Fundamental groups having the whole information of spaces

Conner, Gregory R.; Eda, Katsuya

doi:10.1016/j.topol.2003.05.005

Cited by 21 publications

(28 citation statements)

References 6 publications

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“…This constellation cannot be avoided for certain X. In particular, each handle body construction for a bad set X (in the sense of [4]) gives rise to loops f with incomplete (σ k ([f ])) k≥0 .…”

Section: Proof Of Theorem 32mentioning

confidence: 99%

Fundamental groups of one-dimensional spaces

2013

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Abstract. Let X be a metrizable one-dimensional continuum. In the present paper we describe the fundamental group of X as a subgroup of itsČech homotopy group. In particular, the elements of theČech homotopy group are represented by sequences of words. Among these sequences the elements of the fundamental group are characterized by a simple stabilization condition. This description of the fundamental group is used to give a new algebro-combinatorial proof of a result due to Eda on continuity properties of homomorphisms from the fundamental group of the Hawaiian earring to that of X.

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Section: Proof Of Theorem 32mentioning

confidence: 99%

Fundamental groups of one-dimensional spaces

2013

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“…Specifically the study of homomorphisms has lead to Eda proving that the fundamental group is a perfect invariant of homotopy type for one-dimensional Peano continua [9]. As well, it is the key tool in the proof that set of points at which a space is not semi-locally simply connected is constructible from the fundamental group for one-dimensional Peano continua [5] or planar Peano continua [6].…”

Section: Introductionmentioning

confidence: 99%

Homomorphisms of fundamental groups of planar continua

Kent

2018

Pacific J. Math.

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We prove that every homomorphism from the fundamental group of a planar Peano continuum to the fundamental group of a planar or one-dimensional Peano continuum is induced by a continuous map up to conjugation. This is then used to provide a family of uncountable many planar Peano continua with pairwise non-isomorphic fundamental groups all of which are not homotopy equivalent to a one-dimensional space.of any one-dimensional Peano continuum.Proof. Let {U 1 , U 2 , · · · } be a countable set of disjoint open subsets of (0, 1)×(0, 1). In each U i we can find a subset X i such that X i ⊂ S and X i is homeomorphic to the wedge of i-closed intervals.ForĀ ⊂ N, let XĀ = i∈Ā X i . It is a trivial exercise to show that that XĀ is homeomorphic to XB if and only ifĀ =B.ForĀ ⊂ N, choose A ⊂ N such that K(S A ) = XĀ. The corollary then follows from Proposition 2.14.

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“…The atomic property of the Hawaiian earring group and n-slenderness of free groups play central roles in the study of the fundamental groups of wild spaces and according to them certain spaces are recovered from their fundamental groups [5,7,12]. In addition, the fundamental groups of wild Peano continua also have the atomic property for free products with injective homomorphisms [7].…”

Section: Introductionmentioning

confidence: 99%

Atomic Properties of the Hawaiian Earring Group for HNN Extensions

Nakamura

2015

Communications in Algebra

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In 2011, while investigating fundamental groups of wild spaces, K.Eda [7] showed that the fundamental group of the Hawaiian earring (the Hawaiian earring group, in short) has the property that for any homomorphism h from it to a free product A * B, there exists a natural number N such that is contained in a conjugate subgroup to A or B. In the present article, we prove a corresponding property for certain HNN extensions and amalgamated free products. This allows us to show that some one-relator groups, including Baumslag-Solitar groups, are n-slender.

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Fundamental groups having the whole information of spaces

Cited by 21 publications

References 6 publications

Fundamental groups of one-dimensional spaces

Fundamental groups of one-dimensional spaces

Homomorphisms of fundamental groups of planar continua

Atomic Properties of the Hawaiian Earring Group for HNN Extensions

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