Archipelago groups

Abstract

The classical archipelago is a non-contractible subset of 3 which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, , is the quotient of the topologist's product of , the fundamental group of the shrinking wedge of countably many copies of the circle (the Hawaiian earring), modulo the corresponding free product. We show is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing with arbitrary groups yields the notion of archipelago groups.Surprisingly, every archipelago of countable groups is isomorphic to either ( ) or ( 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 ) is not isomorphic to either.

The harmonic archipelago as a universal locally free group

“…A proof of the proposition is obtained by finding a homotopy equivalence between the archipelago space and a certain mapping cone, and by then applying van Kampen's theorem; see Propositions 13 and 14 in [8], where the construction is also generalized to arbitrary groups serving as factors.…”

“…This space is homeomorphic to a disc but for a single point. Viewed in another way, it is also homotopy equivalent to the reduced suspension of the graph of the topologist's sine curve y = sin( 1 /x) together with the origin (0, 0) as the basepoint, see [8,Proposition 12].…”

“…In a remarkable twist, the group Z could be replaced in this construction by any other countable group with no elements of order 2, without this changing the induced quotient (see [8,Theorem A] for this result, where the topological background is also presented in greater detail).…”

On topological homotopy groups and relation to Hawaiian groups

“…The harmonic archipelago, HA, is a non-simply connected space with small loops. The fundamental group and homology groups of the harmonic archipelago were studied in [8] and [18], respectively. Here, we recall some of their results to use in Example 4.10.…”

“…[8,18]). Let × σ denote the free σ-product of a family of groups, and H N denote the normal closure of the subgroup H in a given group.…”