1,417 research outputs found
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Discrete Morse theory for moment-angle complexes of pairs (D^n,S^{n-1})
For a finite simplicial complex K and a CW-pair (X,A), there is an associated
CW-complex Z_K(X,A), known as a polyhedral product. We apply discrete Morse
theory to a particular CW-structure on the n-sphere moment-angle complexes
Z_K(D^{n}, S^{n-1}). For the class of simplicial complexes with
vertex-decomposable duals, we show that the associated n-sphere moment-angle
complexes have the homotopy type of wedges of spheres. As a corollary we show
that a sufficiently high suspension of any restriction of a simplicial complex
with vertex-decomposable dual is homotopy equivalent to a wedge of spheres.Comment: Corollary 1.2 and 1 reference added. Some formulations and arguments
made more precis
Homotopy Type of the Boolean Complex of a Coxeter System
In any Coxeter group, the set of elements whose principal order ideals are
boolean forms a simplicial poset under the Bruhat order. This simplicial poset
defines a cell complex, called the boolean complex. In this paper it is shown
that, for any Coxeter system of rank n, the boolean complex is homotopy
equivalent to a wedge of (n-1)-dimensional spheres. The number of such spheres
can be computed recursively from the unlabeled Coxeter graph, and defines a new
graph invariant called the boolean number. Specific calculations of the boolean
number are given for all finite and affine irreducible Coxeter systems, as well
as for systems with graphs that are disconnected, complete, or stars. One
implication of these results is that the boolean complex is contractible if and
only if a generator of the Coxeter system is in the center of the group. of
these results is that the boolean complex is contractible if and only if a
generator of the Coxeter system is in the center of the group.Comment: final version, to appear in Advances in Mathematic
- …