`The Topology of 3-Manifolds'
Pisa, June 15 and June 17, 2002
Organized by
Riccardo Benedetti, Carlo Petronio, Dale Rolfsen, and Jeff Weeks


Colin Adams
"The view from the cusp in a hyperbolic 3-manifold"
(joint work with A. Colestock, J. Fowler, D. Gillam, E. Katerman)
There are examples of cusped hyperbolic 3-manifolds, such as the figure-eight knot
complement, where if the cusp is made maximal and one shoots geodesics out from
the cusp, they will all return to the cusp in a uniformly bounded amount of time.
We show that these examples are unusual and that for "almost all" cusped hyperbolic
3-manifolds, there is no upper bound to the return time for geodesics fired from the cusp.

Ian Agol
"Detecting laminar 3-manifolds"
(joint work with Tao Li)
We show that there are algorithms to determine if a 3-manifold contains an essential
lamination or a taut foliation. This work is available at math.GT/0201310.

Silvia Benvenuti
"Presentations for modular groups via the ordered complex of curves"
The ordered complex of curves has been proven to be very
effective to get presentations for the modular groups of surfaces: we
describe those presentations and we apply the results and the tecniques
to the study of the Teichmuller tower of mapping class groups.

Steven Boyer
"Orderable 3-manifold groups"
(joint work with Dale Rolfsen and Bert Wiest)
We investigate the orderability properties of fundamental groups of
3-dimensional manifolds. Many 3-manifold groups support left-invariant
orderings, including all compact P^2-irreducible manifolds with positive
first Betti number. For seven of the eight geometries (excluding hyperbolic)
we are able to characterize which manifolds groups support a left-invariant
or bi-invariant ordering. We also show that manifolds modelled on these
geometries have virtually bi-orderable groups. The question of virtual orderability
of 3-manifold groups in general, and even hyperbolic manifolds, remains open,
and is closely related to conjectures of Waldhausen and others.

Olivier Collin
"Gauge theory invariants and 3-dimensional links of complex singularities"
(joint work with N. Saveliev)
In this talk,  we show how some knowledge of Floer theory invariants for
3-manifolds realized as links of an isolated complex singularity is related to
information about its Milnor fibre.

Daryl Cooper
"The orbifold theorem"
We discuss some aspects of the proof of the orbifold theorem.

Joanna Kania-Bartoszynska
"Applications of quantum invariants to 3-dimensional topology"
I will discuss three applications of quantum topology. First is the criterion for
periodicity of homology spheres in terms of their SO(3)-invariants. Second is a
quantum obstruction to embedding one 3-manifold with boundary into another.
The third is a computation of integrals against the symplectic measure on the
character variety of a cylinder over a compact surface. First application is joint
work with Pat Gilmer and Jozef Przytycki, second and third are joint work with
Charles Frohman.

Sadayoshi Kojima
"Circle packings on surfaces with projective structures"
(joint work with S. Mizushima and S. P. Tan)
A circle (and hence a circle packing) makes sense on a surface with a projective
structure since any projective transformation maps a circle to a circle. We plan
to discuss the local and global deformations of projective structures carrying a
circle packing with a fixed isotopy type of nerves, and to review Andreev-Thurston
rigidity for circle packings on surfaces of constant curvature as a particular instance.

Christine Lescop
"Splitting formulae for all finite-type invariants of homology 3-spheres"
I shall discuss the behaviour of all finite type invariants of homology spheres under
(non-necessarily Torelli) homology handlebodies replacements. The explicit formulae
that I shall present generalize a sum formula that I obtained for the Casson invariant
in 1994. Their proof relies on the Kuperberg-Thurston construction of a universal finite
type invariant of rational homology spheres by means of configuration space integrals.

Paolo Lisca
"Milnor-Wood inequalities, contact structures, and Seifert fibrations"
(joint work with G. Matic)
We give complete criteria for a Seifert fibration over a closed surface to admit contact
structures which are transverse to the fibers. The obstructions to the existence of such
structures can be viewed as a contact version of the classical Milnor-Wood inequalities.

