ABSTRACT
As indicated in the introduction to Chapter 1, the concept of fibration plays a central role in algebraic topology. However, apart from studying some special cases, viz., the covering projection in Chapter 2, a little bit about vector bundles in Chapter 5 and establishing the homotopy exact sequence of a fibration in Chapter 7, we have not studied much about fibrations so far. In this chapter, let us take-up the ever-so-important study of the fibrations again and go one step further. In Section 1, we shall deal with some generalities of fibrations. In Section 2, we shall discuss fibrations with fibres homotopy type of a sphere and establish Thom isomorphism theorem and, as an immediate consequence, the Gysin sequences. In Section 3, we begin the study of fibrations over suspensions, establish Wang homology and cohomology exact sequences by specialising to the case of fibrations over spheres. As a bonus, we derive the well-known homotopy suspension theorem due to Freudenthal. We shall then apply this study to compute the cohomology groups of some classical groups in Section 4. We also include some basic facts about Hopf algebras.
