686 research outputs found
Face vectors of subdivided simplicial complexes
Brenti and Welker have shown that for any simplicial complex X, the face
vectors of successive barycentric subdivisions of X have roots which converge
to fixed values depending only on the dimension of X. We improve and generalize
this result here. We begin with an alternative proof based on geometric
intuition. We then prove an interesting symmetry of these roots about the real
number -2. This symmetry can be seen via a nice algebraic realization of
barycentric subdivision as a simple map on formal power series in two
variables. Finally, we use this algebraic machinery with some geometric
motivation to generalize the combinatorial statements to arbitrary subdivision
methods: any subdivision method will exhibit similar limit behavior and
symmetry. Our techniques allow us to compute explicit formulas for the values
of the limit roots in the case of barycentric subdivision.Comment: 13 pages, final version, appears in Discrete Mathematics 201
gamma-vectors of edge subdivisions of the boundary of the cross polytope
For any flag simplicial complex obtained by stellar subdividing the
boundary of the cross polytope in edges, we define a flag simplicial complex
(dependent on the sequence of subdivisions) whose -vector
is the -vector of . This proves that the -vector of any
such simplicial complex satisfies the Frankl-F\"{u}redi-Kalai inequalities,
partially solving a conjecture by Nevo and Petersen \cite{np}. We show that
when is the dual simplicial complex to a nestohedron, and the sequence
of subdivisions corresponds to a flag ordering as defined in \cite{ai}, that
is equal to the flag simplical complex defined there.Comment: 18 pages, 1 figur
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
Manifolds of isospectral arrow matrices
An arrow matrix is a matrix with zeroes outside the main diagonal, first row,
and first column. We consider the space of Hermitian arrow
-matrices with fixed simple spectrum . We prove
that this space is a smooth -manifold, and its smooth structure is
independent on the spectrum. Next, this manifold carries the locally standard
torus action: we describe the topology and combinatorics of its orbit space. If
, the orbit space is not a polytope, hence
this manifold is not quasitoric. However, there is a natural permutation action
on which induces the combined action of a semidirect product
. The orbit space of this large action is a simple
polytope. The structure of this polytope is described in the paper.
In case , the space is a solid torus with
boundary subdivided into hexagons in a regular way. This description allows to
compute the cohomology ring and equivariant cohomology ring of the
6-dimensional manifold using the general theory developed by
the first author. This theory is also applied to a certain -dimensional
manifold called the twin of . The twin carries a
half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure
Constructing Simplicial Branched Covers
Branched covers are applied frequently in topology - most prominently in the
construction of closed oriented PL d-manifolds. In particular, strong bounds
for the number of sheets and the topology of the branching set are known for
dimension d<=4. On the other hand, Izmestiev and Joswig described how to obtain
a simplicial covering space (the partial unfolding) of a given simplicial
complex, thus obtaining a simplicial branched cover [Adv. Geom. 3(2):191-255,
2003]. We present a large class of branched covers which can be constructed via
the partial unfolding. In particular, for d<=4 every closed oriented PL
d-manifold is the partial unfolding of some polytopal d-sphere.Comment: 15 pages, 8 figures, typos corrected and conjecture adde
A survey of subdivisions and local -vectors
The enumerative theory of simplicial subdivisions (triangulations) of
simplicial complexes was developed by Stanley in order to understand the effect
of such subdivisions on the -vector of a simplicial complex. A key role
there is played by the concept of a local -vector. This paper surveys some
of the highlights of this theory and some recent developments, concerning
subdivisions of flag homology spheres and their -vectors. Several
interesting examples and open problems are discussed.Comment: 13 pages, 3 figures; minor changes and update
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