Bruno Martelli
"Complexity and decomposition of 3-manifolds along tori"
(joint work with Carlo Petronio)
The complexity of a closed 3-manifold M is the minimum number of vertices  in a simple
polyhedron P such that M\P is a ball.  We extend this definition to a manifold with boundary
consisting of tori, each torus marked with a 'theta' graph.  We get a decomposition of
irreducible closed 3-manifolds along such objects, on which the complexity is additive.
Such a decomposition is not unique, but it turns out (experimentally) to be finer than the JSJ
(in particular, it is non-trivial on most atoroidal manifolds).  With these techniques, we have
classified all closed 3-manifolds with complexity up to 9, and all closed Dehn fillings of the
chain link with 3 components that are not negatively curved.

Sergei Matveev
"Finite-type invariants of cubic complexes"
(joint work with M. Polyak)
Cubic complexes are similar to simplicial ones, but they are constructed from
cubes instead of simplices. The faces of any cube are decomposed into pairs of
opposite faces. This additional structure allows one to define the so-called finite
type invariants of cubic complexes. We describe three interesting cubic complexes
consisting of affine cubes in $R^n$, singular knots in $S^3$, and Borromean links
in homology 3-spheres. Their finite type invariants are polynomials, Vassiliev-Goussarov
invariants, and invariants of homology 3-spheres in the sense of M. Goussarov.

Mattia Mecchia
"The number of links with the same hyperbolic 2-fold branched covering"
We consider the problem of how many links in homology 3-spheres can have
the same hyperbolic 3-manifold as their common 2-fold branched covering.
This number depends on the number of components of the links. Using an
algebraic approach M. Reni proved that there exist at most nine knots with
the same hyperbolic 2-fold branched covering and M. Reni and B. Zimmermann
proved that five is an upper bound for the case of links with at least three
components. Using a different and more geometric approach we show that
there exist at most nine 2-component links with the same 2-fold branched
covering. Moreover using Kawauchi's imitation theory, we show how to
construct sets of nine different 2-component links in the 3-sphere with the
same hyperbolic 2-fold branched covering. We show also that the best upper
bound for links with at least three components is three (also in this case we
construct examples with links in the 3-sphere) and for the case of knots we
realize the maximum number nine for knots in homology spheres. The proof
that nine is an upper bound for 2-component links is joint work with M. Reni,
the other results are joint work with B. Zimmermann.

Luisa Paoluzzi
"2-fold branched covers of knots"
We address the following question: "given a knot K, is it possible to recover all knots
with the same 2-fold branched cover, via "modifications" of a planar diagram for K?"
We shall recall some well-known examples where this is known to be true (Montesinos
knots, pi-hyperbolic knots) and we shall describe the different types of diagram
modifications involved. We shall then consider the case of hyperbolic knots with non
trivial Bonahon-Siebenmann decomposition and prove that the question has a positive
answer when restricted to the class of all hyperbolic knots. On the other hand, we shall
exhibit an example of a hyperbolic knot K which shares its 2-fold branched cover with
a non hyperbolic knot which cannot be obtained from K via any of the known diagram

Riccardo Piergallini
"Surgering branched coverings"
We discuss some surgery operations on branched covers of spheres.
In particular, we show how surgery can be used in order to remove
branch set singularities. In this way, we prove that all smooth
orientable closed four-manifolds are covers of the four-sphere
branched over a non-singular surface.


Joan Porti
"The orbifold theorem"
(joint work with M. Boileau and B. Leeb)
We discuss the proof of the orbifold theorem, based on Thurston's
approach using cone manifolds, which are hyperbolic manifolds with
some special singularities. However we have a different point of view
for the analysis of collapses of cone manifolds, which is a key step
in the proof. We will concentrate in the case where the fundamental
group is finite. As a corollary,  finite smooth actions on S^3 are
standard, provided that they are non-free and orientation preserving.
D. Cooper, C. Hodgson and S. Kerckhoff have announced another proof.

