Short, new proofs of Dowker duality

Abstract

This paper presents three short, new proofs of Dowker duality using various poset fiber lemmas. We introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. These relational complexes can be constructed whenever there is a relation between the face posets of simplicial complexes, which includes the context of Dowker duality and covers of simplicial complexes. In this more general setting, we show that the homologies of the simplicial complexes and the relational complexes fit together in a long exact sequence.