684 research outputs found
Higher homotopy groups of complements of complex hyperplane arrangements
We generalize results of Hattori on the topology of complements of hyperplane
arrangements, from the class of generic arrangements, to the much broader class
of hypersolvable arrangements. We show that the higher homotopy groups of the
complement vanish in a certain combinatorially determined range, and we give an
explicit Z\pi_1-module presentation of \pi_p, the first non-vanishing higher
homotopy group. We also give a combinatorial formula for the \pi_1-coinvariants
of \pi_p.
For affine line arrangements whose cones are hypersolvable, we provide a
minimal resolution of \pi_2, and study some of the properties of this module.
For graphic arrangements associated to graphs with no 3-cycles, we obtain
information on \pi_2, directly from the graph. The \pi_1-coinvariants of \pi_2
may distinguish the homotopy 2-types of arrangement complements with the same
\pi_1, and the same Betti numbers in low degrees.Comment: 24 pages, 3 figure
On Stein fillings of contact torus bundles
We consider a large family F of torus bundles over the circle, and we use
recent work of Li--Mak to construct, on each Y in F, a Stein fillable contact
structure C. We prove that (i) each Stein filling of (Y,C) has vanishing first
Chern class and first Betti number, (ii) if Y in F is elliptic then all Stein
fillings of (Y,C) are pairwise diffeomorphic and (iii) if Y in F is parabolic
or hyperbolic then all Stein fillings of (Y,C) share the same Betti numbers and
fall into finitely many diffeomorphism classes. Moreover, for infinitely many
hyperbolic torus bundles Y in F we exhibit non-homotopy equivalent Stein
fillings of (Y,C).Comment: 18 pages, 10 figures. This preprint version differs from the final
version which is to appear in the Bulletin of the London Mathematical Societ
Dimension on Discrete Spaces
In this paper we develop some combinatorial models for continuous spaces. In
this spirit we study the approximations of continuous spaces by graphs,
molecular spaces and coordinate matrices. We define the dimension on a discrete
space by means of axioms, and the axioms are based on an obvious geometrical
background. This work presents some discrete models of n-dimensional Euclidean
spaces, n-dimensional spheres, a torus and a projective plane. It explains how
to construct new discrete spaces and describes in this connection several
three-dimensional closed surfaces with some topological singularities
It also analyzes the topology of (3+1)-spacetime. We are also discussing the
question by R. Sorkin [19] about how to derive the system of simplicial
complexes from a system of open covering of a topological space S.Comment: 16 pages, 8 figures, Latex. Figures are not included, available from
the author upon request. Preprint SU-GP-93/1-1. To appear in "International
Journal of Theoretical Physics
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Toric Topology
Toric topology emerged in the end of the 1990s on the borders of equivariant
topology, algebraic and symplectic geometry, combinatorics and commutative
algebra. It has quickly grown up into a very active area with many
interdisciplinary links and applications, and continues to attract experts from
different fields.
The key players in toric topology are moment-angle manifolds, a family of
manifolds with torus actions defined in combinatorial terms. Their construction
links to combinatorial geometry and algebraic geometry of toric varieties via
the related notion of a quasitoric manifold. Discovery of remarkable geometric
structures on moment-angle manifolds led to seminal connections with the
classical and modern areas of symplectic, Lagrangian and non-Kaehler complex
geometry. A related categorical construction of moment-angle complexes and
their generalisations, polyhedral products, provides a universal framework for
many fundamental constructions of homotopical topology. The study of polyhedral
products is now evolving into a separate area of homotopy theory, with strong
links to other areas of toric topology. A new perspective on torus action has
also contributed to the development of classical areas of algebraic topology,
such as complex cobordism.
The book contains lots of open problems and is addressed to experts
interested in new ideas linking all the subjects involved, as well as to
graduate students and young researchers ready to enter into a beautiful new
area.Comment: Preliminary version. Contains 9 chapters, 5 appendices, bibliography,
index. 495 pages. Comments and suggestions are very welcom
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