91Ó°ÊÓ

Suppose that a CW complex \(X\) contains a subcomplex \(S^{1}\) such that the inclusion \(S^{1} \hookrightarrow X\) induces an injection \(H_{1}\left(S^{1} ; \mathbb{Z}\right) \rightarrow H_{1}(X ; \mathbb{Z})\) with image a direct summand of \(H_{1}(X ; \mathbb{Z}) .\) Show that \(S^{1}\) is a retract of \(X\).

For an H-space \(\left(X, x_{0}\right)\) with multiplication \(\mu: X \times X \rightarrow X,\) show that the group operation in \(\pi_{n}\left(X, x_{0}\right)\) can also be defined by the rule \((f+g)(x)=\mu(f(x), g(x))\).

For a pair \((X, A)\) of path-connected spaces, show that \(\pi_{1}\left(X, A, x_{0}\right)\) can be identified in a natural way with the set of cosets \(\alpha H\) of the subgroup \(H \subset \pi_{1}\left(X, x_{0}\right)\) represented by loops in \(A\) at \(x_{0}\).

Give an example of a weak homotopy equivalence \(X \rightarrow Y\) for which there does not exist a weak homotopy equivalence \(Y \rightarrow X\).