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The classical archipelago is a non-contractible subset of R 3 \mathbb {R}^3 which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, A \mathscr {A} , is the quotient of the topologist’s product of Z \mathbb Z , the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show A \mathscr {A} is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing Z \mathbb Z with arbitrary groups yields the notion of archipelago groups. Surprisingly, every archipelago of countable groups is isomorphic to either A ( Z ) \mathscr {A}(\mathbb Z) or A ( Z 2 ) \mathscr {A}(\mathbb Z_2) , the cases where the archipelago is built from circles or projective planes respectively. We conjecture that these two groups are isomorphic and prove that for large enough cardinalities of G i G_i , A ( G i ) \mathscr {A}(G_i) is not isomorphic to either. |
