Seifert Fibered Research Articles

Computer Calculation of the Degree of Maps into the Poincare Homology Sphere

Let M and P be Seifert 3-manifolds. Is there a degree one map f : M → P? The problem was completely solved by Hayat Legrand, Wang, and Zieschang for all casesexcept when P is the Poincaré homology sphere. We investigate the remaining case by elaborating and implementing a computer algorithm that calculates the degree. As a result, we get an explicit experimental expression for the degree through numerical invariants of the induced homomorphism f# : π1(M) → π1(P).

Generalized Neuwirth Groups and Seifert Fibered Manifolds

The topological properties of the generalized Neuwirth groups, k n are discussed. For example, we demonstrate that the group, k n is the fundamental group of the Seifert fibered space Σ k n . Moreover, discuss some other invariants and algebraic properties of the above groups.

A surgery formula for the [formula omitted] -invariant

We introduce a relative version, μ ̄ ′ , of the μ ̄ -invariant of Neumann and Siebenmann for graph 3-manifolds, and use it to prove a surgery formula for the μ ̄ -invariant. We further identify μ ̄ ′ with the linking number of certain Montesinos links, and relate it to the Floer homology in the special case of Seifert manifolds.

A geometric characteristic splitting in all dimensions

We prove the existence of a geometric characteristic submanifold for non-positively curved manifolds of any dimension greater than or equal to three. In dimension three, our result is a geometric version of the topological characteristic submanifold theorem due to Jaco, Shalen and Johannson. In the 1970's, Jaco and Shalen (JS) and Johannson (J) showed that a closed ori- entable Haken 3-manifold M has a canonical family of disjoint embedded incom- pressible tori, no two of which are parallel, such that the complementary pieces of M are either Seifert bre spaces or are atoroidal. They dened the characteristic submanifold V (M )o f Mto be essentially the union of the Seifert manifold pieces of M. Further, they showed that any essential map of the torus into M is homo- topic into V (M ). Johannson called this last property the Enclosing Property. For brevity, we will refer to these results as the JSJ results. In this paper, we show that if M is a closed manifold of dimension three or more, and if M has a Riemannian metric of non-positive curvature, then either the metric on M is flat or there is a precisely analogous decomposition of M along codimension one submanifolds. Further these submanifolds are totally geodesic in M and are flat in the metric induced from M. Note that in dimension three, a flat manifold must be a Seifert bre space, so that, in particular, our arguments give a new proof of the JSJ results for the special case when M is assumed to have a metric of non-positive curvature. In dimension four or more, a flat manifold need not be a Seifert manifold, see the example near the end of section 1, so this case really is dierent in higher dimensions. We also prove that essentially the same

The cohomology ring of a class of Seifert manifolds

The cohomology groups of the Seifert manifolds are well known. In this article a method is given to compute the cup products in the cohomology ring of any orientable Seifert manifold whose associated orbit surface is S 2 , and for any coefficients. In particular the Z/2 cohomology ring is completely determined. This is applied to determine the existence of degree 1 maps from the Seifert manifold to RP 3 , and to the Lusternik–Schnirelmann category.

Subgroup separability, knot groups and graph manifolds

This paper answers a question of Burns, Karrass and Solitar by giving examples of knot and link groups which are not subgroup-separable. For instance, it is shown that the fundamental group of the square knot complement is not subgroup separable. We characterise the Graph Manifolds with subgroup separable fundamental group as precisely the geometric ones, i.e. the Seifert Fibered 3-manifolds and the Sol manifolds, and show that there is a specific non-subgroup separable group which is a subgroup in all other cases.

Weakly reducible Heegaard splittings of Seifert fibered spaces

We prove that for exceptional Seifert manifolds all weakly reducible Heegaard splittings are reducible. This provides the missing case for the Main Theorem in (Moriah and Schultens, to appear). It follows that for all orientable Seifert fibered spaces which fiber over an orientable base space, irreducible Heegaard splittings are either horizontal or vertical.

Smooth structures on Σ× R

We prove that there are uncountably many smooth structures on the four manifold Σ× R if Σ belongs to one of the following classes: 1. rational homology spheres; 2. Seifert 3-manifolds; 3. almost flat 3-manifolds. Moreover, when Σ is a rational homology sphere, these smooth structures on Σ× R can be embedded into the connected sums of CP 2 .

On the Heegaard genus and tri-genus of non-orientable Seifert 3-manifolds

We calculate the Heegaard genus, h(M), of the closed non-orientable Seifert manifolds.If a 3-manifold M admits a decomposition M=H1∪H2∪H3 into three orientable handlebodies of genera g1,g2,g3, respectively, and g1≤g2≤g3, we call the triple (g1,g2,g3) the tri-genus of M if (g1,g2,g3) is minimal among all such triples ordered lexicographically.We compute the tri-genus (g1,g2,g3) of all non-orientable Seifert manifolds M which admit an S1-bundle structure with fiber an orientable surface. In this case the number g3 is much bigger than h(M) for a fixed M.We obtain also that h(M) is an upper bound for the number g3 in case M is a non-orientable Seifert manifold which does not admit an S1-bundle structure.We see that, although one could expect a relation between the number g3 and the Heegaard genus, this relation, if any, can not be simple.

