Abstract. We prove that the complement of a complexified toric arrangement has the homotopy type of a minimal CW-complex, and thus its homology is torsion-free. To this end, we consider the toric Salvetti complex, a combinatorial model for the arrangement’s complement. Using diagrams of acyclic

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We give a light introduction to selection principles in topology, a young subfield of infinite-combinatorial topology. Emphasis is put on the modern approach to the problems it deals with. Recent results are described, and open problems are stated. Some results which do not appear elsewhere

With any state of a multipartite quantum system its separability polytope is associated. This is an algebro-topological object (nontrivial only for mixed states) which captures the localisation of entanglement of the state. Particular examples of separabilty polytopes for 3-partite systems

Abstract. With a view toward studying the homotopy type of spaces of Boolean formulae, we introduce a simplicial complex, called the theta complex, associated to any hypergraph. In particular, the set of satisfiable formulae in k-conjunctive normal form with ≤ n variables has the homotopy type of Θ(Cube(n, n − k)), where Cube(n, n − k) is a hypergraph associated to the (n − k)-skeleton of an n-cube. We make partial progress in calculating the homotopy type of theta for these cubical hypergraphs, and we also give calculations and examples for other hypergraphs as well. Indeed studying the theta complex of hypergraphs is an interesting problem in its own right. 1.

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Abstract. The paper surveys some new results and open problems connected with such fundamental combinatorial concepts as polytopes, simplicial complexes, cubical complexes, and subspace arrangements. Particular attention is paid to the case of simplicial and cubical subdivisions of manifolds and

such algebras are Koszul. That is not the case and the theorem was recently retracted. We analyze the Koszul property of these algebras for two large classes of graphs associated to finite regular CW complexes, X. Our methods are primarily topological. We solve the Koszul problem by introducing new cohomology

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