The revised and uniform fundamental groups and universal covers of geodesic spaces

Wilkins, J. Ernest

doi:10.1016/j.topol.2013.02.004

Cited by 14 publications

(17 citation statements)

References 31 publications

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“…To show the converse of Proposition 18 holds for locally path connected spaces, we recall the notion of Spanier group [32]. For more related to Spanier groups see [9][22] [24][30] [37]. Our approach is closely related to that in [25].…”

Section: Example 21mentioning

confidence: 99%

On fundamental groups with the quotient topology

Brazas

Fabel

2013

J. Homotopy Relat. Struct.

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The quasitopological fundamental group π qtop 1 (X, x 0 ) is the fundamental group endowed with the natural quotient topology inherited from the space of based loops and is typically non-discrete when X does not admit a traditional universal cover. This topologized fundamental group is an invariant of homotopy type which has the ability to distinguish weakly homotopy equivalent and shape equivalent spaces. In this paper, we clarify various relationships among topological properties of the group π qtop 1 (X, x 0 ) and properties of the underlying space X such as 'π 1 -shape injectivity' and 'homotopically path-Hausdorff. ' A space X is π 1 -shape injective if the fundamental group canonically embeds in the first shape group so that the elements of π 1 (X, x 0 ) can be represented as sequences in an inverse limit. We show a locally path connected metric space X is π 1 -shape injective if and only if π qtop 1 (X, x 0 ) is invariantly separated in the sense that the intersection of all open invariant (i.e. normal) subgroups is the trivial subgroup. In the case that X is not π 1 -shape injective, the homotopically path-Hausdorff property is useful for distinguishing homotopy classes of loops and guarantees the existence of certain generalized covering maps. We show that a locally path connected space X is homotopically path-Hausdorff if and only if π qtop 1 (X, x 0 ) satisfies the T 1 separation axiom. qtop 1 (X, x 0 ). Examples [6] illustrate that π qtop 1 (X, x 0 ) need not be a topological group even if X is a compact metric space [20][21]. 1 arXiv:1304.6453v3 [math.AT] 26 Jun 2013 π qtop 13. Corollary 33: If X is both locally path connected and the inverse limit of nested retracts of polyhedra, then π qtop 1 (X, x 0 ) is T 1 k s + 3 π qtop 1 (X, x 0 ) is T 2 k s + 3 X is π 1 -shape injective. Theorem 34: If πqtop 1 (X, x 0 ) is T 3 , then π qtop 1 (X, x 0 ) is T 4 .

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Section: Example 21mentioning

confidence: 99%

On fundamental groups with the quotient topology

Brazas

Fabel

2013

J. Homotopy Relat. Struct.

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“…If X has a universal cover X, then X is the δ-cover for all sufficiently small δ > 0 and CS is finite. By "universal cover" we mean in a categorical sense, which for compact geodesic spaces is equivalent to finiteness of CS (see Theorem 3.4 in [28] and [35] for related equivalent conditions).…”

Section: Introductionmentioning

confidence: 99%

Spectra related to the length spectrum

Plaut¹

2018

Preprint

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We show how to extend the Covering Spectrum (CS) of Sormani-Wei to two spectra, called the Extended Covering Spectrum (ECS) and Entourage Spectrum (ES) that are new for Riemannian manifolds but defined with useful properties on any metric on a Peano continuum. We do so by measuring in two different ways the "size" of a topological generalization of the δ-covers of Sormani-Wei called "entourage covers". For Riemannian manifolds M of dimension at least 3, we characterize entourage covers as those covers corresponding to the normal closures of finite subsets of π1(M ). We show that CS⊂ES⊂MLS and that for Riemannian manifolds these inclusions may be strict, where MLS is the set of lengths of curves that are shortest in their free homotopy classes. We give equivalent definitions for all of these spectra that do not actually involve lengths of curves. Of particular interest are resistance metrics on fractals for which there are no non-constant rectifiable curves, but where there is a reasonable notion of Laplace Spectrum (LaS). The paper opens new fronts for questions about the relationship between LaS and subsets of the length spectrum for a range of spaces from Riemannian manifolds to resistance metric spaces.

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“…Additional work on the covering spectrum and related notions has been conducted by Bart DeSmit, John Ennis, Ruth Gornet, Conrad Plaut, Craig Sutton, Jay Wilkins and Will Wylie [dSGS10], [dSGS12], [EW06], [PW13], [Wil13], [Wyl06].…”

Section: Introductionmentioning

confidence: 99%

Intrinsic flat convergence of covering spaces

Sinaei¹,

Sormani

2016

Geom Dedicata

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Abstract. We examine the limits of covering spaces and the covering spectra of oriented Riemannian manifolds, M j , which converge to a nonzero integral current space, M ∞ , in the intrinsic flat sense. We provide examples demonstrating that the covering spaces and covering spectra need not converge in this setting. In fact we provide a sequence of simply connected M j diffeomorphic to S 4 that converge in the intrinsic flat sense to a torus S 1 × S 3 . Nevertheless, we prove that if the δ-covers, M

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The revised and uniform fundamental groups and universal covers of geodesic spaces

Cited by 14 publications

References 31 publications

On fundamental groups with the quotient topology

On fundamental groups with the quotient topology

Spectra related to the length spectrum

Intrinsic flat convergence of covering spaces

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