f_∞(x_k) - a_∞ = 2_k 
and y_k ∈^1 be the first point counter-clockwise of w such that 
f_∞(y_k) - b_∞ = 2_k.  
Starting from f_∞(w) and traversing P_∞ we reach f_∞(y_k) before b_∞, so 
f_∞(y_k) ∈ P_∞∖{a_∞,b_∞}.
For u ∈ [w,y_k]_^1, we have 

    f_k(u) - b_∞≥ f_∞(u) - b_∞ - f_k(u) -f_∞(u) > 2_k -_k = _k,

but b_k - b_∞ < _k, so starting from f_k(w) and traversing P_k we reach f_k(y_k) before b_k, which implies
f_k(y_k) ∈ P_k ∖{a_∞,b_∞}.
Similarly, f_k(x_k) ∈ P_k ∖{a_∞,b_∞}.

Let x̃_k, ỹ_k ∈^1 where f_k (x̃_k) = a_k and f_k (ỹ_k) = b_k, and 
let 

    A_k = [x̃_∞,x_k]_^1∪ [x̃_k,x_k]_^1  and   
       B_k = [y_k, ỹ_∞]_^1∪ [y_k, ỹ_k]_^1


    α_k = sup{f_∞(u)-a_∞ : u ∈ A_k }   and  β_k = sup{f_∞(u)-b_∞ : u ∈ B_k }.

Since x_k,x̃_k →x̃_∞ and y_k, ỹ_k →ỹ_∞, we have α_k,β_k → 0. Observe that







    sup{f_k(u)-f_∞(v) : u ∈ (x̃_k,x_k),v ∈ (x̃_∞,x_k) }
         ≤sup{f_k(u) -a_∞ : u ∈ A_k } + sup{f_∞(v) -a_∞ : v ∈ A_k }
          < _k +2α_k,

and similarly
sup{f_k(u)-f_∞(v) : u ∈ (y_k,ỹ_k),v ∈ (y_k,ỹ_∞) } < _k +2β_k.



For k≥ 3, 
let u_k : [1/k,1-1/k]_R→^1 be a counter-clockwise parameterization of the arc from x_k to y_k, and define g_k,h_k : [0,1]_R→^2 as follows.
For t ∈ [1/k,1-1/k]_R, let 

    g_k(t) = f_k ∘ u_k(t)   and   h_k(t) = f_∞∘ u_k(t),
 
and on the rest of [0,1]_R complete g_k and h_k to respective parameterizations of P_k and P_∞.
We now have 

    [ g_k(t)-h_k(t)      _k +2α_k             t     [0,1/k]_R; g_k(t)-h_k(t)            _k             t [1/k,1-1/k]_R; g_k(t)-h_k(t)      _k +2β_k             t   [1-1/k,1]_R ]
 
and α_k,β_k,_k → 0.  Therefore, P_k → P_∞ in Fréchet distance.





First, we show that A* (A,B) = B, since (A,B) is defined by building up a map that sends each cell of B to the corresponding cell of A.
In the case υ∈V_0, this is immediate from the definition.
In the case τ∈V_1, ϕ_B,τ^-1 is defined so that ϕ_B,τ^-1∘_B,σ^-1((B,τ)) = [0,1]_R, and this is bijective, which also means _A,σ^1∘ϕ_A,τ^1([0,1]_R) = (A,τ).
In the case σ∈V_2, (A,B) is a composition of bijections sending (B,σ) to the unit ball, which is then sent to itself, and then to (A,σ).

Next, we show (A,B) ∈(^2).
By definition _A,σ and _A,σ^-1 and ϕ_A,τ are continuous. 
And, ϕ_A,B,σ is a composition of continuous functions except at the origin, where ϕ_A,B,σ is also continuous, since ϕ_A,B,σ(x) = x.  
We just have to check that the definition of (A,B) in a 1-cell agrees with the value (A,B) approaches on the boundary of a 2-cell.  Let σ∈V_2, τ∈X such that τ <_σ, and let x_k ∈(B,σ) such that x_k → x ∈(B,τ).  Then,

    (A,B;x_k)
       →_A,σ^1∘ϕ_A,B,σ^1∘_B,σ^-1(x) 
        = _A,σ^1∘_A,σ^-1∘(A,B)^1∘_B,σ^1∘_B,σ^-1(x) 
        = (A,B;x),

since _B,σ^-1(x)  = 1.  Thus, (A,B) ∈(^2).

Finally, we show that (A,B) depends continuously on A and B.
The vertices of A depend continuously on A by Lemma <ref>, and likewise for B. 
By Lemma <ref>, this implies that each 1-cell of A and B varies continuously, and therefore the conformal maps defining (A,B) vary continuously by Theorem <ref>.




















 §.§ Topology of pseudolinear realization spaces






Let A_k ∈(X) for k ∈{1,2,…,∞}. 
If A_k → A_∞ with respect to , then (A_k,A) →𝕀_^2 and (A_k) →(A_∞), so A_k → A_∞ with respect to .
If A_k → A_∞ with respect to , then each pseudosphere of A_k converges to the corresponding pseudosphere of A_∞ and (A_k) →(A_∞), so A_k → A_∞ with respect to .






By the Topological Representation Theorem, (X) is non-empty <cit.>. 
For A ∈A∈(X), we define a strong deformation retraction ρ from (X) to A.
For each B∈(X), fix B ∈B. Let  
2cm

    ρ : (X) × [0,1]_R→(X)
 

    ρ(B,t) = A*((A,B),t)*_3

Observe that ρ(B,t) does not depend on the choice of B ∈B by Observation <ref> and since  is _3-equivariant with respect to precomposition (right action).  Indeed, for any other choice B' ∈B, there is Q ∈_3 such that B' = B*Q, and we have 

    A*((A,B'),t)*_3 
       = A*((A,B) ∘ Q,t)*_3 
       = A*((A,B),t)*Q*_3 
       = ρ(B,t).
  
Furthermore, ρ is continuous by Lemma <ref> and is a strong deformation retraction by Theorem <ref>.

If (A,B) ∈^+(^2), then ((A,B),t) ∈^+(^2) for all t ∈ [0,1]_R,
so the same deformation also shows that (χ) is contractible.



Note that if  were _3-equivariant with respect to postcomposition (left action), then ρ would not depend on a choice of A ∈A, but the dependence on A is not a problem for our purposes.




































 §.§ Deforming weighted pseudocircle arrangements


We now define the deformation of Theorem <ref>.


We first define several intermediate spaces and deformation retractions between these space.  We then combine these deformations to produce a deformation retraction from _3,n to _3,n.









  §.§.§ Intermediate spaces



For each non-repeating non-empty sequence I = (i_1,…,i_m) 
with entries among [n]_N we define spaces X_I,Y_I,Z_I ⊆_3,n by giving conditions for a weighted pseudosphere arrangement A = (α_1,…,α_n) to be in the space.
For this, let S_i = (α_i)^-1(0). 



For m = |I| ∈{0,1,2,3}, let X_I be the set of arrangements A such that I is an independent set of (A).  That is, 




  * X_() = _3,n; 

  * X_(i_1)⊂ X_() where α_i_1≠ 0;

  * X_(i_1,i_2)⊂ X_(i_1) where α_i_2≠ 0 and S_i_1≠ S_i_2; 

  * X_(i_1,i_2,i_3)⊂ X_(i_1,i_2) where α_i_3≠ 0 and S_i_1∩ S_i_2∩ S_i_3 = ∅.


For m >3, let X_I = X_(i_1,…,i_m)⊂ Y_(i_1,…,i_m-1) such that α_i_m≠ 0.
Informally, we use X_I to first pick a basis and then subsequently pick non-zero weighted pseudocircles as long as one is available.

For m ∈{0,1,2}, let Y_I = X_I. 

For m=3, let Y_(i_1,i_2,i_3)⊂ X_(i_1,i_2,i_3) where S_i_1,S_i_2,S_i_3 are great circles. 
For m >3, let Y_I ⊂ X_I such that S_i_m is antipodally symmetric and is geodesic in each of the 2-cells of the subdivision of ^2 by S_i_1,…,S_i_m-1.  That is, the pseudocircles indicated by the sequence I are symmetric and piecewise geodesic with corners only on the pseudocircles appearing earlier in the sequence.

For m ∈{0,1,2}, let Z_I be the set of spanning weighted great circle arrangements, i.e. vector configurations spanning 3. 
For m≥ 3, let Z_I ⊆ Y_I such that every pseudocircle of A is antipodally symmetric and for all j ∉I, if α_j≠ 0 then S_j is geodesic on each 2-cell of _I(A).  That is, all pseudocircles are piecewise geodesic with corners only on the pseudocircles appearing in the sequence, and those that appear in the sequence have corners only on those earlier in the sequence, and -_I(A) = _I(A) * (-𝕀).


We will define strong equivariant deformation retractions f_I from X_I to Y_I and h_I = h_(i_1,…,i_m) from Z_(i_1,…,i_m) to Z_(i_1,…,i_m-1).  

We then combine these to get strong deformation retractions g_I from Y_I to Z_I.



The deformations f_I and h_I for each I ultimately fit together in equation <ref> on page equation-g (or equation <ref> on page equation-g-infty for n = ∞) to give a deformation g_() from Y_() = _3,n to the space Z_() of vector configurations that span 3.
To complete the deformation of Theorem <ref>, we then perform a strong equivariant deformation retraction from great circles to Parseval frames by a continuous orthonormalization process.

Before defining the deformations we start with some properties of these spaces that will be needed.


