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Journal articles on the topic 'Seifert manifold'

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Published: 4 June 2021

Last updated: 20 February 2023

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1

Ikeda, Toru. "Essential Surfaces in Graph Link Exteriors." Canadian Mathematical Bulletin 52, no. 2 (June 1, 2009): 257–66. http://dx.doi.org/10.4153/cmb-2009-028-9.

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AbstractAn irreducible graph manifold M contains an essential torus if it is not a special Seifert manifold. WhetherM contains a closed essential surface of negative Euler characteristic or not depends on the difference of Seifert fibrations from the two sides of a torus system which splits M into Seifert manifolds. However, it is not easy to characterize geometrically the class of irreducible graph manifolds which contain such surfaces. This article studies this problem in the case of graph link exteriors.

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2

Silver, Daniel S. "Examples of 3-knots with no minimal Seifert manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 3 (November 1991): 417–20. http://dx.doi.org/10.1017/s0305004100070481.

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We work throughout in the smooth category. Homomorphisms of fundamental and homology groups are induced by inclusion. Ann-knot, formn≥ 1, is an embeddedn-sphereK⊂Sn+2. ASeifert manifoldforKis a compact, connected, orientable (n+ 1)-manifoldV⊂Sn+2with boundary ∂V=K. By [9] Seifert manifolds always exist. As in [9] letYdenoteSn+2split alongV; Yis a compact manifold with ∂Y=V0∪V1, whereVt≈V. We say thatVis aminimal Seifert manifoldforKif π1Vt→ π1Yis a monomorphism fort= 0, 1. (Here and throughout basepoint considerations are suppressed.)

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3

COFFEY, JAMES. "THE UNIVERSAL COVER OF 3-MANIFOLDS BUILT FROM INJECTIVE HANDLEBODIES IS ℝ3." Journal of Knot Theory and Its Ramifications 17, no. 10 (October 2008): 1257–80. http://dx.doi.org/10.1142/s0218216508006579.

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This paper gives a proof that the universal cover of a closed 3-manifold built from three π1-injective handlebodies is homeomorphic to ℝ3. This construction is an extension to handlebodies of the conditions for gluing of three solid tori to produce non-Haken Seifert fibered manifolds with infinite fundamental group. This class of manifolds has been shown to contain non-Haken non-Seifert fibered manifolds.

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4

NISHI, HARUKO. "SU(n)-CHERN–SIMONS INVARIANTS OF SEIFERT FIBERED 3-MANIFOLDS." International Journal of Mathematics 09, no. 03 (May 1998): 295–330. http://dx.doi.org/10.1142/s0129167x98000130.

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We derive the formula for the Chern–Simons invariants of irrreducible SU(n)-flat connections on the Seifert fibered 3-manifolds. As an example, we calculate the values explicitly for the Seifert fibered 3-manifold S3/Γ, where Γ is a finite subgroup of SU(2).

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5

Silver, Daniel S. "On the existence of minimal Seifert manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 114, no. 1 (July 1993): 103–9. http://dx.doi.org/10.1017/s0305004100071449.

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AbstractFor n ≥ 3, an n-knot K has a minimal Seifert manifold if and only if its group is isomorphic to an HNN-extension with finitely presented base. In this case, any Seifert manifold for K can be converted to a minimal Seifert manifold for K by some finite sequence of ambient 0- and 1-surgeries.

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6

BRITTENHAM, MARK. "ESSENTIAL LAMINATIONS, EXCEPTIONAL SEIFERT-FIBERED SPACES, AND DEHN FILLING." Journal of Knot Theory and Its Ramifications 07, no. 04 (June 1998): 425–32. http://dx.doi.org/10.1142/s021821659800022x.

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We show how essential laminations can be used to provide an improvement on (some of) the results of the 2π-Theorem; at most 20 Dehn fillings on a hyperbolic 3-manifold with boundary a torus T can yield a reducible manifold, finite π1 manifold, or exceptional Seifert-fibered space. Recent work of Wu allows us to add toroidal manifolds to this list, as well.

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7

Heil, Wolfgang, and Wilbur Whitten. "The Seifert Fiber Space Conjecture and Torus Theorem for Nonorientable 3-Manifolds." Canadian Mathematical Bulletin 37, no. 4 (December 1, 1994): 482–89. http://dx.doi.org/10.4153/cmb-1994-070-7.

