On the geography of symplectic 6-manifolds

The article investigates the geography of closed, connected and simply connected, six-dimensional manifolds. It is proved that any triple of integers satisfying some necessary arithmetical restrictions occurs as the Chern triple of such a manifold. The main tools used for producing the examples are the symplectic connected sum and the symplectic blow-up. Received: 28 May 1998 / Revised version: 22 January 1999     

Regular Legendrian knots and the HOMFLY polynomial of immersed plane curves

We show that every unframed knot type in has a representative obtained by the Legendrian lifting of an immersed plane curve. This gives a positive answer to the question asked by V.I.Arnold in [3]. The Legendrian lifting lowers the framed version of the HOMFLY polynomial [20] to generic plane curves. We prove that the induced polynomial invariant can be completely defined in terms of plane curves only. Moreover it is a genuine, not Laurent, polynomial in the framing variable. This provides an estimate on the Bennequin-Tabachnikov number of a Legendrian knot. Received: 17 April 1996 / Revised: 12 May 1999 / Published online: 28 June 2000     

Structures de contact sur les variétés toriques

Sans résumé

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Re?u le 13 juillet 1998     

On \pi-hyperbolic knots with the same 2-fold branched coverings

We consider the following problem from the Kirby's list (Problem 3.25): Let K be a knot in and M(K) its 2-fold branched covering space. Describe the equivalence class [K] of K in the set of knots under the equivalence relation if is homeomorphic to . It is known that there exist arbitrarily many different hyperbolic knots with the same 2-fold branched coverings, due to mutation along Conway spheres. Thus the most basic class of knots to investigate are knots which do not admit Conway spheres. In this paper we solve the above problem for knots which do not admit Conway spheres, in the following sense: we give upper bounds for the number of knots in the equivalence class [K] of a knot K and we describe how the different knots in the equivalence class of K are related. Received: 3 August 1998 / in final form: 17 June 1999     

Coisotropic actions on compact homogeneous Kähler manifolds

Abstract. The main result of the paper is that a compact homogeneous K?hler manifold admitting an isometric and coisotropic action with a fixed point is isometric to a Hermitian symmetric space. Received: 28 December 2001; in final form: 19 March 2002 / Published online: 14 February 2003 Part of the work on this paper was done during a visit of the second author at the University of Florence that was financially supported by G.N.S.A.G.A. – I.N.d.A.M.     

On closed anti de Sitter spacetimes

This article deals with the structure of the fundamental group of compact anti de Sitter spacetimes, i.e. Lorentz manifolds with constant negative curvature. Algebraically such a manifold is the quotient of the universal cover of the homogeneous space by a discrete group acting properly and co-compactly on it. This exists if and only if is even. Indeed, as this was observed by Kulkarni, is contained in , and acts properly transitively on . It then suffices to take as a co-compact lattice in . The results of the present article give evidence to the question: in dimension , are all compact anti de Sitter spacetimes constructed in this way? Received: 18 May 1996 / Revised version: 3 January 1997     

Representing products of manifolds by edge-coloured graphs: the boundary case

In this paper we generalize the construction - introduced by Gagliardi and Grasselli in the closed case - of a coloured-graph representing the product of two manifolds, starting by two coloured graphs representing the manifolds themselves, to the boundary case. In particular we study the genus of the graph product of low dimensional manifold ( resp. n-spheres ) with m-disks. Received September 28, 1998; in final form January 5, 2000 / Published online October 11, 2000     

Stiefel–Whitney surfaces and the tri-genus of non-orientable 3-manifolds

Every non-orientable 3-manifold M can be expressed as a union of three orientable handlebodies V ₁,V ₂,V ₃ whose interiors are pairwise disjoint. If g _i denotes the genus of ∂V _i and g ₃≤g ₂≤g ₃, then the tri-genus of M is the minimum triple (g ₁,g ₂,g ₃), ordered lexicographically. If the Bockstein of the first Stiefel–Whitney class βw ₁(M)=0, then M has tri-genus (0,2g,g ₃), where g is the minimal genus of a 2-sided Stiefel Whitney surface of M. In this paper it is shown that, if βw ₁(M)&\ne;0, then M has tri-genus (1,2g−1,g ₃), where g is the minimal genus of a (1-sided) Stiefel–Whitney surface. As an application the tri-genus of certain graph manifolds is computed. Received: 28 April 1999     

Surfaces convexes fuchsiennes dans les espaces lorentziens à courbure constante

We show existence and uniqueness of the equivariant isometric immersions of Riemannian surfaces into Lorentz space-forms under conditions implying convexity, when we impose that the associated representations leave a point invariant.