Let (α_1,α_2) be a weighted pseudocircle arrangement.  Then, (α_1,α_2)≤ d= α_1+α_2.  Also, (α_1,α_2) = d if and only if α_1 = -α_2 and these are antipodally symmetric.



We have (α_1,α_2)≤ d immediately from the definition of .
If α_1 = -α_2 are antipodally symmetric, then any parameterizations of S_1 = S_2 with opposite orientations must pass though an antipodal pair of points, so (α_1,α_2) = d.  It remains to show the `only if' implication.

Suppose there were a point x_1 ∈ S_1 such that -x_1 ∉S_2.
Then, there would have to be x_2 ∈ S_2 such that -x_2 ∉S_1, since S_2 cannot be a proper subset of -S_1.  In this case, we would be able to define parameterizations ψ_i : ^1 → S_i such that ψ_1 stays at x_1 while ψ_2 traverses S_2 starting at x_2 and then vise versa, so we would have (α_1,α_2) < d. 

Hence, if (α_1,α_2) = d then S_1 = -S_2 with the antipodal orientations.  Since (S_1,S_2) is a pseudocircle arrangement, they must either coincide or cross transversally at exactly two points with opposite orientation.  If they crossed transversally at a point x, then they would also have to cross at -x with the same orientation, which is a contradiction.  Thus, we have S_1 = S_2 and α_1 = -α_2.




For m = |I| ≥ 3 and A ∈ Y_I, the 2-cells of _I(A) are spherical convex polygons such that each edge is a 1-cell _I(A).



We proceed by induction.  For m=3, each cell is an orthant of the sphere, which is a spherical triangle with 1-cell edges.  Since Y_I ⊂ Y_(i_1,…,i_m-1), the 2-cells of _(i_1,…,i_m-1)(A) are spherical convex polygons with edges that are 1-cells by inductive assumption.  At each 2-cell C such that S_i_m intersects the interior of C, S_i_m must intersect C in a path P, since the restriction of _I(A) is homeomorphic to a 1-dimensional pseudosphere arrangement and P = C ∩ S_i_m is a 1-cell of that arrangement.
From the definition of Y_I, P must be a geodesic arc through C, thereby subdividing C into a pair of spherical convex polygons with edges that are 1-cells. 



For m > 0, X_I is an open proper subset of Y_(i_1,…,i_m-1) in the induced metric topology on Y_(i_1,…,i_m-1).


Note that X_I is not generally open in the topology on _3,n.


Consider some A ∈ Y_(i_1,…,i_m-1), a generic configuration in (3)^n ⊂ Y_(i_1,…,i_m-1) will suffice.  Then, _[n]_N∖ i_m(A) ∈ Y_(i_1,…,i_m-1)∖ X_I, so X_I is a proper subset of Y_(i_1,…,i_m-1).

To show X_I is open, for each A ∈ X_I, we find a radius r>0 sufficiently small that every B ∈ Y_(i_1,…,i_m-1) within distance r of A is in X_I.
For m =1 or m>3, r = α_i_m suffices.
For m ∈{2,3}, we may use 

    r = min{α_i: i∈ I}·inf{ (1/2) x -y: x ∈ P, y ∈ (S_i_1∪…∪ S_i_m) }

where P consists of a point in the interior of each 2-cell of the subdivision of the sphere by S_i_1,…,S_i_m.  
For every B ∈ Y_(i_1,…,i_m-1) within distance r from A, we have that the points of P are each in the corresponding 2-cell of B, which means B ∈ X_I.






















  §.§.§ The deformation f_I from X_I to Y_I




For m ∈{0,1,2}, let f_I(A,t) = A. 

For m ≥ 3, consider A ∈ X_I and assume that Q = ((i_1,i_2,i_3);A) is the identity, otherwise let f_I(A,t) = f_I(A*Q,t)*Q^-1.

let 

    f_I(A,t) = A*((B,C),1-t)

where B = (β_1,…,β_n) = _I(A) and C = (γ_1,…,γ_n) with (C)=(B) depends on m as follows.  

For m=3, let γ_i_1 = α_i_1e_1, while for k ∈{2,3}, γ_i_k∈3 is the unique vector such that 
γ_i_k = α_i_k,
(γ_i_k,γ_i_1) = (α_i_k,α_i_1),
and (γ_i_k,x) = (α_i_k,x) for x ∈(B,(e_i_2)) ∪(B,(e_i_3)).


For m > 3 and for k<m let γ_i_k= β_i_k.
Let γ_i_m be a copy of β_i_k in the upper hemisphere that is straightened in each cell keeping the end points fixed, and a reflected copy in the lower hemisphere.
That is, we let 

    H {(x_1,x_2,x_3) ∈^2 : x_1 > 0  or  (x_1=0  and  x_2 ≥ 0)},


    V ⋃_k=1^m-1 S_i_k.

For v ∈ V, 

    (γ_i_m; v) = - (β_i_m; v)     v ∈ H 
    - (β_i_m; -v)     v ∉H

Let γ_i_m be defined on ^2 ∖ V such that S_i_m is geodesic on all 2-cells of _(i_1,…,i_m-1) A.




f_I is a well defined strong equivariant deformation retraction from X_I to Y_I.



We start with the case m=3.  Since A ∈ X_I, I is a basis of A, so {i_1,i_2,i_3} is an independent set, which implies (α_i_k,α_i_1) < α_i_k +α_i_1 by Claim <ref>.
There is a circle in 3 of points that are distance α_i_k from 0 and distance (α_i_k,α_i_1) from γ_i_1 = α_i_1e_1, and only an antipodal pair among this circle is orthogonal to p_5-k = (B,(e_i_5-k)), and only one can be the vector γ_i_k toward p_k = (B,(e_i_k)), in the sense that 
⟨γ_i_k, p_k⟩ > 0, so C is well defined.
We have (C) = (B) is the set of all sign sequences with support in I, so f_I is a well defined deformation by Lemma <ref>, Theorem <ref>, and Remark <ref> as in the proof of Theorem <ref>. 
Since {γ_i_1,γ_i_2,γ_i_3}⊂3 is a basis, C ∈ Y_I, so f_I deforms X_I to Y_I.  Since _I(f_I(A,t)) remains in the same -realization space throughout the deformation, {i_1,i_2,i_3} is an independent set of f_I(A,t), so f_I(A,t) ∈ X_I, so f_I is a deformation retraction from X_I to Y_I.  
If A ∈ Y_I to start, then {α_i_1,α_i_2,α_i_3}⊂3 is a basis, which means α_i_k = β_i_k = γ_i_k, so B = C, so (B,C) is the identity, which implies f_I(A,t) = A is trivial, since  is strong.  Thus, f_I is strong.

Now consider the case m>3. 
Since the 2-cells of _(i_1,…,i_m-1)(A) are spherical convex polygons by Claim <ref>, the geodesic arc between any pair of points is in that 2-cell, so C is well defined and (B)=(C), so f_I is a well defined deformation.  Since C ∈ Y_I by definition, f_I is a deformation from X_I to Y_I. 
Since the initial sequence of weighted pseudocircles α_i_1,…,α_i_m-1 remain fixed throughout the deformation, we have for all A ∈ X_I and t ∈ [0,1]_R that f_I(A,t) ∈ Y_(i_1,…,i_m-1), and since the norm of α_i_m also remains fixed, we have f_I(A,t) ∈ X_I, which implies that f_I is a deformation retraction.  
If A ∈ Y_I to start, then f_I(A,t) = A is trivial, since β_i_m = γ_i_m, which implies B=C, so the deformation is strong.

Finally, the deformation is equivariant since f_I(A,t) = f_I(A*Q,t)*Q^-1 by definition.




For all p ∈{0,…,m-1} and all j ∈ [n]_N∖{i_1,…,i_p}, 
if A ∈ X_I ∩ X_(i_1,…,i_p,j) then f_I(A,1) ∈ X_(i_1,…,i_p,j).



For p=0, the claim holds since f_I preserves norm.  In particular, for A ∈ X_(j), the norm of the j-th pseudocircle never vanishes throughout the deformation f_I(A,t).
Similarly for p≥ 3, the claim holds since f_I(A,t) ∈ X_I ⊂ Y_(i_1,…,i_m-1)⊆ Y_(i_1,…,i_k), so f_I(A,1) ∈ Y_(i_1,…,i_k), and f_I preserves norm.

For p=1 (or p=2), the claim holds since f_I preserves order type.  In particular, if (i_1,j) (or (i_1,i_2,j)) is an ordered independent set of (A), then it remains independent throughout the deformation.




















  §.§.§ The deformation h_(i_1,…,i_m) from Z_(i_1,…,i_m) to Z_(i_1,…,i_m-1).



We now define a deformation retraction h_I = h_(i_1,…,i_m) from Z_I = Z_(i_1,…,i_m) to Z_(i_1,…,i_m-1).  If m < 3, this is the trivial deformation, so assume m ≥ 3.

Recall from the definition of Z_I that for A ∈ Z_I, S_i_1,S_i_2,S_i_3 are great circles, which means α_i_1, α_i_2,α_i_3∈3.
We project from ^2 to the the real projective plane  (defined as a compactification of R^2) by the 2-to-1 covering map _(α_i_1,α_i_2,α_i_3) : ^2 → that sends S_i_1 to the horizon, and S_i_2 and S_i_3 to the horizontal and vertical axes respectively.
This map is given on x ∈^2 ∖ S_i_1 by

    _(α_i_1,α_i_2,α_i_3; x) = ( ⟨α_i_2, x ⟩⟨α_i_1, x ⟩, ⟨α_i_3, x ⟩⟨α_i_1, x ⟩),

where ⟨·,·⟩ is the inner product on 3 corresponding to the standard inner product on R^3, 
and the map is defined for x ∈ S_i_1 by continuous extension.
We do this projection so that we can make use of the vector space structure of R^2 ⊂ when defining h_I.

