Fundamental groups having the whole information of spaces
The authors propose a new construction of a space from a group, and show that applying this construction to the fundamental group of certain spaces produces the original spaces, which implies that the fundamental group of such spaces has the same information as the topological space itself. Their construction uses homomorphic images of the fundamental group of the Hawaiian earring, and is based on a previous result: ``One-dimensional, locally path-connected, path connected, metric spaces which are not semi-locally simply connected at any point, and have isomorphic fundamental groups, are homeomorphic'' [see \textit{K. Eda}, ibid. 123, No. 3, 479--505 (2002; Zbl 1032.55013)]. According to this, the topologies of one-dimensional fractals which are Peano continua and have nontrivial fundamental groups, i.e., the Sierpinski gasket, the Sierpinski carpet and the Menger sponge, are recovered from their fundamental groups. Finite or countable products of such spaces are also recovered from their fundamental groups.
scientific article; zbMATH DE number 2136473