The homotopy and cohomology of spaces of locally convex curves in the sphere -- I

2009/05/13 by Nicolau C. Saldanha, Saldanha, Nicolau C.
Mathematics · #34B05 #53C42 #57N65 #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0905.2111

openalex publication_date 2009/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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 positive 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 L_-1,c is known to be contractible but the topology of the other two connected components is not well understood. We study the homotopy and cohomology of these spaces. In particular, for L_-1 = L_-1,c \sqcup L_-1,n, we show that dim H^2k(L_(-1)^k, \RR) ≥ 1, that dim H^2k(L_(-1)^(k+1), \RR) ≥ 2, that π₂(L₊₁) contains a copy of Z² and that π_2k(L_(-1)^(k+1)) contains a copy of Z.

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