Results on the homotopy type of the spaces of locally convex curves on S³

2017/03/07 by Alves, Emília, Saldanha, Nicolau C.
#57N12 #57N35 #57N65 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1703.02581

Abstract

A curve γ: [0,1] → Sⁿ of class C^k (k \geqslant n) is locally convex if the vectors γ(t), γ'(t), γ"(t), ⋯, γ⁽ⁿ⁾(t) are a positive orthonormal basis to Rⁿ⁺¹ for all t ∈ [0,1]. Given an integer n ≥ 2 and Q ∈ SO_n+1, let LSⁿ(Q) be the set of all locally convex curves γ: [0,1] → Sⁿ 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 LSⁿ(Q) when Q varies in SO_n+1 (in particular, at most 3 for n=3). For any n \geqslant 2, the homotopy type of one of these spaces is well-known, but not of the others. For n=2, Saldanha determined the homotopy type of the spaces LS²(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 2 other spaces, allowing us to conclude that its connected components are not homeomorphic to the connected components of the known space.

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