Constructing self-dual complexes and self-dual triangulations of manifolds

Abstract

Simplicial complexes K, which are equal to their Alexander dual KΛ are known as self-dual simplicial complexes. We prove that topological and combinatorial properties of any self-dual simplicial complex, are fully determined by topological and combinatorial properties of the link of any of it’s vertices which happens to be sub-dual in smaller combinatorial ambient. Using this observation, we describe a general method for constructing self-dual triangulations of given topological spaces and focus on self-dual triangulations of compact manifolds. We show that there exist only 4 types of self-dual combinatorial manifolds and provide a general method for their construction.