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## Publications

Publications (61)

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if δn=σ1σ2…σn−1 is the dual Garside element and b=δnkP∈Bn is a strongly quas...

We investigate the problem of characterising the family of strongly quasipositive links which have definite symmetrised Seifert forms and apply our results to the problem of determining when such a link can have an L-space cyclic branched cover. In particular, we show that if $\delta_n = \sigma_1 \sigma_2 \ldots \sigma_{n-1}$ is the dual Garside el...

As a consequence of a general result about finite group actions on 3-manifolds, we show that a hyperbolic 3-manifold can be the cyclic branched cover of at most fifteen inequivalent knots in S³ (in fact, a main motivation of the present paper is to establish the existence of such a universal bound). A similar, though weaker, result holds for arbitr...

We prove the Tits alternative for an almost coherent $PD(3)$ group which is not virtually properly locally cyclic. In particular, we show that an almost coherent $PD(3)$ group which cannot be generated by less than four elements always contains a rank 2 free group.

We show that if $L$ is an oriented strongly quasipositive link other than the trivial knot or a link with Alexander polynomial a positive power of $(t-1)$, or it is a quasipositive link with positive smooth $4$-ball genus, then the Alexander polynomial and signature function of $L$ determine an integer $n(L) \geq 1$ such that $\Sigma_n(L)$, the $n$...

We prove for a large class of knots that the meridional rank coincides with the bridge number. This class contains all knots whose exterior is a graph manifold. This gives a partial answer to a question of S. Cappell and J. Shaneson, see problem 1.11 on Kirby's list.

We show that a finite group of orientation preserving diffeomorphisms of a
$3$-manifold which is not the $3$-sphere can contain at most fifteen conjugacy
classes of cyclic subgroups acting with non-empty and connected fixed-point
sets and having the $3$-sphere as space of orbits. A straightforward corollary
of this fact is that a hyperbolic $3$-man...

We show that a graph manifold which is a Z-homology 3-sphere not
homeomorphic to either the 3-sphere or the Poincar\'e homology 3-sphere
admits a horizontal foliation. This combines with known results to show
that the conditions of not being an L-space, of having a left-orderable
fundamental group, and of admitting a co-oriented taut foliation, are...

A knot manifold is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. We first investigate virtual epimorphisms between the fundamental groups of small knot manifolds and prove minimality results for small knot manifolds with respect to nonzero degree maps. These results are applied later in the pape...

We investigate commensurability classes of hyperbolic knot complements in the
generic case of knots without hidden symmetries. We show that such knot
complements which are commensurable are cyclically commensurable, and that
there are at most $3$ hyperbolic knot complements in a cyclic commensurability
class. Moreover if two hyperbolic knots have c...

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics which volume-collapses and whose sectional curvature is locally controlled, then
M is a graph manifold. This is the last step in Perelman’s proof of Thurston’s Geometrisation Conjecture.

We study invertible generating pairs of fundamental groups of graph manifolds, that is, pairs of elements (g,h) for which the map g --> g^{-1}, h --> h^{-1} extends to an automorphism. We show in particular that a graph manifold is of Heegaard genus 2 if and only if its fundamental group has an invertible generating pair.

It is conjectured that for each knot $K$ in $S^3$, the fundamental group of its complement surjects onto only finitely many distinct knot groups. Applying character variety theory we obtain an affirmative solution of the conjecture for a class of small knots that includes 2-bridge knots.

Let k and k′ be two knots in the 3-sphere. Say k 1-dominates k′, if there is a proper degree 1 map f: E(k) → E(k′).
Theorem: Suppose that any companion of k is prime. If k 1-dominates k′ with the same Gromov volume, then k′ can be obtained from k by finitely many de-satellizations.
The condition of "same Gromov volume" clearly cannot be removed....

We construct the first examples of degree one maps between non-homeomorphic closed hyperbolic small 3-manifolds.

Let M be a closed, orientable, irreducible, non-simply connected 3-manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result...

We prove that a prime knot K is not determined by its p-fold cyclic branched cover for at most two odd primes p. Moreover, we show that for a given odd prime p, the p-fold cyclic branched cover of a prime knot K is the p-fold cyclic branched cover of at most one more knot K' non equivalent to K. To prove the main theorem, a result concerning the sy...

In this paper we use character variety methods to study homomorphisms between the fundamental groups of 3-manifolds, in particular those induced by non-zero degree maps. A {\it knot manifold} is a compact, connected, irreducible, orientable 3-manifold whose boundary is an incompressible torus. A {\it virtual epimorphism} is a homomorphism whose ima...

We show that the nonzero roots of the torsion polynomials associated to the infinite cyclic covers of a given compact, connected, orientable 3-manifold M are contained in a compact part of the complex plane a priori determined by M. This result is applied to prove that when M is closed, it dominates at most finitely many Sol manifolds.

We prove that an integral homology 3-sphere is S^3 if and only if it admits four periodic diffeomorphisms of odd prime orders whose space of orbits is S^3. As an application we show that an irreducible integral homology sphere which is not S^3 is the cyclic branched cover of odd prime order of at most four knots in S^3. A result on the structure of...

