# 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…

## 18 Citations

### Cotorsion and wild homology

- Mathematics
- 2017

The classical concept of cotorsion of an abelian group is here characterized in the style of algebraic compactness, namely by the existence of solutions of certain systems of equations. This approach…

### Archipelago groups are locally free, Corrigendum to: “Cotorsion and wild homology”

- MathematicsIsrael Journal of Mathematics
- 2022

An Archipelago group is the quotient of the topologist’s product $$G = {\circledast_{i \ge 1}}{G_i}$$ G = ⊛ i ≥ 1 G i of a sequence $${({G_i})_{i \ge 1}}$$ ( G i ) i ≥ 1 of groups modulo the normal…

### Adding Limit Points to Bass-Serre Graphs of Groups

- Mathematics
- 2018

Adding Limit Points to Bass-Serre Graphs of Groups Alexander Jin Shumway Department of Mathematics, BYU Master of Science We give a brief overview of Bass-Serre theory and introduce a method of…

### On the failure of the first Čech homotopy group to register geometrically relevant fundamental group elements

- MathematicsBulletin of the London Mathematical Society
- 2020

We construct a space P for which the canonical homomorphism π1(P,p)→π̌1(P,p) from the fundamental group to the first Čech homotopy group is not injective, although it has all of the following…

### Scattered products in fundamental groupoids

- MathematicsProceedings of the American Mathematical Society
- 2019

Infinitary operations, such as products indexed by countably infinite linear orders, arise naturally in the context of fundamental groups and groupoids. We prove that the well-definedness of products…

### The Griffiths double cone group is isomorphic to the triple.

- Mathematics
- 2020

It is shown that the fundamental group of the Griffiths double cone space is isomorphic to that of the triple cone. More generally if $\kappa$ is a cardinal such that $2 \leq \kappa \leq…

### Transfinite product reduction in fundamental groupoids

- Mathematics
- 2020

Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called “transfinite” if the index orders are permitted to…

### Elements of higher homotopy groups undetectable by polyhedral approximation

- Mathematics
- 2022

When non-trivial local structures are present in a topological space X , a common approach to characterizing the isomorphism type of the n -th homotopy group π n p X, x 0 q is to consider the image…

## References

SHOWING 1-10 OF 23 REFERENCES

### SINGULAR HOMOLOGY OF ONE-DIMENSIONAL SPACES

- Mathematics
- 1959

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

- Mathematics
- 2013

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

- Mathematics
- 1999

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

- Mathematics
- 2012

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

- Physics
- 2009

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…