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Generalized Hawaiian Earrings, The Big Free Groups, and The Big Fundamental Group


Jim Cannon,
Brigham Young University

Generalized Hawaiian Earrings, The Big Free Groups, and The Big Fundamental Group

Abstract: The Hawaiian earring H is the union of the countably many planar circles Cn, with n = 1, 2, ..., where Cn has radius 1/n and is tangent to the x-axis at the origin. The earring H is famous since it has no simply connected covering space and since its fundamental group BF is neither countable nor free. We shall give natural descriptions of H and BF that reveal them as the countable cases of constructions H(C) and BF(C) valid for every cardinal C, H(C) being the generalized Hawaiian earring and BF(C) the big free group. These generalized objects are connected by a new topological invariant $\Pi_1$ called the big fundamental group, which satisfies $BF(C) = \Pi_1(H(C))$.