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32
Volume collapsed threemanifolds with a lower curvature
"... Abstract. In this paper we determine the topology of threedimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small. 1. ..."
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Abstract. In this paper we determine the topology of threedimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small. 1.
Tight contact structures on Seifert Manifolds over T² with one singular fiber
, 2003
"... In this article we classify up to isotopy the tight contact structures on Seifert manifolds over the torus with one singular fibre. ..."
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Cited by 10 (3 self)
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In this article we classify up to isotopy the tight contact structures on Seifert manifolds over the torus with one singular fibre.
Crystallographic Topology and Its Applications
, 1996
"... Geometric topology and structural crystallography concepts are combined to define a new area we call Structural Crystallographic Topology, which may be of interest to both crystallographers and mathematicians. In this paper, we represent crystallographic symmetry groups by orbifolds and crystal stru ..."
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Geometric topology and structural crystallography concepts are combined to define a new area we call Structural Crystallographic Topology, which may be of interest to both crystallographers and mathematicians. In this paper, we represent crystallographic symmetry groups by orbifolds and crystal structures by Morse functions. The Morse function uses mildly overlapping Gaussian thermalmotion probability density functions centered on atomic sites to form a critical net with peak, pass, pale, and pit critical points joined into a graph by density gradientflow separatrices. Critical net crystal structure drawings can be made with the ORTEPIII graphics program. An orbifold consists of an underlying topological space with an embedded singular set that represents the Wyckoff sites of the crystallographic group. An orbifold for a point group, plane group, or space group is derived by gluing together equivalent edges or faces of a crystallographic asymmetric unit. The criticalnetonorbifol...
Explicit horizontal open books on some plumbings
 BURAK OZBAGCI
"... ABSTRACT. We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for nonpositive plumbings, the open books we construct are horizontal and the corresponding compatible contact structures are Stein fillable and hence tight. In p ..."
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Cited by 8 (4 self)
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ABSTRACT. We describe explicit open books on arbitrary plumbings of oriented circle bundles over closed oriented surfaces. We show that, for nonpositive plumbings, the open books we construct are horizontal and the corresponding compatible contact structures are Stein fillable and hence tight. In particular, we describe horizontal open books on some Seifert fibered 3–manifolds. As another application we describe horizontal open books isomorphic to Milnor open books for some complex surface singularities. Moreover we give examples of tight contact 3–manifolds supported by planar open books. As a consequence the Weinstein conjecture is satisfied for these tight contact structures [ACH]. 1.
Transverse contact structures on Seifert 3–manifolds
 Algebr. Geom. Topol
"... Abstract We characterize the oriented Seifert–fibered three–manifolds which admit positive, transverse contact structures. AMS Classification 57R17 Keywords transverse contact structures, Seifert three–manifolds 1 Introduction and statement of results Foliations and contact structures are arguably a ..."
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Abstract We characterize the oriented Seifert–fibered three–manifolds which admit positive, transverse contact structures. AMS Classification 57R17 Keywords transverse contact structures, Seifert three–manifolds 1 Introduction and statement of results Foliations and contact structures are arguably at opposite ends of the spectrum of the possible 2plane fields ξ on a 3manifold. While foliations are integrable plane fields, contact structures are totally nonintegrable. If the planes in the distribution ξ are given as kernels of a one form α, then they form a foliation
Completion of the proof of the geometrization conjecture
"... This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a ..."
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This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a locally homogeneous Riemannian metric of finite volume. Recall that a Riemannian manifold is homogeneous if its isometry group acts transitively on the underlying manifold; a locally homogeneous Riemannian manifold is the quotient of a homogeneous Riemannian manifold by a discrete group of isometries acting freely. Recall also that a prime 3manifold is one which is not diffeomorphic to S 3 and which is not a connected sum of two manifolds neither of which is diffeomorphic to S 3. It is a classic result in 3manifold topology, see [12] that every 3manifold is a connected sum of a finite number of prime 3manifolds, and this decomposition is unique up to the order of the factors. The main part of this paper is devoted to giving a proof of Theorem 7.4 stated
Infinitely many universally tight contact manifolds with trivial Ozsváth–Szabó contact invariants
, 2006
"... In this article we present infinitely many 3–manifolds admitting infinitely many universally tight contact structures each with trivial Ozsváth–Szabó contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable. ..."
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Cited by 5 (2 self)
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In this article we present infinitely many 3–manifolds admitting infinitely many universally tight contact structures each with trivial Ozsváth–Szabó contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.
Analytic asymptotic expansions of the Reshetikhin–Turaev invariants of Seifert 3–manifolds for sl2(C
"... Abstract. We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asy ..."
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Abstract. We calculate the large quantum level asymptotic expansion of the RT–invariants associated to SU(2) of all oriented Seifert 3–manifolds X with orientable base or nonorientable base with even genus. Moreover, we identify the Chern–Simons invariants of flat SU(2)– connections on X in the asymptotic formula thereby proving the socalled asymptotic expansion conjecture (AEC) due to J. E. Andersen [An1], [An2] for these manifolds. For the case of Seifert manifolds with base S 2 we actually prove a little weaker result, namely that the asymptotic formula has a form as predicted by the AEC but contains some extra terms which should be zero according to the AEC. We prove that these ‘extra ’ terms are indeed zero if the number of exceptional fibers n is less than 4 and conjecture that this is also the case if n≥4. For the case of Seifert fibered rational homology spheres we identify the Casson–Walker invariant in the asymptotic formula. Our calculations demonstrate a general method for calculating the large r asymptotics of a finite sum Σ r k=1f(k), where f is a meromorphic function depending on the integer parameter r and satisfying certain symmetries. Basically the method, which is due to Rozansky [Ro1], [Ro3], is based on a limiting version of the Poisson summation formula together with an application of the steepest descent method from asymptotic analysis. Contents
Cusp areas of Farey manifolds and applications to knot theory
, 2008
"... We find explicit, combinatorial estimates for the cusp areas of once–punctured torus bundles, 4–punctured sphere bundles, and 2–bridge link complements. Applications include volume estimates for the hyperbolic 3manifolds obtained by Dehn filling these bundles, for example estimates on the volume of ..."
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We find explicit, combinatorial estimates for the cusp areas of once–punctured torus bundles, 4–punctured sphere bundles, and 2–bridge link complements. Applications include volume estimates for the hyperbolic 3manifolds obtained by Dehn filling these bundles, for example estimates on the volume of closed 3–braid complements in terms of the complexity of the braid word. We also relate the volume of a closed 3–braid to certain coefficients of its Jones polynomial.