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AbstractThe Seifert-fiber-space conjecture for nonorientable 3-manifolds states that if M denotes a compact, irreducible, nonorientable 3-manifold that is not a fake P2 x S1, if π1M is infinite and does not contain Z2 * Z2 as a subgroup, and if π1M does however contain a nontrivial, cyclic, normal subgroup, then M is a Seifert bundle. In this paper, we construct all compact, irreducible, nonorientable 3-manifolds (that do not contain a fake P2 × I) each of whose fundamental group contains Z2 * Z2 and an infinité cyclic, normal subgroup; none of these manifolds admits a Seifert fibration, but they satisfy Thurston's Geometrization Conjecture. We then reformulate the statement of the (nonorientable) SFS-conjecture and obtain a torus theorem for nonorientable manifolds.

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8

Millett, Kenneth, and Dale Rolfsen. "A theorem of Borsuk—Ulam type for Seifert-fibred 3-manifolds." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 3 (November 1986): 523–32. http://dx.doi.org/10.1017/s0305004100066251.

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Let M denote a compact topological 3-manifold, without boundary, foliated by topological circles in the sense of Seifert's gefaserter Räume[8]. This will be called a Seifert structure, or Seifert fibration on M; the leaves of the foliation are called (regular or exceptional) fibres. Our main result is the following theorem, reminiscent of the theorem of Borsuk—Ulam [1] stating that every continuous function from S3 to S2 takes at least one pair of antipodal points to the same value.

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9

Peet, Benjamin. "Finite, Fiber- and Orientation-Preserving Group Actions on Totally Orientable Seifert Manifolds." Annales Mathematicae Silesianae 33, no. 1 (September 1, 2019): 235–65. http://dx.doi.org/10.2478/amsil-2019-0007.

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AbstractIn this paper we consider the finite groups that act fiber- and orientation-preservingly on closed, compact, and orientable Seifert manifolds that fiber over an orientable base space. We establish a method of constructing such group actions and then show that if an action satisfies a condition on the obstruction class of the Seifert manifold, it can be derived from the given construction. The obstruction condition is refined and the general structure of the finite groups that act via the construction is provided.

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10

Kropholler, P. H. "A note on centrality in 3-manifold groups." Mathematical Proceedings of the Cambridge Philosophical Society 107, no. 2 (March 1990): 261–66. http://dx.doi.org/10.1017/s0305004100068523.

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Centralizers in fundamental groups of 3-manifolds are well understood because of their relationship with Seifert fibre spaces. Jaco and Shalen's book [4] provides detailed information about Seifert fibre spaces in 3-manifolds, and consequently about centralizers in their fundamental groups. It is the purpose of this note to record two group-theoretic properties, both easily deduced from results of Jaco and Shalen. Doubtless many other authors could have established the same results had they needed them. Our motivation for writing this paper is that these properties can be used as a basis for group-theoretic proofs of certain fundamental results in 3-manifold theory: Proposition 1 below can be used as a basis for a proof of the Torus Theorem (cf. [8]) and Proposition 2 for the Torus Decomposition Theorem (cf. [9]).

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11

Grines, Vyacheslav Z., Elena Ya Gurevich, and Evgenii Iv Yakovlev. "On topology of manifolds admitting gradient-like calscades with surface dynamics and on growth of the number of non-compact heteroclinic curves." Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva 23, no. 4 (December 30, 2021): 379–93. http://dx.doi.org/10.15507/2079-6900.23.202104.379-393.

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We consider a class GSD(M3) of gradient-like diffeomorphisms with surface dynamics given on closed oriented manifold M3 of dimension three. Earlier it was proved that manifolds admitting such diffeomorphisms are mapping tori under closed orientable surface of genus g, and the number of non-compact heteroclinic curves of such diffeomorphisms is not less than 12g. In this paper, we determine a class of diffeomorphisms GSDR(M3)⊂GSD(M3) that have the minimum number of heteroclinic curves for a given number of periodic points, and prove that the supporting manifold of such diffeomorphisms is a Seifert manifold. The separatrices of periodic points of diffeomorphisms from the class GSDR(M3) have regular asymptotic behavior, in particular, their closures are locally flat. We provide sufficient conditions (independent on dynamics) for mapping torus to be Seifert. At the same time, the paper establishes that for any fixed g geq1, fixed number of periodic points, and any integer n≥12g, there exists a manifold M3 and a diffeomorphism f∈GSD(M3) having exactly n non-compact heteroclinic curves.

