`Braids in Cortona'
Cortona (Arezzo), Italy, June 19-22, 2002
International Conference Organized
by
Riccardo Benedetti, Carlo Petronio,
and Mario Salvetti
Abstracts
MINI-COURSES
Stephen
Bigelow
"Representations of braid groups"
The representation theory of the
symmetric group is well understood over a
field of characteristic zero. This
corresponds to the fact that the Hecke
algebra is well understood when
the parameter q is not a root of unity. I
will explain how to obtain representations
of the Hecke algebra by letting
the braid group act on the homology
of certain spaces. This gives new
topological insight into the mysterious
behaviour at roots of unity.
Michel
Boileau
"Non-free actions of finite groups
on S^3"
The goal of this minicourse is
to explain the proof of the following result:
a finite subgroup G of Diff^+(S^3)
which acts non-freely on S^3
is conjugated into SO(4). We will
introduce the notion of hyperbolic cone
manifold structures on the orbifold
quotient S^3/G and study their deformations
to get an elliptic structure on
S^3/G.
Dale
Rolfsen
"Braids, knots, 3-manifolds and
orderable groups"
It was discovered by Dehornoy,
about ten years ago, that the
Artin braid groups are left-orderable.
That is, there is a total ordering
of its elements which is invariant
under left-multiplication. The pure
braid groups are bi-orderable,
having an ordering invariant under
multiplication on both sides.
This minicourse will discuss these results
and the algebraic consequences
of orderability of groups. Moreover, it will
be shown that many groups which
arise in topology are orderable. For
example all knot and link groups
are left-orderable and some are
bi-orderable. We will also
discuss orderability of fundamental groups of
3-manifolds and relation with other
questions in 3-manifold theory, such
as the virtual Haken conjecture
and the existence of taut foliations.
Alexander
Varchenko
"Multidimensional hypergeometric
functions and representation theory"
The Knizhnik-Zamolodchikov and
quantum Knizhnik-Zamolodchikov equations
are differential and respectively
difference equations of
mathematical physics and representation
theory with many connections to
statistical mechanics, topology,...
The modern theory of multidimensional
hypergeometric functions is a broad
generalization of the classical theory
of the Gauss hypergeometric and
q-hypergeometric functions. It turned out
that the KZ and qKZ equations can
be realized geometrically: a class of
hypergeometric functions was distinguished
which satisfy the KZ and qKZ
equations. This fact enriches both
sides of this interaction: the
jorepresentation theory and the
theory of hypergeometric functions.
In these lectures the geometric
theory of the KZ and qKZ equations will
be discussed.
TALKS
Joan Birman
"Transverse knots"
A classical knot in 3-space is
said to be {\it transverse}
if at every point on the knot the
tangent vector is transverse
to the 2-planes of the standard
tight contact structure in S^3.
Its {\it transversal knot type}
is its equivalence class under
isotopy through transverse knots.
This is a sharper notion than
topological knot type. I
will discuss a theorem proved in new
joint work with William Menasco:
Theorem: There exist infinitely
many examples of pairs of transverse
knots (TK_1, TK_2) such that TK_1
and TK_2 have the same classical knot
type and the same Thurston-Bennequin
invariant, but have distinct
transversal knot types.
Daniel
Cohen
"Gauss-Manin connections for arrangements"
(joint work with Peter Orlik)
We study the Gauss-Manin connection
for the moduli space of an
arrangement of complex hyperplanes
in the cohomology of a complex rank one
local system. For an arrangement
of points in the complex line, the
moduli space is a configuration
space, and the connection is closely
related to the classical Gassner
representation of the pure braid group.
For an arrangement of hyperplanes
in general position (and certain local
systems), the Gauss-Manin connection
was determined by Aomoto and Kita.
We show how this result may be
used to determine the Gauss-Manin
connection for an arbitrary arrangement
(and certain local systems).
Fred
Cohen
"Representations of pure braid groups and
associated vector bundles"
Let G denote a discrete group together with either a real orthogonal or
complex unitary representation. Vector bundles associated to such
representations over the space K(G,1) are considered here in the
special cases for which G is the pure braid group or the fundamental
group of a K(G,1) hyperplane arrangement.
The main results in this case are as follows:
(1) The vector bundle associated to a real orthogonal representation of G
is trivial if and only it is a Spin representation.
(2) All non-trivial bundles which arise from a representation of the pure
braid group are realized through the cohomology algebra (as elucidated in the
lecture). The subgroup of the real K-theory of K(G,1) generated by
orthogonal representations is an elementary abelian 2-group determined by
the cohomology of G.
(3) Properties of some other representations are discussed.
The above is based on joint work with Alex Adem, and Dan Cohen.
