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Making spaces wild (simply-connected case). (English) Zbl 1506.55012

This paper is a continuation of a series of papers by the author of the present paper and others [K. Eda, Topology Appl. 123, No. 3, 479–505 (2002; Zbl 1032.55013); Proc. Am. Math. Soc. 130, No. 5, 1515–1522 (2002; Zbl 0991.57003); G. Conner and K. Eda, Topology Appl. 146–147, 317–328 (2005; Zbl 1063.55011); K. Eda, Comment. Math. Univ. St. Pauli 32, 131–135 (1983; Zbl 0518.20054); Fundam. Math. 209, No. 1, 27–42 (2010; Zbl 1201.55002); J. Math. Soc. Japan 63, No. 3, 769–787 (2011; Zbl 1229.55008)], in which they research the fundamental group of some wild spaces (not semi-locally simply connected) and how it can recover the space under certain conditions.
In the present paper, the author considers a metric separable space \(X\) and a dense subset \(D\in X\). Then, an earring space \(E(X,D)\) is formed by attaching infinitely many copies of a circle to \(D\), making it wild. Then, the following results are obtained:
1. If \(X\) is simply-connected and locally simply-connected, then the fundamental group of \(E(X,D)\) is isomorphic to a subgroup of the Hawaiian earring group.
2. If \(X\) is also locally path-connected, \(X\) can be (topologically) recovered from the fundamental group of \(E(X,D)\).
The proofs rely heavily on the investigation of the structure of the fundamental group of \(E(X,D)\), using a notion of words of infinite lenght (infinitary words).
The non-simply-connected case is postponed for a different paper [K. Eda, “Making spaces wild (non-simply-connected case)”, Preprint, arXiv:2001.01874], since the structure of the fundamental group of \(E(X,D)\) becomes more complicated.

MSC:

55Q20 Homotopy groups of wedges, joins, and simple spaces
55Q70 Homotopy groups of special types
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
54F50 Topological spaces of dimension \(\leq 1\); curves, dendrites
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References:

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