Homotopy and cohomology of spaces of locally convex curves in the sphere
نویسنده
چکیده
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,c is contractible, X1 and X−1 are simply connected, π2(X−1) contains a copy of Z and π2(X1) contains a copy of Z2. Also, H(X1,R) and H (X−1,R) are nontrivial for all even n. More, dimH(X1,R) ≥ 2 and dimH(X−1,R) ≥ 2 for all positive n.
منابع مشابه
The homotopy and cohomology of spaces of locally convex curves in the sphere — I Nicolau C . Saldanha
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 stud...
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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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تاریخ انتشار 2008