The Reidemeister zeta function with applications to Nielsen theory and a connection with Reidemeister torsion

In this paper we study, the Reidemeister zeta function. We prove rationality and functional equations of the Reidemeister zeta function of an endomorphism of finite group. We also obtain these results for eventually commutative endomorphisms. These results are applied to the theory of Reidemeister and Nielsen numbers of self-maps of topological spaces. Our method is to identify the Reidemeister number of a group endomorphism with the number of fixed points in the unitary dual. As a consequence, we show that the Reidemeister torsion of the mapping torus of the unitary dual is a special value of the Reidemeister zeta function. We also prove certain congruences for Reidemeister numbers which are equivalent to a Euler product formula for the Reidemeister zeta function. The congruences are the same as those found by Dold for Lefschetz numbers.     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

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.     

Dehn surgery on knots in solid tori creating essential annuli

Let be a -manifold obtained by performing a Dehn surgery on a knot in a solid torus. In the present paper we study when contains a separating essential annulus. It is shown that does not contain such an annulus in the majority of cases. As a corollary, we prove that symmetric knots in the -sphere which are not periodic knots of period satisfy the cabling conjecture. This is an improvement of a result of Luft and Zhang. We have one more application to a problem on Dehn surgeries on knots producing a Seifert fibred manifold over the -sphere with exactly three exceptional fibres.

     

Reidemeister torsion of -representations

We compute the -Reidemeister torsion of torus bundles over which monodromies are hyperbolic elements in .

     

Reidemeister torsion and the K-theory of von Neumann algebras

We introduce a topological-type invariant for a cocompact properly discontinuous action of a discrete group on a Riemannian manifold generalizing classical notions of Reidemeister torsion. It takes values in the weak algebraic K-theory of the von Neumann algebra of . We give basic tools for its computation like sum and product formulas and calculate it in several cases. It encompasses, for instance, the Alexander polynomial and is related to analytic torsion.     

Scott's rigidity theorem for Seifert fibered spaces; revisited

We will present a new proof of the rigidity theorem for Seifert fibered spaces of infinite by Scott (1983) in the case when the base of the fibration is a hyperbolic triangle 2-orbifold. Our proof is based on arguments in the rigidity theorem for hyperbolic 3-manifolds by Gabai (1997).

     

Toroidal Dehn surgery on hyperbolic knots and hitting number

In this paper, we prove that for any positive even integer m, there exists a hyperbolic knot such that its longitudinal Dehn surgery yields a 3-manifold containing a unique separating, incompressible torus, which meets the core of the attached solid torus in m points minimally.     

Dehn surgeries on strongly invertible knots which yield lens spaces

In this article we show no Dehn surgery on nontrivial strongly invertible knots can yield the lens space for any integer . In order to do that, we determine band attaches to -torus links producing the trivial knot.

     

Classification of tight contact structures on small Seifert 3-manifolds with

We classify positive, tight contact structures on closed Seifert fibered 3-manifolds with base , three singular fibers and .

     

Reidemeister torsion and lens surgeries on knots in homology 3-spheres II

We introduce the norm and the order of a polynomial and of a homology lens space. We calculate the norm of the cyclotomic polynomials, and apply it to lens surgery problem for a knot whose Alexander polynomial is the same as an iterated torus knot.     

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of ₁ (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL ₂-analytic andL ₂-Reidemeister torsions are equal.The first three authors were supported by NSF. The first two authors wish to thank the Erwin-Schrödinger-Institute, Vienna, for hospitality and support during the summer of 1993 when part of this work was done.     

Seifert-fibered surgeries which do not arise from primitive/Seifert-fibered constructions

We construct two infinite families of knots each of which admits a Seifert fibered surgery with none of these surgeries coming from Dean's primitive/Seifert-fibered construction. This disproves a conjecture that all Seifert-fibered surgeries arise from Dean's primitive/Seifert-fibered construction. The -pretzel knot belongs to both of the infinite families.

     

Nonzero degree maps between closed orientable three-manifolds

This paper adresses the following problem: Given a closed orientable three-manifold , are there at most finitely many closed orientable three-manifolds 1-dominated by ? We solve this question for the class of closed orientable graph manifolds. More precisely the main result of this paper asserts that any closed orientable graph manifold 1-dominates at most finitely many orientable closed three-manifolds satisfying the Poincaré-Thurston Geometrization Conjecture. To prove this result we state a more general theorem for Haken manifolds which says that any closed orientable three-manifold 1-dominates at most finitely many Haken manifolds whose Gromov simplicial volume is sufficiently close to that of .

     

Distance between toroidal surgeries on hyperbolic knots in the -sphere

For a hyperbolic knot in the -sphere, at most finitely many Dehn surgeries yield non-hyperbolic -manifolds. As a typical case of such an exceptional surgery, a toroidal surgery is one that yields a closed -manifold containing an incompressible torus. The slope corresponding to a toroidal surgery, called a toroidal slope, is known to be integral or half-integral. We show that the distance between two integral toroidal slopes for a hyperbolic knot, except the figure-eight knot, is at most four.

     

Synchronism of an incompressible non-free Seifert surface for a knot and an algebraically split closed incompressible surface in the knot complement

We give a necessary and sufficient condition for knots to bound incompressible non-free Seifert surfaces.

     

Dehn surgery on the figure 8 knot: Immersed surfaces

It is known that about 70% of surgeries on the figure 8 knot give manifolds which contain immersed incompressible surfaces. We improve this to about 80% by giving a very simple proof that all even surgeries give manifolds containing such a surface. Moreover, we give a quick proof that every surgery is virtually Haken, thereby partially dealing with some exceptional cases in Baker's results.

     

Contact Dehn surgery, symplectic fillings, and Property P for knots

These are notes of a talk given at the Mathematische Arbeitstagung 2005 in Bonn. Following ideas of Özbağcı–Stipsicz, a proof based on contact Dehn surgery is given of Eliashberg's concave filling theorem for contact 3-manifolds. The role of the theorem in the Kronheimer–Mrowka proof of Property P for nontrivial knots is sketched.     

Positively oriented ideal triangulations on hyperbolic three-manifolds

Let M³ be a non-compact hyperbolic 3-manifold that has a triangulation by positively oriented ideal tetrahedra. We show that the gluing variety defined by the gluing consistency equations is a smooth complex manifold with dimension equal to the number of boundary components of M³. Moreover, we show that the complex lengths of any collection of non-trivial boundary curves, one from each boundary component, give a local holomorphic parameterization of the gluing variety. As an application, some estimates for the size of hyperbolic Dehn surgery space of once-punctured torus bundles are given.