Combinatorics of CAT(0) cubical complexes, crossing complexes and co-skeletons

Abstract

This thesis consists of three papers about cubical complexes: Chapter 1 is [Rowlands 22a], Chapter 2 is [Rowlands 23], and Chapter 3 is [Rowlands 22b]. Chapter 1 extends a result by Dancis to cubical complexes: Dancis proved that any d-dimensional simplicial manifold can be reconstructed from its (floor(d/2) + 1)-skeleton, and we prove an analogous result for d-dimensional cubical manifolds that can be embedded as a subcomplex into a cube I^N. Chapter 2 studies CAT(0) cubical complexes, using the framework of a poset with inconsistent pairs developed by Ardila et al. We introduce a simplicial complex called the "crossing complex" associated to each CAT(0) cubical complex, and study its properties. We deduce that this crossing complex holds much of the combinatorial information contained in the cubical complex: our main results relate their f-vectors, hyperplane/link structure, and balancedness. Finally, Chapter 3 studies the topology of complements of skeletons in polytopal complexes: we derive a long exact sequence involving homology of skeleton complements and links, and we characterise various topological properties of spaces in terms of skeleton complements. Our main application of this machinery is to CAT(0) cubical complexes: we conclude that these complexes also share several topological properties with their crossing complexes.