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The geometry and fundamental groups of solenoid complements. (English) Zbl 1334.57002

A solenoid is an inverse limit of circles \(S^1\). It may be embedded into \(S^3\) as an intersection of a sequence of nested solid tori. Such an embedding is called unknotted if the solid tori used in the construction are unknotted in \(S^3\), and is called knotted otherwise. In this paper the authors study the fundamental group \(G=\pi_1(S^3 \setminus S)\) of the complement of a solenoid \(S\) embedded into \(S^3\) using the techniques of knot theory. Note that the complement \(S^3 \setminus S\) is an open manifold.
The main results of the paper are the following:
\(\bullet\)
For an unknotted embedding, \(G\) is an Abelian group (in fact it is a subgroup of \(\mathbb Q\)).
\(\bullet\)
For a knotted embedding, \(G\) is a non-Abelian group.
\(\bullet\)
Each subgroup of \(\mathbb Q\) may be realized as such a \(G\), i.e., as the fundamental group of the complement of an embedded solenoid in \(S^3\).
\(\bullet\)
There exist uncountably many inequivalent embeddings (i.e., embeddings with non-homeomorphic complement) of any given solenoid into \(S^3\).
Related work may be found in [B. Jiang et al., Fundam. Math. 214, No. 1, 57–75 (2011; Zbl 1260.57037)].
Reviewer: Ziga Virk (Litija)

MSC:

57M05 Fundamental group, presentations, free differential calculus
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57N12 Topology of the Euclidean \(3\)-space and the \(3\)-sphere (MSC2010)
57N35 Embeddings and immersions in topological manifolds
57N65 Algebraic topology of manifolds

Citations:

Zbl 1260.57037

Software:

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References:

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