Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$

Alves, Emília; Saldanha, Nicolau C.

doi:10.5802/aif.3267

Results on the homotopy type of the spaces of locally convex curves on

𝕊^{3}

[Résultats sur le type d’homotopie des espaces de courbes localement convexes en

𝕊^{3}

]

Alves, Emília ¹ ; Saldanha, Nicolau C.¹

¹ Departamento de Matemática, PUC-Rio Rua Marquês de São Vicente 225, Rio de Janeiro, RJ 22451-900 (Brazil)

Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1147-1185.

Résumé
Abstract

La courbe $γ : [0, 1] \to 𝕊^{n}$ de classe $C^{k}$ ( $k ⩾ n$ ) est localement convexe si les vecteurs $γ (t), γ^{'} (t), γ^{''} (t), \dots, γ^{(n)} (t)$ forment une base positive de $ℝ^{n + 1}$ pour chaque $t \in [0, 1]$ . Pour un entier $n \geq 2$ et $Q \in {SO}_{n + 1}$ , soit $ℒ 𝕊^{n} (Q)$ l’ensemble de toutes les courbes localement convexes $γ : [0, 1] \to 𝕊^{n}$ avec repères de Frenet initial et final fixes $ℱ_{γ} (0) = I$ et $ℱ_{γ} (1) = Q$ . Saldanha et Shapiro ont demontré qu’il n’y a qu’un nombre fini d’espaces non-homéomorphes parmi les $ℒ 𝕊^{n} (Q)$ avec $Q \in {SO}_{n + 1}$ (en particulier, au plus $3$ pour $n = 3$ ). Pour $n ⩾ 2$ , ils demontrent qu’un de ces espaces est homéomorphe à l’espace (bien compris) des courbes génériques (défini ci-dessous) mais on connaît très peu les autres espaces. Pour $n = 2$ , Saldanha a déterminé le type d’homotopie des espaces $ℒ 𝕊^{2} (Q)$ . Le but de ce travail est d’étudier le cas $n = 3$ . On obtient des informations sur le type d’homotopie d’un de ces autres deux espaces, ce qui nous permet de déduire qu’aucune des composantes connexes de $ℒ 𝕊^{3} (- I)$ n’est homéomorphe à une composante connexe de l’espace des courbes génériques.

A curve $γ : [0, 1] \to 𝕊^{n}$ of class $C^{k}$ ( $k ⩾ n$ ) is locally convex if the vectors $γ (t), γ^{'} (t), γ^{''} (t), \dots, γ^{(n)} (t)$ are a positive basis to $ℝ^{n + 1}$ for all $t \in [0, 1]$ . Given an integer $n \geq 2$ and $Q \in {SO}_{n + 1}$ , let $ℒ 𝕊^{n} (Q)$ be the set of all locally convex curves $γ : [0, 1] \to 𝕊^{n}$ with fixed initial and final Frenet frame $ℱ_{γ} (0) = I$ and $ℱ_{γ} (1) = Q$ . Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among $ℒ 𝕊^{n} (Q)$ when $Q$ varies in ${SO}_{n + 1}$ (in particular, at most $3$ for $n = 3$ ). For any $n ⩾ 2$ , one of these spaces is proved to be homeomorphic to the (well understood) space of generic curves (see below), but very little is known in general about the others. For $n = 2$ , Saldanha determined the homotopy type of the spaces $ℒ 𝕊^{2} (Q)$ . The purpose of this work is to study the case $n = 3$ . We will obtain information on the homotopy type of one of these two other spaces, allowing us to conclude that none of the connected components of $ℒ 𝕊^{3} (- I)$ is homeomorphic to a connected component of the space of generic curves.

Reçu le : 2017-04-12
Révisé le : 2017-10-16
Accepté le : 2018-02-02
Publié le : 2019-06-03

Zbl

DOI : 10.5802/aif.3267

Classification : 57N12, 57N35, 57N65
Keywords: Locally convex curves, homotopy type, Bruhat decomposition
Mot clés : Courbes localement convexes, type d’homotopie, décomposition de Bruhat

Affiliations des auteurs :

Alves, Emília ¹ ; Saldanha, Nicolau C. ¹

¹ Departamento de Matemática, PUC-Rio Rua Marquês de São Vicente 225, Rio de Janeiro, RJ 22451-900 (Brazil)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     pages = {1147--1185},
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Alves, Emília; Saldanha, Nicolau C. Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$. Annales de l'Institut Fourier, Tome 69 (2019) no. 3, pp. 1147-1185. doi : 10.5802/aif.3267. https://aif.centre-mersenne.org/articles/10.5802/aif.3267/

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