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The harmonic archipelago is a standard example of a two-dimensional
space with unusual properties, regarding its algebraic topology. The space
is homeomorphic to a disc but for a single point and can be described as the
reduced suspension of the graph of the topologist's sine curve \(y = \sin(1/x)\).
On the other hand it also has a natural interpretation as a mapping cone
over a wedge of circles.
We will see how these equivalences come about topologically, then turn to
the similarly curious algebraic mapping properties of its fundamental group
\(G\). For example, every countable locally free group embeds in \(G\) as a
subgroup, and in turn, every separable profinite group is an epimorphic
image.
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