Collapsing along monotone poset maps

We introduce the notion of nonevasive reduction, and show that for any monotone poset map $\phi:P\to P$, the simplicial complex $\Delta(P)$ {\tt NE}-reduces to $\Delta(Q)$, for any $Q\supseteq{\text{\rm Fix}}\phi$. As a corollary, we prove that for any order-preserving map $\phi:P\to P$ satisfying $\phi(x)\geq x$, for any $x\in P$, the simplicial complex $\Delta(P)$ collapses to $\Delta(\phi(P))$. We also obtain a generalization of Crapo's closure theorem.Comment: To appear in the International Journal of Mathematics and Mathematical Science

Shellability of generalized Dowling posets

A generalization of Dowling lattices was recently introduced by Bibby and Gadish, in a work on orbit configuration spaces. The authors left open the question as to whether these posets are shellable. In this paper we prove EL-shellability and use it to determine the homotopy type. We also show that subposets corresponding to invariant subarrangements are not shellable in general

Linear colorings of simplicial complexes and collapsing

A vertex coloring of a simplicial complex $\Delta$ is called a linear coloring if it satisfies the property that for every pair of facets $(F_1, F_2)$ of $\Delta$, there exists no pair of vertices $(v_1, v_2)$ with the same color such that $v_1\in F_1\backslash F_2$ and $v_2\in F_2\backslash F_1$. We show that every simplicial complex $\Delta$ which is linearly colored with $k$ colors includes a subcomplex $\Delta'$ with $k$ vertices such that $\Delta'$ is a strong deformation retract of $\Delta$. We also prove that this deformation is a nonevasive reduction, in particular, a collapsing.Comment: 18 page

Hom complexes and homotopy theory in the category of graphs

We investigate a notion of $\times$-homotopy of graph maps that is based on the internal hom associated to the categorical product in the category of graphs. It is shown that graph $\times$-homotopy is characterized by the topological properties of the \Hom complex, a functorial way to assign a poset (and hence topological space) to a pair of graphs; \Hom complexes were introduced by Lov\'{a}sz and further studied by Babson and Kozlov to give topological bounds on chromatic number. Along the way, we also establish some structural properties of \Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. Graph $\times$-homotopy naturally leads to a notion of homotopy equivalence which we show has several equivalent characterizations. We apply the notions of $\times$-homotopy equivalence to the class of dismantlable graphs to get a list of conditions that again characterize these. We end with a discussion of graph homotopies arising from other internal homs, including the construction of `$A$-theory' associated to the cartesian product in the category of reflexive graphs.Comment: 28 pages, 13 figures, final version, to be published in European J. Com

Equivariant Piecewise-Linear Topology and Combinatorial Applications

For G a finite group, we develop some theory of G-equivariant piecewise-linear topology and prove characterization theorems for G-equivariant regular neighborhoods. We use these results to prove a conjecture of Csorba that the Lovász complex Hom(C5,Kn) of graph multimorphisms from the 5-cycle C5 to the complete graph Kn is equivariantly homeomorphic to the Stiefel manifold, Vn-1,2, the space of (ordered) orthonormal 2-frames in Rn-1 with respect to an action of the cyclic group of order 2