4,564 research outputs found
The FAn Conjecture for Coxeter groups
We study global fixed points for actions of Coxeter groups on nonpositively
curved singular spaces. In particular, we consider property FA_n, an analogue
of Serre's property FA for actions on CAT(0) complexes. Property FA_n has
implications for irreducible representations and complex of groups
decompositions. In this paper, we give a specific condition on Coxeter
presentations that implies FA_n and show that this condition is in fact
equivalent to FA_n for n=1 and 2. As part of the proof, we compute the
Gersten-Stallings angles between special subgroups of Coxeter groups.Comment: This is the version published by Algebraic & Geometric Topology on 19
November 200
A counterexample to the Hirsch conjecture
The Hirsch Conjecture (1957) stated that the graph of a -dimensional
polytope with facets cannot have (combinatorial) diameter greater than
. That is, that any two vertices of the polytope can be connected by a
path of at most edges.
This paper presents the first counterexample to the conjecture. Our polytope
has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope
with 48 facets which violates a certain generalization of the -step
conjecture of Klee and Walkup.Comment: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by
referees. This version has been accepted in the Annals of Mathematic
Topology of geometric joins
We consider the geometric join of a family of subsets of the Euclidean space.
This is a construction frequently used in the (colorful) Carath\'eodory and
Tverberg theorems, and their relatives. We conjecture that when the family has
at least sets, where is the dimension of the space, then the
geometric join is contractible. We are able to prove this when equals
and , while for larger we show that the geometric join is contractible
provided the number of sets is quadratic in . We also consider a matroid
generalization of geometric joins and provide similar bounds in this case
Unimodality Problems in Ehrhart Theory
Ehrhart theory is the study of sequences recording the number of integer
points in non-negative integral dilates of rational polytopes. For a given
lattice polytope, this sequence is encoded in a finite vector called the
Ehrhart -vector. Ehrhart -vectors have connections to many areas of
mathematics, including commutative algebra and enumerative combinatorics. In
this survey we discuss what is known about unimodality for Ehrhart
-vectors and highlight open questions and problems.Comment: Published in Recent Trends in Combinatorics, Beveridge, A., et al.
(eds), Springer, 2016, pp 687-711, doi 10.1007/978-3-319-24298-9_27. This
version updated October 2017 to correct an error in the original versio
Stellar theory for flag complexes
Refining a basic result of Alexander, we show that two flag simplicial
complexes are piecewise linearly homeomorphic if and only if they can be
connected by a sequence of flag complexes, each obtained from the previous one
by either an edge subdivision or its inverse. For flag spheres we pose new
conjectures on their combinatorial structure forced by their face numbers,
analogous to the extremal examples in the upper and lower bound theorems for
simplicial spheres. Furthermore, we show that our algorithm to test the
conjectures searches through the entire space of flag PL spheres of any given
dimension.Comment: 12 pages, 2 figures. Notation unified and presentation of proofs
improve
Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture
We introduce a new combinatorial abstraction for the graphs of polyhedra. The
new abstraction is a flexible framework defined by combinatorial properties,
with each collection of properties taken providing a variant for studying the
diameters of polyhedral graphs. One particular variant has a diameter which
satisfies the best known upper bound on the diameters of polyhedra. Another
variant has superlinear asymptotic diameter, and together with some
combinatorial operations, gives a concrete approach for disproving the Linear
Hirsch Conjecture.Comment: 16 pages, 4 figure
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