`Braids in Cortona'

Cortona (Arezzo), Italy, June 19-22, 2002

International Conference Organized by
Riccardo Benedetti, Carlo Petronio, and Mario Salvetti



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.


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
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"

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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Page last updated on June 13, 2002.
For comments and suggestions please contact Carlo Petronio.