605 research outputs found
Order complexes of noncomplemented lattices are nonevasive
We reprove and generalize in a combinatorial way the result of A. Bj\"orner
[J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order
complexes of noncomplemented lattices are contractible, namely by showing that
these simplicial complexes are in fact nonevasive, in particular collapsible
Moduli spaces of metric graphs of genus 1 with marks on vertices
In this paper we study homotopy type of certain moduli spaces of metric
graphs. More precisely, we show that the spaces , which parametrize
the isometry classes of metric graphs of genus 1 with marks on vertices are
homotopy equivalent to the spaces , which are the moduli spaces of
tropical curves of genus 1 with marked points. Our proof proceeds by
providing a sequence of explicit homotopies, with key role played by the
so-called scanning homotopy. We conjecture that our result generalizes to the
case of arbitrary genus.Comment: Topology and its Applications, In Press, Corrected Proof, Available
online 3 August 200
Witness structures and immediate snapshot complexes
In this paper we introduce and study a new family of combinatorial simplicial
complexes, which we call immediate snapshot complexes. Our construction and
terminology is strongly motivated by theoretical distributed computing, as
these complexes are combinatorial models of the standard protocol complexes
associated to immediate snapshot read/write shared memory communication model.
In order to define the immediate snapshot complexes we need a new combinatorial
object, which we call a witness structure. These objects are indexing the
simplices in the immediate snapshot complexes, while a special operation on
them, called ghosting, describes the combinatorics of taking simplicial
boundary. In general, we develop the theory of witness structures and use it to
prove several combinatorial as well as topological properties of the immediate
snapshot complexes.Comment: full paper version of the 1st part of the preprint arXiv:1402.4707;
to appear in DMTC
Collapsing along monotone poset maps
We introduce the notion of nonevasive reduction, and show that for any
monotone poset map , the simplicial complex {\tt
NE}-reduces to , for any .
As a corollary, we prove that for any order-preserving map
satisfying , for any , the simplicial complex
collapses to . We also obtain a generalization of
Crapo's closure theorem.Comment: To appear in the International Journal of Mathematics and
Mathematical Science
- β¦