(4) A bridge between further families representations of the pure braid
group, homotopy theory, knots, links, and homotopy links will be addressed.
This part is based on joint work with Jie Wu.
Corrado
De Concini
"Some facts on the cohomology of Artin and Coxeter
groups"
In this talk we shall review some results which have been
obtained in recent years together with Salvetti and also with Procesi
and Stumbo, regarding the construction of some explicit complexes
which can be used to compute the cohomology of Artin and Coxeter
groups. Some explicit calculations will be explained.
Patrick
Dehornoy
"Homology of Gaussian groups"
(joint work with Yves Lafont)
We give one (actually two) explicit
method(s) for computing
the homology of a Gaussian group,
or, more generally, of a
monoid where division is Noetherian
and least common multiples
exist when common multiples do.
We construct two exact chain
complexes, one relying on the greedy
normal form, and the
other on using some preordering
after Kobayashi.
Roger Fenn
"The birack: an invariant of (virtual)
knots and links"
(joint work with M. Jordan, L.
Kauffman, G. Wraith)
The birack is a generalisation of the rack and as such occupies a position
with respect to virtual knots and links as the rack does with classical
knots and links. It is also thought that its importance occupies a
similarly exalted position. Using a particular birack composed of
quaternions a polynomial can be defined. This work is currently very much
in progress but computations have been made which show that the above
polynomial is new.
Emmanuel
Giroux
"Open books, closed braids and
contact structures"
We will show that, on any closed
three-manifold V, isotopy classes of contact
structures are in one-to-one correspondence
with open books in V up to isotopy
and positive plumbings. We will
then discuss several applications of this result as
well as some related questions.
John
Guaschi
"Roots of the full twist in surface
braid groups"
(joint work with Daciberg Gonçalves)
Let M be a compact, connected surface
without boundary different from RP^2. We
study the roots of the full twist
braid Delta_n in the surface braid groups B_n(M) of M. If
M is not S^2 then \Delta_n possesses
a k-th root if and only if k divides n
or n-1, while if M=S^2, it possesses
a k-th root if and only if k divides
n, n-1 or n-2. We also show that
the Artin pure braid groups P_n(D^2) and the
sphere pure braid groups P_n(S^2)
admit a splitting as a direct sum.
Simon King
"Links and the geometry of triangulations
of S^3"
Let L be a link, formed by edges
of a triangulation T of S^3 with
n tetrahedra. By work of Armentrout
and Lickorish, the bridge number b(L)
is bounded from above by a linear
function of n, provided T or its dual is
shellable.
We prove by a series of examples
that in general there is no
subexponential upper bound for
b(L) in terms of L, if one drops the
shellability assumption. Thus,
there are triangulations of S^3 that are
"far" from being shellable. By
a study of the Rubinstein-Thompson
recognition algorithm of S^3, we
establish an upper bound for b(L) that is
exponential in n^2.
By the results of Armentrout and
Lickorish, the presence of links
in T^(1) with a big bridge number
implies the absence of geometric
properties of T. We prove a partial
converse: We define a numerical
invariant p(T), called "polytopality",
that can be seen as a bridge number
of the dual graph of T. We show
how to transform T into a polytopal
triangulation (i.e., a triangulation
that is isomorphic to the boundary of
a convex 4-dimensional polytope)
by inserting new vertices, whose number
is bounded from above by a quadratic
function of p(T). Thus, although
p(T) is defined in terms of topology,
it is a measure for the geometric
complexity of T.
Our results yield a conceptually
very simple recognition algorithm of S^3.
Toshitake
Kohno
"Surface braids and the moduli of
flat connections"
We construct a universal flat connection on the
configuration spaces of ordered points on a surface and
describe its holonomy representation with values in the
space of horizontal chord diagrams.
We also discuss an application to a deformation
quantization of the Poisson algebra of functions on the
moduli space of flat connections on surfaces.
Mustafa
Korkmaz
"The second homology groups of
mapping class groups of orientable surfaces"
(joint work with A. Stipsicz)
We first give an elementary computation
for the second homology
groups of mapping class groups
of closed orientable surfaces of genus at
least 4. This computation uses
only the presentation of the mapping class
group and the Hopf theorem which
gives the second homology of a group
from a given presentation. We then
use Harer's homology stability theorem
and the Hochshild-Serre spectral
sqeuence for group extensions to give a
new proof of Harer's theorem, by
extending to genus 4 case, on the second
homology groups of mapping class
groups.
Gregor
Masbaum
"Alexander-Conway polynomial, Milnor
numbers, and a new matrix-tree theorem"
(joint work with A. Vaintrob)
The lowest degree coefficient of
the Alexander-Conway
polynomial of an algebraically
split link can be expressed via
Milnor's triple linking numbers
in two different ways. One way is
via a determinantal expression
due to Levine. Using the
Alexander-Conway weight system,
we give another expression in
terms of spanning trees on a 3-graph.
