| Popis: |
Polyhedral products are topological spaces which unify and generalise many fundamental constructions. Their study is of significant interest in homotopy theory, where they underpin the combinatorics inherent in a broad variety of key objects, such as the join product, the Cartesian product, the smash product, and the Whitehead filtration. More broadly, polyhedral products are of interest in fields including algebraic geometry, geometric group theory, number theory, and representation theory, where they enable the study of familiar objects through underlying combinatorial structures.In this thesis, we study homotopy-theoretic properties of polyhedral products. Our results lie within two major areas. The first of these is the study of duality. Duality phenomena are exhibited by fundamental objects across mathematics, such as manifolds in topology, generalised homology spheres in combinatorics, and Gorenstein rings in commutative algebra. Through a study of polyhedral products known as moment-angle complexes, we equivocate homotopy-theoretic, combinatorial, and algebraic dualities. We show that this equivocation has both algebraic and homotopy-theoretic applications, to the study of moment-angle complexes and to broader families of polyhedral products.The second area in which our results lie is the study of homotopy classes of maps. Generalising known results, we show that a new family of relations is satisfied by certain homotopy classes of maps of polyhedral products. This brings new insight into the rich algebraic structures behind sets of homotopy classes of maps, and as an application we show that this work enables us to analyse the algebraic structure of the homotopy groups of odd spheres from an entirely new angle. |
