Dynamical incoherence for a large class of partially hyperbolic diffeomorphisms

We show that if a partially hyperbolic diffeomorphism of a Seifert manifold induces a map in the base which has a pseudo-Anosov component then it cannot be dynamically coherent. This extends [C. Bonatti, A. Gogolev, A. Hammerlindl and R. Potrie. Anomalous partially hyperbolic diffeomorphisms III: Abundance and incoherence. Geom. Topol., to appear] to the whole isotopy class. We relate the techniques to the study of certain partially hyperbolic diffeomorphisms in hyperbolic 3-manifolds performed in [T. Barthelmé, S. Fenley, S. Frankel and R. Potrie. Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part I: The dynamically coherent case. Preprint, 2019, arXiv:1908.06227; Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, part II: Branching foliations. Preprint, 2020, arXiv: 2008.04871]. The appendix reviews some consequences of the Nielsen–Thurston classification of surface homeomorphisms for the dynamics of lifts of such maps to the universal cover.

Finite, Fiber- and Orientation-Preserving Group Actions on Totally Orientable Seifert Manifolds

In 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.

Coverings of torus knots in S2 × S1 and universals

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.

Open-book decompositions of 𝕊5 and real singularities

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.

Bundles over Quantum RealWeighted Projective Spaces

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.

Essential Surfaces in Graph Link Exteriors

An 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.

On the existence of minimal Seifert manifolds

For 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.

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.)