We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes for which the corresponding moment-angle complex <!-- MATH $\mathcal Z_{K}$ --> has the homotopy type of a wedge of spheres or a connected sum of sphere products. When is flag, we identify in algebraic and combinatorial terms those for which <!-- MATH $\mathcal Z_{K}$ --> is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Jöllenbeck on Golod rings and homotopy theoretical results of the first and third authors. We also establish a connection between minimally non-Golod rings and moment-angle complexes <!-- MATH $\mathcal Z_{K}$ --> which are homotopy equivalent to a connected sum of sphere products. We go on to show that for any flag complex the loop spaces <!-- MATH $\Omega \mathcal Z_{K}$ --> and <!-- MATH $\Omega \mbox {$D{\mskip -1mu}J\/$}(K)$ --> are homotopy equivalent to a product of spheres and loops on spheres when localised rationally or at any prime .

We establish a new theory which unifies various aspects of topological approaches for data science, by being applicable both to point cloud data and to graph data, including networks beyond pairwise interactions. We generalize simplicial complexes and hypergraphs to super-hypergraphs and establish super-hypergraph homology as an extension of simplicial homology. Driven by applications, we also introduce super-persistent homology.

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex .