Basic Algebraic Topology

Chapter 8 Homology of Manifolds

We shall discuss the homology and cohomology groups of a manifold. In the first section, we shall discuss the concept of orientability (independently of triangulations). In Section 8.2, we shall prsent various forms of duality theorems. In Section 8.3, some applications of duality are presented. The notions of degree and cobordism are two important topics here. In Section 8.4, we recall some basic facts about differential forms, integration, the Stokes’ theorem and the Poincaré lemma, etc., introduce the de Rham cohomology of a smooth oriented manifold and then show that it is canonically isomorphic to singular cohomology with real coefficients.

8.1 Orientability

In this section we shall study the notion of orientability ...