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12

Rasmussen, Sarah Dean. "L-space intervals for graph manifolds and cables." Compositio Mathematica 153, no. 5 (April 4, 2017): 1008–49. http://dx.doi.org/10.1112/s0010437x16008319.

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We present a graph manifold analog of the Jankins–Neumann classification of Seifert fibered spaces over$S^{2}$admitting taut foliations, providing a finite recursive formula to compute the L-space Dehn-filling interval for any graph manifold with torus boundary. As an application of a generalization of this result to Floer simple manifolds, we compute the L-space interval for any cable of a Floer simple knot complement in a closed three-manifold in terms of the original L-space interval, recovering a result of Hedden and Hom as a special case.

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13

WHITTEN, WILBUR. "RECOGNIZING NONORIENTABLE SEIFERT BUNDLES." Journal of Knot Theory and Its Ramifications 01, no. 04 (December 1992): 471–75. http://dx.doi.org/10.1142/s0218216592000240.

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Roughly speaking, a compact, orientable, irreducible 3-manifold M with infinite fundamental group is a Seifert fiber space, if either 1) π1M contains a nontrivial, cyclic, normal subgroup (the so-called Seifert-fiber-space conjecture), 2) M is finitely covered by a Seifert fiber space, or 3) π1M is isomorphic to the group of a Seifert fiber space. Excluding a fake P2 × S1 where necessary, we show in this paper that similar results hold when M is nonorientable.

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14

KANG, ENSIL. "SEIFERT SURFACES IN KNOT COMPLEMENTS." Journal of Knot Theory and Its Ramifications 16, no. 08 (October 2007): 1053–66. http://dx.doi.org/10.1142/s0218216507005622.

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In the ordinary normal surface for a compact 3-manifold, any incompressible, ∂-incompressible, compact surface can be moved by an isotopy to a normal surface [9]. But in a non-compact 3-manifold with an ideal triangulation, the existence of a normal surface representing an incompressible surface cannot be guaranteed. The figure-8 knot complement is presented in a counterexample in [12]. In this paper, we show the existence of normal Seifert surface under some restriction for a given ideal triangulation of the knot complement.

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15

Lozano, María Teresa, and José María Montesinos-Amilibia. "On continuous families of geometric Seifert conemanifold structures." Journal of Knot Theory and Its Ramifications 25, no. 14 (December 2016): 1650083. http://dx.doi.org/10.1142/s0218216516500838.

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In this paper, dedicated to Prof. Lou Kauffman, we determine the Thurston’s geometry possesed by any Seifert fibered conemanifold structure in a Seifert manifold with orbit space [Formula: see text] and no more than three exceptional fibers, whose singular set, composed by fibers, has at most three components which can include exceptional or general fibers (the total number of exceptional and singular fibers is less than or equal to three). We also give the method to obtain the holonomy of that structure. We apply these results to three families of Seifert manifolds, namely, spherical, Nil manifolds and manifolds obtained by Dehn surgery on a torus knot [Formula: see text]. As a consequence we generalize to all torus knots the results obtained in [Geometric conemanifolds structures on [Formula: see text], the result of [Formula: see text] surgery in the left-handed trefoil knot [Formula: see text], J. Knot Theory Ramifications 24(12) (2015), Article ID: 1550057, 38pp., doi: 10.1142/S0218216515500571] for the case of the left handle trefoil knot. We associate a plot to each torus knot for the different geometries, in the spirit of Thurston.

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16

HILLMAN, J. A., and J. HOWIE. "SEIFERT FIBERED KNOT MANIFOLDS." Journal of Knot Theory and Its Ramifications 22, no. 14 (December 2013): 1350082. http://dx.doi.org/10.1142/s021821651350082x.

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We consider the question of when is the closed manifold obtained by elementary surgery on an n-knot Seifert fibered over a 2-orbifold. The possible bases are strongly constrained by the fact that knot groups have weight 1. We present a new family of 2-knots with solvable groups, overlooked in earlier work. The knots in this new family are neither invertible nor amphicheiral, and the weight orbits for the knot groups are parametrized by ℤ. There are no known examples in higher dimensions.

