The harmonic archipelago as a universal locally free group

Hojka, Wolfram

doi:10.1016/j.jalgebra.2015.04.014

Cited by 8 publications

(4 citation statements)

References 14 publications

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“…Finding a subgroup isomorphic to is similar in spirit to [BZ12] where the same is shown for Griffiths' double cone space. It is also a corollary to the theorem in [Hoj13] that every countable locally free group embeds in the archipelago.…”

Section: Further Resultsmentioning

confidence: 94%

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Archipelago groups

Conner

Hojka²,

Meilstrup

2015

Proc. Amer. Math. Soc.

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The classical archipelago is a non-contractible subset of 3 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 ( 2 ), 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 cardinalities of G i , (G i ) is not isomorphic to either.

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Section: Further Resultsmentioning

confidence: 94%

“…Proof. We will content ourselves with the basic idea of finding a subgroup isomorphic to , a more general construction can be found in [Hoj13]. Consider as an element in n the infinite word w := a 1 (a 2 (a 3 (a 4 (.…”

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confidence: 99%

Archipelago groups

Conner

Hojka²,

Meilstrup

2015

Proc. Amer. Math. Soc.

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“…Example 5.40 (Harmonic Archipelago). The harmonic archipelago HA is the space obtained by attaching a 2-cell e n to H along the loop n ¨ ń`1 for all n P N. Although π 1 pHA, b 0 q is uncountable and locally free [26], every loop α P ΩpHA, b 0 q is homotopic to a loop in every neighborhood of b 0 . Let A 1 " H and A m " H Y e 1 Y e 2 Y ¨¨¨Y e m´1 so that tA m u is a k ω -decomposition for HA.…”

Section: Proof the Function Hmentioning

confidence: 99%

Infinitary commutativity and fundamental groups of topological monoids

Brazas¹,

Gillespie²

2020

Preprint

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The well-known Eckmann-Hilton Principle may be applied to prove that fundamental groups of H-spaces are commutative. In this paper, we identify an infinitary analogue of the Eckmann-Hilton Principle that applies to fundamental groups of all topological monoids and slightly more general objects called pre-∆-monoids. In particular, we show that every pre-∆-monoid M is "transfinitely π 1 -commutative" in the sense that permutation of the factors of any infinite loop-concatenation indexed by a countably infinite order and based at the identity e P M is a homotopy invariant action. We also give a detailed account of fundamental groups of James reduced products and apply transfinite π 1 -commutativity to make several computations.

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“…The group A is isomorphic to the fundamental group of the harmonic archipelago mentioned in [1], which is now known to be isomorphic to the fundamental group of the Griffiths space constructed in [7] (see [3]). Another consequence is that, for {H n } n∈ω as in the theorem, A({H n } n∈ω ) is locally free and every countable locally free group embeds into it (see [8,Theorem 2]).…”

Section: Introductionmentioning

confidence: 99%

The nonabelian product modulo sum

Corson¹

2022

Preprint

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It is shown that if {Hn}n∈ω is a sequence of groups without involutions, with 1 < |Hn| ≤ 2 ℵ 0 , then the topologist's product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if {An}n∈ω is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum n∈ω An/ n∈ω An is dependent on the sequence.

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The harmonic archipelago as a universal locally free group

Cited by 8 publications

References 14 publications

Archipelago groups

Archipelago groups

Infinitary commutativity and fundamental groups of topological monoids

The nonabelian product modulo sum

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