Archipelago groups

@inproceedings{Conner2014ArchipelagoG,
  title={Archipelago groups},
  author={Gregory R. Conner and Wolfram Hojka and Mark H. Meilstrup},
  year={2014}
}
The classical archipelago is a non-contractible subset of R3 which is homeomorphic to a disk except at one non-manifold point. Its fundamental group, A , is the quotient of the topologist’s product of 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 is locally free, not indicable, and has the rationals both as a subgroup and a quotient group. Replacing Z with arbitrary groups yields… 

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References

SHOWING 1-10 OF 23 REFERENCES

The harmonic archipelago as a universal locally free group

SINGULAR HOMOLOGY OF ONE-DIMENSIONAL SPACES

If X is a one-dimensional separable metric space, then wk(X) = 0 for all k > 1 (see [2]). Hence, such a space X is a K(r, 1), and w1(X) determines the singular homology of X. The principal result of

Fundamental groups of one-dimensional spaces

Let X be a metrizable one-dimensional continuum. In the present paper we describe the fundamental group of X as a subgroup of its Cech homotopy group. In particular, the elements of the Cech homotopy

Weighted Combinatorial Group Theory and Wild Metric Complexes

In this paper, we develop the low dimensional homotopy theory required for weighted combinatorial group theory. In [S97], the usual concepts of generators and relators of group presentations are

On the homology of the Harmonic Archipelago

We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is

HOMOTOPY (LIMITS AND) COLIMITS

A laser includes a laser medium, a light source for emitting light which pumps the laser medium, a pair of mirrors which are disposed on opposite sides of the laser medium and form a resonator, an

Free σ-products and noncommutatively slender groups