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17

Roushon, Sayed K. "The isomorphism conjecture for 3-manifold groups and K-theory of virtually poly-surface groups." Journal of K-Theory 1, no. 1 (November 30, 2007): 83–93. http://dx.doi.org/10.1017/is007011012jkt006.

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AbstractThis article has two purposes. In [15] we showed that the FIC (Fibered Isomorphism Conjecture for pseudoisotopy functor) for a particular class of 3-manifolds (we denoted this class by C) is the key to prove the FIC for 3-manifold groups in general. And we proved the FIC for the fundamental groups of members of a subclass of C. This result was obtained by showing that the double of any member of this subclass is either Seifert fibered or supports a nonpositively curved metric. In this article we prove that for any M ε C there is a closed 3-manifold P such that either P is Seifert fibered or is a nonpositively curved 3-manifold and π1(M) is a subgroup of π1(P). As a consequence it is obtained that the FIC is true for any B-group (see definition 4.2 in [15]). Therefore, the FIC is true for any Haken 3-manifold group and hence for any 3-manifold group (using the reduction theorem of [15]) provided we assume the Geometrization conjecture. The above result also proves the FIC for a class of 4-manifold groups (see [14]).The second aspect of this article is to relax a condition in the definition of strongly poly-surface group ([13]) and define a new class of groups (we call them weak strongly poly-surface groups). Then using the above result we prove the FIC for any virtually weak strongly poly-surface group. We also give a corrected proof of the main lemma of [13].

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18

Cheng, Miranda C. N., Francesca Ferrari, and Gabriele Sgroi. "Three-manifold quantum invariants and mock theta functions." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 378, no. 2163 (December 9, 2019): 20180439. http://dx.doi.org/10.1098/rsta.2018.0439.

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Mock modular forms have found applications in numerous branches of mathematical sciences since they were first introduced by Ramanujan nearly a century ago. In this proceeding, we highlight a new area where mock modular forms start to play an important role, namely the study of three-manifold invariants. For a certain class of Seifert three-manifolds, we describe a conjecture on the mock modular properties of a recently proposed quantum invariant. As an illustration, we include concrete computations for a specific three-manifold, the Brieskorn sphere Σ (2, 3, 7). This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

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19

Núñez, Víctor, Enrique Ramírez-Losada, and Jesús Rodríguez-Viorato. "Coverings of torus knots in S2 × S1 and universals." Journal of Knot Theory and Its Ramifications 26, no. 08 (March 31, 2017): 1750044. http://dx.doi.org/10.1142/s0218216517500444.

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Let [Formula: see text] be an ordinary fiber of a Seifert fibering of [Formula: see text] with two exceptional fibers of order [Formula: see text]. We show that any Seifert manifold with Euler number zero is a branched covering of [Formula: see text] with branching [Formula: see text] if [Formula: see text]. We compute the Seifert invariants of the Abelian covers of [Formula: see text] branched along a [Formula: see text]. We also show that [Formula: see text], a non-trivial torus knot in [Formula: see text], is not universal.

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20

Shanti Caillat-Gibert and Daniel Matignon. "Existence of Taut Foliations on Seifert Fibered Homology 3-spheres." Canadian Journal of Mathematics 66, no. 1 (February 2014): 141–69. http://dx.doi.org/10.4153/cjm-2013-011-4.

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AbstractThis paper concerns the problem of existence of taut foliations among 3-manifolds. From the work of David Gabai we know that a closed 3-manifold with non-trivial second homology group admits a taut foliation. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite different if they are integral or rational but non-integral homology 3-spheres. Concerning integral homology 3-spheres, we can see that all but the 3-sphere and the Poincaré 3-sphere admit a taut foliation. Concerning non-integral homology 3-spheres, we prove there are infinitely many that admit a taut foliation, and infinitely many without a taut foliation. Moreover, we show that the geometries do not determine the existence of taut foliations on non-integral Seifert fibered homology 3-spheres.

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21

Préaux, Jean-Philippe. "A Survey on Seifert Fiber Space Theorem." ISRN Geometry 2014 (March 5, 2014): 1–9. http://dx.doi.org/10.1155/2014/694106.

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22

KOLPAKOV, ALEXANDER. "EXAMPLES OF RIGID AND FLEXIBLE SEIFERT FIBRED CONE-MANIFOLDS." Glasgow Mathematical Journal 55, no. 2 (February 25, 2013): 411–29. http://dx.doi.org/10.1017/s0017089512000651.

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23

Kim, Jin Hong. "Examples of simply-connected K-contact non-Sasakian manifolds of dimension 5." International Journal of Geometric Methods in Modern Physics 12, no. 03 (February 27, 2015): 1550027. http://dx.doi.org/10.1142/s0219887815500279.

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The existence of compact simply-connected K-contact, but not Sasakian, manifolds has been unknown only for dimension 5. The aim of this paper is to show that the Kollár's simply-connected example which is a Seifert bundle over the complex projective space ℂℙ2 and does not carry any Sasakian structure is actually a K-contact manifold. As a consequence, we affirmatively answer the above existence problem in dimension 5, establishing that there are infinitely many compact simply-connected K-contact manifolds of dimension 5 which do not carry a Sasakian structure.

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24

Beliakova, Anna, and Thang T. Q. Lê. "Integrality of quantum 3-manifold invariants and a rational surgery formula." Compositio Mathematica 143, no. 6 (November 2007): 1593–612. http://dx.doi.org/10.1112/s0010437x07003053.

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AbstractWe prove that the Witten–Reshetikhin–Turaev (WRT) SO(3) invariant of an arbitrary 3-manifold M is always an algebraic integer. Moreover, we give a rational surgery formula for the unified invariant dominating WRT SO(3) invariants of rational homology 3-spheres at roots of unity of order co-prime with the torsion. As an application, we compute the unified invariant for Seifert fibered spaces and for Dehn surgeries on twist knots. We show that this invariant separates Seifert fibered integral homology spaces and can be used to detect the unknot.

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CASALI, MARIA RITA, and PAOLA CRISTOFORI. "COMPUTING MATVEEV'S COMPLEXITY VIA CRYSTALLIZATION THEORY: THE BOUNDARY CASE." Journal of Knot Theory and Its Ramifications 22, no. 08 (July 2013): 1350038. http://dx.doi.org/10.1142/s0218216513500387.

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The notion of Gem–Matveev complexity (GM-complexity) has been introduced within crystallization theory, as a combinatorial method to estimate Matveev's complexity of closed 3-manifolds; it yielded upper bounds for interesting classes of such manifolds. In this paper, we extend the definition to the case of non-empty boundary and prove that for each compact irreducible and boundary-irreducible 3-manifold it coincides with the modified Heegaard complexity introduced by Cattabriga, Mulazzani and Vesnin. Moreover, via GM-complexity, we obtain an estimation of Matveev's complexity for all Seifert 3-manifolds with base 𝔻2 and two exceptional fibers and, therefore, for all torus knot complements.

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Allenby, R. B. J. T., Goansu Kim, and C. Y. Tang. "Conjugacy separability of certain Seifert 3-manifold groups." Journal of Algebra 285, no. 2 (March 2005): 481–507. http://dx.doi.org/10.1016/j.jalgebra.2004.10.022.

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Trace, Bruce. "A note concerning Seifert manifolds for 2-knots." Mathematical Proceedings of the Cambridge Philosophical Society 100, no. 1 (July 1986): 113–16. http://dx.doi.org/10.1017/s0305004100065907.

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AbstractElementary observations yield new classes of knotted 2-spheres in S4 which do not admit Punct (# S1 × S2) as a Seifert manifold. This provides a rather painless proof which re-establishes the existence of non-ribbon 2-knots.

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Blau, Matthias, Kaniba Mady Keita, K. S. Narain, and George Thompson. "Chern–Simons theory on a general Seifert $3$-manifold." Advances in Theoretical and Mathematical Physics 24, no. 2 (2020): 279–304. http://dx.doi.org/10.4310/atmp.2020.v24.n2.a2.

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Closset, Cyril, and Heeyeon Kim. "Three-dimensional 𝒩 = 2 supersymmetric gauge theories and partition functions on Seifert manifolds: A review." International Journal of Modern Physics A 34, no. 23 (August 20, 2019): 1930011. http://dx.doi.org/10.1142/s0217751x19300114.

