scholarly journals Universal covers of topological modules and a monodromy principle

2003 ◽  
Vol 34 (4) ◽  
pp. 299-308
Author(s):  
Osman Mucuk ◽  
Mehmet Ozdemir
Keyword(s):  
Universal Cover ◽  
Topological Ring ◽  
Universal Covers ◽  
Topological Modules ◽  
Simply Connected ◽  
Path Connected

Let $R$ be a simply connected topological ring and $M$ be a topological left $R$-module in which the underling topology is path connected and has a universal cover. In this paper, we prove that a simply connected cover of $M$ admits the structure of a topological left $R$-module, and prove a Monodromy Principle, that a local morphism on $M$ of topological left $R$-modules extends to a morphism of topological left $R$-modules.

l′-Isolated Maps and Localizations

1983 ◽  
Vol 35 (2) ◽  
pp. 193-217
Author(s):  
Sara Hurvitz
Keyword(s):  
Finite Number ◽  
Free Algebra ◽  
Nilpotent Groups ◽  
Homotopy Groups ◽  
Homology Groups ◽  
Rational Cohomology ◽  
Locally Nilpotent Groups ◽  
Definition Of ◽  
Simply Connected ◽  
Path Connected

Let P be the set of primes, l ⊆ P a subset and l′ = P – l Recall that an H0-space is a space the rational cohomology of which is a free algebra.Cassidy and Hilton defined and investigated l′-isolated homomorphisms between locally nilpotent groups. Zabrodsky [8] showed that if X and Y are simply connected H0-spaces either with a finite number of homotopy groups or with a finite number of homology groups, then every rational equivalence f : X → Y can be decomposed into an l-equivalence and an l′-equivalence.In this paper we define and investigate l′-isolated maps between pointed spaces, which are of the homotopy type of path-connected nilpotent CW-complexes. Our definition of an l′-isolated map is analogous to the definition of an l′-isolated homomorphism. As every homomorphism can be decomposed into an l-isomorphism and an l′-isolated homomorphism, every map can be decomposed into an l-equivalence and an l′-isolated map.

scholarly journals THE UNIVERSAL COVER OF AN AFFINE THREE-MANIFOLD WITH HOLONOMY OF SHRINKABLE DIMENSION ≤ TWO

2000 ◽  
Vol 11 (03) ◽  
pp. 305-365 ◽  
Author(s):  
SUHYOUNG CHOI
Keyword(s):  
Affine Connection ◽  
Holonomy Group ◽  
Singular Locus ◽  
Universal Cover ◽  
Affine Structure ◽  
Affine Manifold ◽  
Cone Type ◽  
Parallel Volume ◽  
A Cell ◽  
Simply Connected

An affine manifold is a manifold with an affine structure, i.e. a torsion-free flat affine connection. We show that the universal cover of a closed affine 3-manifold M with holonomy group of shrinkable dimension (or discompacité in French) less than or equal to two is diffeomorphic to R3. Hence, M is irreducible. This follows from two results: (i) a simply connected affine 3-manifold which is 2-convex is diffeomorphic to R3, whose proof using the Morse theory takes most of this paper; and (ii) a closed affine manifold of holonomy of shrinkable dimension less or equal to d is d-convex. To prove (i), we show that 2-convexity is a geometric form of topological incompressibility of level sets. As a consequence, we show that the universal cover of a closed affine three-manifold with parallel volume form is diffeomorphic to R3, a part of the weak Markus conjecture. As applications, we show that the universal cover of a hyperbolic 3-manifold with cone-type singularity of arbitrarily assigned cone-angles along a link removed with the singular locus is diffeomorphic to R3. A fake cell has an affine structure as shown by Gromov. Such a cell must have a concave point at the boundary.

scholarly journals Riemannian manifolds with local symmetry

2014 ◽  
Vol 06 (02) ◽  
pp. 211-236 ◽  
Author(s):  
Wouter van Limbeek
Keyword(s):  
Riemannian Manifolds ◽  
Local Symmetry ◽  
Universal Cover ◽  
Connected Components ◽  
Curved Manifolds ◽  
Locally Homogeneous ◽  
Simply Connected ◽  
Positively Curved ◽  
Aspherical Manifolds

We give a classification of many closed Riemannian manifolds M whose universal cover [Formula: see text] possesses a nontrivial amount of symmetry. More precisely, we consider closed Riemannian manifolds M such that [Formula: see text] has noncompact connected components. We prove that in many cases, such a manifold is as a fiber bundle over a locally homogeneous space. This is inspired by work of Eberlein (for non-positively curved manifolds) and Farb-Weinberger (for aspherical manifolds), and generalizes work of Frankel (for a semisimple group action). As an application, we characterize simply-connected Riemannian manifolds with both compact and finite volume noncompact quotients.

scholarly journals Moduli Spaces of Metrics of Positive Scalar Curvature on Topological Spherical Space Forms

