Seifert Fibered Research Articles

Reidemeister torsion, the Alexander polynomial and U (1,1) Chern-Simons theory

We show that the U (1,1) (super) Chern-Simons theory is one-loop exact. This provides a direct proof of the relation between the Alexander polynomial and analytic and Reidemeister torsions. We then compute explicitly the torsions of Lens spaces and Seifert manifolds using surgery and the S and T matrices of the U (1,1) Wess-Zumino-Witten model recently determined, with complete agreement with known results. U (1,1) quantum field theories and the Alexander polynomial thus provide “toy” models with a non-trivial topological content, where all ideas put forward by Witten for SU (2) and the Jones polynomial can be explicitly checked, at finite k. Some simple but presumably generic aspects of non-compact groups, like the modified relation between Chern-Simons and Wess-Zumino-Witten theories, are also illustrated. We comment on the closely related case of GL (1,1).

Co-Hopficity of Seifert-bundle groups

A group $G$ is cohopfian, if every monomorphism $G \to G$ is an automorphism. In this paper, we answer the cohopficity question for the fundamental groups of compact Seifert fiber spaces (or Seifert bundles, in the current vernacular). If $M$ is a closed Seifert bundle, then the following are equivalent: (a) ${\pi _1}M$ is cohopfian; (b) $M$ does not cover itself nontrivially; (c) $M$ admits a geometric structure modeled on ${S^3}$ or on ${\tilde {\text {SL}_2\mathbf {R}}}$. If $M$ is a compact Seifert bundle with nonempty boundary, then ${\pi _1}M$ is not cohopfian.

Homotopy and isotopy in dimension three

Let M be a closed p2-irreducible 3-manifold. It is a long standing problem to decide if homotopic homeomorphisms of M must be isotopic. The answer is now known to be affirmative if M is Haken, [Wal], see also [L], or if M is a Seifert fiber space [Ho-R] [Bon] [B-R] [A] [R] [Scl] [B-O], and for a few other special manifolds [B-R]. Thus is now seems reasonable to conjecture that the answer is always affirmative. However, if one considers reducible manifolds, there is a counter example [F W]. In this paper, we further enlarge the class of 3-manifolds for which the above conjecture can be proved. If a closed P2-irreducible 3-manifold is non-orientable, it must be Haken, so we consider only orientable 3-manifolds in the rest of this paper. Let M be an orientable 3-manifold, let F be a closed orientable surface not S 2 and let f: F ~ M be an immersion which injects n~ (F). Let Mr denote the cover of M such that nj (Mr) equals f,(nl(F)) and let M denote the universal cover of M. We will suppose that the lift of f into MF is an embedding. (Note that this is automatic iff is least area in the smooth or PL sense.) Thus the pre-image in M of f(F) consists of an embedded plane /7 which covers F in M,, and the translates of // by hi(M). We will say that f has the k-plane property if, given k distinct translates of H, some pair is disjoint. In this paper we will consider the case when k equals 3. A map with the 3-plane property has no transverse triple points. We will say that f has the 1-line-intersection property if two distinct translates of /7 are disjoint or intersect transversely in a single line. The main result of this paper is THEOREM 1.1. Let M be a closed orientable irreducible 3-manifold which is neither Haken nor a Seifert fiber space. If there is a closed orientable surface F, not S z, and an immersion f:F~M which injects nl(F) and has the 3-plane and l-line-intersection properties, then homotopic homeomorphisms of M are isotopic. In [H-S], we show that if M satisfies the hypotheses of this theorem, and M is homotopy equivalent to an irreducible 3-manifold N, then M and N are homeomor

On the existence of minimal Seifert manifolds

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.

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT 2

We describe a cut-and-paste method for computing Chern-Simons invariant of flatG-connections on 3-manifolds decomposed along tori, especially forG=SU(2) andSL(2,C). We use this method to make computations ofSU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stern's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flatSL(2,C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.

Vafa's formula and equivariant K-theory

Vafa's formula of the Euler characteristic of Calabi-Yau hypersurface in weighted projective four-space is identified with the Euler characteristic of S 1-equivariant K-theory of the Seifert fibration over it.

Expansive flows on Seifert manifolds and on torus bundles

We show that any expansive flow on a 3-manifold which is a Seifert fibration or a torus bundle overS1 is topologically equivalent to a transitive Anosov flow. This is achieved by analyzing the trace of the stable foliation (with singularities) of the flow on incompressible tori embedded in such a manifold.

