91Ó°ÊÓ

Show that \(f: X \rightarrow Y\) is a homotopy equivalence if there exist maps \(g, h: Y \rightarrow X\) such that \(f g \simeq \mathbb{1}\) and \(h f \simeq \mathbb{1} .\) More generally, show that \(f\) is a homotopy equivalence if \(f g\) and \(h f\) are homotopy equivalences.

Show that a homotopy equivalence \(f: X \rightarrow Y\) induces a bijection between the set of path-components of \(X\) and the set of path-components of \(Y,\) and that \(f\) restricts to a homotopy equivalence from each path-component of \(X\) to the corresponding pathcomponent of \(Y\). Prove also the corresponding statement with components instead of path-components. Deduce from this that if the components and path-components of a space coincide, then the same is true for any homotopy equivalent space.

Show that a CW complex is contractible if it is the union of two contractible subcomplexes whose intersection is also contractible.

If \(X\) is a CW complex with components \(X_{\alpha},\) show that the suspension \(S X\) is homotopy equivalent to \(Y \vee_{\alpha} S X_{\alpha}\) for some graph \(Y .\) In the case that \(X\) is a finite graph, show that \(S X\) is homotopy equivalent to a wedge sum of circles and 2 -spheres.