The geometry and topology of three-manifolds

Book•

The geometry and topology of three-manifolds

01 Jan 1979-

About: The article was published on 1979-01-01 and is currently open access. It has received 1566 citations till now. The article focuses on the topics: Extension topology & Computational topology.

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Journal Article•DOI•

Moduli spaces of local systems and higher Teichmüller theory

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V. V. Fock¹, Alexander Goncharov²•Institutions (2)

Institute on Taxation and Economic Policy¹, Brown University²

01 Jun 2006-Publications Mathématiques de l'IHÉS

TL;DR: The moduli space of positive representations is a topologically trivial open domain in the space of all representations as discussed by the authors, and all positive representations of the fundamental group of S to G(R) are faithful, discrete and positive hyperbolic.

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Abstract: Let G be a split semisimple algebraic group over Q with trivial center. Let S be a compact oriented surface, with or without boundary. We define positive representations of the fundamental group of S to G(R), construct explicitly all positive representations, and prove that they are faithful, discrete, and positive hyperbolic; the moduli space of positive representations is a topologically trivial open domain in the space of all representations. When S have holes, we defined two moduli spaces closely related to the moduli spaces of G-local systems on S. We show that they carry a lot of interesting structures. In particular we define a distinguished collection of coordinate systems, equivariant under the action of the mapping class group of S. We prove that their transition functions are subtraction free. Thus we have positive structures on these moduli spaces. Therefore we can take their points with values in any positive semifield. Their positive real points provide the two higher Teichmuller spaces related to G and S, while the points with values in the tropical semifields provide the lamination spaces. We define the motivic avatar of the Weil–Petersson form for one of these spaces. It is related to the motivic dilogarithm.

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858 citations

Book•

Lectures on Spaces of Nonpositive Curvature

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Werner Ballmann

01 Jan 1995

TL;DR: In this paper, the Hadamard Cartan Theorem and the Hopf-Rinow Theorem for rank rigidity of geodesic flows on metric spaces are discussed.

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Abstract: I. On the interior geometry of metric spaces.- 1. Preliminaries.- 2. The Hopf-Rinow Theorem.- 3. Spaces with curvature bounded from above.- 4. The Hadamard-Cartan Theorem.- 5. Hadamard spaces.- II. The boundary at infinity.- 1. Closure of X via Busemann functions.- 2. Closure of X via rays.- 3. Classification of isometries.- 4. The cone at infinity and the Tits metric.- III. Weak hyperbolicity.- 1. The duality condition.- 2. Geodesic flows on Hadamard spaces.- 3. The flat half plane condition.- 4. Harmonic functions and random walks on ?.- IV. Rank rigidity.- 1. Preliminaries on geodesic flows.- 2. Jacobi fields and curvature.- 3. Busemann functions and horospheres.- 4. Rank, regular vectors and flats.- 5. An invariant set at infinity.- 6. Proof of the rank rigidity.- Appendix. Ergodicity of geodesic flows.- 1. Introductory remarks.- Measure and ergodic theory preliminaries.- Absolutely continuous foliations.- Anosov flows and the Ho continuity of invariant distributions.- Proof of absolute continuity and ergodicity.

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614 citations

Journal Article•DOI•

A New Cohomology Theory of Orbifold

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Weimin Chen¹, Weimin Chen², Yongbin Ruan²•Institutions (2)

Tulane University¹, University of Wisconsin-Madison²

04 May 2004-Communications in Mathematical Physics

TL;DR: In this paper, a new cohomology ring for almost complex orbifolds is constructed based on the string theory model in physics, and the key theorem is the associativity of this new ring.

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Abstract: Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

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596 citations

Cites background from "The geometry and topology of three-..."

...on g1 ···gk= 1. Finally, we set (4.1.4) Xeo k:= G (g)∈To k X(g). In order to deﬁne the orbifold cup product, we need a digression on a few classical results about reduced 2-dimensional orbifolds (cf. [Th], [Sc]). Every closed orbifold of dimension 2 is complex, whose underlying topological space is a closed Riemann surface. More concretely, a closed, reduced 2-dimensional orbifold consists of the foll...
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Journal Article•DOI•

Gauge Theories Labelled by Three-Manifolds

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Tudor Dimofte, Davide Gaiotto, Sergei Gukov¹, Sergei Gukov²•Institutions (2)

California Institute of Technology¹, Max Planck Society²

01 Jan 2014-Communications in Mathematical Physics

TL;DR: In this article, a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories is proposed, which can be seen as a way of describing boundary conditions and duality walls in four-dimensional SCFTs.

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Abstract: We propose a dictionary between geometry of triangulated 3-manifolds and physics of three-dimensional N=2 gauge theories. Under this duality, standard operations on triangulated 3-manifolds and various invariants thereof (classical as well as quantum) find a natural interpretation in field theory. For example, independence of the SL(2) Chern-Simons partition function on the choice of triangulation translates to a statement that S^3_b partition functions of two mirror 3d N=2 gauge theories are equal. Three-dimensional N=2 field theories associated to 3-manifolds can be thought of as theories that describe boundary conditions and duality walls in four-dimensional N=2 SCFTs, thus making the whole construction functorial with respect to cobordisms and gluing.

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526 citations

Journal Article•DOI•

Relatively Hyperbolic Groups

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Benson Farb¹•Institutions (1)

University of Chicago¹

01 Nov 1998-Geometric and Functional Analysis

TL;DR: In this article, the authors propose a method to solve the problem of the problem: without abstracts, without abstractions, and without abstract sentences, without the abstracts of abstracts.

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Abstract: Abstract. ((Without abstract))

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518 citations

Cites background or result from "The geometry and topology of three-..."

...subgroup of . T hen is hyperbolic relative to H and the pair ( ;H )has the BCP property....
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...Since xy and xS(x) lie in the complement of T ,t he geodesic yS(x) lies within a Hausdor distance of S. On the other hand, this Hausdor distance is pinched between the corresponding distances in the spaces of constant curvatures a 2 and b 2 ([HI, Lemma 4.2])....
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...Corollary 3.3. Suppose that N T hen Gis hyperbolic relative to H if and only if G is hyperbolic relative to N....
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...graph of Gi with respect to Hi, and assume that there exists D> 0so that hd(f(H1);H 2 )D . T hen b 1is quasi-isometric to b 2; in particular, b 1 is negatively curved if and only if b 2 is negatively curved....
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...Let H denote the stabilizer subgroup of the endpoint on @X of a lift of this ray to X . T hen Gis said to be hyperbolic relative to H in the Gromov denition....
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