Pisa, June 15 and June 17, 2002

Organized by

Riccardo Benedetti, Carlo Petronio, Dale Rolfsen, and Jeff Weeks

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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Page last updated on April 1, 2002.

For comments and suggestions please contact Carlo Petronio.