Recall that a pseudoline in  is a simple closed curve in the real projective plane that cannot be deformed to a point.
Let  denote the space of all pseudolines in . 
We give  the metric that is the pull-back by _(e_1,e_2,e_3) of Fréchet distance on pseudocircles in ^2.
By pseudoline arrangement we mean a collection of pseudolines such that any two either coincide or intersect at a single point.  
We say L is a pseudoline extension of a pseudoline arrangement (L_i_1,…,L_i_m) when (L_i_1,…,L_i_m,L) is a pseudoline arrangement, i.e. either L = L_i_j for some j, or L is a pseudoline that intersects each L_i_j exactly once.


Since the pseudocircles of an arrangement in Z_I are antipodally symmetric, these project to pseudolines in , which we denote L_i = L_i(A) = _(α_i_1,α_i_2,α_i_3;S_i).  
We will define the deformation h_I by deforming the pseudolines L_i and lifting these to deformations of weighted pseudocircles.
An important feature of the deformations that we will use is that each pseudoline L_j for j ∈I = [n]_N∖ I deforms in a way that depends only on L_i_1,…,L_i_m, and the initial position of L_j, but distinct pseudolines deform independently of each other.
Claim <ref> below shows that this is a valid way to define such a deformation.



Let

    L_(j_1,…,j_k)(A)  = (L_j_1(A),…,L_j_k(A)) 
      and  Z̃_I  = L_I(Z_I).

We call a partial map

    λ̃: Z̃_I ×× [0,1]_R↛

an extension deformation process on Z̃_I when, for all Ã∈Z̃_I, λ_0 ∈, and t ∈ [0,1]_R, the following hold.


  * λ̃(Ã, λ_0, 0) = λ_0.

  * λ̃(Ã, λ_0, t) is defined, 

provided that λ_0 is a pseudoline extension of Ã and λ_0 is linear in each cell of Ã.

  * λ̃ is continuous on the domain where it is defined.

We say η: Z_I × [0,1]_R→_3,n is a deformation of Z_I induced by the extension deformation process λ̃ on Z_I when, for all A ∈ Z_I, t ∈ [0,1]_R, and j ∈I, the following hold.


  * (η(A,t)) = (A). 

  * _I(η(A,t)) = _I(A). 

  * η(A,t) is antipodally symmetric and L_j(η(A,t)) = λ̃(L_I(A),L_j(A),t).



If λ̃ is an extension deformation process on Z̃_I such that

    (λ̃(Ã, L_1, t),…,λ̃(Ã, L_n, t) )
 
is a pseudoline arrangement for every Ã∈Z̃_I, t ∈ [0,1]_R, and every pseudoline arrangement (L_1,…,L_n) where (L_i_1,…,L_i_m) = Ã, 
then there is a unique deformation η of Z_I induced by λ̃ and η is _3-equivariant. 




We first demonstrate that we do not need for λ̃ to keep track of the orientations of the pseudocircles, as these can be tracked throughout the deformation, and thereby show uniqueness.
Let 

    W_A = {B ∈_3,n: -B = B*(-𝕀), (B) = (A), _I(B) = _I(A) }

be the space of symmetric weighted pseudocircle arrangements where the weight of each pseudocircle is fixed to match that of A,
and the pseudocircles indexed by I coincide with those of A.
Let J = {j ∈I: α_j≠ 0 } be the indices of the non-zero elements of A.  
For B = (β_1,…,β_n) ∈ W_A, we have (β_j,- β_j) = 2α_j >0, so the Z_2-action reversing the orientation of the j-th element is free on W_A, provided that j ∈ J.
Therefore, W_A is a (Z_2)^J-fiber bundle over the space of pseudoline arrangements extending L_I(A) by |J| pseudolines. 
Let Ã = L_I(A) and 

    Λ(B,t) = ( λ̃(Ã, L_1(B), t),…, λ̃(Ã, L_n(B),t) ).

By the hypotheses of the claim, Λ(B,t) is a pseudoline arrangement, so there is a non-empty subset of W_A that projects to Λ(B,t).
Since fiber bundles have the homotopy lifting property, the deformation Λ lifts to a continuous deformation η_A of Z_I ∩ W_A in W_A.  

Let η be the function η(A,t) = η_A(A,t) for each A ∈ Z_I.
By definition, if there is a deformation η' of Z_I induced by λ, then η'(A,t) ∈ W_A and each pseudocircle projects to a pseudoline that deforms according to λ̃, so η' = η is the lift of Λ.  
Hence η is the unique deformation induced by λ̃, provided η is continuous.

Next, we show that η is continuous where it is defined.
Let η_A,i be the i-th weighted pseudocircle of η_A so that  
η_A = η_A,1×…×η_A,n.
Note that some of the weighted pseudocircles may vanish and reappear elsewhere as A varies, but L_i_1,…,L_i_m cannot vanish by the definition of Z_I.
Consider A_k → A_∞∈ Z_I and t_k → t_∞∈ [0,1]_R.  Let

    A_k = (α_k,1,…,α_k,n) = ((r_k,1,θ_k,1),…,(r_k,n,θ_k,n)),

and let M_k be the linear transformation sending α_∞,i_1,α_∞,i_2,α_∞,i_3 to α_k,i_1,α_k,i_2,α_k,i_3, and let N_k ∈(^2) be M_k followed by projecting radially to the sphere.
Recall that {α_k,i_1,α_k,i_2,α_k,i_3} is a basis for the vector space 3 by definition of Z_I, so M_k is a well defined invertible linear transformation and M_k → M_∞=𝕀 in the operator norm.

For each i such that r_∞,i > 0, we have k large enough that r_k,i > 0, so we may let 

    B_k = (β_k,1,…,β_k,n) ∈ W_A_∞

where 

    β_k,i = 
    (r_∞,i,θ_k,i∘ N_k)     r_∞,i > 0 
    
    0     r_∞,i = 0.

Since θ_k,i→θ_∞,i and N_k →𝕀, we have B_k → A_∞, and since η_A_∞ is a continuous deformation in W_A_∞, we have η(B_k,t_k) = η_A_∞(B_k,t_k) →η(A_∞,t_∞).


The distance between the i-th element α of η(A_k,t_k) and β of η(B_k,t_k) is, by definition of Fréchet distance, the infimum over maps from α_k,iS ⊂R^3 to β_k,iT = α_∞,iT ⊂R^3 respecting the orientations of S and T, where S and T are the kernel of (α) and (β).
Let ·_op denote the operator norm of a linear transformation.
We can obtain α_∞,iT from α_k,iS by first scaling by α_∞,i/α_k,i, which moves each point at most | α_∞,i - α_k,i|,  then applying M_k, which moves each point at most α_∞,i·M_k -𝕀_op, and then projecting radially to α_∞,i^2, which moves each point at most α_∞,i· | M_k _op -1 |.  Hence, we can bound the distance to β from α as the sum of the distances for each of these three steps, which gives,

    (η(B_k,t_k), η(A_k,t_k)) ≤max_i ∈ [n]_N( | α_∞,i - α_k,i| +α_∞,i( M_k -𝕀_op +| M_k _op  -1 | ) ).

Since α_k,i→α_∞,i and M_k →𝕀, we have (η(B_k,t_k), η(A_k,t_k)) → 0, so 
η(A_k,t_k) →η(A_∞,t_∞), which means η is continuous.
Thus, η is the deformation of Z_I induced by λ̃.


Finally, we show that η is _3-equivariant. For Q ∈_3, we have 

    L_i(A*Q) 
    = {( ⟨ Q^*α_i_2, x ⟩⟨ Q^*α_i_1, x ⟩, ⟨ Q^*α_i_3, x ⟩⟨ Q^*α_i_1, x ⟩) : x ∈ Q^* S_i } 
    = L_i(A).

That is, applying an orthogonal transformation to a weighted pseudocircle arrangement in A ∈ Z_I, does not change its image in the projective plane, since the projection _(α_i_1,α_i_2,α_i_3) also changes by the same transformation.
Therefore, L_j(η(A,t)*Q) = L_j(η(A,t)),
and λ̃(L_I(A*Q),L_j(A*Q),t) = λ̃(L_I(A),L_j(A),t), 
so L_j(η(A*Q,t)) = L_j(η(A,t)), which means L_j(η(A*Q,t)) = L_j(η(A,t)*Q).
Also _I(η(A*Q,t)) = _I(A*Q) = _I(η(A,t)*Q), 
so η(A*Q,t) = η(A,t)*Q, which means η is _3-equivariant.




We will define h_I as the deformation of Z_I induced by an extension deformation process λ̃ as in Claim <ref>.
We first choose some arbitrary A ∈ Z_I and let Ã = L_I(A) ∈Z̃_I, and choose an arbitrary pseudoline extension λ(0) that is linear on the cells of Ã.  To define λ̃, we define a deformation λ(t) of the pseudoline λ(0), and then let λ̃(Ã,λ(0),t) = λ(t).  Note that λ(t) may depend on Ã = (L_i_1,…,L_i_m) and λ(0).




