The fundamental groups of one-dimensional spaces

Cotorsion-free groups from a topological viewpoint

“…This homomorphism is often injective. For example, it is injective for one-dimensional spaces [15], for subsets of surfaces [19] and for certain trees of manifolds [18]. Hence, for such X, we have π s (X, x) = 1.…”

On the fundamental groups of trees of manifolds

“…Other classes of spaces Z for which ϕ : π 1 (Z , z 0 ) →π 1 (Z , z 0 ) has been shown to be injective, include (i) one-dimensional compacta [Curtis and Fort 1959;Eda and Kawamura 1998;Cannon and Conner 1998] and (ii) subsets of closed surfaces [Fischer and Zastrow 2005].…”

The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

“…Here we show that these continuous surjections induce injective homomorphisms between fundamental groups. For a one-dimensional space X, a canonical homomorphism from the fundamental group to the firstČech homotopy group is injective [6]. On the other hand there are natural isomorphisms between the firstČech homotopy groups of (A1) and (B1), and (A2) and (B2), respectively, which are induced from f 1 and f 2 and hence commute with the induced homomorphisms f 1 * and f 2 * .…”