5,497 research outputs found
Regular cell complexes in total positivity
This paper proves a conjecture of Fomin and Shapiro that their combinatorial
model for any Bruhat interval is a regular CW complex which is homeomorphic to
a ball. The model consists of a stratified space which may be regarded as the
link of an open cell intersected with a larger closed cell, all within the
totally nonnegative part of the unipotent radical of an algebraic group. A
parametrization due to Lusztig turns out to have all the requisite features to
provide the attaching maps. A key ingredient is a new, readily verifiable
criterion for which finite CW complexes are regular involving an interplay of
topology with combinatorics.Comment: accepted to Inventiones Mathematicae; 60 pages; substantially revised
from earlier version
The Morse theory of \v{C}ech and Delaunay complexes
Given a finite set of points in and a radius parameter, we
study the \v{C}ech, Delaunay-\v{C}ech, Delaunay (or Alpha), and Wrap complexes
in the light of generalized discrete Morse theory. Establishing the \v{C}ech
and Delaunay complexes as sublevel sets of generalized discrete Morse
functions, we prove that the four complexes are simple-homotopy equivalent by a
sequence of simplicial collapses, which are explicitly described by a single
discrete gradient field.Comment: 21 pages, 2 figures, improved expositio
The combinatorial Mandelbrot set as the quotient of the space of geolaminations
We interpret the combinatorial Mandelbrot set in terms of \it{quadratic
laminations} (equivalence relations on the unit circle invariant under
). To each lamination we associate a particular {\em geolamination}
(the collection of points of the circle and edges of convex
hulls of -equivalence classes) so that the closure of the set of all of
them is a compact metric space with the Hausdorff metric. Two such
geolaminations are said to be {\em minor equivalent} if their {\em minors}
(images of their longest chords) intersect. We show that the corresponding
quotient space of this topological space is homeomorphic to the boundary of the
combinatorial Mandelbrot set. To each equivalence class of these geolaminations
we associate a unique lamination and its topological polynomial so that this
interpretation can be viewed as a way to endow the space of all quadratic
topological polynomials with a suitable topology.Comment: 28 pages; in the new version a few typos are corrected; to appear in
Contemporary Mathematic
On the distance between the expressions of a permutation
We prove that the combinatorial distance between any two reduced expressions
of a given permutation of {1, ..., n} in terms of transpositions lies in
O(n^4), a sharp bound. Using a connection with the intersection numbers of
certain curves in van Kampen diagrams, we prove that this bound is sharp, and
give a practical criterion for proving that the derivations provided by the
reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also
show the existence of length l expressions whose reversing requires C l^4
elementary steps
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