Homotopy and cohomology of spaces of locally convex curves in the sphere

2004/07/23 by Nicolau C. Saldanha, Saldanha, Nicolau C.
Mathematics · #58B05 #58D10 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT #msc:58B05 #msc:58D10

paper · pdf · doi:10.48550/arxiv.math/0407410

25 pages, 19 figures

arxiv created 2004/07/23 · arxiv updated 2009/12/01

Abstract

We discuss the homotopy type and the cohomology of spaces of locally convex parametrized curves gamma: [0,1] -> S², i.e., curves with positive geodesic curvature. The space of all such curves with gamma(0) = gamma(1) = e₁ and gamma'(0) = gamma'(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, pi₂(X_-1) contains a copy of Z and pi₂(X₁) contains a copy of Z². Also, Hⁿ(X₁, R) and Hⁿ(X_-1, R) are nontrivial for all even n. More, dim H^4n-2(X₁, R) >= 2 and dim H⁴ⁿ(X_-1, R) >= 2 for all positive n.

Related

• The homotopy and cohomology of spaces of locally convex curves in the sphere -- I
• The homotopy and cohomology of spaces of locally convex curves in the\n sphere -- II
• Topology of the space of locally convex curves on the 3-sphere
• Results on the homotopy type of the spaces of locally convex curves on S³
• Homotopy type of spaces of locally convex curves in the sphere S³