Witten-Reshetikhin-Turaev Invariants of�Seifert Manifolds

For Seifert homology spheres, we derive a holomorphic function of K whose value at integer K is the sl2 Witten–Reshetikhin–Turaev invariant, ZK, at q= exp 2πi/K. This function is expressed as a sum of terms, which can be naturally corresponded to the contributions of flat connections in the stationary phase expansion of the Witten–Chern–Simons path integral. The trivial connection contribution is found to have an asymptotic expansion in powers of K−1 which, for K an odd prime power, converges K-adically to the exact total value of the invariant ZK at that root of unity. Evaluations at rational $K$ are also discussed. Using similar techniques, an expression for the coloured Jones polynomial of a torus knot is obtained, providing a trivial connection contribution which is an analytic function of the colour. This demonstrates that the stationary phase expansion of the Chern–Simons–Witten theory is exact for Seifert manifolds and for torus knots in S3. The possibility of generalising such results is also discussed.

Essential laminations in Seifert-fibered spaces: Boundary behavior

We show that an essential lamination in a Seifert-fibered space M rarely meets the boundary of M in a Reeb-foliated annulus.

Discriminant of a Germ Φ: (C2 , 0)→(C2 , 0) and Seifert Fibred Manifolds

Let f and g be two analytic function germs without common branches. We define the Jacobian quotients of (g, f), which are ‘first order invariants’ of the discriminant curve of (g, f), and we prove that they only depend on the topological type of (g, f). We compute them with the help of the topology of (g, f). If g is a linear form transverse to f, the Jacobian quotients are exactly the polar quotients of f and we affirm the results of D. T. Lê, F. Michel and C. Weber.

Toroidal Dehn surgeries on knots in lens spaces

Let M be a compact, orientable 3-manifold with ∂ M a torus. If r is a slope on ∂ M (the isotopy class of an unoriented essential simple loop), then we can form the closed 3-manifold M ( r ) by gluing a solid torus V r to M along their boundaries in such a way that r bounds a disc in V r . We say that M ( r ) is obtained from M by r -Dehn filling. Assume now that M contains no essential sphere, disc, torus or annulus. Then, by Thurston's Geometrization Theorem for Haken manifolds [ T1 , T2 ], M is hyperbolic, in the sense that int M has a complete hyperbolic structure of finite volume. Furthermore, M ( r ) is hyperbolic for all but finitely many r [ T1 , T2 ] and the precise nature of the set of exceptional slopes E ( M )={ r : M ( r ) is not hyperbolic} has been the subject of a considerable amount of investigation. The maximal observed value of e ( M )=[mid ] E ( M )[mid ] (the cardinality of E ( M )) is 10, realized, apparently uniquely, by the exterior of the figure eight knot [ T1 ]. Let Δ( r 1 , r 2 ) denote as usual the minimal geometric intersection number of two slopes r 1 and r 2 . If [Sscr ] is any set of slopes, then clearly any upper bound for Δ([Sscr ])=max{Δ( r 1 , r 2 ): r 1 , r 2 ∈[Sscr ]} gives one for [mid ][Sscr ][mid ]. For example, one can check (using [ GLi , lemma 2·1]) that for 1[les ]Δ([Sscr ])[les ]10, the maximum values of [mid ][Sscr ][mid ] are as given in Table 1. In particular, any upper bound for Δ( M )=Δ( E ( M )) gives a corresponding bound for e ( M ). (The maximal observed value of Δ( M ) is 8, realized by the figure eight knot exterior and the figure eight sister manifold [ T1 , HW ].) If M ( r ) is not hyperbolic, then it is either reducible (contains an essential sphere), toroidal (contains an essential torus), a small Seifert fibre space (one with base S 2 and at most three singular fibres), or a counterexample to the Geometrization Conjecture [ T1 , T2 ]. A survey of the presently known upper bounds on the distances Δ( r 1 , r 2 ) between various classes of exceptional slopes r 1 and r 2 , and the maximal values realized by known examples, is given in [ Go2 ]. (See also [ Wu2 ] for a discussion of the additional cases that arise when M has more than one boundary component.) In the present note we prove the following theorem, which deals with one further pair of possibilities.