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We give a pedagogical introduction to the study of supersymmetric partition functions of 3D [Formula: see text] supersymmetric Chern–Simons-matter theories (with an [Formula: see text]-symmetry) on half-BPS closed three-manifolds — including [Formula: see text], [Formula: see text], and any Seifert three-manifold. Three-dimensional gauge theories can flow to nontrivial fixed points in the infrared. In the presence of 3D [Formula: see text] supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

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BAADER, SEBASTIAN, KAI CIELIEBAK, and THOMAS VOGEL. "LEGENDRIAN RIBBONS IN OVERTWISTED CONTACT STRUCTURES." Journal of Knot Theory and Its Ramifications 18, no. 04 (April 2009): 523–29. http://dx.doi.org/10.1142/s0218216509006999.

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We show that a null-homologous transverse knot K in the complement of an overtwisted disk in a contact 3-manifold is the boundary of a Legendrian ribbon if and only if it possesses a Seifert surface S such that the self-linking number of K with respect to S satisfies sl (K, S) = -χ(S). In particular, every null-homologous topological knot type in an overtwisted contact manifold can be represented by the boundary of a Legendrian ribbon. Finally, we show that a contact structure is tight if and only if every Legendrian ribbon minimizes genus in its relative homology class.

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Scardua, Bruno. "A Measurable Stability Theorem for Holomorphic Foliations Transverse to Fibrations." International Journal of Differential Equations 2012 (2012): 1–6. http://dx.doi.org/10.1155/2012/585298.

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We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.

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Planat, Michel, Raymond Aschheim, Marcelo M. Amaral, and Klee Irwin. "Quantum Computing, Seifert Surfaces, and Singular Fibers." Quantum Reports 1, no. 1 (April 24, 2019): 12–22. http://dx.doi.org/10.3390/quantum1010003.

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The fundamental group π 1 ( L ) of a knot or link L may be used to generate magic states appropriate for performing universal quantum computation and simultaneously for retrieving complete information about the processed quantum states. In this paper, one defines braids whose closure is the L of such a quantum computer model and computes their braid-induced Seifert surfaces and the corresponding Alexander polynomial. In particular, some d-fold coverings of the trefoil knot, with d = 3 , 4, 6, or 12, define appropriate links L, and the latter two cases connect to the Dynkin diagrams of E 6 and D 4 , respectively. In this new context, one finds that this correspondence continues with Kodaira’s classification of elliptic singular fibers. The Seifert fibered toroidal manifold Σ ′ , at the boundary of the singular fiber E 8 ˜ , allows possible models of quantum computing.

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Mejía, Maria E., and Reinaldo R. Rosa. "An Alternative Manifold for Cosmology Using Seifert Fibered and Hyperbolic Spaces." Applied Mathematics 05, no. 06 (2014): 1013–28. http://dx.doi.org/10.4236/am.2014.56096.

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Allenby, R. B. J. T., Goansu Kim, and C. Y. Tang. "Conjugacy separability of Seifert 3-manifold groups over non-orientable surfaces." Journal of Algebra 323, no. 1 (January 2010): 1–9. http://dx.doi.org/10.1016/j.jalgebra.2009.10.003.

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35

Savelyev, Yasha. "Extended Fuller index, sky catastrophes and the Seifert conjecture." International Journal of Mathematics 29, no. 13 (December 2018): 1850096. http://dx.doi.org/10.1142/s0129167x18500969.

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We extend the classical Fuller index, and use this to prove that for a certain general class of vector fields [Formula: see text] on a compact smooth manifold, if a homotopy of smooth non-singular vector fields starting at [Formula: see text] has no sky catastrophes as defined by the paper, then the time 1 limit of the homotopy has periodic orbits. This class of vector fields includes the Hopf vector field on [Formula: see text]. A sky catastrophe is a kind of bifurcation originally discovered by Fuller. This answers a natural question that existed since the time of Fuller’s foundational papers. We also put strong constraints on the kind of sky-catastrophes that may appear for homotopies of Reeb vector fields.

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Allenby, R. B. J. T., Goansu Kim, and C. Y. Tang. "Outer automorphism groups of Seifert 3-manifold groups over non-orientable surfaces." Journal of Algebra 322, no. 4 (August 2009): 957–68. http://dx.doi.org/10.1016/j.jalgebra.2009.05.015.

