Results on the homotopy type of the spaces of locally convex curves on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>𝕊</mml:mi> <mml:mn>3</mml:mn> </mml:msup></mml:math>

Alves, Emília; Saldanha, Nicolau C.

doi:10.5802/aif.3267

Cited by 8 publications

(30 citation statements)

References 16 publications

(23 reference statements)

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“…This combinatorial approach has already allowed further progress on the subject, which we expect to cover in a forthcoming paper [3]. A sample may be found in the conjectures stated in our final remarks (Section 18).…”

Section: Introductionmentioning

confidence: 77%

“…For a fixed positive integer n ≥ 2, consider the n-sphere S n ⊂ R n+1 with respect to the usual Euclidean metric of R n+1 . A parametric curve γ : [0, 1] → S n of class C n is said to be nondegenerate [32,33,34,40,59,65] or locally convex [2,51,52,53,54] if and only if its derivatives up to n th order γ(t), γ (t), • • • , γ (n) (t) span a complete flag γ(t), γ (t), • • • , γ (n) (t) of R n+1 for all t ∈ [0, 1]. Without loss of generality, we shall consider only positive nondegenerate curves, i.e., those satisfying det γ(t), γ (t), • • • , γ (n) (t) > 0.…”

Section: Introductionmentioning

confidence: 99%

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Combinatorialization of spaces of nondegenerate spherical curves

Goulart¹,

Saldanha²

2018

Preprint

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A parametric curve γ of class C n on the n-sphere is said to be nondegenerate (or locally convex) when det γ(t), γ (t), • • • , γ (n) (t) > 0 for all values of the parameter t. We orthogonalize this ordered basis to obtain the Frenet frame Fγ of γ assuming values in the orthogonal group SO n+1 (or its universal double cover, Spin n+1 ), which we decompose into Schubert or Bruhat cells. To each nondegenerate curve γ we assign its itinerary: a word w in the alphabet S n+1 {e} that encodes the succession of non open Schubert cells pierced by the complete flag of R n+1 spanned by the columns of Fγ . Without loss of generality, we can focus on nondegenerate curves with initial and final flags both fixed at the (non oriented) standard complete flag. For such curves, given a word w, the subspace of curves following the itinerary w is a contractible globally collared topological submanifold of finite codimension. By a construction reminiscent of Poincaré duality, we define abstract cell complexes mapped into the original space of curves by weak homotopy equivalences. The gluing instructions come from a partial order in the set of words. The main aim of this construction is to attempt to determine the homotopy type of spaces of nondegenerate curves for n > 2. The reader may want to contrast the present paper's combinatorial approach with the geometry-flavoured methods of previous works.

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Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

Combinatorialization of spaces of nondegenerate spherical curves

Goulart¹,

Saldanha²

2018

Preprint

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“…For q ∈ {±1} we have: For n = 3 we have Spin 4 = S 3 ×S 3 for S 3 ⊂ H. We also have that Quat 4 ⊂ Q 8 × Q 8 is generated by (1, −1), (i, i) and (j, j). This point of view is used in [2] to obtain partial results concerning the homotopy type of L 3 . The following conjecture promises a more complete result.…”

Section: Final Remarksmentioning

confidence: 99%

“…Recall that, in the notation from [2], we have: It seems to be perhaps within reach but certainly harder to use this combinatorial approach to determine the homotopy type of L n (1; q) for n > 3 and q ∈ Z(Quat n+1 ). We hope to be able to prove at least the following claim, which should be contrasted with Corollary 1.1.…”

Section: Final Remarksmentioning

confidence: 99%

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A CW complex homotopy equivalent to spaces of locally convex curves

Goulart¹,

Saldanha²

2021

Preprint

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Locally convex (or nondegenerate) curves in the sphere S n (or the projective space) have been studied for several reasons, including the study of linear ordinary differential equations of order n + 1. Taking Frenet frames allows us to obtain corresponding curves Γ in the group Spin n+1 ; recall that Π : Spin n+1 → Flag n+1 is the universal cover of the space of flags. Determining the homotopy type of spaces of such curves Γ with prescribed initial and final points appears to be a hard problem. Due to known results, we may focus on L n , the space of (sufficiently smooth) locally convex curves Γ : [0, 1] → Spin n+1 with Γ(0) = 1 and Π(Γ(1)) = Π(1). Convex curves form a contractible connected component of L n ; there are 2 n+1 other components, corresponding to non convex curves, one for each endpoint. The homotopy type of L n has so far been determined only for n = 2 (the case n = 1 is trivial). This paper is a step towards solving the problem for larger values of n.The itinerary of a locally convex curve Γ : [0, 1] → Spin n+1 belongs to W n , the set of finite words in the alphabet S n+1 {e}. The itinerary of a curve lists the non open Bruhat cells crossed by the curve. Itineraries yield a stratification of the space L n . We construct a CW complex D n which is a kind of dual of L n under this stratification: the construction is similar to Poincaré duality. The CW complex D n is homotopy equivalent to L n . The cells of D n are naturally labeled by words in W n so that D n is infinite but locally finite. Explicit glueing instructions are described for lower dimensions.As an application, we describe an open subset Y n ⊂ L n , a union of strata of L n . In each non convex component of L n , the intersection with Y n is connected and dense. Most connected components of L n are contained in Y n . For n > 3, in the other components the complement of Y n has codimension at least 2. We prove that Y n is homotopically equivalent to the disjoint union of 2 n+1 copies of Ω Spin n+1 . In particular, for all n ≥ 2, all connected components of L n are simply connected.

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