Duality, Manifolds and Some Algebraic Topology

This is a short note intended to explore the applications of duality theory to the study of manifolds. I discuss Alexander duality, Lefschetz duality and Poincare duality, along with applications to the study of compact connected orientable manifolds. 1. Manifolds and points 1.1. The core question. One of the questions we shall be interested in is: Given two manifolds M and N , what are the ways in which N embeds as a submanifold of M? In other words, what are the submanifolds of M homeomorphic to N? Roughly speaking, we want to know how N “sits inside” M merely from the data of what M and N look like abstractly. First, we need to define what it means for “ways in which N embeds”. Definition (Equivalently embedded subsets). Give a topological space X and subspaces Y1 and Y2, we say that Y1 and Y2 are i equivalently embedded subsets(defined)f there is a homeomorphism of X under which Y1 maps homeomorphically to Y2, or equivalently, there is a homeomorphism of the pair (X,Y1) and (X,Y2). The question we want to ask, more precisely is: what are the various ways of embedding one manifold inside another, upto equivalence? The answer in general could be lots. Moreover, non-equivalent embeddings may also look very similar to the algebraic topologist. Algebraic topology tries to partially solve this problem by looking at an “invariant” of embeddings: Given a topological space X and a subspace Y , describe all possibilities for the induced maps of homology: