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2015 The homotopy type of spaces of locally convex curves in the sphere
Nicolau C Saldanha
Geom. Topol. 19(3): 1155-1203 (2015). DOI: 10.2140/gt.2015.19.1155


A smooth curve γ: [0,1] S2 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 1,c, +1, 1,n. The space 1,c is known to be contractible. We prove that +1 and 1,n are homotopy equivalent to (ΩS3) S2 S6 S10 and (ΩS3) S4 S8 S12 , respectively. As a corollary, we deduce the homotopy type of the components of the space Free(S1, S2) of free curves γ: S1 S2 (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces Free([0,1], S2) with fixed initial and final frames.


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Nicolau C Saldanha. "The homotopy type of spaces of locally convex curves in the sphere." Geom. Topol. 19 (3) 1155 - 1203, 2015.


Received: 8 August 2012; Revised: 29 August 2013; Accepted: 9 April 2014; Published: 2015
First available in Project Euclid: 16 November 2017

zbMATH: 1318.53066
MathSciNet: MR3352233
Digital Object Identifier: 10.2140/gt.2015.19.1155

Primary: 53C42, 57N65
Secondary: 34B05

Rights: Copyright © 2015 Mathematical Sciences Publishers


Vol.19 • No. 3 • 2015
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