We show that the $L^{\infty}$-norm of the contact Hamiltonian induces a non-degenerate right-invariant metric on the group of contactomorphisms of any closed contact manifold. This contact Hofer metric is not left-invariant, but rather depends naturally on the choice of a contact form $\alpha,$ whence its restriction to the subgroup of $\alpha$-strict contactomorphisms is bi-invariant. The non-degeneracy of this metric follows from an analogue of the energy-capacity inequality. We show furthermore that this metric has infinite diameter in a number of cases by investigating its relations to previously defined metrics on the group of contact diffeomorphisms. We study its relation to Hofer's metric on the group of Hamiltonian diffeomorphisms, in the case of prequantization spaces. We further consider the distance in this metric to the Reeb one-parameter subgroup, which yields an intrinsic formulation of a small-energy case of Sandon's conjecture on the translated points of a contactomorphism. We prove this Chekanov-type statement for contact manifolds admitting a strong exact filling.

https://www.intlpress.com/site/pub/files/_fulltext/journals/jsg/2017/0015/0004/JSG-2017-0015-0004-a007.pdf

The Hofer norm of a contactomorphism

We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [MO07] or a contact form [BS12]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools are an energy-capacity inequality we prove for contact diffeomorphisms, combined with techniques from measure theory on oriented manifolds. We establish non-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [BD06]. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C^0-rigidity of contact and strictly contact diffeomorphisms.

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Topological contact dynamics I: symplectization and applications of the energy-capacity inequality

In this paper the interrelations between several natural morphisms on the π1 of groups of Hamiltonian diffeomorphisms are investigated. As an application, the equality of the (nonlinear) Maslov index of loops of quantomorphisms of prequantizations of ℂPn and the Calabi–Weinstein invariant is shown, settling affirmatively a conjecture by Givental. We also prove, in the wake of a remark by Woodward, the proportionality of the mixed action-Maslov morphism and the Futaki invariant on loops of Hamiltonian biholomorphisms of Fano Kahler manifolds. Finally, a family of generalized action-Maslov invariants is computed for toric manifolds, on loops coming from the torus action, via barycenters of their moment polytopes, with an application to mass-linear functions recently introduced by McDuff and Tolman. In addition, we reinterpret the quasimorphism of Py on the universal cover of the group of Hamiltonian diffeomorphisms of monotone symplectic manifolds.

Remarks on invariants of hamiltonian loops

Using a "Hodge decomposition" of symplectic isotopies on a compact symplectic manifold $(M,\omega)$, we construct a norm on the identity component in the group of all symplectic diffeomorphisms of $(M,\omega)$ whose restriction to the group $Ham(M,\omega)$ of hamiltonian diffeomorphisms is bounded from above by the Hofer norm. Moreover, $Ham(M,\omega)$ is closed in $Symp(M,\omega)$ equipped with the topology induced by the extended norm. We give an application to the $C^0$ symplectic topology. We also discuss extensions of Oh's spectral distance.

A Hofer-like metric on the group of symplectic diffeomorphisms

The study of Vassiliev knot invariants arose from Vassiliev’s work on singularity theory and from the perturbative Chern-Simons theory of Witten. One reason for studying Vassiliev invariants is that they give topological ways of looking at “quantum” knot invariants — that is invariants which arise by generalizing the Jones polynomial. This thesis contains various results on Vassiliev invariants: common themes running through include their polynomial nature, their functoriality, and the use of Gaus diagrams. The first chapter examines the functoriality of Vassiliev invariants and describes how they can be defined on different types of knotty objects such as knots, framed knots and braids, and how algebraic structure naturally arises. An explicit form of the relationship between the framed and unframed knot theory is given. Chapter 2 considers the important question of whether a Vassiliev invariant can be naively obtained from a combinatorial object called a weight system. A partial answer to this is given by showing how “half” of the steps in such a transition can be performed canonically and explicitly. In Chapter 3 the first two non-trivial invariants for knots, evaluated on prime knots up to twelve crossing are examined, and some surprising graphs are obtained by plotting them. A number of results for torus knots are proved, relating unknotting number and crossing number to the first two Vassiliev invariants. The second half of the thesis is concerned primarily with Vassiliev invariants of pure braids and their connection with de Rham homotopy theory. In Chapter 4 a simple derivation is given showing the relationship between Vassiliev invariants and the lower central series of the pure braid groups. This is used to obtain closed formulae for the actual number of invariants of each type. Chapter 5 is a digression on de Rham homotopy theory and explains the geometric connections between Chen’s iterated integrals, higher order Albanese manifolds, and Sullivan’s 1-minimal models. A method of Chen’s for obtaining integral invariants of elements of the fundamental group from a 1-minimal model is given, and in Chapter 6 this is used to find Vassiliev invariants of pure braids at low order: this extends work of M. A. Berger. Finally, a similar method using currents is employed to obtain a combinatorial formula for a type two invariant which is independent of winding numbers.

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On the Vassiliev invariants for knots and for pure braids

Using the Hofer metric, we construct, under a certain condition, a bi-invariant distance on the identity component in the group of strictly contact diffeomorphisms of a compact regular contact manifold. We also show that the Hofer metric on Ham(M) has a right-invariant (but not left invariant) extension to the identity component in the groups of symplectic diffeomorphisms of certain symplectic manifolds.

Lengths of contact isotopies and extensions of the hofer metric

Starting from a Weight-System denoted by P and defined on the n-Chord-Diagrams with values in an arbitrary Q–module, we give an explicit combinatorial formula for an invariant of (n–1)-singular knots which has P as its derivative. The formula is defined for regular knot projections. Its invariance under singular Reidemeister moves is then proved.

Paul Donato

Papers

Lengths of contact isotopies and extensions of the hofer metric

Integrating a weight system of order n to an invariant of (n−1)-singular knots

Geometrie des orbites coadjointes des groupes de diffeomorphismes