Fundamental groups of locally connected subsets of the plane

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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“…all of which lie in U , violating the choice of . Therefore, as in the proof of [8,Lemma 5.5], the loop δ contracts within X; a contradiction.…”

Section: Generalized Covering Projections and The Homotopically Hausdmentioning

confidence: 66%

“…(Such is the case for the Hawaiian Earring.) However, for all one-dimensional spaces and for all planar spaces, these monodromies have discrete graphs; a fact implicitly used in the work of Eda [14,15] and Conner-Kent [8]. If any two spaces with this property (cf.…”

Section: Introductionmentioning

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 Peano continuity is defined in the compact metric space, which is a connected space (including the locally connectedness property). There exists homeomorphism between a fundamental group in one-dimensional Peano continuum and another fundamental group in the Peano continuum on a plane if the map is continuous [7]. Moreover, a set of homotopy fixed points derived from a planar Peano continuum coincides with a point set representing a space, which is not a (locally) simply connected space.…”

Section: Introductionmentioning

confidence: 99%

“…In one-dimensional wild space with Peano continuum, the fundamental group determines the respective homeomorphism type. Note that the loops of fundamental groups in a planar space (set) are homotopy rigid [7].…”

Section: Introductionmentioning

confidence: 99%

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Bagchi

2020

Symmetry

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The fundamental groups and homotopy decompositions of algebraic topology have applications in systems involving symmetry breaking with topological excitations. The main aim of this paper is to analyze the properties of homotopy decompositions in quotient topological spaces depending on the connectedness of the space and the fundamental groups. This paper presents constructions and analysis of two varieties of homotopy decompositions depending on the variations in topological connectedness of decomposed subspaces. The proposed homotopy decomposition considers connected fundamental groups, where the homotopy equivalences are relaxed and the homeomorphisms between the fundamental groups are maintained. It is considered that one fundamental group is strictly homotopy equivalent to a set of 1-spheres on a plane and as a result it is homotopy rigid. The other fundamental group is topologically homeomorphic to the first one within the connected space and it is not homotopy rigid. The homotopy decompositions are analyzed in quotient topological spaces, where the base space and the quotient space are separable topological spaces. In specific cases, the decomposed quotient space symmetrically extends Sierpinski space with respect to origin. The connectedness of fundamental groups in the topological space is maintained by open curve embeddings without enforcing the conditions of homotopy classes on it. The extended decomposed quotient topological space preserves the trivial group structure of Sierpinski space.

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“…The use of such infinite products is ubiquitous in the progressive literature on the homotopy and homology groups of such spaces, e.g. [6,8,9,13,16,23]. For example, fundamental group(oid)s of one-dimensional spaces [14,15,17,21] are very much infinitary extensions of fundamental group(oid)s of graphs since path-homotopy classes have "reduced" representatives that are unique up to reparameterization [5].…”

Section: Introductionmentioning

confidence: 99%

Scattered products in fundamental groupoids

Brazas

2019

Proc. Amer. Math. Soc.

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Infinite products, indexed by countably infinite linear orders, arise naturally in the context of fundamental groupoids. Such products are called "transfinite" if the index orders are permitted to contain a dense suborder and are called "scattered" otherwise. In this paper, we prove several technical lemmas related to the reduction (i.e. combining of factors) of transfinite products in fundamental groupoids. Applying these results, we show that if the transfinite fundamental group operations are well-defined in a space X with a scattered algebraic 1-wild set aw(X), then all transfinite fundamental groupoid operations are also well-defined. arXiv:2001.06733v1 [math.AT]

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Fundamental groups of locally connected subsets of the plane

Cited by 12 publications

References 18 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

Analysis of Homotopy Decomposition Varieties in Quotient Topological Spaces

Scattered products in fundamental groupoids

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