16 research outputs found
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
Maximal admissible faces and asymptotic bounds for the normal surface solution space
The enumeration of normal surfaces is a key bottleneck in computational
three-dimensional topology. The underlying procedure is the enumeration of
admissible vertices of a high-dimensional polytope, where admissibility is a
powerful but non-linear and non-convex constraint. The main results of this
paper are significant improvements upon the best known asymptotic bounds on the
number of admissible vertices, using polytopes in both the standard normal
surface coordinate system and the streamlined quadrilateral coordinate system.
To achieve these results we examine the layout of admissible points within
these polytopes. We show that these points correspond to well-behaved
substructures of the face lattice, and we study properties of the corresponding
"admissible faces". Key lemmata include upper bounds on the number of maximal
admissible faces of each dimension, and a bijection between the maximal
admissible faces in the two coordinate systems mentioned above.Comment: 31 pages, 10 figures, 2 tables; v2: minor revisions (to appear in
Journal of Combinatorial Theory A