  
The case m>3.
For m >3, h_I 
is the deformation of Z_I induced by the extension deformation process λ̃ where λ(t) = λ̃(Ã, λ(0), t) is defined as follows.
Let λ(t) = λ(0) be fixed unless λ(0) intersects L_i_m at a single point p(0) that is in the interior of a 2-cell C of the subdivision of R^2 by L_i_2,…,L_i_m-1.
Otherwise, let λ be fixed on the complement of C, and let λ evolve in C as follows; see Figure <ref>.
Let a,b be the points where λ(0) meets the boundary of C, and
p(1) be the point where the segment [a,b]_C intersects L_i_m,
and 

    p(t) = t p(1) + (1-t) p(0), 
    λ(t) ∩ C = [a,p(t),b]_C.








2cm

For m >3, λ(t) is well defined and is a pseudoline extension of L_i_1,…,L_i_m.



We may assume that λ(0) is a pseudoline extension of (L_i_1,…,L_i_m-1) that passes though the interior of the cell C where it intersects L_i_m, otherwise the deformation is trivial.
Since λ(0) is a pseudoline extension, λ(0) intersects C in a single connected component, so λ(0) ∩ C is a polygonal path with a pair of well defined endpoints a,b, and at most one of these points may be on the horizon L_i_1.
By Claim <ref> and Z_I ⊂ Y_I we know that S_i_1,…,S_i_m-1 subdivide ^2 into spherical convex polygons, so C is a convex polygonal region of R^2 = ∖ L_i_1.  Note that C might be unbounded.  
From the definition of Z_I, we know that L_i_m∩ C is a segment through C that subdivides C into two convex polygonal regions. 
From the definition of C, we know that λ(0) crosses L_i_m at a single point p(0) ∈ C, which implies that a,b are separated in C by L_i_m, and therefore the segment [a,b]_C intersects L_i_m at a single point p(1) ∈ C.
Hence p(t) ∈ (L_i_m∩ C) is well defined, and λ(t) ∩ C is a well defined polygonal path that intersects L_i_m at a single point.  Since λ(t) is fixed outside of C and λ(t) is a path in C between fixed endpoints on the boundary of C, λ(t) is a simple closed curve.  Since λ(t) is a deformation of the pseudoline λ(0), λ(t) is a pseudoline. 
By definition, λ(0) starts as a pseudoline extension of L_i_1,…,L_i_m, and λ(t) only deforms in the interior of the cell C of (L_i_1,…,L_i_m-1) where it meets L_i_m at a single point p(t), so λ(t) is a pseudoline extension of (L_i_1,…,L_i_m-1) throughout the deformation.




For m >3, λ̃ is continuous on the domain where it is defined.  Hence, λ̃ is an extension deformation process on Z̃_I.



Let Ã_k = (L_k,i_1,…,L_k,i_m) ∈Z̃_I, λ_k(0) ∈ be a pseudoline extension that is linear on cells of Ã_k, and t_k ∈ [0,1]_R such that Ã_k →Ã_∞, λ_k(0) →λ_∞(0), and t_k → t_∞.
Note that L_k,i_1 = L_i_1, L_k,i_2 = L_i_2, and L_k,i_3 = L_i_3 are fixed at the horizon, horizontal axis, and vertical axis respectively.
Let λ_k(t) = λ̃(Ã_k,λ_k(0),t) be defined as above.
Our goal is to show that λ_k(t_k) →λ_∞(t_∞).

Since there are only finitely many rank 3 sign hyperfield-vector sets on n elements, we may assume that Ã_k has the same covector set for all k ∈N, otherwise partition into finitely many subsequences by the covector sets of the Ã_k and show convergence to λ_∞(t_∞) for each subsequence separately. 

If λ_k(0) = L_k,i_m, then λ_k is fixed throughout the deformation, so the limit converges and we are done.
Otherwise, let p_k(0) = λ_k(0) ∩ L_k,i_m.
If p_k(t) is on one of the pseudolines L_k,i_1,…,L_k,i_m-1, then λ_k is fixed again and we are done.
Otherwise, let C_k be the 2-cell of the subdivision of R^2 by L_k,i_2,…,L_k,i_m-1 that contains p_k(t) in its interior.

By Lemma <ref>, the vertices of Ã_k converge to the corresponding vertices of Ã_∞.  
Consequently, by Lemma <ref> the 1-cells of Ã_k converge to corresponding 1-cells or vertices of Ã_∞.
In particular, p_k(0) → p_∞(0) and {a_k,b_k}→{a_∞,b_∞} where these are defined in the same way as a,b above.
Since L_k,i_m∩ C_k is a segment approaching L_∞,i_m∩ C_∞ and λ_k(0) ∩ C_k is a distinct segment approaching L_∞,i_m∩ C_∞, we have p_k(1) → p_∞(1), since the intersection point of a pair of non-parallel segments depends continuously on their end points. 
Since λ_k(t) is defined continuously in terms of λ_k(0), a_k, b_k, p_k(0), and p_k(1), we have λ_k(t_k) →λ_∞(t_∞).
Thus, λ̃ is continuous on the domain where it is defined, and therefore is an extension deformation process on Z̃_I.


Let Λ(A,t) = (λ̃(Ã,L_1,t),…,λ̃(Ã,L_n,t)). 



For m>3, Λ(A,t) is a pseudoline arrangement.  Hence, h_I is a well defined _3-equivariant deformation of Z_I.






Let J = { j ∈I: α_j > 0 }, and let λ_j(t) = λ̃(Ã, L_j,t).  
Since the pseudolines λ_j(t) deform according to their initial positions, any initially identical pairs of pseudolines remain identical throughout the deformation.  Consider L_j,L_ distinct.  We may assume that at least one of these is deforming, and it suffices to show that the number of crossings does not change in the cell where it deforms.
Assume L_j crosses L_i_m at a point p(0) in the interior of a 2-cell C of L_i_1,…,L_i_m-1.


Suppose L_j,L_ meet in the interior of the cell.  Then, L_j,L_ alternate around the boundary of C, and since points on the boundary of C remain fixed, λ_j(t),λ_(t) meet in C for all t.  Since λ_j(t),λ_(t) are linear segments in the cells C_1,C_2 ⊂ C divided by L_i_m, they can meet in at most one point in each cell C_1,C_2.  If λ_j(t),λ_(t) met at a point in both C_1 and C_2, then they would alternate around the boundary of both cells, which would imply that they do not alternate around the boundary of C.
Thus, λ_j(t),λ_(t) meet at only a single point in C for all t.

Now suppose L_j,L_ do not meet in the interior C ∈C.  
Then, they do not alternate around the boundary of C, C_1, or C_2,  
and we may assume L_j,L_ both meet the segment L_i_m∩ C, otherwise they would never alternate around C, C_1, or C_2 as a result of the deformation. 
Since λ_j(t),λ_(t) are fixed on the boundary of C, they still do not alternate around the boundary of C at t=1, so λ_j(1),λ_(1) do not meet in C, which implies that they also do not alternate around the boundary of C_1 or C_2.  
Therefore, the order of the points p_j(t),p_(t) where λ_j(t),λ_(t) meet the segment L_i_m∩ C is the same for t=0 as for t=1. 
That is, p_j(1)-p_(1)= r (p_j(0)-p_(0)) for some r >0, so 
p_j(t)-p_(t)= (tr +(1-t)) (p_j(0)-p_(0)), which implies λ_j(t),λ_(t) do not alternate around the boundary of C_1 or C_2 for all t.  Hence, λ_j(t),λ_(t) never meet in C throughout the deformation.

Thus, each pair of pseudolines λ_j(t),λ_(t) either coincide or cross exactly once throughout the deformation, so Λ(A,t) is a pseudoline arrangement, and therefore by Claims <ref> and <ref>, h_I is well defined as the equivariant deformation induced by λ̃.





For m >3, h_I = h_(i_1,…,i_m) is a strong _3-equivariant deformation retraction from Z_(i_1,…,i_m) to Z_(i_1,…,i_m-1).



We have already that h_I is an _3-equivariant deformation of Z_I by Claim <ref>.
Since λ(t) is linear in each cell of Ã = (L_i_1,…,L_i_m), we have h_I(A,t) ∈ Z_I, 
and since λ(1) is linear in each cell of (L_i_1,…,L_i_m-1), we have h_I(A,1) ∈ Z_(i_1,…,i_m-1),
so h_I is a deformation retraction from Z_I to Z_(i_1,…,i_m-1). 
For A ∈ Z_(i_1,…,i_m-1) and λ(0) = L_j, we have that λ(0) is linear in each cell of (L_i_1,…,L_i_m-1), so p(1) = p(0), which implies λ(t) = λ(0) is a trivial deformation, and therefore h_I(A,t) = h_I(A,0) is trivial.  Thus, h_I is a strong deformation retraction.





  
The case m=3.
For m=3, let h_I be the deformation of Z_I induced by the extension deformation process λ̃ defined by λ(t) = λ̃(Ã, λ(0), t)
for a pseudoline extension λ(0) of {Ã} = {(L_i_1,L_i_2,L_i_3)} = Z̃_I as follows.

If λ(0) is a straight line, 
then let λ be fixed. 
Equivalently, λ is fixed unless λ(0) intersects L_i_1, L_i_2, L_i_3 at three distinct points that are not collinear. 
Assume this is so, and let p_k(0) = λ(0) ∩ L_i_k for k ∈{1,2,3}.  


We define λ(t) as a polygonal path with moving points p_k(t) as vertices.
For points a,b ∈, let [a,b]_⊕ denote the segment between a,b contained in a single quadrant of  when such a segment exists and is unique, and let [a,b,c,…]_⊕ = [a,b]_⊕∪ [b,c]_⊕∪….  With this we have λ(0) = [p_1(0),p_2(0),p_3(0),p_1(0)]_⊕. 


