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Anatoly Libgober

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
ANATOLY LIBGOBER
Preface
Tome XXIII, no 2 (2014), p. i-ii.
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Preface
This volume is a collection of surveys and original contributions by par-
ticipants in a workshop which took place at University of Pau on June
11-15, 2012. This workshop was part of “Arrangements in Pyrénées, School
on Arrangements of Hyperplanes and Related Topics” which gathered al-
most 100 graduate students, recent PhD graduates and experts in this field.
It was organized by the Algebra and Geometry team at the University of
Pau, with D. Faenzi, V. Florens and J. Valles as members of the organizing
committee. The workshop included several minicourses, research talks and
poster presentations.
Works in this volume represent the highlights of the meeting and were
submitted by invitation. In particular material of all five minicourses given
during the workshop appear in the papers below. As for the workshop itself,
the unifying theme here was recent developments in the topology of the
complements to arrangements of affine hyperplanes in complex vector space
and their relations with geometry and combinatorics.
The papers by E. Artal Bartolo, L. Paris and A. Suciu consider differ-
ent aspects of the topology of complements. E. Artal Bartolo describes the
method of braid monodromy for finding presentations of the fundamental
groups of the complements and properties of the invariant of the fundamen-
tal group called the “characteristic variety”. The latter is a generalization of
the classical Alexander polynomial, playing prominent role in knot theory
as well as in the study of the topology of complements to plane algebraic
curves. Characteristic varieties, introduced at the beginning of the 90s by
this editor, have rich connections with geometry, combinatorics, cohomol-
ogy of local systems etc... The paper by L. Paris focuses on the following
long-standing problem: when complements to arrangements of hyperplanes
corresponding to classical root systems are Eilenberg-MacLane spaces? The
paper of A. Suciu describes the topology of boundary manifolds of arrange-
ments, the topology of Milnor fibers of arrangements and the boundaries
of these Milnor fibers. The characteristic varieties and cohomology of local
systems play a key role in this theory as well. All these papers not only are
comprehensive surveys but do contain numerous results appearing for the
first time.
– i –
Preface
The work of M. Yoshinaga provides an introduction to the theory of
free arrangements and to its connection with vector bundles. Ample exam-
ples are given including establishing freeness of extended Catalan and Shi
arrangements.
The work of G. Denham describes compactifications of the complements
to arrangements, in particular bringing ideas from tropical geometry
F. Callegaro surveys spectral sequence calculations of the cohomology
of Artin groups with constant as well as twisted coefficients. The paper by
S. Manfredini and S. Settepanella considers homotopy groups of configura-
tion spaces associated with Grassmannians. Finally the work of C. Dunn,
M. Miller, M. Wakefield and S. Zwicknagl settles several cases of an im-
portant open problem connecting topology of the complement and com-
binatorics by proving non existence of some 4-nets in complex projective
plane.
These papers should be useful to novices at this area as well as to ac-
tive researchers studying the topology and geometry of the complements to
complex arrangements.
Anatoly Libgober
– ii –


Preface

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