Computational aspects of affine representations for torsion free nilpotent groups via the Seifert construction

We study and describe faithful affine representations of a certain canonical form for torsion free, finitely generated nilpotent groups. The algebraic set-up used is that of the Seifert Fiber Space Construction conceived by Kyung Bai Lee and Frank Raymond, and can be considered as suitable for an iteration procedure. The representations obtained realise such a group as the fundamental group of a compact, complete affinely flat manifold. We investigate computational aspects for explicit constructions of these representations. An equivalent description for canonical form representations, in terms of matrices over polynomial rings, is presented. As these groups are uniform lattices in nilpotent Lie groups, it is interesting to draw the related picture on the Lie algebra level. An example (due to Dan Segal and Fritz Grunewald) is used to illustrate that the iteration aspect of this set-up should be well understood and will need a very careful treatment. Finally, we present canonical representations for several infinite families of groups, thus also giving positive evidence for the nilpotent case of a conjecture of Milnor's.

Flat conformal structures on 3-manifolds. I. Uniformization of closed Seifert manifolds

Flat conformal structures on 3-manifolds. I. Uniformization of closed Seifert manifolds

S- and T-matrices for the super U (1,1) WZW model application to surgery and 3-manifolds invariants based on the Alexander-Conway polynomial

We carry on (in a self-contained fashion) the study of the Alexander-Conway invariant from the quantum field theory point of view started earlier. We investigate for that purpose various aspects of WZW models on supergroups. We first discuss in details S- and T-matrices for the U(1,1) super WZW model and obtain, for the level k an integer, new finite-dimensional representations of the modular group. These have the remarkable property that some of the S-matrix elements are infinite (we show how to properly handle such divergences). Moreover, typical and atypical representations as well as indecomposable blocks are mixed: truncation to maximally atypical representations, as advocated in some recent papers, is not consistent. Using our approach, multivariable Alexander invariants for links in S 3 can now be fully computed by surgery. Examples of torus and cable knots are discussed. Consistency with classical results provides independent checks of the solution of the U(1,1) WZW model. The main topological application of this work is the computation of Alexander invariants for 3-manifolds and more generally for links in 3-manifolds. Invariants of 3-manifolds themselves seem to depend trivially on the level k, but still contain interesting topological information. For Seifert manifolds for instance, they essentially coincide with the order (number of elements) of the first homology group. Examples of invariants of links in 3-manifolds are given. They exhibit interesting arithmetic properties.

RECOGNIZING NONORIENTABLE SEIFERT BUNDLES

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.

COMBINATORIAL CALCULATION OF THE VARIOUS NORMALIZATIONS OF THE WITTEN INVARIANTS FOR 3-MANIFOLDS

Polynomial formulae for the Witten invariant of 3-manifolds for manifolds which bound plumbed 4-manifolds are developed via the combinatorial approach of Lickorish. A number of tables of calculations of the various normalizations of this invariant are presented for a variety of Seifert fiber spaces over S2.

Homotopy equivalence and hemeomorphism of 3-manifolds

IN THIS paper we extend the class of 3-manifolds which are determined up to homeomorphism by their fundamental groups to the class of closed orientable irreducible 3-manifolds containing a singular surface satisfying two properties, the l-line-intersection property and the 4-plane property. A basic problem in the classification of 3-dimensional manifolds is to decide to what extent the homotopy type of a closed manifold determines the manifold up to homeomorphism. In the case of 3-manifolds with finite fundamental groups, it is known that there are homotopy equivalent manifolds which are not homeomorphic, but there are no known examples of closed orientable irreducible 3-manifolds with isomorphic infinite fundamental groups which are not homcomorphic. Waldhausen [30] and Heil Cl23 proved that if M and M’ are Haken 3-manifolds which have isomorphic fundamental groups then they are homcomorphic. If M and M’ arc hyperbolic then the Mostow rigidity theorem implies the same result [19]. but if only M is assumed to be hyperbolic then it is unknown. Boehme [3] extended Waldhausen’s theorem to certain non-Haken Seifert fiber spaces. Scott [27] showed that a closed orientable irreducible 3-manifold which is homotopy equivalent to a Seifcrt fiber space with infinite fundamental group is homeomorphic to that Seifert fiber space. Many of these Scifert fiber spaces are non-Haken. To date however, Seifert fiber spaces have provided the only examples of non-Haken 3-manifolds which are known to be determined up to homeomorphism by their fundamental groups. For manifolds with boundary there are simple examples of non-homemorphic homotopy equivalent Haken manifolds, such as the product with the circle of a thricepunctured sphere and the product with the circle of a once-punctured torus. Such examples are well understood in terms of the characteristic decomposition of the 3-manifold [17] [16]. Another possible source of troublesome examples comes by taking the connected sum of a 3-manifold with a fake homotopy 3-sphere, resulting in a homotopy equivalent non-homeomorphic 3-manifold. This possibility would be ruled out by a successful solution to the Poincari conjecture. To get around this potential problem we work with irreducible 3-manifolds, in which any 2-sphere bounds a bail. Since non-orientable P2-irreducible 3-manifolds are always Haken, we restrict our attention to the orientable case. For simplicity of notation, we sometimes follow the convention of not distinguishing between a map of a surface into a manifold M and the image of the map. When it is

