Hatcher, Algebraic Topology, Chapter 0
16. Show that is contractible.
Proof. Dr. Hatcher himself gives the proof of this in Example 1B.3 on Page 88, a proof the ideas of which are replicated here.
The goal is to show that is contractible, i.e. that the identity map is homotopic to a constant map. It will be shown that where is the map for all . First, note that is the collection of all points for which : This fact will be used to build the desired homotopy, the construction of which will take place in two parts.
Let be the homotopy
for all . This is a straight-line homotopy between the point and the point from which we define the restriction for all . Clearly, then, is a homotopy between points and . Next, let be given by
As with , is a straight-line homotopy, this time between the points and . As above, defining the restriction for all gives a chain of homotopies
In particular, then, the composition is the desired homotopy between and the constant map , where