Abstracts
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
modifications.
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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