Related papers: Topology of complements of skeletons
The essential subtoposes of a fixed topos form a complete lattice, which gives rise to the notion of a level in a topos. In the familiar example of simplicial sets, levels coincide with dimensions and give rise to the usual notions of…
We survey interactions between the topology and the combinatorics of complex hyperplane arrangements. Without claiming to be exhaustive, we examine in this setting combinatorial aspects of fundamental groups, associated graded Lie algebras,…
The article is devoted to a structure of topological spaces related with topological quasigroups. Regular and complete spaces over topological quasigroups are studied. Separations and embeddings are also investigated for them. Their…
The face numbers of simplicial complexes without missing faces of dimension larger than $i$ are studied. It is shown that among all such $(d-1)$-dimensional complexes with non-vanishing top homology, a certain polytopal sphere has the…
Given a symmetric monoidal category $C$ with product $\sqcup$, where the neutral element for the product is an initial object, we consider the poset of $\sqcup$-complemented subobjects of a given object $X$. When this poset has finite…
We compute the rational homology of the moduli stack $\mathcal{M}$ of objects in the derived category of certain smooth complex projective varieties $X$ including toric varieties, flag varieties, curves, surfaces, and some 3- and 4-folds.…
We prove that if a simplicial complex is shellable, then the intersection lattice for the corresponding diagonal arrangement is homotopy equivalent to a wedge of spheres. Furthermore, we describe precisely the spheres in the wedge, based on…
A simplicial complement P is a sequence of subsets of [m] and the simplicial complement P corresponds to a unique simplicial complex K with vertices in [m]. In this paper, we defined the homology of a simplicial complement…
Random shapes arise naturally in many contexts. The topological and geometric structure of such objects is interesting for its own sake, and also for applications. In physics, for example, such objects arise naturally in quantum gravity, in…
Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice…
We study the higher dimensional geometry of Berkovich spaces using virtual open disks, which are given by fibration of relative dimension $1$. Inspired by birational geometry, we conjecture that the Berkovich skeleton is the complement of…
There is a canonical way to associate two simplicial complexes K, L to any relation $R\subset X\times Y$. Moreover, the geometric realizations of K and L are homotopy equivalent. This was studied in the fifties by C.H. Dowker. In this…
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a…
In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let $K$ be a complete non-Archimedean field, and let $X$ be a closed subscheme of a toric variety over $K$. We define the…
In this paper, we survey constructions of and nonexistence results on combinatorial/geometric structures which arise from unions of cyclotomic classes of finite fields. In particular, we survey both classical and recent results on…
Graphs with given k vertices generate an (acyclic) simplicial complex. We describe the homology of its quotient complex, formed by all connected graphs, and demonstrate its applications to the topology of braid groups, knot theory,…
We classify the complementary vectors of doubly Cohen-Macaulay complexes. This proves a conjecture of Swartz, negatively answers a question of Athanasiadis and Tzanaki, and gives new bounds on the number of independent sets in a matroid.…
Skeletal polyhedra are discrete connected structures consisting of finite (planar or skew) or infinite (linear, planar, or spatial) polygons as faces, with two faces on each edge and a circular vertex figure at each vertex. The present…
A notion of an $i$-banner simplicial complex is introduced. For various values of $i$, these complexes interpolate between the class of flag complexes and the class of all simplicial complexes. Examples of simplicial spheres of an arbitrary…
We survey results on the topological complexity of classical configuration spaces of distinct ordered points in orientable surfaces and related spaces, including certain orbit configuration spaces and Eilenberg-Mac Lane spaces associated to…