Hatcher 0.10

Hatcher, Algebraic Topology, Chapter 0

10. Show that a space X is contractible if and only if every map f:X\to Y, for arbitrary Y, is null-homotopic. Similarly, show X is contractible if and only if every map f:Y\to X is null-homotopic.

Proof. First, recall what the terms in this problem mean:

  • A map f:X\to Y is null-homotopic if it is homotopic to a constant map.
  • A space X is contractible if it is homotopy equivalent to a point \{x_0\}, a useful equivalent of which is to say that X is contractible whenever the identity map id_X:X\to X is homotopic to a constant map.

Throughout, let g:X\to x_0 and h:x_0\to X be such that gh\simeq 1_{x_0} and hg\simeq 1_X. For convenience, consider the first claim to be part (a) and the second part (b).

(a) \left(\Longrightarrow\right) Suppose that X is contractible and let f:X\to Y be a map for Y arbitrary. Intuitively, X homotopy equivalent to x_0 implies that X can be “continuously deformed” to x_0, whereby it seems logical that the map f:X\to Y can be likewise “continuously deformed” into a map \widehat{f}:x_0\to Y, that is, a constant map. To be precise, define \widehat{f}=f\circ h:x_0\to Y where h is the half of the homotopy equivalence mentioned above. Clearly, then, \widehat{f} is constant, and because \widehat{f}\circ g = (f\circ h)\circ g\simeq f\circ id_X=f, it follows that f is homotopic to a constant map.

\left(\Longleftarrow\right) Suppose, now, that f:X\to Y is null-homotopic (that is, homotopic to a constant map) for every map f and for arbitrary Y. In particular for Y=X and f=id_X, it follows that id_X:X\to X is homotopic to a constant map, i.e. that X is contractible.

(b) \left(\Longrightarrow\right) Suppose that X is contractible and that f:Y\to X is a map for Y arbitrary. The argument here is similar to the one in part (a): Define \widehat{f}=gf:Y\to x_0 and note that \widehat{f} is necessarily constant and that h\circ\widehat{f}=h\circ(g\circ f)\simeq id_X\circ f = f. Hence, f is homotopic to a constant map.

\left(\Longleftarrow\right) Suppose now that f:Y\to X is null-homotopic for every Y and for every f. Again, choosing Y=X and f=id_X implies that id_X:X\to X is null-homotopic and hence that X is contractible.   \square

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