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  ]
Annales de l'Institut Fourier, à paraître, 39 p.

La courbe γ:[0,1]𝕊 n de classe C k (kn) est localement convexe si les vecteurs γ(t),γ ' (t),γ '' (t),,γ (n) (t) forment une base positive de n+1 pour chaque t[0,1]. Pour un entier n2 et QSO n+1 , soit 𝕊 n (Q) l’ensemble de toutes les courbes localement convexes γ:[0,1]𝕊 n avec repères de Frenet initial et final fixes γ (0)=I et γ (1)=Q. Saldanha et Shapiro ont demontré qu’il n’y a qu’un nombre fini d’espaces non-homéomorphes parmi les 𝕊 n (Q) avec QSO n+1 (en particulier, au plus 3 pour n=3). Pour n2, 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 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 𝕊 3 (-I) n’est homéomorphe à une composante connexe de l’espace des courbes génériques.

A curve γ:[0,1]𝕊 n of class C k (kn) is locally convex if the vectors γ(t),γ ' (t),γ '' (t),,γ (n) (t) are a positive basis to n+1 for all t[0,1]. Given an integer n2 and QSO n+1 , let 𝕊 n (Q) be the set of all locally convex curves γ:[0,1]𝕊 n with fixed initial and final Frenet frame γ (0)=I and γ (1)=Q. Saldanha and Shapiro proved that there are just finitely many non-homeomorphic spaces among 𝕊 n (Q) when Q varies in SO n+1 (in particular, at most 3 for n=3). For any n2, 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 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 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 : 2019-03-08
Classification:  57N12,  57N35,  57N65
Mots clés: Courbes localement convexes, type d’homotopie, décomposition de Bruhat
@unpublished{AIF_0__0_0_A31_0,
     author = {Alves, Em\'\i lia and Saldanha, Nicolau C.},
     title = {Results on the homotopy type of the spaces of locally convex curves on $\protect \mathbb{S}^3$},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
}
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, à paraître, 39 p.

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