The homotopy type of spaces of locally convex curves in the sphere

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...

On the dual of certain locally convex function spaces

In this paper, we first introduce some function spaces, with certain locally convex topologies, closely  related to the space of real-valued continuous functions on $X$,  where $X$ is a $C$-distinguished topological space. Then, we show that their dual spaces can be identified in a natural way with certain spaces of Radon measures.

Topological number for locally convex topological spaces with continuous semi-norms

In this paper we introduce the concept of topological number for locally convex topological spaces and prove some of its properties. It gives some criterions to study locally convex topological spaces in a discrete approach.

Approximate Convexity and Submonotonicity in Locally Convex Spaces

The homotopy and cohomology of spaces of locally convex curves in the sphere — II

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...