ON SEIFERT CIRCLES AND FUNCTORS FOR TANGLES

The properties of the Seifert circles in an oriented tangle diagram are exploited to prove a theorem that asserts that every (n,n)-tangle diagram is isotopic to a partially closed braid, and a second one that facilitates the assignment of wrong-way edges, one on each Seifert circle, in a tangle diagram. These result are used to identify the structure of an abstract algebra on which a functor for the isotopy of general tangles may be constructed. Any finite dimensional irreducible representation of a quasitriangular Hopf algebra is a realization of this algebra.

Finite group actions on handlebodies and equivariant Heegaard genus for 3-manifolds

Every finite group of symmetries (homeomorphisms) of a compact bounded surface of algebraic genus g acts, by taking the product with the interval [0,1], also on the 3-dimensional handlebody V g of genus g. In both cases, the maximal possible order of such a group is 12( g−1), and we call such a group a maximal bounded surface respectively handlebody group. Here we construct the first examples of maximal handlebody groups which are not maximal bounded surface groups. Our examples lead in a natural way to the notion of an equivariant Heegaard genus for finite group actions on 3-manifolds; we compute this genus for some classes of interesting examples. The notion of an equivariant Heegaard genus gives a certain hierarchy for finite group actions on a 3-manifold and the notion of a maximally symmetric 3-manifold; we show that the irreducible maximally symmetric 3-manifolds belong to the hyperbolic geometry, or are Seifert fibered of a very special type.

CR-structures on Seifert manifolds

CR-structures on Seifert manifolds

Examples of 3-knots with no minimal Seifert manifolds

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.)

Computer calculation of Witten's 3-manifold invariant

Witten's 2+1 dimensional Chern-Simons theory is exactly solvable. We compute the partition function, a topological invariant of 3-manifolds, on generalized Seifert spaces. Thus we test the path integral using the theory of 3-manifolds. In particular, we compare the exact solution with the asymptotic formula predicted by perturbation theory. We conclude that this path integral works as advertised and gives an effective topological invariant.

Genus 2 Heegaard decompositions of small Seifert manifolds

Les scindements et décompositions de Heegaard de genre 2 des variétés de Seifert de base S ayant 3 fibres exceptionnelles sont classifiés à isotopie et homéomorphisme près. En général il y a 3 classes d’isotopie distinctes de scindements de Heegaard et 6 classes d’isotopie distinctes de décompositions de Heegaard. De plus on détermine précisément les classes d’homéomorphie qui ne sont pas des classes d’isotopie.

Instanton Homology of Seifert Fibred Homology Three Spheres

For an oriented integral homology 3-sphere 2, A. Casson has introduced an integer invariant A(2) that is defined by using the space £%(2) of conjugacy classes of irreducible representations of ^ ( 2 ) into SU(2) (see [1]). This invariant A(2) can be computed from a surgery or Heegard description of 2 and satisfies A(2) = JU(2) (mod 2), where /i(2) is the Kervaire-Milnor-Rochlin invariant of 2. This powerful new invariant was used to settle an outstanding problem in 3-manifold topology; namely, showing that if 2 is a homotopy 3-sphere, then ju(2) = O. C. Taubes [17], utilizing gauge-theoretic considerations, has reinterpreted Casson's invariant in terms of flat connections. Refining this approach, A. Floer [13] has recently defined another invariant of 2 , its 'instanton homology', which takes the form of an abelian group /*(2) with a natural Z8-grading that is an enhancement of A(2) in that