arXiv subjects
Publications and source records attributed to Michel Boileau.
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.
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 finite groups of odd order acting on integral homology spheres is also obtained.
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.
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 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 gives an alternative approach for the last step in Perelman's proof of the Geometrisation Conjecture for aspherical 3-manifolds.
Let $k$ and $k'$ be two knots in 3-sphere. Say $k$ 1--dominates $k'$, if there is a proper degree 1 map $f\co E(k)\to E(k')$, between knot exterior of $k_i$. 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 can not be removed. We also give a new construction of 1-domination between knots with same Gromov volume to show that the condition "any companion of $k$ is prime" can not be removed.
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 image is of finite index in its range. We show that the existence of such homomorphisms places constraints on the algebraic decomposition of a knot manifold's $PSL_2(\mathbb C)$-character variety and consequently determine a priori bounds on the number of virtual epimorphisms between the fundamental groups of small knot manifolds with a fixed domain. In the second part of the paper we fix a small knot manifold $M$ and investigate various sets of characters of representations with discrete image in $PSL_2(\mathbb C)$. The topology of these sets is intimately related to the algebraic structure of the $PSL_2(\mathbb C)$-character variety of $M$ as well as dominations of manifolds by $M$ and its Dehn fillings. In particular, we apply our results to study families of non-zero degree maps $f_n: M(α_n) \to V_n$ where $M(α_n)$ is the $α_n$-Dehn filling of $M$ and $V_n$ is either a hyperbolic manifold or $\widetilde{SL_2}$ manifold. We show that quite often, up to taking a subsequence, there is a knot manifold $V$, slopes $β_j$ on $\partial V$ such that $V_j \cong V(β_j)$, and a non-zero degree map $M \to V$ which induces $f_j$ up to homotopy. The work of the first part of the paper is then applied to construct infinite families of small, closed, connected, orientable 3-manifolds which do not admit non-zero degree maps, other than homeomorphisms, to any hyperbolic manifold, or even manifolds with infinite fundamental groups.
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.
Let $F',F$ be any two closed orientable surfaces of genus $g'>g\ge 1$, and $f:F\to F$ be any pseudo-Anosov map. Then we can "extend" $f$ to be a pseudo-Anosov map $f':F'\to F'$ so that there is a fiber preserving degree one map $M(F',f')\to M(F,f)$ between the hyperbolic surface bundles. Moreover the extension $f'$ can be chosen so that the surface bundles $M(F',f')$ and $M(F,f)$ have the same first Betti numbers.
This paper has been withdrawn, because its material has been revised and became part of paper math.GT/0010184
The purpose of this article is to give a proof of the Orbifold Theorem announced by Thurston in late 1981: If $O$ is a compact, connected, orientable, irreducible and topologically atoroidal 3-orbifold with non-empty ramification locus, then $O$ is geometric. As a corollary, any smooth orientation preserving non-free finite group action on $S^3$ is conjugate to an orthogonal action.
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 being a C-boundary in a strictly pseudoconvex S. We make several conjectures.
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 and their finite sheeted covering spaces you might want to finish this off. Thanks.