Jonathan Bowden

Approximating C 0-foliations by contact structures

We show that any co-orientable foliation of dimension two on a closed orientable 3-manifold with continuous tangent plane field can be C0-approximated by both positive and negative contact structures unless all leaves of the foliation are simply connected. As applications we deduce that the existence of a taut C0-foliation implies the existence of universally tight contact structures in the same homotopy class of plane fields and that a closed 3-manifold that admits a taut C0-foliation of codimension-1 is not an L-space in the sense of Heegaard-Floer homology.

The topology of symplectic circle bundles

We consider circle bundles over compact three-manifolds with symplectic total spaces. We show that the base of such a space must be irreducible or the product of the two-sphere with the circle. We then deduce that such a bundle admits a symplectic form if and only if it admits one that is invariant under the circle action in three special cases: namely if the base is Seifert bered, has vanishing Thurston norm, or if the total space admits a Lefschetz bration.

Contact structures, deformations and taut foliations

Using deformations of foliations to contact structures as well as rigidity properties of Anosov foliations we provide infinite families of examples which show that the space of taut foliations in a given homotopy class of plane fields need not be path connected. Similar methods also show that the space of representations of the fundamental group of a hyperbolic surface to the group of smooth diffeomorphisms of the circle with fixed Euler class is in general not path connected. As an important step along the way we resolve the question of which universally tight contact structures on Seifert fibred spaces are deformations of taut or Reebless foliations when the genus of the base is positive or the twisting number of the contact structure in the sense of Giroux is non-negative.

The topology of Stein fillable manifolds in high dimensions I

We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashbergs h-principle for Stein manifolds in the setting of Krecks modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

Symplectic 4-manifolds with fixed point free circle actions

We show that recent results of Friedl-Vidussi and Chen imply that a symplectic manifold admits a fixed point free circle action if and only if it admits a symplectic circle action and we give a complete description of the symplectic cone in this case. This then completes the characterisation of symplectic 4-manifolds that admit non-trivial circle actions.

Flat structures on surface bundles

We show that there exist flat surface bundles with closed leaves having nontrivial normal bundles. This leads us to compute the abelianisation of surface diffeomorphism groups with marked points. We also extend a formula of Tsuboi that expresses the Euler class of a flat circle bundle in terms of the Calabi invariant of certain Hamiltonian diffeomorphisms to surfaces of higher genus and derive a similar formula for the first MMM‐class of surface bundles with punctured fibre. 37E30, 57R30, 57R50; 57R17 In this paper we study properties of the horizontal foliations of certain kinds of flat bundles. The paper divides into two main parts, the first of which is concerned with closed leaves of flat surface bundles and the second of which deals with fillings of flat circle bundles.

Exactly fillable contact structures without Stein fillings

We give examples of contact structures which admit exact symplectic fillings, but no Stein fillings, answering a question of Ghiggini.

The homology of surface diffeomorphism groups and a question of Morita

We answer a question posed by Morita concerning the non-triviality of certain secondary characteristic classes for surface bundles. In doing so we are naturally led to show that a form of Harer stability holds for surface diffeomorphism groups in homology of small degree.

ON CLOSED LEAVES OF FOLIATIONS, MULTISECTIONS AND STABLE COMMUTATOR LENGTHS

We give examples of foliations that answer two questions posed by Mitsumatsu and Vogt about the genus minimising properties of closed leaves of 2-dimensional foliations on 4-manifolds. By studying stable commutator lengths in certain stable mapping class groups, we also answer an asymptotic version of another question of theirs concerning bounds on self-intersection numbers of multisections in surface bundles.

Asymptotic properties of MMM-classes

We study geometric properties of characteristic classes of surfaces bundles. In particular, we show that oriented surface bundles over bases with amenable fundamental groups and dimension at least 2 have trivial simplicial volume. We show furthermore that all MMM-classes are hyperbolic in the sense of Gromov, verifying a weakened version of a conjecture due to Morita. Finally we consider surface bundles over products and restrictions on their characteristic classes