Marta Rampichini
"An algorithm for recognizing the unknot"
(joint work with J. S. Birman, P. Boldi, S. Vigna)
A knot in $S^3$ is trivial if and only if it bounds an embedded disc. If the
knot is presented as a closed braid, its Seifert surfaces can be studied by
means of the `braid foliation', the foliation induced on the surface by the
standard fibration of the complement of the braid axis. In 1998 Birman and
Hirsch presented a then new algorithm for recognizing the unknot. The first
part of the algorithm required the systematic enumeration of all discs which
support a `braid foliation' and are embeddable in three-space. The boundaries
of these `foliated embeddable discs' (FED's) are the collection of all closed
braid representatives of the unknot, up to conjugacy. The second part of the
algorithm produces a word in the generators of the braid group whose closure
represents the boundary of the previously listed FED's. The third part tests
whether a given braid is conjugate to some of the braids so found. We describe
implementations of the first and second part of the algorithm. We also give some
of the data which we obtained. The data suggest that the FED's have unexplored
and interesting structure. The third part of the algorithm was studied by Birman,
Ko and Lee and implemented by S. J. Lee.

Martin Scharlemann
"Thinning genus two Heegaard spines in the 3-sphere"
(joint work with Abigail Thompson)
We study those trivalent graphs in $S^{3}$ which have closed complement a
genus two handlebody. We show that such a graph, when put in thin position,
has a simple (i. e. non-loop) level edge, without the need for any edge-slides.

Abigail Thompson
"Invariants of immersed curves in the projective plane"
There is an elegant relation, due to Fabricius-Bjerre, among the double tangent
lines, crossings, inflections points, and cusps of a singular immersed curve in
the plane. This has previously been been generalized to curves in the 2-sphere
and, in some sense, to curves in real projective space. I'll describe a new
generalization to immersed curves in the projective plane. Noting that the quantities
involved in the formula are naturally dual to each other in the projective plane
yields a dual formula. The work specializes to give new relations among invariants
of immersed curves in the plane.

Bruno Zimmermann
"On finite groups acting on homology 3-spheres"
It is part of Thurston's geometrization conjecture in dimension three that
every finite group action on the 3-sphere is equivalent (conjugate) to an orthogonal
action. For nonfree actions, this follows from the orbifold geometrization theorem, but
for free actions it is still open (and equivalent to the question if every irreducible
3-manifold with finite fundamental group is spherical); in particular, it is still not
known if only the finite subgroups of the orthogonal group SO(4) occur.
In the present work we consider homology 3-spheres, i.e. closed orientable 3-manifolds
with the same integer homology as  the 3-sphere: we are interested in the possible finite
groups $G$ which admit orientation-preserving actions on homology 3-spheres. If such
an action is free, than the group $G$ has periodic cohomology (of period four), and
the classification of such groups is well-known from the Zassenhaus-Suzuki theorem. We
are interested in arbitrary actions, i.e. possibly with fixed points. Continuing work of
Marco Reni, we give a characterization of the finite simple and nonsolvable groups which
admit orientation-preserving actions on homology 3-spheres. We find exactly the finite
nonsolvable subgroups of the orthogonal group SO(4), plus two extra families of groups for
which the question remains open, for the moment. In particular, the only finite nonabelian
simple group occuring is the alternating or dodecahedral group $A_5$, and we suspect
that also in the nonsolvable case only the finite subgroups of SO(4) occur (for the solvable
case, some finite groups are known which admit free actions on homology 3-spheres but
which are not subgroups of SO(4); the situation is not completely understood here, even for
the case of free actions, in particular it is still open for the most interesting case of the
3-sphere itself).

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Page last updated on April 1, 2002.
For comments and suggestions please contact Carlo Petronio.