Chapter 15

Show that \(\pi(X, x)=0\) if \(X\) is a finite topological space with the discrete topology.

Prove that if \(\mathrm{A}\) is a strong deformation retract of \(\mathrm{X}\) then the inclusion map i: \(\mathrm{A} \rightarrow \mathrm{X}\) induces an isomorphism $$ \mathrm{i}_{*}: \pi(\mathrm{A}, \mathrm{a}) \rightarrow \pi(\mathrm{X}, \mathrm{a}) $$ for any point \(\mathrm{a} \in \mathrm{A}\).

For \(S^{1} \subseteq C\) define \(\mu: \mathrm{S}^{1} \times \mathrm{S}^{1} \rightarrow \mathrm{S}^{1}\) by \(\mu\left(\mathrm{z}_{1}, \mathrm{~L}_{2}\right)=\mathrm{z}_{1} \mathrm{z}_{2}\), and \(\nu\) : \(\mathrm{S}^{1} \rightarrow \mathrm{S}^{1}\) by \(\nu(\mathrm{z})=\mathrm{z}^{-1}\). Prove that \(\mathrm{S}^{1}\) is a topological group. Deduce that \(\pi\left(S^{1}, 1\right)\) is an abelian group.