2020 ◽  
Vol 63 (4) ◽  
pp. 901-908
Author(s):  
Philipp Reiser
Keyword(s):  
Scalar Curvature ◽  
Moduli Spaces ◽  
Riemannian Metrics ◽  
Universal Cover ◽  
Positive Scalar Curvature ◽  
Spherical Space ◽  
Space Forms ◽  
Positive Scalar ◽  
Space Of Riemannian Metrics ◽  
Simply Connected

AbstractLet $M$ be a topological spherical space form, i.e., a smooth manifold whose universal cover is a homotopy sphere. We determine the number of path components of the space and moduli space of Riemannian metrics with positive scalar curvature on $M$ if the dimension of $M$ is at least 5 and $M$ is not simply-connected.

scholarly journals Covering groupoids of categorical rings

Filomat ◽  
2015 ◽  
Vol 29 (1) ◽  
pp. 39-49 ◽  
Author(s):  
Osman Mucuk ◽  
Serap Demir
Keyword(s):  
Topological Group ◽  
Covering Spaces ◽  
Categorical Group ◽  
Fundamental Groupoid ◽  
Group Object ◽  
Literature Group ◽  
Simply Connected ◽  
Path Connected ◽  
Covering Groups

A categorical group is a kind of categorization of group and similarly a categorical ring is a categorization of ring. For a topological group X, the fundamental groupoid ?X is a group object in the category of groupoids, which is also called in literature group-groupoid or 2-group. If X is a path connected topological group which has a simply connected cover, then the category of covering groups of X and the category of covering groupoids of ?X are equivalent. Recently it was proved that if (X, x0) is an H-group, then the fundamental groupoid ?X is a categorical group and the category of the covering spaces of (X, x0) is equivalent to the category of covering groupoids of the categorical group ?X. The purpose of this paper is to present similar results for rings and categorical rings.

scholarly journals Group extensions are quasi-simply-filtrated

1994 ◽  
Vol 50 (1) ◽  
pp. 21-27 ◽  
Author(s):  
S.G. Brick ◽  
M.L. Mihalik
Keyword(s):  
Fundamental Group ◽  
Universal Cover ◽  
Infinite Group ◽  
Finite Complex ◽  
Group Extensions ◽  
Finitely Presented Group ◽  
Simply Connected ◽  
Finitely Presented

A finitely presented group G is quasi-simply-filtrated (abbreviated qsf) if, given a finite complex with fundamental group G, the universal cover of the complex can be “approximated” by simply connected finite complexes. This notion is a generalisation of a concept of Casson's used in the study of three-manifolds.In this paper we show that any extension of a finitely presented infinite group by a finitely presented infinite group is qsf.

scholarly journals Algebraic varieties with quasi-projective universal cover

2013 ◽  
Vol 2013 (679) ◽  
pp. 207-221 ◽  
Author(s):  
Benoît Claudon ◽  
Andreas Höring ◽  
János Kollár
Keyword(s):  
Abelian Variety ◽  
Fiber Bundle ◽  
Projective Variety ◽  
Universal Cover ◽  
Algebraic Varieties ◽  
Simply Connected ◽  
Connected Fiber

Abstract We prove that the universal cover of a normal projective variety X is quasi-projective iff a finite étale cover of X is a fiber bundle over an Abelian variety with simply connected fiber.

scholarly journals Holonomy, extendibility, and the star universal cover of a topological groupoid

2003 ◽  
Vol 4 (1) ◽  
pp. 79
Author(s):  
Osman Mucuk ◽  
Ilhan Icen
Keyword(s):  
Universal Cover ◽  
Topological Groupoid ◽  
Universal Covers

<p>Let G be a groupoid and W be a subset of G which contains all the identities and has a topology. With some conditions on G and W, the pair (G;W) is called a locally topological groupoid. We explain a criterion for a locally topological groupoid to be extendible to a topological groupoid. In this paper we apply this result to get a topology on the monodromy groupoid MG which is the union of the universal covers of Gx's.</p>

Coverings and Ring-Groupoids

1998 ◽  
Vol 5 (5) ◽  
pp. 475-482
Author(s):  
Osman Mucuk
Keyword(s):  
Ring Structure ◽  
Covering Space ◽  
Topological Ring ◽  
Homotopy Classes ◽  
Connected Covering ◽  
Simply Connected

Abstract We prove that the set of homotopy classes of the paths in a topological ring is a ring object (called ring groupoid). Using this concept we show that the ring structure of a topological ring lifts to a simply connected covering space.

CASSON'S IDEA ABOUT 3-MANIFOLDS WHOSE UNIVERSAL COVER IS R3

1991 ◽  
Vol 01 (04) ◽  
pp. 395-406 ◽  
Author(s):  
JOHN R. STALLINGS ◽  
S. M. GERSTEN
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Universal Cover ◽  
Group Presentations ◽  
Simply Connected

Metric conditions "Cα" and "[Formula: see text]" are defined for finite group presentations. If the fundamental group of a closed aspherical 3-manifold has some presentation which satisfies C2 or [Formula: see text], then its universal cover is simply connected at infinity. These ideas are derived from work by A. Casson and V. Poénaru.

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