We prove a finiteness result for the $\partial$-patterned guts decomposition
of all 3-manifolds obtained by splitting a given orientable, irreducible and
$\partial$-irreducible 3-manifold along a closed incompressible surface. Then
using the Thurston norm, we deduce that the JSJ-pieces of all 3-manifolds
dominated by a given compact 3-manifold belo...

Erratum to: Math. Z. (2007) 256:913–923 DOI 10.1007/s00209-007-0113-8In the original publication, the proof of Lemma 4 was incorrect and it is now corrected.In [1, Lemma 4], we constructed two homotopic pinches between closed surfaces, such that the boundary circles of the two regions we pinched are not homotopic. However, as pointed out by Tao Li,...

This paper is devoted to the proof of the orbifold theorem: If O is a compact connected orientable irreducible and topologically atoroidal 3-orbifold with nonempty ramification locus, then O is geometric (i.e. has a metric of constant curvature or is Seifert fibred). As a corollary, any smooth orientationpreserving nonfree finite group action on S3...

We provide a structure theorem for 3-manifolds with 2-generated fundamental group and non-trivial JSJ-decomposition. We further give a number of applications.

The workshop on Low-Dimensional was organized by M. Boileau (Toulouse), K. Johannson (Frankfurt) and P. Scott (Ann Arbor). The aim was to bring together an international audience in order to expose some of the recent results in Low-Dimensional Topology and related areas. There were altogether 22 talks from a broad range of topics including Heegaard...

This is a list of open problems on invariants of knots and 3-manifolds with expositions of their history, background, significance, or importance. This list was made by editing open problems given in problem sessions in the workshop and seminars on `Invariants of Knots and 3-Manifolds' held at Kyoto in 2001.

We prove a rigidity theorem for degree one maps between small 3-manifolds using Heegaard genus, and provide some applications and connections to Heegaard genus and Dehn surgery problems.

An oriented link L in a 3-sphere S in complex 2-space is a C-boundary if it bounds a piece of algebraic curve in the 4-ball bounded by S. Using Kronheimer and Mrowka's proof of the Thom Conjecture, we construct many oriented knots which are not concordant to a C-boundary. We use the two-variable HOMFLY polynomial to give an obstruction to a knot's...

We prove a structure theorem for 3-manifolds with non-trivial JSJ-decomposition and 2-generated fundamental group. We deduce a variety of Corollaries. Note this is not a complete classification of such manifolds. In particular we believe that one of the families in our list is empty. If you know something about hyperbolic 2-bridge knot exteriors an...

A quasipositive braid is any product of conjugates of the standard generators of the braid group, and a quasipositive link in S3 is isotopic to the closure of quasipositive braid. In this Note we prove that the boundary of an analytic curve in a pseudoconvex 4-ball is a quasipositive link. It was conjectured by Lee Rudolph.

It is proved that there is, up to isotopy, a unique irreducible Heegaard splitting in an orientable, closed,
connected Seifert 3-manifold with an orientable elliptical or euclidean orbifold basis. Using Hamilton's and
Lawson's results, the topological uniqueness is obtained of closed orientable minimal surfaces of a given
genus g [gt-or-equal, s...

Knot theory has been known for a long time to be a powerful tool for the study of the topology of local isolated singular points of a plane algebraic curve. However it is rather recently that knot theory has been used to study plane algebraic curves in the large. Given a reduced plane algebraic curve Γ − ℂ2 passing through the origin, let Lr =Γ∩ ∂B...

We prove a uniformization theorem for small compact orientable 3-orbifolds, that implies Thurston's orbifold theorem.

Our main result is the construction of an infinite tower of covers of hyperbolic integral homology spheres.

This paper has been withdrawn, because its material has been revised and became part of paper math.GT/0010184

vour remain wide open and are central conjectures in 3-dimensional topology: The Generalized Poincar'e Conjecture that 3-manifolds with finite fundamental group are spherical, and the Hyperbolization Conjecture that compact irreducible atoroidal 3-manifolds with incompressible boundary are hyperbolic, cf. [Sc]. Thurston's Hyperbolisation Theorem as...

We give a complete proof of Thurston's Orbifold Theorem for very good 3-orbifolds of cyclic type. An orbifold is said to be very good when it has a finite cover which is a manifold. A 3-orbifold is of cyclic type if the singular set is a non-empty 1-manifold transverse to the boundary.

We give sufficient conditions for a π-hyperbolic knot to be determined, up to homeomorphism in S3, by the topological types of its 2-fold and 4-fold cyclic branched coverings. We obtain also sufficient conditions on a π-hyperbolic link to be determined by its 2-fold branched covering.RésuméOn donne des conditions suffisantes pour qu'un noeud π-hype...

The genus 2 Heegaard splittings and decompositions of Seifert manifolds over $S$ with 3 exeptional fibres are classified with respect to isotopies and homeomorphisms. In general there are 3 different isotopy classes of Heegaard splittings and 6 different isotopy classes of Heegaard decompositions. Moreover, we determine when a homeomorphism class i...

We prove that there is, up to isotopy, a unique Heegaard splitting of a given genus g ≥ 3 in the 3-torus T3.Using Meek's results, this classification theorem gives that two minimal surfaces of a given genus g ≥ 3 in a flat torus are isotopic. This implies, in particular, the topological uniqueness of triply periodic minimal surfaces in R3. © 1990,...

Thesis (doctoral)--Université Paris-Sud, 1986.