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On the connectivity of finite subset spaces

Jacob Mostovoy, Rustam Sadykov (2012)

Fundamenta Mathematicae

We prove that the space e x p k S m + 1 of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space e x p k X is (m+k-2)-connected for any m-connected cell complex X.

On the finiteness of the fundamental group of a compact shrinking Ricci soliton

Zhenlei Zhang (2007)

Colloquium Mathematicae

Myers's classical theorem says that a compact Riemannian manifold with positive Ricci curvature has finite fundamental group. Using Ambrose's compactness criterion or J. Lott's results, M. Fernández-López and E. García-Río showed that the finiteness of the fundamental group remains valid for a compact shrinking Ricci soliton. We give a self-contained proof of this fact by estimating the lengths of shortest geodesic loops in each homotopy class.

Open subgroups of free topological groups

Jeremy Brazas (2014)

Fundamenta Mathematicae

The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for topological groups: Every open subgroup of a free Graev topological group is a free Graev topological group. An open subgroup of a free Markov topological group is a free Markov topological group if and only if it is disconnected.

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