Received: 2 December 1995     

On a contact 3-structure

A contact 3-structure consists of three contact metric structures which satisfy the relation (2.1). On a product manifold of the real line and a manifold with a contact 3-structure, we can construct three almost Hermitian structures satisfying the quaternionic identities. From this view point we discuss a contact 3-structure. Owing to Hitchin's well known Lemma concerning to hyperk?hler structure (Lemma H), we show that a contact 3-structure is necessarily a Sasakian 3-structure. Received: 26 August 1999; in final form: 2 May 2000 / Published online: 4 May 2001     

On the existence of closed timelike geodesics in compact spacetimes

In [19], Tipler has shown that a compact spacetime having a regular globally hyperbolic covering space with compact Cauchy surfaces necessarily contains a closed timelike geodesic. The restriction to compact spacetimes with just regular globally hyperbolic coverings (i.e., the Cauchy surfaces are not required to be compact) is still an open question. Here, we shall answer this question negatively by providing examples of compact flat Lorentz space forms without closed timelike geodesics, and shall give some criterion for the existence of such geodesics. More generally, we will show that in a compact spacetime having a regular globally hyperbolic covering, each free timelike homotopy class determined by a central deck transformation must contain a closed timelike geodesic. Whether or not a compact flat spacetime contains closed nonspacelike geodesics is, as far as we know, an open question. We shall answer this question affirmatively. We shall also introduce the notion of timelike injectivity radius for a spacetime relative to a free timelike homotopy class and shall show that it is finite whenever the corresponding deck transformation is central. Received: 9 November 1999; in final form: 19 September 2000 / Published online: 25 June 2001     

Constructing infinitely many smooth structures on $3\mathbb{CP}^2 \# n\overline{\mathbb{CP}}^2$

Using Seiberg-Witten theory and rational blow-down procedures of R. Fintushel and R.J. Stern, we construct infinitely many irreducible smooth structures, both symplectic and non-symplectic, on the four-manifold for each integer n lying in the interval . Received: 17 January 2000 / Published online: 18 January 2002     

Handle attaching in symplectic homology and the Chord Conjecture

Arnold conjectured that every Legendrian knot in the standard contact structure on the 3-sphere possesses a characteristic chord with respect to any contact form. I confirm this conjecture if the know has Thurston-Bennequin invariant −1. More generally, existence of chords is proved for a standard Legendrian unknot on the boundary of a subcritical Stein manifold of any dimension. There is also a multiplicity result which implies in some situations existence of infinitely many chords.?The proof relies on the behaviour of symplectic homology under handle attaching. The main observation is that symplectic homology only changes in the presence of chords. Received July 14, 2000 / final version received June 1, 2001?Published online August 1, 2001     

Annuli in generalized Heegaard splittings and degeneration of tunnel number

Abstract. We analyze how a family of essential annuli in a compact 3-manifold will induce, from a strongly irreducible generalized Heegaard splitting of the ambient manifold, generalized Heegaard splittings of the complementary components. There are specific applications to the subadditivity of tunnel number of knots, improving somewhat bounds of Kowng [Kw]. For example, in the absence of 2-bridge summands, the tunnel number of the sum of n knots is no less than the sum of the tunnel numbers. Received: 10 November 1999 / Published online: 28 June 2000     

On symplectic cobordisms

In this note we make several observations concerning symplectic cobordisms. Among other things we show that every contact 3-manifold has infinitely many concave symplectic fillings and that all overtwisted contact 3-manifolds are “symplectic cobordism equivalent”. Received: 26 March 2001 / Revised version: 1 May 2001 / Published online: 28 February 2002