Let p_1 = p_1(t) = p_1(0) be fixed throughout the deformation. 
Let P be the line through the origin 0 that is perpendicular to line though {0,p_1}.
Let q_0 be the point where the line P meets λ(0); see Figure <ref>.

We define λ(t) as the polygonal path consisting of a segment in one quadrant that pivots around the point q_0 and then extends beyond the axis as rays in the fixed direction of p_1.
We pivot at such a rate that the exterior angle at the vertices of λ(t) is (π/2)(1-t) once this becomes smaller than the initial exterior angle at t=0; see Figure <ref>.
Specifically, let λ(1) be the line though {q_0,p_1}, let ϕ(0) ∈ (-π/2,π/2)_R be the signed angle from λ(1) to the line though {p_2(0),p_3(0)} and ϕ(t) = min(ϕ(0),(π/2)(1-t)), let Q(t) be the line though q_0 at angle ϕ(t) from λ(1), and let p_2(t) = Q(t) ∩ L_i_2 and p_3(t) = Q(t) ∩ L_i_3.
Finally, let

    λ(t) = [p_1,p_2(t),p_3(t),p_1]_⊕.










5cm

For m =3, λ(t) is well defined and is a pseudoline extension of L_i_1,L_i_2,L_i_3.



Assume that we are in the case where λ(0) is not straight and that (π/2)(1-t) < |δ|, since the claim is trivial otherwise. 
We may assume by symmetry that p_3(0) is above the origin on the vertical axis, and p_2(0) is to the right of the origin on the horizontal axis.

We show that q_0 is well defined and is on the segment between p_2(0) and p_3(0).  
By our assumption, λ(0) is an implicit function with a finite negative slope in the upper-right quadrant.
Also, our assumption implies that λ(0) is unbounded on R^2 in the upper-left and lower-right quadrants, and λ(0) is neither vertical nor horizontal there since p_1 is not on L_i_2 or L_i_3, so λ(0) is an implicit function with a finite negative slope in both of these quadrants as well.  

Since lines through p_1 have finite negative slope, P is an implicit function with a finite positive slope, and therefore, λ(0) and P meet at a unique point q_0 in the upper-right quadrant, so the point q_0 is between p_2(0) and p_3(0) on λ(0).

Any line through q_0 with finite negative slope intersects L_i_2 in R^2 to the right of the origin and intersects L_i_3 in R^2 above the origin.
Therefore, the points p_2(t) and p_3(t) move along L_i_2 and L_i_3 respectively without crossing the origin or leaving the plane.
Hence, λ(t) is a well defined path consisting of a segment in the upper-left, upper-right, and lower-right quadrants each.
Furthermore, we now have that λ(t) crosses L_i_1, L_i_2, and L_i_3 once each, so this is a pseudoline arrangement.




For m =3, λ̃ is continuous on the domain where it is defined.  Hence, λ̃ is an extension deformation process on Z̃_I.



For j ∈{1,…,∞}, let λ_j(t) be as above and t_j ∈ [0,1]_R such that λ_j(0) →λ_∞(0) and t_j → t_∞.  
Recall that we use the metric on  induced from Fréchet distance on the sphere.  

If λ_j(0) is straight for all j ∈N large enough, then λ_j(t) = λ_j(0) →λ_∞(0) = λ_∞(t') and we are done.
Otherwise we may restrict to a subsequence that is not straight.
Therefore, assume that λ_j(0) is not a straight line for j≠∞. 

Let p_1,j, p_2,j(t), p_3,j(t), P_j, q_0,j, and Q_j(t) be defined as above for j < ∞ and also for j=∞ where appropriate.


We have four cases to consider, λ_∞(0) intersects L_i_1,L_i_2,L_i_3 at 3 distinct points, or λ_∞(0) is vertical or horizontal, or λ_∞(0) is a straight line through the origin that is neither vertical nor horizontal, or λ_∞(0) = L_i_1 is the horizon.

Suppose that λ_∞(0) intersects L_i_1,L_i_2,L_i_3 at 3 distinct points.  Then by Lemma <ref>, we have p_k,j(0) → p_k,∞(0), so q_0,j→ q_0,∞, so λ_j(1) →λ_∞(1), so Q_j(t_j) → Q_∞(t_∞), so p_k,j(t_j) → p_k,∞(t_∞), so λ_j(t_j) →λ_∞(t_∞) since these are defined continuously in terms of each other in succession.

Suppose that λ_∞(0) is vertical.  Then, p_1 and p_3,j(0) both converge to (L_i_1∩ L_i_3), so P_j → L_i_2, so q_0,j→ p_2,∞(0) = (λ_∞(0) ∩ L_i_2), so λ_j(t_j) converges to the vertical line through p_2,∞(0), which is λ_∞(t_∞) = λ_∞(0).
The argument for λ_∞(0) horizontal is essentially the same.

Suppose that λ_∞(0) contains the origin and is neither vertical nor horizontal.  Then, p_2,j(0) → 0 and p_3,j(0) → 0, so q_0,j→ 0.  Since p_1,j→ p_1,∞ = (λ_∞(0) ∩ L_i_1), we have p_1,j bounded away from L_i_2 and L_i_3 for j large enough, so p_k,j(1) → 0 for k ∈{2,3}. Since p_k,j(t_k) is on L_i_k between p_k,j(0) and p_k,j(1) in R^2, we have p_k,j(t_j) → 0, so λ_j(t_j) converges to the line through the origin and p_1,∞, which is λ_∞(t_∞) = λ_∞(0).

Suppose that λ_∞(0) is the horizon.


Then, min{x : x ∈λ_j(0)}→∞, so 
although q_0,j might not converge, q_0,j→∞.
Since q_0,j = (P_j∩λ_j(1)), and P_j and λ_j(1) are perpendicular, we have min{x : x ∈λ_j(1)} = q_0,j→∞.
Since λ_j(t) pivots about the point q_0,j in one quadrant, and is parallel in two other quadrants, λ_j(t_j) is separated from the origin by the lower envelope of λ_j(0) and λ_j(1), so min{x : x ∈λ_j(t_j)}→∞, which means λ_j(t_j) converges to the horizon.  


Let Λ(A,t) = (λ̃(Ã,L_1,t),…,λ̃(Ã,L_n,t)). 


For m=3, Λ(A,t) is a pseudoline arrangement.  Hence, h_I is a well defined _3-equivariant deformation of Z_I.



Let λ_j(t)=λ̃(Ã, L_j,t), and again let p_1,j, p_2,j(t), p_3,j(t), P_j, q_0,j, and ϕ_j(t) be defined for λ_j as above.

Assume for the sake of contradiction that Λ(A,t) is not a pseudoline arrangement for some t∈ [0,1]_R.
Since we start with a pseudoline arrangement Λ(A,0), there is some minimum t_0 > 0 where Λ(A,t_0) is not a pseudoline arrangement, and there is some pair j,∈ [n]_N∖ I distinct such that λ_j(t_0) and λ_(t_0) intersect at more than 1 point.







Since we start with a pseudoline arrangement Λ(A,0), λ_j(0) and λ_(0) meet at a single point.
Since λ_j(t) and λ_(t) are pseudolines, they must meet in at least 1 point, so there is some minimum time t_0 > 0 where the pseudolines meet at more than 1 point, and one of these points x ∈ (λ_j(t_0) ∩λ_(t_0)) is not a limit point of λ_j(t) ∩λ_(t) for t → t_0 from below.
If x is in the interior of a segment, then either the segments coincide or they cross at x, but the segments cannot cross, since that would make x a limit point of λ_j(t) ∩λ_(t) for t → t_0.
Therefore, we may assume that x is a vertex of λ_j(t_0) and of λ_(t_0).


We cannot have x = p_1,j(t_0) = p_1,(t_0), since these points are fixed throughout the deformation.  Therefore, x is one of the other vertices, and by symmetry we may assume that x = p_2,j(t_0) = p_2,(t_0).

We will show that exterior angles of λ_j(t_0) and λ_(t_0) at x cannot be equal. 
If the exterior angles of λ_j(t_0) and λ_(t_0) at x were equal, then either they would cross at x, which is impossible, or they would coincide along the segments on both sides of x, which would imply that λ_j(t_0) = λ_(t_0).
If we had λ_j(t_0) = λ_(t_0), then p_1,j(0) = p_1,(0), so P_j = P_, so q_0,j = (P_j ∩λ_j(t_0)) = (P_∩λ_(t_0)) = q_0,, but then λ_j(0) and λ_(0) would meet at p_1,j and at q_0,j, which contradicts that λ_j(0) and λ_(0) meet at a single point.
Therefore, we may assume that the exterior angle of λ_j(t_0) at x is strictly greater than that of λ_(t_0).  Thus, λ_j(t_0) has already begun deforming by time t_0, while λ_(t_0) has been fixed up to time t_0.




Let C_j and C_ be the cones emanating from x that are respectively generated by the segments of λ_j(t_0) and λ_(t_0) incident to x. 
Since the exterior angle of λ_j(t_0) at x is strictly greater than that of λ_(t_0), the interior angle of λ_j(t_0) is strictly less than that of λ_(t_0), so these cones are nested C_⊂ C_j.

We now have that t_0 < 1, since λ_j(1) and λ_(1) are both lines, and therefore have the same exterior angle, namely 0.  Also, we know that λ_j(t_0) cannot be a line, since it has exterior angle strictly greater than that of λ_(t_0), which is at least 0.

