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Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

100%

Zastrow A.

Open Mathematics

2003

tom 1

nr 1

1-35

The space which is composed by embedding countably many circles in such a way into the plane that their radii are given by a null-sequence and that they all have a common tangent point is called “The Hawaiian Earrings”. The fundamental group of this space is known to be a subgroup of the inverse limit of the finitely generated free groups, and it is known to be not free. Within the recent move of trying to get hands on the algebraic invariants of non-tame (e.g. non-triangulable) spaces this space usually serves as the simplest example in this context. This paper contributes to understanding this group and corresponding phenomena by pointing out that several subgroups that are constructed according to similar schemes partially turn out to be free and not to be free. Amongst them is a countable non-free subgroup, and an uncountable free subgroup that is not contained in two other free subgroups that have recently been found. This group, although free, contains infinitely huge “virtual powers”, i.e. elements of the fundamental group of that kind that are usually used in proofs that this fundamental group is not free, and, although this group contains all homotopy classes of paths that are associated with a single loop of the Hawaiian Earrings, this system of ‘natural generators’ can be proven to be not contained in any free basis of this free group.

Generalized universal covering spaces and the shape group

63%

Fischer H., Zastrow A.

Fundamenta Mathematicae

2007

tom 197

nr 1

167-196

If a paracompact Hausdorff space X admits a (classical) universal covering space, then the natural homomorphism φ: π₁(X) → π̌₁(X) from the fundamental group to its first shape homotopy group is an isomorphism. We present a partial converse to this result: a path-connected topological space X admits a generalized universal covering space if φ: π₁(X) → π̌₁(X) is injective. This generalized notion of universal covering p: X̃ → X enjoys most of the usual properties, with the possible exception of evenly covered neighborhoods: the space X̃ is path-connected, locally path-connected and simply-connected and the continuous surjection p: X̃ → X is universally characterized by the usual general lifting properties. (If X is first countable, then p: X̃ → X is already characterized by the unique lifting of paths and their homotopies.) In particular, the group of covering transformations $G = Aut(X̃ \stackrel{p}{→} X)$ is isomorphic to π₁(X) and it acts freely and transitively on every fiber. If X is locally path-connected, then the quotient X̃/G is homeomorphic to X. If X is Hausdorff or metrizable, then so is X̃, and in the latter case G can be made to act by isometry. If X is path-connected, locally path-connected and semilocally simply-connected, then p: X̃ → X agrees with the classical universal covering. A necessary condition for the standard construction to yield a generalized universal covering is that X be homotopically Hausdorff, which is also sufficient if π₁(X) is countable. Spaces X for which φ: π₁(X) → π̌₁(X) is known to be injective include all subsets of closed surfaces, all 1-dimensional separable metric spaces (which we prove to be covered by topological ℝ-trees), as well as so-called trees of manifolds which arise, for example, as boundaries of certain Coxeter groups. We also obtain generalized regular coverings, relative to some special normal subgroups of π₁(X), and provide the appropriate relative version of being homotopically Hausdorff, along with its corresponding properties.

Ograniczanie wyników

1 Fundamenta Mathematicae

1 Open Mathematics

2 Zastrow A.

1 Fischer H.

1 2007

1 2003

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Free and non-free subgroups of the fundamental group of the Hawaiian Earrings

Generalized universal covering spaces and the shape group