Let X be a one-dimensional metric space and ℍ be the Hawaiian earring.(1) Each homomorphism from π1(ℍ) to π1(X) is induced from a continuous map up to the base-point-change isomorphism on π1(X).(2) Let X be a one-dimensional Peano continuum. Then X has the same homotopy type as that of ℍ if and only if π1(X) is isomorphic to π1(ℍ), if and only if X has a unique point at which X is not semi-locally simply connected. (3) Let X and Y be one-dimensional Peano continua which are not semi-locally simply connected at any point. Then, X and Y are homeomorphic if and only if π1(X) and π1(Y) are isomorphic. Moreover, each isomorphism from π1(X) to π1(Y) is induced by a homeomorphism from X to Y up to the base-point-change-isomorphism.
ASJC Scopus subject areas
- Geometry and Topology