Archipelago groups
Gregory R. Conner, Wolfram Hojka, Mark Meilstrup

TL;DR
This paper investigates the fundamental groups of classical archipelago spaces, revealing their properties and classifying archipelago groups built from countable groups, with conjectures on their isomorphisms.
Contribution
It characterizes the fundamental group of archipelago spaces and classifies archipelago groups formed from countable groups, establishing isomorphism results and conjectures.
Findings
The fundamental group is locally free and not indicable.
Archipelago groups from countable groups are isomorphic to either () or (_2).
For large cardinalities, (G_i) is not isomorphic to these cases.
Abstract
The classical archipelago is a non-contractible subset of 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 , 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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Advanced Topology and Set Theory
