We introduce a large self-dual class of simplicial complexes about which we show that each complex in it is contractible or homotopy equivalent to a sphere. Examples of complexes in this class include independence and dominance complexes of forests, pointed simplicial complexes, and their combinatorial Alexander duals.

In this article, we present various new results on Cauchy tensors and Hankel tensors. We first introduce the concept of generalized Cauchy tensors which extends Cauchy tensors in the current literature, and provide several conditions characterizing positive semi-definiteness of generalized Cauchy tensors with nonzero entries. Furthermore, we prove that all even order generalized Cauchy tensorsw...

We prove that the homotopy theory of parametrized spaces embeds fully and faithfully in the homotopy theory of simplicial presheaves, and that its essential image consists of the locally homotopically constant objects. This gives a homotopy-theoretic version of the classical identification of covering spaces with locally constant sheaves. We also prove a new version of the classical result that...

We verify the Upper Bound Conjecture (UBC) for a class of odddimensional simplicial complexes that in particular includes all Eulerian simplicial complexes with isolated singularities. The proof relies on a new invariant of simplicial complexes — a short simplicial h-vector.

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining nonisomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. More...

We use an extension of Sunada's theorem to construct a nonisometric pair of isospectral simply connected domains in the Euclidean plane, thus answering negatively Kac's question, "can one hear the shape of a drum?" In order to construct simply connected examples, we exploit the observation that an orbifold whose underlying space is a simply connected manifold with boundary need not be simply co...

We say that a simplicial complex is shrinkable if there exists a sequence of admissible edge contractions that reduces the complex to a single vertex. We prove that it is NP-complete to decide whether a (two-dimensional) simplicial complex is shrinkable. Along the way, we describe examples of contractible complexes that are not shrinkable.

We introduce a notion of join for (augmented) simplicial sets gen-eralising the classical join of geometric simplicial complexes. The definition comes naturally from the ordinal sum on the base simplicial category ∆.

Motivated by the question of the polytopal realizability of the simplicial complex Γn,k of (k + 1)-crossing-free sets of diagonals of the convex n-gon, we study the first non-obvious case, namely the simplicial complex Γ8,2 of 3-crossing-free sets of diagonals of the convex octagon. We give a complete description of the space of symmetric realizations of Γ8,2, that is, of the polytopes P whose ...

We study face vectors of barycentric subdivisions of simplicial homology manifolds. Recently, Kubitzke and Nevo proved that the g-vector of the barycentric subdivision of a Cohen–Macaulay simplicial complex is an M -vector, which in particular proves the g-conjecture for barycentric subdivisions of simplicial homology spheres. In this paper, we prove an analogue of this result for Buchsbaum sim...