Since λ_j(t_0) is not a line, we have q_0,j≠ x, but q_0,j is on the boundary of C_j.
Since λ_j(t_0) crosses L_i_2 at x, the axis L_i_2 passes through the interior of C_j.
We now have p_2,j(1) ∈ C_j, since λ_j(1) passes through p_2,j(1), is parallel to one side of C_j, and intersects C_j at q_0,j.


Since λ_j(0) and λ_(0) do not intersect at x ∈λ_(0), λ_j must have started deforming at an earlier time s < t_0.
For t ∈ [s,1]_R^2, the angle of λ_j(t) at q_0,j changes monotonically at a uniform speed, so the point p_2,j(t) moves continuously and monotonically from p_2,j(s) = p_2,j(0) to p_2,j(1) along L_i_2.
Hence, the point p_2,j(t) is outside C_ for time t<t_0, and then crosses into to C_ at time t_0, and then stays in C_ for time t ≥ t_0. 


Let R(t) = [p_2,j(t),p_1,j)_⊕.
The rays R(t) are all parallel
and ray R(t_0) is in the interior of C_, so each ray R(t) intersects C_.
In particular, the rays R(t) for t < t_0 extend from the point p_2,j(t) ∉C_ and cross into C_ at a point y(t) ∈λ_j(t_0)∩λ_(t_0), and y(t) → y(t_0) = x as t → t_0 from below.
But this is a contradiction, since x is not a limit point of λ_j(t) ∩λ_(t) for t → t_0.
Thus, our assumption cannot hold, so Λ(A,t) must be a pseudoline arrangement for all t ∈ [0,1]_R.

Therefore, by Claims <ref> and <ref>, h_I is well defined as the equivariant deformation induced by λ̃.




h_(i_1,i_2,i_3) is a strong _3-equivariant deformation retraction from Z_(i_1,i_2,i_3) to Z_(i_1,i_2) = Z_().



Since λ_j(t) is geodesic in each cell of L_i_1,L_i_2,L_i_3 throughout the deformation, we have h_I(t) ∈ Z_I, and since λ_j(1) is a line, we have h_I(1) ∈ Z_().
Therefore, h_I is a deformation retraction from Z_I to Z_().
Since λ is the trivial deformation if λ(0) is a line, h_I is a strong deformation retraction.


Recall that for m < 3, h_I(A,t) = A is trivial.


For all p ∈{0,…,m-1} and all j ∈ [n]_N∖{i_1,…,i_p}, 
if A ∈ Z_I ∩ X_(i_1,…,i_p,j) then h_I(A,1) ∈ X_(i_1,…,i_p,j).



For p = 0 or p ≥ 3, the claim holds since the deformation h_I preserves norms.
For p = 1, if S_i_1 and S_j are distinct, then they remain distinct throughout the deformation, so the claim holds.
For p = 2 and m > 3, the pseudocircle S_j can only deform in interiors of 2-cells of S_i_1,…,S_i_m, so the intersections of S_j with S_i_1 and with S_i_2 are preserved throughout the deformation, so the claim holds.
For p = 2 and m = 3, the intersection of λ(t) with L_i_1 is fixed throughout the deformation, so the points where S_i_2 and S_j meet S_i_1 are preserved throughout the deformation, so the claim holds.






























  §.§.§ The deformation g_I from Y_I to Z_I for n < ∞




Let (·) denote the operation augmenting a sequence, (x_1, …, x_k) · y = (x_1, …, x_k, y).
Let this also denote concatenation of deformations,

    ϕ_2 ·ϕ_1 (x,t) {[          ϕ_1(x, 2t)      t ∈ [0, 1/2]_R; ϕ_2(ϕ_1(x,1), 2t-1)       t ∈ [1/2,1]_R;                     ].

with composition left associative.  Note that ϕ_2 must be a deformation of the range of ϕ_1 at t=1.
We use ∏_j ∈ Jϕ_j to denote the concatenation of the deformations ϕ_j for j ∈ J in increasing order from right to left.

We next define the deformations g_I using the f_I and h_I.  
Here we assume that n ∈{3,…}.  The infinite case will be dealt with in the next subsubsection. 


Our situation so far is this. 
If we were only concerned with the space of arrangements where none of the weighted pseudocircles vanish, and the first three α_1,α_2,α_3 are always a basis, then we could straighten all pseudocircles with the deformation given by


    h_(1)· h_(1,2)⋯ h_(1,…,n)· f_(1,…,n)⋯ f_(1,2)· f_(1),



but these assumptions only hold on some proper subset of _3,n. 
What we need is a deformation retraction from Y_() = _3,n to Z_().
Moreover, f_(1,…,m-1) is a deformation retraction to Y_(1,…,m-1), and the next deformation we would like to use f_(1,…,m) is only defined on X_(1,…,m), which is a proper subset of Y_(1,…,m-1).
Even worse, the spaces X_I· j for j ∉I do not even cover Y_I.
To deal with this, we will use the fact that these deformations approach the trivial deformation near the complement of the union of the X_I· j.
We will also continuously shift between the deformations corresponding to different sequences I · j by stopping certain deformations early.


Let

    U_I = ⋃_j∈I X_I· j 
    = { (α_1,…,α_n) ∈ Y_I: ∃ j ∈I. α_j > 0 }.

Note that the second equality above holds since every A ∈_3,n has a basis.  Moreover, since every independent set can be completed to a basis, we have that U_I = Y_I for m < 3.


For spaces X ⊆ Y, let

    :  (X × [0,1]_R→ Y) × [0,1]_R→ (X × [0,1]_R→ Y) , 
    
       (ϕ,s;x,t) = ϕ(x,min(s,t)).

That is, (ϕ,s) is the deformation of X that coincides with ϕ up to the stopping time s and then remains fixed thereafter. 
We allow (ϕ,0) to be the trivial deformation even for input that is outside the domain of ϕ.
That is, we let (ϕ,0; y,t) = y for y ∈ Y.










Recall m=|I|.
If m=n, then g_I is just the trivial deformation g_I(A,t) = A.
Otherwise for m<n, g_I is defined recursively by 

    g_I(A,t) = ( ∏_j ∈I(h_I · j· g_I · j· f_I · j, s_I · j (A) )) (A,t),

where s_I · j is defined for j ∈I as follows,


    r_I· j(A) = inf{(B,A): B ∈ Y_I∖ X_I· j},


    s_I · j(A) =  
    0     A ∉U_I
    (2(  r_I· j(A)/max_k ∈I r_I· k(A) ) -1 )^+     A ∈ U_I,

where (ϕ(x))^+ = max(0,ϕ(x)) is the positive part of a function ϕ.

Here, we let (h_I · j· g_I · j· f_I · j, s ) be the trivial deformation on all of Y_I when s=0.  Note that if s > 0, then this deformation is only defined on X_I· j.














For all A ∈ Y_I and j ∈I, s_I· j(A) is well defined, and s_I· j is continuous on U_I.


Note that s_I· j is not continuous on all of Y_I.


If A ∉U_I, then s_I· j(A)=0 is well defined.
Alternatively, if A ∈ U_I, then there is some j_0 ∈I such that A ∈ X_I· j_0, which implies that r_I· j_0(A) > 0 by Claim <ref>.  
Therefore, max_k ∈I r_I· k(A) > 0, which implies that s_I· j(A) is well defined for all j ∈I.

For the second part, observe that distance to a closed set in a metric space is continuous, so each r_I· j is continuous on Y_I.  We have that max_k ∈I r_I· k(A) is strictly positive on A ∈ U_I, so (max_k ∈I r_I· k(A))^-1 is continuous there, which implies that s_I· j is continuous on U_I.






g_I is a well defined strong equivariant deformation retraction from Y_I to Z_I.

Moreover, let I = {j_1,…,j_n-m} and for e ∈{0,…,n-m} let 

    g̃_I, e(A,t) = ( ∏_j =j_1^j_e(h_I · j· g_I · j· f_I · j, s_I · j (A) ) ) (A,t)

be the initial part of the deformation g_I up to the e-th element of I.
For all A ∈ Y_I, t ∈ [0,1]_R, and e ∈{0,…,n-m}, we have the following.




  * 
g̃_I, e(A,t) ∈ Y_I is well defined and equivariant.

  * 
g̃_I, e is continuous.

  * 
If A ∈ Z_I, then g̃_I,e(A,t) ∈ Z_I.

  * 
For all p ∈{0,…,m} and j ∈ [n]_N∖{i_1,…,i_p}, 
if A ∈ X_(i_1,…,i_p,j) then g̃_I,e(A,1) ∈ X_(i_1,…,i_p,j).

  * 
_I(g̃_I,e(A,t)) = _I(A) and g̃_I,e preserves norms.







We proceed by nested induction arguments on m decreasing from m=n and on e increasing from e=0.  To show continuity, we consider A_k → A and t_k → t.

For m = n, g_I is just the trivial deformation and Z_I = Y_I, so the claim holds.  Let m < n.  Our first inductive assumption is that for all j∈I, the claim holds for g_I· j = g̃_I· j,n-m-1.

Also for e=0, g̃_I,e is just the trivial deformation, so the parts of the claim for g̃_I,e hold.
Let e > 0, and let A' = g̃_I,e-1(A,1).  Our second inductive assumption is that the claim holds for g̃_I,e-1.



Suppose first that A ∈ X_I· j_e.

