Basic Algebraic Topology

Chapter 5 Topology of Manifolds

Manifolds are central objects of study in topology. Earlier we introduced CW-complexes and simplicial complexes. These are in some sense a slight generalization of manifolds and help us in understanding manifolds better.

In Section 5.1, let us get familiar with the general concept of a topological manifold and some of its point-set topological properties. The topological invariance of domain is a crucial result which helps us to ‘define’ manifolds with boundary in an unambiguous way.

In Section 5.2, we shall study triangulability of manifolds. For a triangulated manifold X, Poincaré introduced the concept of ‘a dual cell decomposition’ using which he showed that the i^th Betti number β_i(X) is equal to β_n–i(X). ...