The homotopy and cohomology of spaces of locally convex curves in the\n sphere -- II

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) #Geometric and Algebraic Topology #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0905.2116

openalex publication_date 2009/05/13 · openalex created_date 2022/09/15 · openalex updated_date 2026/07/28

Abstract

A smooth curve \γ: [0,1] \→ S² is locally convex if its geodesic\ncurvature is positive at every point. J. A. Little showed that the space of all\nlocally positive curves \γ with \γ(0) = \γ(1) = e₁ and\n\γ'(0) = \γ'(1) = e₂ has three connected components L_-1,c,\nL₊₁, L_-1,n. The space L_-1,c is known to be contractible but the\ntopology of the other two connected components is not well understood. We prove\nthat all connected components of L_I are simply connected, that\nH²(L₊₁;Z) = Z² and H²(L_-1,n;Z) = Z.\n

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