Vector bundles over the two-sphere

Exercise 9.1.7 (Vector bundles over the two-sphere). Compute that \(\pi _1(\GL _1(\C ))\cong \Z \), with \(m\in \Z \) represented by the clutching function \(g_m\colon S^1\to \C ^\times \), \(z\mapsto z^m\), and construct an explicit representative \(L_m\) for the corresponding line bundle over \(S^2\). Next, show that the inclusion \(U(n) \hookrightarrow \GL _n(\C )\) is a homotopy equivalence. Using the fiber sequence \[ U(n-1) \hookrightarrow U(n) \twoheadrightarrow S^{2n-1}, \] deduce that \(\pi _1(\GL _n(\C )) \cong \Z \) for all \(n \geq 1\). Conclude that \(\pi _0\Vect _k(S^2) \cong \Z \) for all \(k \geq 1\), with every rank \(k\) complex vector bundle over \(S^2\) isomorphic to \[ L_m \oplus \ul {\C }^{k-1} \] for a unique \(m \in \Z \).