Homotopy theory

Abstract

In the first part, the fundamental group is defined using loops in topological spaces, which is the first of a series of invariants called homotopy groups. Unlike other homotopy groups, these groups are non-Abelian. However, these are computable in many examples. In this chapter, we discuss some properties of Fundamental groups and some computations.

Higher dimensional analogues of the above involve maps out of higher dimensional spheres and the resulting invariants are called homotopy groups. In the second part, we define homotopy groups and list some of the main computations.

In a tutorial section, the fundamental groups of spheres and real projective spaces are worked out and a few examples of group action discussed.