Wrapped branes and confined momentum

We present string-like soliton solutions of three-dimensional gravity, coupled to a compact scalar field x 11 and Kaluza-Klein reduced on a circle. These solitons carry fractional magnetic flux with respect to the Kaluza-Klein gauge field. Summing over such “Kaluza-Klein flux tubes” is shown to imply summing over a subclass of three-dimensional topologies (Seifert manifolds). It is also argued to imply an area law for the Wilson loop of the Kaluza-Klein gauge field; the confined charge is nothing but Kaluza-Klein momentum. Applied to the membrane of M-theory, this is interpreted in terms of “wrapping” of the M-brane around its eleventh embedding dimension x 11.

Generalized graph manifolds and their effective recognition

A generalized graph manifold is a three-dimensional manifold obtained by gluing together elementary blocks, each of which is either a Seifert manifold or contains no essential tori or annuli. By a well-known result on torus decomposition each compact three-dimensional manifold with boundary that is either empty or consists of tori has a canonical representation as a generalized graph manifold. A short simple proof of the existence of a canonical representation is presented and a (partial) algorithm for its construction is described. A simple hyperbolicity test for blocks that are not Seifert manifolds is also presented.

3-dimensional Seiberg-Witten invariants and non-Kählerian geometry

The paper is concerned with Seiberg-Witten invariants of closed 3-manifolds and their relation to the 4-dimensional invariants. The main points are: 1. The introduction of parameter-dependent Seiberg-Witten invariants for 3-manifolds with b1 = 0. These invariants depend on a complicated chamber structure defined by a differential geometric Brill-Noether locus in the parameter space. 2. A comparison theorem which shows that the Seiberg-Witten invariants of 3-manifolds M with b_1(M ) ≥ 1 are simply the restrictions of the 4-dimensional invariants of the product S^1 × M to the set of pull-back Spinc-structures. As an immediate consequence one gets the wall crossing formula for the invariants of 3-dimensional manifolds with b1 = 1 from the 4-dimensional wall crossing formula for manifolds with b+ = 1. 3. A novel method to compute the Seiberg-Witten invariants of 3-manifolds M using (non-Kahlerian) complex geometry on S 1 × M , provided this product has such a structure which is compatible with a product metric and orientation induced from M . This method relies essentially on a Kobayashi-Hitchin correspondence, which is new in the non-Kahler case. 4. A complete description of those 3-manifolds to which our method applies. These are Riemannian manifolds which admit an oriented Riemannian foliation by geodesics. Such manifolds are always diffeomorphic to Seifert manifolds, but the foliation has not to be Seifert; it can have non-closed leaves. 5. Explicit computations of Seiberg-Witten invariants for the 3-sphere S^3 using non-Seifert Riemannian foliations. We show that the chamber structure induced by the Brill-Noether locus is very complicated. There exist countably many dif- ferent chambers, and the Seiberg-Witten invariant takes on all integer values.

Polyhedral fundamental domains for discrete subgroups of PSL (2, [formula omitted

In his Ph.D. thesis [4], Thomas Fischer suggested how to construct a fundamental domain for the action of a discrete cocompact subgroup Γ of I +( H 2) , the group of orientation-preserving isometries of the hyperbolic plane H 2, on this three-dimensional Liegroup I +( H 2) by left translation. We recall Fischer’s construction and discuss properties of the tiling of I +( H 2) by fundamental domains of Γ. In particular, its symmetries and their relation to the Seifert fibration of Γ\\I +( H 2) are considered.

Algebraic Criteria to Decide if a Finite Group Acts Effectively on a Model Aspherical Manifold

For an aspherical manifoldM, arising from a Seifert fiber space construction, it is known that, under some additional conditions onM, a finite abstract kernel ψ:F→Out(π1(M)) can be (effectively) geometrically realized by a group of fiber preserving homeomorphisms ofMif and only if ψ can be realized by an (admissible) group extension 1→π1(M)→E→F→1. Hence, the study of the symmetry of such a manifold (in terms of finite effective actions onM) can be converted into a group-theoretical study of realizing (algebraically) finite abstract kernelsF→Out(π1(M)). This question, conceptually, is well understood: there exists an extension realizing a given abstract kernel if and only if the corresponding third cohomology class, called the obstruction, vanishes. Unfortunately, a straightforward computation of this obstruction can be extremely hard. In this paper, we present some criteria which solve this problem for certain finite abstract kernels. Instead of using cohomological arguments, a completely different, rather technical and computational approach, based on the Reidemeister–Schreier method for presenting subgroups of finite index in a given finitely presented group, is followed. Not only the actual results but also this approach is of interest since it certainly allows one to produce similar criteria for other finite groups.

ESSENTIAL LAMINATIONS, EXCEPTIONAL SEIFERT-FIBERED SPACES, AND DEHN FILLING

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.

Exceptional Seifert-fibered spaces and Dehn surgery on 2-bridge knots

We show that non-integer surgery on a non-torus 2-bridge knot can never yield an exceptional Scifert-fibered space. In most cases, no surgery will yield an exceptional Scifert-fibered space.