Basic Algebraic Topology

Chapter 10 Homotopy Theory

This chapter may be called the heart of the book. Throughout this chapter we shall be working with the category of pointed topological spaces. In Section 10.1, we begin with a general discussion on the homotopy sets [X, Y] which leads us to the notion of H-spaces and their duals. In particular we get some canonical abelian group structures on [$\mathbb{S}$ⁿ, Y], n ≥ 2, which we call higher homotopy groups. In Section 10.2, we make an earnest beginning of the study of these groups. The central result here is the exact sequence of homotopy groups for pointed topological pairs and the same for fibrations. As a consequence we derive the non triviality of π₃($\mathbb{S}$²) which is a milestone in homotopy theory and was a big surprise to the ...