995 research outputs found
k-irreducible triangulations of 2-manifolds
This thesis deals with k-irreducible triangulations of closed, compact 2-manifolds without boundary. A triangulation is k-irreducible, if all its closed cycles of length less than k are nullhomotopic and no edge can be contracted without losing this property. k-irreducibility is a generalization of the well-known concept of irreducibility, and can be regarded as a measure of how closely the triangulation approximates a smooth version of the underlying surface.
Research follows three main questions: What are lower and upper bounds for the minimum and maximum size of a k-irreducible triangulation? What are the smallest and biggest explicitly constructible examples? Can one achieve complete classifications for specific 2-manifolds, and fixed k
Geometry of the Complex of Curves I: Hyperbolicity
The Complex of Curves on a Surface is a simplicial complex whose vertices are
homotopy classes of simple closed curves, and whose simplices are sets of
homotopy classes which can be realized disjointly. It is not hard to see that
the complex is finite-dimensional, but locally infinite. It was introduced by
Harvey as an analogy, in the context of Teichmuller space, for Tits buildings
for symmetric spaces, and has been studied by Harer and Ivanov as a tool for
understanding mapping class groups of surfaces.
In this paper we prove that, endowed with a natural metric, the complex is
hyperbolic in the sense of Gromov. In a certain sense this hyperbolicity is an
explanation of why the Teichmuller space has some negative-curvature properties
in spite of not being itself hyperbolic: Hyperbolicity in the Teichmuller space
fails most obviously in the regions corresponding to surfaces where some curve
is extremely short. The complex of curves exactly encodes the intersection
patterns of this family of regions (it is the "nerve" of the family), and we
show that its hyperbolicity means that the Teichmuller space is "relatively
hyperbolic" with respect to this family. A similar relative hyperbolicity
result is proved for the mapping class group of a surface.
We also show that the action of pseudo-Anosov mapping classes on the complex
is hyperbolic, with a uniform bound on translation distance.Comment: Revised version of IMS preprint. 36 pages, 6 Figure
Upper and Lower Bounds on Long Dual-Paths in Line Arrangements
Given a line arrangement with lines, we show that there exists a
path of length in the dual graph of formed by its
faces. This bound is tight up to lower order terms. For the bicolored version,
we describe an example of a line arrangement with blue and red lines
with no alternating path longer than . Further, we show that any line
arrangement with lines has a coloring such that it has an alternating path
of length . Our results also hold for pseudoline
arrangements.Comment: 19 page
Moduli space actions on the Hochschild Co-Chains of a Frobenius algebra I: Cell Operads
This is the first of two papers in which we prove that a cell model of the
moduli space of curves with marked points and tangent vectors at the marked
points acts on the Hochschild co--chains of a Frobenius algebra. We also prove
that a there is dg--PROP action of a version of Sullivan Chord diagrams which
acts on the normalized Hochschild co-chains of a Frobenius algebra. These
actions lift to operadic correlation functions on the co--cycles. In
particular, the PROP action gives an action on the homology of a loop space of
a compact simply--connected manifold.
In this first part, we set up the topological operads/PROPs and their cell
models. The main theorems of this part are that there is a cell model operad
for the moduli space of genus curves with punctures and a tangent
vector at each of these punctures and that there exists a CW complex whose
chains are isomorphic to a certain type of Sullivan Chord diagrams and that
they form a PROP. Furthermore there exist weak versions of these structures on
the topological level which all lie inside an all encompassing cyclic
(rational) operad.Comment: 50 pages, 7 figures. Newer version has minor changes. Some material
shifted. Typos and small things correcte
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