Manifolds and Poincaré Duality

Abstract

This chapter deals with some of the basic homological properties of topological manifolds. Since the main result is the Poincaré duality theorem, we begin with a simple example to establish an intuitive feeling for this classical result. This is followed by material on topological manifolds and a detailed proof of the theorem. The approach used follows the excellent treatment of Samelson [1965, pp. 323–336] and proceeds by way of the Thorn isomorphism theorem. Several applications of the theorem follow, including the determination of the cohomology rings of projective spaces and results on the index of topological manifolds and cobordism.