The homotopy and cohomology of spaces of locally convex curves in the sphere — I Nicolau C . Saldanha
نویسنده
چکیده
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⊔L−1,n, we show that dimH(L(−1)k ,R) ≥ 1, that dimH(L(−1)(k+1) ,R) ≥ 2, that π2(L+1) contains a copy of Z2 and that π2k(L(−1)(k+1)) contains a copy of Z.
منابع مشابه
Homotopy and cohomology of spaces of locally convex curves in the sphere
We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves γ : [0, 1] → S2, i.e., curves with positive geodesic curvature. The space of all such curves with γ(0) = γ(1) = e1 and γ′(0) = γ′(1) = e2 is known to have three connected components X−1,c, X1, X−1. We show several results concerning the homotopy type and cohomology of these spaces. In particular, X−1...
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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 prov...
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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 L−1,c, L+1, L−1,n. The space L−1,c is known to be contractible. In this paper we prove that L+1 and L−1,n are homotopy equivalent to ΩS3 ∨ S2 ∨ S6 ∨ S10 ...
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