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37

Tosun, Bülent. "Stein domains in ℂ² with prescribed boundary." Advances in Geometry 22, no. 1 (January 1, 2022): 9–22. http://dx.doi.org/10.1515/advgeom-2021-0035.

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Abstract We give an overview of the research related to the topological characterization of Stein domains in complex two-dimensional space, and an instance of their many important connections to smooth manifold topology in dimension four. One goal is to motivate and explain the following remarkable conjecture of Gompf: no Brieskorn integral homology sphere (other than S 3) admits a pseudoconvex embedding in ℂ2, with either orientation. We include some new examples and results that consider the conjecture for families of rational homology spheres which are Seifert fibered, and integral homology spheres which are hyperbolic.

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BARBOT, THIERRY, and SÉRGIO R. FENLEY. "Classification and rigidity of totally periodic pseudo-Anosov flows in graph manifolds." Ergodic Theory and Dynamical Systems 35, no. 6 (July 3, 2014): 1681–722. http://dx.doi.org/10.1017/etds.2014.9.

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In this article we analyze totally periodic pseudo-Anosov flows in graph 3-manifolds. This means that in each Seifert fibered piece of the torus decomposition, the free homotopy class of regular fibers has a finite power which is also a finite power of the free homotopy class of a closed orbit of the flow. We show that each such flow is topologically equivalent to one of the model pseudo-Anosov flows which we previously constructed in Barbot and Fenley (Pseudo-Anosov flows in toroidal manifolds.Geom. Topol. 17(2013), 1877–1954). A model pseudo-Anosov flow is obtained by glueing standard neighborhoods of Birkhoff annuli and perhaps doing Dehn surgery on certain orbits. We also show that two model flows on the same graph manifold are isotopically equivalent (i.e. there is a isotopy of$M$mapping the oriented orbits of the first flow to the oriented orbits of the second flow) if and only if they have the same topological and dynamical data in the collection of standard neighborhoods of the Birkhoff annuli.

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39

STITZ, CARL J. "A COMBINATORIAL APPROACH TO LINKING NUMBERS IN RATIONAL HOMOLOGY SPHERES." Journal of Knot Theory and Its Ramifications 09, no. 05 (August 2000): 703–11. http://dx.doi.org/10.1142/s0218216500000396.

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In this paper we find a method to compute the classical Seifert-Threlfall linking number for rational homology spheres without using 2-chains bounded by the curves in question. By using a Heegaard diagram for the manifold, we describe link isotopy combinatorially using the three traditional Reidemeister moves along with a fourth move which is essentially a Kirby move along the characteristic curves. This result is mathematical folklore which we set in print. We then use this combinatorial description of link isotopy to develop and prove the invariance of linking numbers. Once the linking numbers are in place, matrix invariants such as the Alexander polynomial can be computed.

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40

Huisman, Johannes, and Frédéric Mangolte. "Every orientable Seifert 3-manifold is a real component of a uniruled algebraic variety." Topology 44, no. 1 (January 2005): 63–71. http://dx.doi.org/10.1016/j.top.2004.03.003.

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OHTSUKI, TOMOTADA. "INVARIANTS OF KNOTS DERIVED FROM EQUIVARIANT LINKING MATRICES OF THEIR SURGERY PRESENTATIONS." International Journal of Mathematics 20, no. 07 (July 2009): 883–913. http://dx.doi.org/10.1142/s0129167x09005583.

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The quantum U(1) invariant of a closed 3-manifold M is defined from the linking matrix of a framed link of a surgery presentation of M. As an equivariant version of it, we formulate an invariant of a knot K from the equivariant linking matrix of a lift of a framed link of a surgery presentation of K. We show that this invariant is determined by the Blanchfield pairing of K, or equivalently, determined by the S-equivalent class of a Seifert matrix of K, and that the "product" of this invariant and its complex conjugation is presented by the Alexander module of K. We present some values of this invariant of some classes of knots concretely.

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42

Aguilar-Cabrera, Haydée. "Open-book decompositions of 𝕊5 and real singularities." International Journal of Mathematics 25, no. 09 (August 2014): 1450085. http://dx.doi.org/10.1142/s0129167x14500852.