The equivalence of the
two answers is explained by a new
matrix-tree theorem, relating
enumeration of spanning trees in
a 3-graph and the Pfaffian of a
certain skew-symmetric matrix associated
with it. Similar results
for the lowest degree coefficient
of the Alexander-Conway
polynomial exist if all Milnor
numbers up to a given order vanish.
Hugh Morton
"Choosing a basis for Homfly decorations"
(joint work with R. Hadji)
The Homfly polynomial of a knot
decorated by a diagram in the annulus,
such as the (2,1) cable or a Whitehead
double, depends on the diagram as
an element of the Homfly skein
of the annulus.
I shall present a basis for the
full skein of the annulus, including
reverse-string diagrams. The basis
elements Q_{\lambda,\mu} depend on
a pair of partitions \lambda and
\mu, and are eigenvectors of many
natural endomorphisms of the skein.
Specialisations of the Homfly polynomial
of a knot K decorated by
Q_{\lambda,\mu} can be identified
with quantum invariants of K coloured
by irreducible sl(N)_q modules.
Stefan
Papadima
"Braids and Koszulness"
(joint work with Alex Suciu)
We extend previously known relations,
between pure braid groups and various other
groups, associated to loop spaces
of higher configuration spaces.
Our generalization is based on
the Koszul property from homological algebra.
Luis Paris
"On a theorem of Artin"
(joint work with Arjeh M. Cohen)
Let B_n denote the braid group
on n strings, and let Sym_n denote the n-th
symmetric group. In 1947, Artin
proved that, with two exceptions for n=4, any
epimorphism of B_n onto Sym_n
is the standard epimorphism, up to
automorphism of Sym_n, and proved
that the pure braid group PB_n is a
characteristic subgroup of B_n.
The aim of this lecture is to explain Artin's
arguments, and to show how to extend
them to the other irreducible Artin
groups of spherical type
Bernard
Perron
"A new definition of the Casson
invariant"
Using a new definition of the Johnson-Morita
homomorphisms for the
mapping class group M(g,1) of a
surface of genus g with one boundary
component, we define a rational
invariant D(f) for any f in M(g,1).
For f in T(g,1) the Johnson
subgroup of M(g,1) (in fact a subgroup of the
Torelli group) ,we prove,using
the Casson surgery formula, that D(f) coincides
with the Casson invariant of the
homology sphereS(f) obtained by gluing
two handlebodies along f.
Reversing the point
of view, we can prove directly (i.e without reference to
Casson) thatD(f), for f in T(g,1),
depends only onS(f), by showing that D(f) is
invariant under Reidemeister- Singer
equivalence. The surgery formula, which
is a difficult poit in Casson version,
follows almost immediatly from the definition
D(f). This gives an independent
point of view of Casson invariant.
Claudio
Procesi
"Braid versus
symmetric group cohomology"
We discuss the problem of computing the genus of the covering of
polynomials with distinct roots by the roots, and describe a universal
obstruction and some explicit computations.
Kioji Saito
"The polyhedron
dual to the Weyl chamber system"
The polyhedron dual to the Weyl chamber system is a basic object in
a combinatorial and topological study of the braid groups. Namely,
it was used 1. to show the K(\pi,1) property of the configuration
space ('74 Deligne) and 2. to determine the braid relation of the
fundamental group of the configuration space ('70 Brieskorn).
In this talk, I shall show that the dual polyhedron is closely
related to the flat sturcure (Frobenius manifold) on the quotient
variety by the Weyl group action, and then, in application, give
a geometric construction of the generators of the fundamental group
satisfying the braid relation, which answers to a question of Deligne.
Mina
Teicher
"Computational problems related
to braid monodromy type"
Abstract
Miguel
Xicotencatl
"Orbit configuration spaces and
surface braid groups"
Given a n-manifold M, define its
pure braid group on k strands,
P_k(M), as \pi_1 F(M,k), where
F(M,k) is the configuration space
of k distinct points in M. It is
well known that the natural
homomorphism \varphi: P_k(M)
\to (\pi_1 M)^k
is an isomorphism for n>2 and epimorphism
for n=2.
In this talk we describe the kernel
of \varphi in the case M = S_g,
a compact orientable surface of
genus g>1. Namely, if S_g is the
quotient of the upper-half plane
H^2 by the subgroup \Gamma \leq SL(2,R),
then \ker \varphi \cong \pi_1 F_{\gamma}(H^2,k)
(the fundamental group
of an orbit configuration space).
As a consequence we obtain a spectral
sequence converging to the homology
of P_k(S_g).
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