#### Volume 19, issue 3 (2015)

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

### Nicolau C Saldanha

Geometry & Topology 19 (2015) 1155–1203
##### Abstract

A smooth curve $\gamma :\left[0,1\right]\to {\mathbb{S}}^{2}$ is locally convex if its geodesic curvature is positive at every point. J A Little showed that the space of all locally convex curves $\gamma$ with $\gamma \left(0\right)=\gamma \left(1\right)={e}_{1}$ and ${\gamma }^{\prime }\left(0\right)={\gamma }^{\prime }\left(1\right)={e}_{2}$ has three connected components ${\mathsc{ℒ}}_{-1,c}$, ${\mathsc{ℒ}}_{+1}$, ${\mathsc{ℒ}}_{-1,n}$. The space ${\mathsc{ℒ}}_{-1,c}$ is known to be contractible. We prove that ${\mathsc{ℒ}}_{+1}$ and ${\mathsc{ℒ}}_{-1,n}$ are homotopy equivalent to $\left(\Omega {\mathbb{S}}^{3}\right)\vee {\mathbb{S}}^{2}\vee {\mathbb{S}}^{6}\vee {\mathbb{S}}^{10}\vee \cdots$ and $\left(\Omega {\mathbb{S}}^{3}\right)\vee {\mathbb{S}}^{4}\vee {\mathbb{S}}^{8}\vee {\mathbb{S}}^{12}\vee \cdots$, respectively. As a corollary, we deduce the homotopy type of the components of the space $Free\left({\mathbb{S}}^{1},{\mathbb{S}}^{2}\right)$ of free curves $\gamma :{\mathbb{S}}^{1}\to {\mathbb{S}}^{2}$ (ie curves with nonzero geodesic curvature). We also determine the homotopy type of the spaces $Free\left(\left[0,1\right],{\mathbb{S}}^{2}\right)$ with fixed initial and final frames.

##### Keywords
convex curves, topology in infinite dimension, periodic solutions of linear ODEs
##### Mathematical Subject Classification 2010
Primary: 53C42, 57N65
Secondary: 34B05