Fundamental groups of one-dimensional spaces

Dorfer, Gerhard; Thuswaldner, Jörg Μ.; Winkler, R.

doi:10.4064/fm223-2-2

Cited by 9 publications

(9 citation statements)

References 15 publications

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“…for sufficiently large k, and obtain commutative diagrams with injective functions ω n = (ω n,k ) k n−1 that output stabilized word sequences. (See, for example, [12] or [24].) For k 2n + 1, let D n : W + n,k → W + n+1,k be the monomorphism that replaces every occurrence of the letter n by ρ n,1 2n ρ −1 n,1 ρ n,2 2n+1 ρ −1 n,2 and every occurrence of the letter…”

Section: An Inverse Limit Of Finitely Generated Free Monoidsmentioning

confidence: 99%

“…For every

n \in N

x \in π_{1} false(H_{n}^{+}, b_{0} false)

, and

m ⩾ n - 1

, the sequence

{false(R_{n, m} \circ R_{n, m + 1} \circ \dots \circ R_{n, k - 1} (g_{n, k} false(x false)) false)}_{k ⩾ m}

of (unreduced) words is eventually constant, so that we may define functions

ω_{n, m} : π_{1} false(H_{n}^{+}, b_{0} false) \to {scriptW}_{n, m}^{+}

ω_{n, m} false(x false) = R_{n, m} \circ R_{n, m + 1} \circ \dots \circ R_{n, k - 1} false(g_{n, k} (x) false),

for sufficiently large

k

, and obtain commutative diagrams

with injective functions

ω_{n} = {(ω_{n, k})}_{k ⩾ n - 1}

that output stabilized word sequences. (See, for example, [12] or [24]. )…”

Section: An Inverse Limit Of Finitely Generated Free Monoidsmentioning

confidence: 99%

“…. , ±1 n } with R n−1 deleting the letters n and −1 n from every word [12,24]. This raises the following:…”

Section: Introductionmentioning

confidence: 99%

“…As far as the algebraic structure of fundamental groups of low‐dimensional spaces is concerned, we recall that the fundamental group of an arbitrary planar Peano continuum (not necessarily homotopy equivalent to a one‐dimensional space) is isomorphic to a subgroup of the fundamental group of some one‐dimensional planar Peano continuum [6]. In turn, fundamental groups of one‐dimensional path‐connected separable metric spaces are locally free [10] and, in the compact case, have structures similar to that of the Hawaiian Earring

double-struckH

, where the injective function

π_{1} false(double-struckH false) ↪ {\overset{π}{true}}_{1} false(double-struckH false)

factors through the limit

scriptM

of an inverse system

{scriptM}_{1} \overset{R_{1}}{⟵} {scriptM}_{2} \overset{R_{2}}{⟵} {scriptM}_{3} \overset{R_{3}}{⟵} \dots

of free monoids

M_{n}

{ℓ_{1}^{\pm 1}, ℓ_{2}^{\pm 1}, \dots, ℓ_{n}^{\pm 1}}

with

R_{n - 1}

deleting the letters

ℓ_{n}

and

ℓ_{n}^{- 1}

from every word [12, 24].…”

Section: Introductionmentioning

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: An Inverse Limit Of Finitely Generated Free Monoidsmentioning

confidence: 99%

“…For every

n \in N

x \in π_{1} false(H_{n}^{+}, b_{0} false)

, and

m ⩾ n - 1

, the sequence

{false(R_{n, m} \circ R_{n, m + 1} \circ \dots \circ R_{n, k - 1} (g_{n, k} false(x false)) false)}_{k ⩾ m}

of (unreduced) words is eventually constant, so that we may define functions

ω_{n, m} : π_{1} false(H_{n}^{+}, b_{0} false) \to {scriptW}_{n, m}^{+}

ω_{n, m} false(x false) = R_{n, m} \circ R_{n, m + 1} \circ \dots \circ R_{n, k - 1} false(g_{n, k} (x) false),

for sufficiently large

k

, and obtain commutative diagrams

with injective functions

ω_{n} = {(ω_{n, k})}_{k ⩾ n - 1}

that output stabilized word sequences. (See, for example, [12] or [24]. )…”

Section: An Inverse Limit Of Finitely Generated Free Monoidsmentioning

confidence: 99%

“…. , ±1 n } with R n−1 deleting the letters n and −1 n from every word [12,24]. This raises the following:…”

Section: Introductionmentioning

confidence: 99%

double-struckH

, where the injective function

π_{1} false(double-struckH false) ↪ {\overset{π}{true}}_{1} false(double-struckH false)

factors through the limit

scriptM

of an inverse system

{scriptM}_{1} \overset{R_{1}}{⟵} {scriptM}_{2} \overset{R_{2}}{⟵} {scriptM}_{3} \overset{R_{3}}{⟵} \dots

of free monoids

M_{n}

{ℓ_{1}^{\pm 1}, ℓ_{2}^{\pm 1}, \dots, ℓ_{n}^{\pm 1}}

with

R_{n - 1}

deleting the letters

ℓ_{n}

and

ℓ_{n}^{- 1}

from every word [12, 24].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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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“…n } with R n−1 deleting the letters n and −1 n from every word [12,22]. This raises the following Question: Is there a space which has all of the above properties, with the exception of π 1 injecting into π1 ?…”

Section: Introductionmentioning

confidence: 99%

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

Brazas,

Fischer

2019

Preprint

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We construct a space P for which the canonical homomorphism π 1 (P, p) → π1 (P, p) from the fundamental group to the first Čech 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-UV 0 ; (6) P admits a simply connected generalized covering space with monodromies between fibers that have discrete graphs; (7) π 1 (P, p) naturally injects into the inverse limit of finitely generated free monoids otherwise associated with the Hawaiian Earring.

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