Semicoverings: a generalization of covering space theory

Brazas, Jeremy

doi:10.4310/hha.2012.v14.n1.a3

Cited by 29 publications

(84 citation statements)

References 22 publications

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“…it corresponds to the subgroup Hom(Λ,Ẑ), which is clearly open and compact. We then set H = Gal(F /E) and let κ : Gal(F /E) → Hom(V /Λ, µ ∞ ) be the map induced by the Kummer pairing, as in (12). From (13) we see that the H-action on Hom exp (V, µ ∞ ) is indeed as described.…”

Section: Hence For the Subsetmentioning

confidence: 99%

Topological Realisations of Absolute Galois Groups

Kucharczyk

Scholze

2018

Cohomology of Arithmetic Groups

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Let F be a field of characteristic 0 containing all roots of unity. We construct a functorial compact Hausdorff space X F whose profinite fundamental group agrees with the absolute Galois group of F , i.e. the category of finite covering spaces of X F is equivalent to the category of finite extensions of F .The construction is based on the ring of rational Witt vectors of F . In the case of the cyclotomic extension of Q, the classical fundamental group of X F is a (proper) dense subgroup of the absolute Galois group of F . We also discuss a variant of this construction when the field is not required to contain all roots of unity, in which case there are natural Frobenius-type automorphisms which encode the descent along the cyclotomic extension.

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Section: Hence For the Subsetmentioning

confidence: 99%

Topological Realisations of Absolute Galois Groups

Kucharczyk

Scholze

2018

Cohomology of Arithmetic Groups

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“…It suffices to show that f α H is continuous. Semicovering maps are introduced by Brazas [5]. A semicovering map p :X → X is a local homeomorphism with continuous lifting of paths and homotopies [7].…”

Section: Proofmentioning

confidence: 99%

Small loop transfer spaces with respect to subgroups of fundamental groups

Pashaei

Mashayekhy

Torabi

et al. 2017

Topology and its Applications

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Let H be a subgroup of π 1 (X, x 0 ). In this paper, we extend the concept of X being SLT space to H-SLT space at x 0 . First, we show that the fibers of the endpoint projection p H :X H → X are topological group when X is H-SLT space at x 0 and H is a normal subgroup. Also, we show that under these conditions the concepts of homotopically path Hausdorff relative to H and homotopically Hausdorff relative to H coincide. Moreover, among other things, we show that the endpoint projection map p H has the unique path lifting property if and only if H is a closed normal subgroup of π qtop 1 (X, x 0 ) when X is SLT at x 0 . Second, we present conditions under which the whisker topology is agree with the quotient of compact-open topology onX H . Also, we study the relationship between open subsets of π wh 1 (X, x 0 ) and π qtop 1 (X, x 0 ).

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“…Therefore, by Theorem 2.3.2 of [10], p is a fibration. Moreover, it is easy to see that p has upl, but p is not a covering map (see [2,Example 3.8]). (2, ⇒): Refer to [10].…”

Section: Proposition 41 I) Composition Of Two Maps With Wuphl Is a mentioning

confidence: 99%

Unique Path Lifting from Homotopy Point of View

2017

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The paper is devoted to introduce some notions extending the unique path lifting property from a homotopy viewpoint and to study their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting property and find all possible relationships between them. Moreover, we supplement the full relationships of these new notions in the presence of fibrations. Second, we deduce some results in the category of fibrations with these notions instead of unique path lifting such as the existence of products and coproducts. Also, we give a brief comparison of these new categories to some categories of the other generalizations of covering maps. Finally, we present two subgroups of the fundamental group related to the fibrations with these notions and compare them to the subgroups of the fundamental group related to covering and generalized covering maps.

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Semicoverings: a generalization of covering space theory

Cited by 29 publications

References 22 publications

Topological Realisations of Absolute Galois Groups

Topological Realisations of Absolute Galois Groups

Small loop transfer spaces with respect to subgroups of fundamental groups

Unique Path Lifting from Homotopy Point of View

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