We start by showing parts <ref>, <ref>, and <ref>.
Here is the crucial reason why we need part <ref> of the claim to show that g_I is well defined: by the second inductive assumption for part <ref> of the claim, we have that A' ∈ X_I· j_e.  This implies that f_I· j_e(A',t) ∈ X_I· j_e is well defined, equivariant, and trivial on Y_I· j_e by Claim <ref>, and f_I· j_e(A',1) ∈ Y_I· j_e.
Therefore, g_I· j_e· f_I· j_e(A',t) ∈ Y_I· j_e is well defined, equivariant, and trivial on Z_I· j_e by the first inductive assumption, and g_I· j_e· f_I· j_e(A',1) ∈ Z_I· j_e, so h_I· j_e· g_I· j_e· f_I· j_e(A',t)  ∈ Z_I· j_e is well defined, equivariant, and trivial on Z_I by Claims <ref> and <ref>. 


Recall that Z_I ⊂ Z_I· j_e⊆ Y_I· j_e⊂ X_I· j_e⊂ Y_I, so g̃_I, j_1(A,t) ∈ Y_I is well defined, equivariant, and trivial on Z_I, which means parts <ref> and <ref> hold.

Similarly, we have part <ref> by the inductive assumptions and the definitions of f_I· j_e and h_I· j_e.

Next we show part <ref>.
Consider now the case where A ∈ X_(i_1,…,i_p,j).
Again, A' ∈ X_(i_1,…,i_p,j) by the second inductive assumption, so f_I· j_e(A',1) ∈ X_(i_1,…,i_p,j) by Claim <ref>, and so g_I· j_e· f_I· j_e(A',1) ∈ X_(i_1,…,i_p,j) by the first inductive assumption, and so h_I· j_e· g_I· j_e· f_I· j_e(A',1) ∈ X_(i_1,…,i_p,j) by Claim <ref>.
Thus, part <ref> holds.



Next we show part <ref>.
By the second inductive assumption, we have A'_k = g_I,e-1(A,1) → A' ∈ X_I· j_e. 
Since X_I· j_e is an open subset of Y_I by Claim <ref>, A'_k ∈ X_I· j_e for k sufficiently large, and since f_I· j_e is continuous on X_I· j_e, we have f_I,e(A'_k,t_k) → f_I,e(A',t).
Since g_I· j_e, h_I· j_e, and s_I· j_e are respectively continuous on Y_I· j_e, Z_I· j_e, and U_I⊃ X_I· j_e, we have that g̃_I,e is continuous. Thus, part <ref> holds.

Suppose next that A ∉X_I· j_e.

Then, s_I· j_e(A) = 0, so g̃_I,e(A,t) is the concatenation of g̃_I,e-1(A,t) with a trivial deformation.
Specifically, g̃_I,e(A,t) = g̃_I,e-1(A,τ(t)) where τ(t) = min(2t,1) 

is a non-decreasing surjective transformation of the unit interval, which means that parts <ref>, <ref>, <ref>, and <ref> hold by the second inductive assumption.  It remains to show part <ref> in this case.

Suppose also that A ∈ U_I.  Then, there is some j such that A ∈ X_I· j, so r_I· j(A_k) is bounded below by some r > 0 for k sufficiently large, whereas r_I· j_e(A_k) → 0, which implies that s_I· j_e(A_k) = 0 for k sufficiently large.  Therefore, g̃_I,e(A_k,t_k) = g̃_I,e-1(A_k,τ(t_k)), so part <ref> holds.

Suppose instead that A ∉U_I.  
This implies that for all j ∈I, s_I· j(A) = 0, so g̃_I· e(A,t) = A.
Since every A ∈_3,n includes a basis among its elements, and any independent set can be completed to a basis, we have U_I = Y_I for m ≤ 2.  Therefore, we must have m ≥ 3 in this case, which means that each of the X_I· j consists of the weighted pseudocircle arrangements in Y_I where the j-th element does not vanish.  Since A is in none of the X_I· j, we have _I(A) = A, which implies that _I(A_k) → A. 

We have already shown that part <ref> of the claim holds for g̃_I,e(A_k,t_k) (in both cases A_k ∉X_I· j_e and A_k ∈ X_I· j_e), so _I(g̃_I,e(A_k,t_k)) = _I(A_k) and g̃_I,e preserves norms.  Therefore, for all j ∈I, the j-th element of g̃_I,e(A_k,t_k) converges to 0, since the j-th element of A_k converges to 0.  This implies that 
    (g̃_I,e(A_k,t_k),_I(A_k)) = (g̃_I,e(A_k,t_k),_I(g̃_I,e(A_k,t_k))) → 0,

so g̃_I,e(A_k,t_k) → A = g̃_I· e(A,t).
Thus, part <ref> holds.

This completes the induction on e.  So far we have established parts <ref>, <ref>, <ref>, <ref>, and <ref> of the claim.  

It remains to show that g_I(A,1) ∈ Z_I. 
If A ∉U_I, then we must have m ≥ 3, which means α_j = 0 for all j ∈I, so A ∈ Z_I trivially, so g_I(A,1) ∈ Z_I.
Alternatively, if A ∈ U_I, then there is some j_e ∈I such that r_I· j_e(A) is maximal among all r_I· j(A) for j ∈I.
Therefore, s_I· j_e(A) = 1, so g̃_I· j_e(A,1) ∈ Z_I by Claims <ref> and <ref>.  Since f_I· j, g_I· j, and h_I· j are all trivial on Z_I, once g_I(A,t) attains a value in Z_I for some t ∈ [0,1]_R, it is trivial thereafter.  That is, g_I(A,t') = g_I(A,t) ∈ Z_I for all t' ∈ [t,1]_R, so we have g_I(A,1) ∈ Z_I. 

Finally, parts <ref>, <ref>, and <ref> applied to g̃_I,n-m = g_I together with g_I(A,1) ∈ Z_I imply that g_I is a strong equivariant deformation retraction from Y_I to Z_I.

























  §.§.§ The deformation g_() from _3,∞ to Z_() for n = ∞




We will define a strong equivariant deformation retraction g_() from _3,∞ to Z_().
For any finite non-repeating sequence I of natural numbers, we have X_I,Y_I,Z_I ⊂_3,∞ defined in the same way as in Subsubsection <ref> for n = ∞.
We also have the deformations f_I from X_I to Y_I from Subsubsection <ref> and the deformation h_I from Z_I to Z_(i_1,…,i_m-1) from Subsubsection <ref> in the infinite case.

For n̂≥ 3, we define strong equivariant deformation retractions ĝ_I,n̂ from Y_I ∩_3,n̂ to Z_I ∩_3,n̂, which will be a little different from the deformation of Subsubsection <ref>.
For m = |I| ≥n̂, we again let ĝ_I,n̂ be the trivial deformation.  Otherwise, ĝ_I,n̂ is defined recursively by 

    ĝ_I, n̂(A,t) = ( ∏_j ∈N∖ I (h_I · j·ĝ_I · j, n̂· f_I · j, s_I · j (A) )) (A,t),

where s_I· j is the same as in Subsubsection <ref> with n = ∞.
Finally, let g_()(A,t) = ĝ_(),n̂(A,t) for A ∈_3,n̂⊂_3,∞.


Note that the composition of deformations is defined to be left associative, so that an infinite composition of deformations is well defined for t<1.  Specifically, a composition of deformations 

    ϕ = (∏_j ∈Nϕ_j) (A,t)

performes the deformation ϕ_1 twice as fast up to time t = 1/2, and then ϕ_2 four times as fast up to time t = 3/4, etcetera.
When ϕ(A,t) converges as t → 1 from below, the final state is defined as the limit ϕ(A,1) = lim_t → 1ϕ(A,t).  


ĝ_I,n̂ is a well defined strong equivariant deformation retraction from Y_I ∩_3,n̂ to Z_I ∩_3,n̂.
Also, for n̂_1 < n̂_2 and A ∈ Y_I ∩_3,n̂_1, ĝ_I,n̂_1(A,t) = ĝ_I,n̂_2(A,t).



First observe that if there is some i_k ∈ I such that i_k > n̂, then Y_I ∩_3,n̂ = Z_I ∩_3,n̂ = ∅, so we may assume all entries of I are no greater than n̂.
For j > n̂ and A ∈ Y_I ∩_3,n̂, we have A ∉X_I· j, so r_I· j(A) = 0, which makes s_I· j(A) = 0.  Hence, ĝ_I,n̂ is a composition of deformations that become trivial after j > n̂, and therefore ĝ_I,n̂ is a well defined equivariant deformation retraction by the same argument as in Claim <ref>.  

For the second part, we proceed by induction on n̂_1 -m.
For m = n̂_1, we have that I is a permutation of [n̂_1]_N, so A ∈ (Y_I ∩_3,n̂_1) = (Z_I ∩_3,n̂_1).  Since ĝ_I,n̂_2 is a strong deformation retraction, ĝ_I,n̂_2(A,t) = A = ĝ_I,n̂_1(A,t).  For m < n_1, we have ĝ_I,n̂_2(A,t) = ĝ_I,n̂_1(A,t) by induction.




g_() is a well defined strong equivariant deformation retraction from _3,∞ to Z_().



