The fundamental groups of one-dimensional spaces and spatial homomorphisms

Fundamental groups of one-dimensional spaces

“…Next we mention the following result of Eda [11,Lemma 3.15]. It is an immediate consequence of Lemma 5.1 and the fact that π(X, x 0 ) ֒→π(X, x 0 ) (see [12]).…”

“…In this section we provide a new proof of a result of Eda [11,Theorem 1.1] which states that each homomorphism h from the fundamental group of the Hawaiian Earring E to π(X, x 0 ) is induced by a continuous map from E to X. The methods we have developed in the previous sections enable us to give an almost purely algebro-combinatorial proof of this result (though topological intuitions are helpful to understand the idea).…”

“…In the second part of the paper our description of the fundamental group is applied in order to give a quite elementary algebro-combinatorial proof of a result due to Eda [11]. We show that each homomorphism from the fundamental group of the Hawaiian earring E to the fundamental group of a metrizable one-dimensional continuum X is induced by a continuous mapping ψ : E → X (Theorem 5.10).…”

“…Meanwhile, properties of fundamental groups of one-dimensional (cf. for instance [1,3,10,11]) and planar (see [5,13]) spaces were derived. Especially the description of such fundamental groups in terms of words turned out to be useful.…”

Uniform universal covers of uniform spaces

“…Since Δ S and C S are non-homeomorphic one-dimensional Peano continua that are not semi-locally simply connected at any point, then by a result of [9], π 1 (Δ S ) and π 1 (C S ) are not isomorphic as abstract groups. Finally, note that the conditions of Proposition 82 are satisfied and hence the image of the fundamental group with respect to λ is dense in δ 1 (X) in each case.…”