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In this article, we study the topology of the family of real analytic germs F : (ℂ3, 0) → (ℂ, 0) given by [Formula: see text] with p, q, r ∈ ℕ, p, q, r ≥ 2 and (p, q) = 1. Such a germ has an isolated singularity at 0 and gives rise to a Milnor fibration [Formula: see text]. Moreover, it is known that the link LF is a Seifert manifold and that it is always homeomorphic to the link of a complex singularity. However, we prove that in almost all the cases the open-book decomposition of 𝕊5 given by the Milnor fibration of F cannot come from the Milnor fibration of a complex singularity in ℂ3.

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43

Brzeziński, Tomasz, and Simon A. Fairfax. "Bundles over Quantum RealWeighted Projective Spaces." Axioms 1, no. 2 (September 17, 2012): 201–25. http://dx.doi.org/10.3390/axioms1020201.

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The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.

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44

Martino, Armando. "A proof that all Seifert 3-manifold groups and all virtual surface groups are conjugacy separable." Journal of Algebra 313, no. 2 (July 2007): 773–81. http://dx.doi.org/10.1016/j.jalgebra.2006.07.003.

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45

BISWAS, INDRANIL, MAHAN MJ, and HARISH SESHADRI. "3-MANIFOLD GROUPS, KÄHLER GROUPS AND COMPLEX SURFACES." Communications in Contemporary Mathematics 14, no. 06 (October 8, 2012): 1250038. http://dx.doi.org/10.1142/s0219199712500381.

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Let G be a Kähler group admitting a short exact sequence [Formula: see text] where N is finitely generated. (i) Then Q cannot be non-nilpotent solvable. (ii) Suppose in addition that Q satisfies one of the following: (a) Q admits a discrete faithful non-elementary action on ℍn for some n ≥ 2. (b) Q admits a discrete faithful non-elementary minimal action on a simplicial tree with more than two ends. (c) Q admits a (strong-stable) cut R such that the intersection of all conjugates of R is trivial. Then G is virtually a surface group. It follows that if Q is infinite, not virtually cyclic, and is the fundamental group of some closed 3-manifold, then Q contains as a finite index subgroup either a finite index subgroup of the three-dimensional Heisenberg group or the fundamental group of the Cartesian product of a closed oriented surface of positive genus and the circle. As a corollary, we obtain a new proof of a theorem of Dimca and Suciu in [Which 3-manifold groups are Kähler groups? J. Eur. Math. Soc.11 (2009) 521–528] by taking N to be the trivial group. If instead, G is the fundamental group of a compact complex surface, and N is finitely presented, then we show that Q must contain the fundamental group of a Seifert-fibered 3-manifold as a finite index subgroup, and G contains as a finite index subgroup the fundamental group of an elliptic fibration. We also give an example showing that the relation of quasi-isometry does not preserve Kähler groups. This gives a negative answer to a question of Gromov which asks whether Kähler groups can be characterized by their asymptotic geometry.

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Hayat-Legrand, Claude, Shicheng Wang, and Heiner Zieschang. "Minimal Seifert manifolds." Mathematische Annalen 308, no. 4 (August 1, 1997): 673–700. http://dx.doi.org/10.1007/s002080050096.

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47

HILLMAN, J. A. "-MANIFOLDS." Journal of the Australian Mathematical Society 105, no. 1 (December 4, 2017): 46–56. http://dx.doi.org/10.1017/s1446788717000258.

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We show that closed $\mathbb{S}\text{ol}^{3}\times \mathbb{E}^{1}$-manifolds are Seifert fibred, with general fibre the torus, and base one of the flat 2-orbifolds $T,Kb,\mathbb{A},\mathbb{M}b,S(2,2,2,2),P(2,2)$ or $\mathbb{D}(2,2)$, and outline how such manifolds may be classified.

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48

Barbot, Thierry. "Flag structures on Seifert manifolds." Geometry & Topology 5, no. 1 (March 23, 2001): 227–66. http://dx.doi.org/10.2140/gt.2001.5.227.

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49

Kamishima, Yoshinobu, and Takashi Tsuboi. "CR-structures on Seifert manifolds." Inventiones mathematicae 104, no. 1 (December 1991): 149–63. http://dx.doi.org/10.1007/bf01245069.

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Mijatović, Aleksandar. "Triangulations of Seifert fibred manifolds." Mathematische Annalen 330, no. 2 (June 23, 2004): 235–73. http://dx.doi.org/10.1007/s00208-004-0547-9.

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