Let ι_n̂ : _3,n̂↪_3,∞ denote inclusion.
By the second part of Claim <ref>, there is a unique function g_() that makes the following diagram commute for all n̂∈{3,…},

    (m) [matrix of math nodes, row sep=4em,
    column sep=5em, text height=1.5ex, text depth=0.25ex]
    _3,n̂× [0,1]_R   _3,n̂
    _3,∞× [0,1]_R   _3,∞
    ;
    [->]
    (m-1-1) edge node[auto] ĝ_(),n̂ (m-1-2)
    (m-1-2) edge node[auto] ι_n̂ (m-2-2)
    (m-1-1) edge node[auto] ι_n̂×𝕀 (m-2-1)
    (m-2-1) edge node[auto] g_() (m-2-2) 
    ;
    [dashed,->]
    (m-1-1) edge (m-2-2)
    ;

so g_() is well defined. By the first part, it only remains to show that g_() is continuous.
We have g_()∘ (ι_n̂×𝕀) = ι_n̂∘ĝ_(),n̂ is continuous by the universal property of the subspace topology, since ĝ_(),n̂ is continuous.  Therefore, g_() is continuous by the universal property of the direct limit topology.






























  §.§.§ Orthonormalization




To complete the deformation, we perform a continuous orthonormalization.  This may be accomplished in a variety of ways, of which a continuous analog of the Gram–Schmidt process may be the most familiar.  For our purposes, a continuous deformation using the polar decomposition of a matrix is more directly relevant, so that is what we do here.


We may represent A = (α_1,…,α_n) ∈ Z_()⊂ (3)^n ≃R^n× 3 as the (n× 3)-matrix (also denoted A) where the entries of the j-th row are given by the coordinates of ^-1(α_j) ∈R^3.
In this way, we will simply treat Z_() as the space of all full-rank (n× 3)-matrices where it is convenient to do so, and make use of the standard matrix operations of matrix multiplication and taking roots of symmetric positive definite matrices.

For n = ∞, we treat Z_() as the union of the ascending chain of spaces of full-rank (n̂× 3)-matrices with the direct limit topology.  Here matrices that differ by a tail of rows of zeros are identified.



Let q be the deformation of Z_() by

    q(A,t) = A (t(A^*A)^-1/2+(1-t)),

where  is the identity (3× 3)-matrix.


q is a well defined strong equivariant deformation retraction from Z_() to _3,n.



Since A is full-rank, A^*A is positive definite symmetric, so it has a well defined square root that is also positive definite symmetric, so q is well defined. Also, q is defined by a composition of continuous functions for n < ∞ so q is continuous.  For n=∞, q is continuous on Z_(), since q is continuous on each subspace of (n̂× 3)-matrices.


if A ∈_3,n, then A has orthonormal rows, so A^*A =, and ^-1/2 =, which gives q(A,t) = A.


For Q ∈_3, we have 

    q(A*Q,t) 
       = AQ (t(Q^*A^*AQ)^-1/2+(1-t)) 
       = AQ (tQ^*(A^*A)^-1/2Q+(1-t)) 
       = A (t(A^*A)^-1/2+(1-t)) Q 
       = q(A,t)*Q.



Let A_1 = q(A,1).  We have 

    A_1^* A_1 
    = (A^*A)^-1/2 A^* A (A^*A)^-1/2 
    = (A^*A)^-1/2 (A^* A)^1/2 (A^* A)^1/2 (A^*A)^-1/2 
    = ,

so q(A,1) ∈_3.

Finally, since (A^*A)^-1/2 and  are both symmetric positive definite, positive linear combinations of (A^*A)^-1/2 and  are also symmetric positive definite, which implies that (t(A^*A)^-1/2+(1-t)) is full-rank, so q(A,t) is full-rank, so q(A,t) ∈ Z_().
Thus, q is a strong equivariant deformation retraction from Z_() to _3,n.




The deformation q · g_() is a strong _3-equivariant deformation retraction from _3,n to _3,n by Claims <ref> and <ref> for n < ∞ or <ref> for n = ∞.
Hence, taking the quotient of this deformation by the _3-action on _3,n provides a strong deformation retraction from _3,n to _3,n, and the quotient by _3 provides a strong deformation retraction from _3,n to _3,n.



































 §.§ Universal vector bundles and classifying spaces.



Recall that a real rank k vector bundle is a space that locally has the structure of a product with R^k;
for a precise definition see <cit.>. 


One way to obtain a vector bundle is as the quotient space (X ×R^k)/_3 where X is a locally trivial principal _3-bundle <cit.>.




















_3,n is a locally trivial principal _3-bundle. 
Hence, the projection maps 

    ξ_3,n: _3,n→_3,n  and  ξ_3,n: _3,n→_3,n
 
are respectively a vector bundle and oriented vector bundle. 




As a consequence of Lemma <ref>, _3,n is a free _3-space.  To see this, let Q ∈_3 and A = A*Q ∈_3,n, and I be an independent set of A.  Then, (I;A) = (I;A*Q) = Q^* (I;A), so Q = 𝕀.
This implies that there is a translation function

    τ : {(A,A*Q): A ∈_3,n, Q ∈_3 }→_3
 
where τ(A,B) is the unique orthogonal transformation such that B = A*τ(A,B).
To check uniqueness, if  B = A*Q_1 = A*Q_2, then A = A*(Q_2 Q_1^-1), and since the _3 action is free Q_2 Q_1^-1 = 𝕀, so Q_2=Q_1.

Next we will verify the continuity of τ and define local sections associated to an open cover of _3,n.
Let I be the set of non-repeating ordered triples with entries among [n]_N, and for each I = (i_1,i_2,i_3) ∈I let U_I denote the subset of _3,n where I is a basis.
If pseudocircles S_i_1,S_i_2,S_i_3 have no common point of intersection, then any triple of pseudocircles that are sufficiently close in Fréchet distance will also not have a common point of intersection, so U_I is open.  Since every A ∈_3,n has a basis, {U_I : I ∈I} is an open cover of _3,n.  

For B = A*τ(A,B) with basis I, we have (I;B) = (I;A*τ(A,B)) = τ(A,B)^*(I;A), so τ(A,B) = (I;A)(I;B)^*, which is continuous on A,B ∈ U_I, so τ is continuous.
Thus, _3,n is a principal _3-bundle. 

Let U_I = U_I/_3, and define a local section on U_I by

    s_I : U_I → U_I,    s_I(A) = A *(I;A)  for  A ∈A,

Observe that s_I does not depend on the choice of A ∈A, since for B = A*Q, we have
B *(I;B) = A*Q (I;A*Q) = A*Q Q^* (I;A) = A*(I;A), 
and s_I is continuous since (I) is continuous and by definition of the topology on _3,n. 
We may equivalently let s_I(A) be the unique element of A such that (I;s_I(A)) = 𝕀.








Since _3,n has local sections, _3,n is locally trivial, which implies that _3,n is locally trivial <cit.>. 
Specifically, we have a local trivialization on ξ_3,n^-1(U_I) ⊂_3,n given by 

    h_I : ξ_3,n^-1(U_I) →U_I ×R^3,    h_I(X) = (ξ_3,n(X),(I;A)^*x)  for  (A,x) ∈X

Observe that h_I(X) does not depend on the choice of (A,x), since for (B,y) ∈X, we have B = A*Q and y = Q^* x, so 
(I;B)^*y = (I;A*Q)^*Q^*x = (Q^*(I;A))^*Q^*x = (I;A)^*QQ^*x = (I;A)^*x.  Equivalently, we may choose h_I(X)=(A,x) such that (s_I(A),x) ∈X.  Again, h_I is continuous since (I) is continuous, and we have a continuous inverse given by 
h_I^-1(A,x) ∈_3,n such that (s_I(A),x) ∈ h_I^-1(A,x).  
For A∈U_I ∩U_J, we have h_I ∘ h_J^-1(A,x) = (A,y) where y = (I;s_J(A))^*x, so h_I ∘ h_J^-1 acts as a linear isometry on fibers. 
Thus, ξ_3,n is a vector bundle.  

The same argument with _3 instead of _3 shows that ξ_3,n is an oriented vector bundle.


Recall that for a principal _3-bundle ξ: E → B (or vector bundle or oriented vector bundle) and a continuous map f: B' → B between paracompact spaces, there is a pullback bundle (in the same category) f^*(ξ) with base space B', and if f_0,f_1 are homotopy equivalent then their pull back bundles f_0^*(ξ),f_1^*(ξ) are isomorphic. 
This induces a map pb(ξ,B') from homotopy classes of maps B' → B to isomorphism classes of bundles with base space B' (in the same category as ξ), and we say ξ is universal when pb(ξ,B') is a bijection for every paracompact space B'.
In particular, the canonical bundles on _3,∞, _3,∞, and œ_3,∞ are respectively universal for principal _3-bundles, vector bundles, and oriented vector bundles <cit.>.


Since _3,∞ is a subspace of _3,∞, the map pb(ξ_3,∞,B) is surjective for every paracompact space B.  That is every vector bundle over B is isomorpic to the pullback of a map f : B →_3,∞, which is also a map into _3,∞. 
Suppose the pullbacks of a pair of maps f_1,f_4 : B →_3,∞ are isomorphic vector bundles.  
Since _3,∞ is a deformation retract of _3,∞, f_1,f_4 are respectively homotopic to a pair f_2,f_3 : B →_3,∞, which are homotopic to each other since _3,∞ is a universal vector bundle, so f_1,f_4 are homotopic to eachother.
Thus, _3,∞ is a universal vector bundle. 
The same argument applies to _3,∞ and _3,∞.































§ ACKNOWLEDGMENTS


The author would like to thank 
Laura Anderson,
Pavle Blagojević,
Kenneth Clarkson,
Andreas Holmsen,
and Alfredo Hubard 
for helpful discussions and insights.
This research was partly supported by the National Research Foundation of Korea NRF grant 2011-0030044, SRC-GAIA.













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