22 research outputs found
Deletion-restriction in toric arrangements
Deletion-restriction is a fundamental tool in the theory of hyperplane
arrangements. Various important results in this field have been proved using
deletion-restriction. In this paper we use deletion-restriction to identify a
class of toric arrangements for which the cohomology algebra of the complement
is generated in degree . We also show that for these arrangements the
complement is formal in the sense of Sullivan.Comment: v2: typos fixed, 11 pages. Accepted for publication in Journal of
Ramanujan Mathematical Societ
Arrangements of Submanifolds and the Tangent Bundle Complement
Drawing parallels with the theory of hyperplane arrangements, we develop the theory of arrangements of submanifolds. Given a smooth, finite dimensional, real manifold we consider a finite collection \A of locally flat codimension submanifolds that intersect like hyperplanes. To such an arrangement we associate two posets: the \emph{poset of faces} (or strata) \FA and the \emph{poset of intersections} L(\A). We also associate two topological spaces to \A. First, the complement of the union of submanifolds in which we call the \emph{set of chambers} and denote by \Ch. Second, the complement of union of tangent bundles of these submanifolds inside which we call the \emph{tangent bundle complement} and denote by M(\A). Our aim is to investigate the relationship between combinatorics of the posets and topology of the complements.
We generalize the Salvetti complex construction in this setting and also charcterize its connected covers using incidence relations in the face poset. We also demonstrate some calculations of the fundamental group and the cohomology ring. We apply these general results to study arrangements of spheres, projective spaces, tori and pseudohyperplanes. Finally we generalize Zaslavsky\u27s classical result in order to count the number of chambers
Coxeter transformation groups and reflection arrangements in smooth manifolds
Artin groups are a natural generalization of braid groups and are
well-understood in certain cases. Artin groups are closely related to Coxeter
groups. There is a faithful representation of a Coxeter group as a linear
reflection group on a real vector space . The group acts properly and fixes
a union of hyperplanes. The -action extends as the covering space action to
the complexified complement of these hyperplanes. The fundamental groups of the
complement and the orbit space are the pure Artin group and the Artin group
respectively. For the Artin groups of finite type Deligne proved that the
associated complement is aspherical. Using the Coxeter group data Salvetti gave
a construction of a cell complex which is a -equivariant deformation retract
of the complement. This construction was independently generalized by Charney
and Davis to the Artin groups of infinite type. A lot of algebraic properties
of these groups were discovered using combinatorial and topological properties
of this cell complex.
In this paper we represent a Coxeter group as a subgroup of diffeomorphisms
of a smooth manifold. These so-called Coxeter transformation groups fix a union
of codimension- (reflecting) submanifolds and permute the connected
components of the complement. Their action naturally extends to the tangent
bundle of the ambient manifold and fixes the union of tangent bundles of these
reflecting submanifolds. Fundamental group of the tangent bundle complement and
that of its quotient serve as the analogue of pure Artin group and Artin group
respectively. The main aim of this paper is to prove Salvetti's theorems in
this context. We show that the combinatorial data of the Coxeter transformation
group can be used to construct a cell complex homotopy equivalent to the
tangent bundle complement and that this homotopy equivalence is equivariant.Comment: 18 pages, 2 figures. V2: minor changes, typos fixed. final versio