Fibrations

Abstract

The reader familiar with homology will have noticed that in Chapter 3 we proved the analogs for π _n (X, A, x₀) of the first five Eilenberg-Steenrod axioms and in place of the seventh axiom (“dimension axiom”) we have ${\pi _{n}}\left( {\left\{ {{x_{0}}} \right\},{x_{0}}} \right) = 0$ for all n ≥ 0. It is the sixth axiom, the “excision axiom”, however, which makes homology computable for such a wide class of spaces (including the spheres Sⁿ), and we shall see in Chapter 6 that excision holds for homotopy only under very restricted circumstances. Homotopy does have this redeeming feature, though: it behaves well with respect to fibrations. This chapter is devoted to a proof of this fact and a brief investigation of its consequences. We begin with a simple case.