Cofibrations and Fibrations

Abstract

The notions of cofibration and fibration are central to homotopy theory. We show that the defining property of a cofiber inclusion map i : A → X is equivalent to the homotopy extension property of the pair (X,A). Thus the inclusion map of a subcomplex into a CW complex is a cofiber map, and so this concept is widespread in topology. We study cofiber maps in Section 3.2 and introduce pushout squares and mapping cones. In Section 3.3 we treat fiber maps as well as pullback squares and homotopy fibers and obtain some of their basic properties. We also consider fiber bundles which are defined by maps p : E → B for which there is a locally trivializing cover of B. We prove that fiber bundles are fibrations and thus obtain examples of fiber maps by exhibiting fiber bundles. In this way we obtain many diverse fibrations in which spheres, topological groups, Stiefel manifolds and Grassmannians appear. This is done in Section 3.4, and these results will be used in Chapter 5 to calculate homotopy groups. In the last section we introduce the mapping cylinder and its dual and use them to show that any map can be factored into the composition of a cofiber map followed by a homotopy equivalence or into the composition of a homotopy equivalence followed by a fiber map. These are important techniques and highlight the major role of cofiber and fiber maps in homotopy theory.