ABSTRACT
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 [Sn, 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(S
2) which is a milestone in homotopy theory and was a big surprise to the mathematical community when Hopf discovered it. Section 10.3 takes care of the study of the effect of change of base points on homotopy groups. In Section 10.4, we deal with a landmark result known as the Hurewicz isomorphism theorem, which relates homotopy groups with homology groups. As a consequence, we get a complete hold on the homotopy type of CW-complexes, a result due to J. H. C. Whitehead. It is high time now that we came back to one of the basic problems of extension and lifting. Sections 10.5-10.8 deal with these problems. This study goes under the generic name ‘obstruction theory’. It has its own gems such as Eilenberg-Mac Lane spaces and representability of cohomology groups. We end this chapter with some computation of homotopy groups of classical groups, which in turn, have many diverse applications. In Section 10.9, we carry out some elementary computation of homotopy groups of classical groups. Section 10.10 contains a brief introduction to homology with local coefficients and relates it to the equivariant homology.
