Cotorsion-free groups from a topological viewpoint

Eda, Katsuya; Fischer, Hanspeter

doi:10.1016/j.topol.2016.09.010

Cited by 12 publications

(9 citation statements)

References 29 publications

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“…Recall that if the fundamental group of a Peano continuum does not (canonically) inject into the firstČech homotopy group, then it is not residually n-slender [16]. However, since P is not a Peano continuum, we verify this separately: Definition 3.7 (Noncommutatively slender [13]).…”

Section: Non-injectivity Into the Firstčech Homotopy Groupmentioning

confidence: 99%

“…Recall that if the fundamental group of a Peano continuum does not (canonically) inject into the first Čech homotopy group, then it is not residually n‐slender [16]. However, since

double-struckP

is not a Peano continuum, we verify this separately: Definition A group

G

is called noncommutatively slender ( n‐slender for short) if for every homomorphism

h : π_{1} false(double-struckH, b_{0} false) \to G

, there is a

k \in N

such that

h ([α]) = 1

for all loops

α : false([0, 1], {0, 1} false) \to false(H_{k}, b_{0} false)

.…”

Section: Non‐injectivity Into the First čEch Homotopy Groupmentioning

confidence: 99%

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

Brazas

Fischer

2020

Bull. London Math. Soc.

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We construct a space P for which the canonical homomorphism π1(P, p) →π1(P, p) from the fundamental group to the firstČech homotopy group is not injective, although it has all of the following properties: (1) P \ {p} is a 2-manifold with connected non-compact boundary; (2) P is connected and locally path connected; (3) P is strongly homotopically Hausdorff; (4) P is homotopically path Hausdorff; (5) P is 1-UV0; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π1(P, p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring; (8) π1(P, p) is locally free.

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Section: Non-injectivity Into the Firstčech Homotopy Groupmentioning

confidence: 99%

“…Recall that if the fundamental group of a Peano continuum does not (canonically) inject into the first Čech homotopy group, then it is not residually n‐slender [16]. However, since

double-struckP

is not a Peano continuum, we verify this separately: Definition A group

G

is called noncommutatively slender ( n‐slender for short) if for every homomorphism

h : π_{1} false(double-struckH, b_{0} false) \to G

, there is a

k \in N

such that

h ([α]) = 1

for all loops

α : false([0, 1], {0, 1} false) \to false(H_{k}, b_{0} false)

.…”

Section: Non‐injectivity Into the First čEch Homotopy Groupmentioning

confidence: 99%

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

Brazas

Fischer

2020

Bull. London Math. Soc.

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“…The fundamental group is freely indecomposable and includes a copy of the additive group of the rationals and of the fundamental group of the Hawaiian earring. This group has found use in defining cotorsion-free groups in the non-abelian setting [10] and continues to serve as a counterexample [16] and as a test model for notions of infinitary abelianization [3].…”

Section: Introductionmentioning

confidence: 99%

“…fundamental group, and when κ > 2 ℵ0 one has |π 1 (GS κ )| > 2 ℵ0 = |π 1 (GS 2 )| (Theorem 2.13). Using techniques of [10] or [12] one can compute the abelianizations of π 1 (GS 2 ) and π 1 (GS 3 ) and see that these abelianizations are isomorphic.…”

Section: Introductionmentioning

confidence: 99%

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The Griffiths double cone group is isomorphic to the triple

Corson

2020

Preprint

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It is shown that the fundamental group of the Griffiths double cone space is isomorphic to that of the triple cone. More generally if κ is a cardinal such that 2 ≤ κ ≤ 2 ℵ 0 then the κ-fold cone has the same fundamental group as the double cone. The isomorphisms produced are non-constructive, and no isomorphism between the fundamental group of the 2-and of the κ-fold cones, with 2 < κ, can be realized via continuous mappings. We also prove a conjecture of James W. Cannon and Gregory R. Conner which states that the fundamental group of the Griffiths double cone space is isomorphic to that of the harmonic archipelago.The bounds on κ in the statement of Theorem A are the best possible. The spaces GS 0 and GS 1 both strongly deformation retract to a point and therefore have trivial

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Automatic continuity of ℵ1-free groups

Corson

2020

Isr. J. Math.

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Cotorsion-free groups from a topological viewpoint

Cited by 12 publications

References 29 publications

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

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

The Griffiths double cone group is isomorphic to the triple

Automatic continuity of ℵ1-free groups

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