In this paper we will present a theory for simplicial diffeomorphims, that is, diffeomorphims that preserve the incidence relations of a simplicial complex. Simplicial diffeomorphisms can be regarded as curvilinear barycentric coordinates. Using the combination of piecewise linear functions on complexes with simplicial diffeomorphisms, we also propose a new representation of curves and surfaces...

When a cone is added to a simplicial complex ∆ over one of its faces, we investigate the relation between the arithmetical ranks of the StanleyReisner ideals of the original simplicial complex and the new simplicial complex ∆′. In particular, we show that the arithmetical rank of the Stanley-Reisner ideal of ∆′ equals the projective dimension of the Stanley-Reisner ring of ∆′ if the correspondi...

A large variety of interacting complex systems are characterized by interactions occurring between more than two nodes. These systems are described by simplicial complexes. Simplicial complexes are formed by simplices (nodes, links, triangles, tetrahedra etc.) that have a natural geometric interpretation. As such simplicial complexes are widely used in quantum gravity approaches that involve a ...

We introduce a two player topological game and study the relationship of the existence of winning strategies to base properties and covering properties of the underlying space. The existence of a winning strategy for one of the players is conjectured to be equivalent to the space have countable network weight. In addition, connections to the class of D-spaces and the class of hereditarily Linde...

For a finite real reflection group W with Coxeter element γ we give a uniform proof that the closed interval, [I, γ] forms a lattice in the partial order on W induced by reflection length. The proof involves the construction of a simplicial complex which can be embedded in the type W simplicial generalised associahedron.

The 4-dimensional abstract Kummer variety K4 with 16 nodes leads to the K3 surface by resolving the 16 singularities. Here we present a simplicial realization of this minimal resolution. Starting with the minimal 16-vertex triangulation (K)16 we resolve its 16 isolated singularities – step by step – by simplicial blowups. A key step is the construction of a triangulated version of the mapping c...

We show that any accordion complex associated to a dissection of a convex polygon is isomorphic to the support τ -tilting simplicial complex of an explicit finite dimensional algebra. To this end, we prove a property of some induced subcomplexes of support τ -tilting simplicial complexes of finite dimensional algebras.

In this paper, we use subword complexes to provide a uniform approach to finite-type cluster complexes and multi-associahedra. We introduce, for any finite Coxeter group and any nonnegative integer k, a spherical subword complex called multi-cluster complex. For k = 1, we show that this subword complex is isomorphic to the cluster complex of the given type. We show that multi-cluster complexes ...

Two theorems are proved. One concerns coverings of a simplicial complex D by subcomplexes. It is shown that if every t-wise intersection of these subcomplexes is ðk t þ 1Þ-connected, then for jpk there are isomorphisms pjðDÞDpjðNÞ of homotopy groups of D and of the nerve N of the covering. The other concerns poset maps f : P-Q: It is shown that if all fibers f ðQpqÞ are kconnected, then f induc...

The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zon...