The homotopy type of spaces of locally convex curves in the sphere

2012/07/31 by Nicolau C. Saldanha · 1 citation
Mathematics · #math.CA #math.GT #msc:34B05 #msc:53C42 #msc:57N65

paper · pdf · doi:10.2140/gt.2015.19.1155

published as Geom. Topol. 19 (2015) 1155-1203 · 47 pages, 13 figures

arxiv created 2013/08/29 · arxiv updated 2015/06/03

Abstract

A smooth curve γ: [0,1] → \Ss² is locally convex if its geodesic curvature is positive at every point. J. A. Little showed that the space of all locally convex curves γ with γ(0) = γ(1) = e₁ and γ'(0) = γ'(1) = e₂ has three connected components L_-1,c, L₊₁, L_-1,n. The space \cL_-1,c is known to be contractible. We prove that \cL₊₁ and \cL_-1,n are homotopy equivalent to (Ω\Ss³) \vee \Ss² \vee \Ss⁶ \vee \Ss¹⁰ \vee ⋯ and (Ω\Ss³) \vee \Ss⁴ \vee \Ss⁸ \vee \Ss¹² \vee ⋯, respectively. As a corollary, we deduce the homotopy type of the components of the space \Free(\Ss¹,\Ss²) of free curves γ: \Ss¹ → \Ss² (i.e., curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces \Free([0,1], \Ss²) with fixed initial and final frames.

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