Hatcher, Algebraic Topology, Chapter 0
10. Show that a space is contractible if and only if every map , for arbitrary , is null-homotopic. Similarly, show is contractible if and only if every map is null-homotopic.
Proof. First, recall what the terms in this problem mean:
- A map is null-homotopic if it is homotopic to a constant map.
- A space is contractible if it is homotopy equivalent to a point , a useful equivalent of which is to say that is contractible whenever the identity map is homotopic to a constant map.
Throughout, let and be such that and . For convenience, consider the first claim to be part (a) and the second part (b).
(a) Suppose that is contractible and let be a map for arbitrary. Intuitively, homotopy equivalent to implies that can be “continuously deformed” to , whereby it seems logical that the map can be likewise “continuously deformed” into a map , that is, a constant map. To be precise, define where is the half of the homotopy equivalence mentioned above. Clearly, then, is constant, and because , it follows that is homotopic to a constant map.
Suppose, now, that is null-homotopic (that is, homotopic to a constant map) for every map and for arbitrary . In particular for and , it follows that is homotopic to a constant map, i.e. that is contractible.
(b) Suppose that is contractible and that is a map for arbitrary. The argument here is similar to the one in part (a): Define and note that is necessarily constant and that . Hence, is homotopic to a constant map.
Suppose now that is null-homotopic for every and for every . Again, choosing and implies that is null-homotopic and hence that is contractible.