I do not know any book on homotopy theory which covers all the material to which I need to refer, but one useful introduction is May's book [89].

Definitions and basic properties

A continuous map X × I →Y is said to be a homotopy between the maps X →Y given by its restrictions to X × ﹛0﹜ and X × ﹛1﹜. The relation of homotopy between maps is an equivalence relation. A major concern of homotopy theory is the set of homotopy equivalence classes of maps X →Y, which in this appendix we denote by [X : Y]. Unless otherwise stated we fix base points in X and Y and require maps and homotopies to respect the base point. The base point is usually denoted ∗, but is often suppressed from the notation. A map X →Y homotopic to the constant map X → ∗ is said to be nullhomotopic. We write X+ for the disjoint union of X and a point, taken as base point.

An important type of homotopy occurs when B ⊂ A, h : A × I → A satisfies h(x, 0) = x for all x ∈ A, h(x, t) = x for all x ∈ B, t ∈ I and h(A × ﹛1﹜) = B: B is then called a deformation retract of A and h is a deformation retraction. A simple example is when A is a square and B the union of three sides.

Two spaces are said to be homotopy equivalent if there are maps and such that each composite is homotopic to the identity map.

If f : Sn−1 → X is a continuous map, we define a space: as a set, we have the disjoint union of X and the map g : Dn → X ∪f en is given by the identity on and by f on Sn−1; and we declare a subset to be open if its preimages by both g and the inclusion of X are open. This process is called attaching an n-cell to X. We can allow n = 0: S−1 is the empty set, so X ∪f e0 = X+ is the disjoint union of X and a point.