Hatcher, Algebraic Topology, Chapter 0
3. (a) Show that the composition of homotopy equivalences and is a homotopy equivalence . Deduce that homotopy equivalence is an equivalence relation. (b) Show that the relation of homotopy among maps is an equivalence relation. (c) Show that a map homotopic to a homotopy equivalence is a homotopy equivalence.
Proof. Recall that a map is called a homotopy equivalence if there exists a map such that and . In this case, will be referred to as the homotopy inverse of . Also, one says that two maps are homotopic if there exists a homotopy connecting to . Here, one writes .
(a) Note that the main claim here is that homotopy equivalence is a transitive relation, the proof of which will verify that homotopy equivalence is an equivalence relation due to the fact that is a homotopy equivalence (where is its own homotopy inverse), and that is homotopy equivalent to if and only if is homotopy equivalent to .
To that end, suppose that are topological spaces and that , are homotopy equivalences with homotopy inverses , respectively. By definition, then, , and , . It suffices to show that the map is a homotopy equivalence, which is immediate due to the fact that it’s continuous (the composition of continuous maps is continuous) and that is a map for which
Hence, the result.
(b) To show that the relationship of homotopy among maps is an equivalence relation, note again that the reflexive and symmetric properties are free. Indeed, if are maps which are homotopic by way of a homotopy , then by way of the identity homotopy and by way of the homotopy . So, again, it suffices to prove that the relation is transitive.
To that end, suppose that is a homotopy connecting to and that is a homotopy connecting to , where . Define, then, a map which takes on the value for and the value for . Continuity of is immediate due to the fact that is made continuous at (the only point of concern), whereby it follows that .
(c) Finally, suppose that is a homotopy equivalence with homotopy inverse and suppose that is homotopic to $f$, i.e. that there exists a homotopy connecting to . Said a different way, says that there exists a family connecting to , a fact which immediately implies that the family connects to . Thus, because by the homotopy inverse property, and because because by (b), it follows that is also a homotopy equivalence.